what statistical analysis should i use?

What statistical analysis should I use? Running

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1Saturday 22 April 2023 08:36 AM

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Introduction

5Index End

For a useful general guide see Policy: Twenty tips for interpreting scientific claims : Nature News & Comment William J. Sutherland, David Spiegelhalter and Mark Burgman Nature 503 335–337 2013.

Some criticism has been made of their discussion of p values, see Replication, statistical consistency, and publication bias G. Francis, Journal of Mathematical Psychology 57(5) 153–169 2013.

http://www.nature.com/news/policy-twenty-tips-for-interpreting-scientific-claims-1.14183

http://www.nature.com/news/policy-twenty-tips-for-interpreting-scientific-claims-1.14183

http://www.sciencedirect.com/science/article/pii/S002224961300014X

Introduction

6Index End

These examples are loosely based on a UCLA tutorial sheet. All can be realised via the syntax window. Appropriate command strokes are also indicated. The guidelines to the APA reporting style is motivated by Using SPSS for Windows and Macintosh: Analyzing And Understanding Data Samuel B. Green and Neil J. Salkind. Much information is available on the web on the APA style. The source text is Publication Manual of the American Psychological Association, Sixth Edition a useful summary is Reporting Statistics in APA Style.

These pages show how to perform a number of statistical tests using SPSS. Each section gives a brief description of the aim of the statistical test, when it is used, an example showing the SPSS commands and SPSS (often abbreviated) output with a brief interpretation of the output.

http://www.ats.ucla.edu/stat/spss/whatstat/whatstat.htm

http://www.pearsonhighered.com/educator/product/Using-SPSS-for-Windows-and-Macintosh/9780205958603.page

http://www.pearsonhighered.com/educator/product/Using-SPSS-for-Windows-and-Macintosh/9780205958603.page

http://www.apastyle.org/manual/

http://www.apastyle.org/manual/

http://my.ilstu.edu/~jhkahn/apastats.html

About the A data file

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Most of the examples in this document will use a data file called A, high school and beyond. This data file contains 200 observations from a sample of high school students with demographic information about the students, such as their gender (female), socio-economic status (ses) and ethnic background (race). It also contains a number of scores on standardized tests, including tests of reading (read), writing (write), mathematics (math) and social studies (socst).


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Variable Position Label Value Label id 1

.00 Male female 2 1.00 Female

1.00 Hispanic 2.00 Asian 3.00 african-amer

race 3

4.00 White 1.00 Low 2.00 Middle ses 4 3.00 High 1.00 Public

schtyp 5 type of school 2.00 private

1.00 general 2.00 academic prog 6 type of program 3.00 vocation

read 7 reading score write 8 writing score math 9 math score science 10 science score socst 11 social studies score

Syntax:-display dictionary /VARIABLES id female race ses schtyp prog read write math science socst.


9Index End

One sample t-test

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A one sample t-test allows us to test whether a sample mean (of a normally distributed interval variable) significantly differs from a hypothesized value. For example, using the A data file, say we wish to test whether the average writing score (write) differs significantly from 50. Test variable writing score (write), Test value 50. We can do this as shown below.

Menu selection:- Analyze > Compare Means > One-Sample T test

Syntax:- t-test /testval = 50 /variable = write.

One sample t-test

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Note the test value of 50 has been selected

One sample t-test

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One-Sample Statistics

N Mean Std. Deviation Std. Error Mean

writing score 200 52.7750 9.47859 .67024

One-Sample Test

Test Value = 50

t df Sig. (2-tailed)

Mean

Difference

writing score 4.140 199 .000 2.77500

One-Sample Test

Test Value = 50

95% Confidence Interval of the

Difference

Lower Upper

writing score 1.4533 4.0967

The mean of the variable write for this particular sample of students is 52.775, which is statistically significantly (p<.001) different from the test value of 50. We would conclude that this group of students has a significantly higher mean on the writing test than 50. This is consistent with the reported confidence interval (1.45,4.10) that is (51.45,54.10) which excludes 50, of course the mid-point is the mean.

Confidence intervalCrichton, N.Journal Of Clinical Nursing 8(5) 618-618 1999

http://www.blackwellpublishing.com/specialarticles/jcn_8_618.pdf

One sample t-testEffect Size Statistics

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SPSS supplies all the information necessary to compute an effect size, d, given by:

d = Mean Difference / SDwhere the mean difference and standard deviation are reported in the SPSS output. We can also compute d from the t value by using the equation

where N is the total sample size. d evaluates the degree that the mean on the test variable differs from the test value in standard deviation units. Potentially, d can range in value from negative infinity to positive infinity. If d equals 0, the mean of the scores is equal to the test value. As d deviates from 0, we interpret the effect size to be stronger. What is a small versus a large d is dependent on the area of investigation. However, d values of .2, .5 and .8, regardless of sign, are by convention interpreted as small, medium, and large effect sizes, respectively.

Nt

d

One sample t-testAn APA Results Section

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A one-sample t test was conducted to evaluate whether the mean of the writing scores was significantly different from 50, the accepted mean. The sample mean of 52.78 ( SD = 9.48) was significantly different from 50, t(199) = 4.14, p < .001. The 95% confidence interval for the writing scores mean ranged from 51.45 to 54.10. The effect size d of .29 indicates a medium effect.

Index End

One sample median test

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A one sample median test allows us to test whether a sample median differs significantly from a hypothesized value. We will use the same variable, write, as we did in the one sample t-test example above. But we do not need to assume that it is interval and normally distributed (we only need to assume that write is an ordinal variable).

Menu selection:- Analyze > Nonparametric Tests > One Sample

Syntax:- nptests /onesample test (write) wilcoxon(testvalue = 50).


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17

Choose customize analysis


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Only retain writing score


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Choose tests tick “compare median…” and enter 50 as the desired value.Finally select the “run” button


20Index End

We would conclude that this group of students has a significantly higher median (calculated median 54) on the writing test than 50.

Binomial test

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A one sample binomial test allows us to test whether the proportion of successes on a two-level categorical dependent variable significantly differs from a hypothesized value. For example, using the A data file, say we wish to test whether the proportion of females (female) differs significantly from 50%, i.e., from .5. We can do this as shown below.

Two alternate approaches are available.

EitherMenu selection:- Analyze > Nonparametric Tests > One Sample

Syntax:- npar tests /binomial (.5) = female.Two-sided confidence intervals for the single proportion: Comparison of seven methodsNewcombe, R.G.Statistics In Medicine 1998 17(8) 857-872DOI: 10.1002/(SICI)1097-0258(19980430)17:8<857::AID-SIM777>3.0.CO;2-E

http://dx.doi.org/10.1002/(SICI)1097-0258(19980430)17:8%3C857::AID-SIM777%3E3.0.CO;2-E

Binomial test

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Binomial test

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Choose customize analysis

Binomial test

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Only retain female

Binomial test

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Choose tests tick “compare observed…” and under options

Binomial test

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enter .5 as the desired value.Finally select the “run” button

Binomial test

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OrMenu selection:- Analyze > Nonparametric Tests > Legacy Dialogs > Binomial

Syntax:- npar tests /binomial (.5) = female.

Binomial test

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Select female as the test variable, the default test proportion is .5Finally select the “OK” button

Binomial test

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Binomial Test

Category N

Observed Prop. Test Prop.

Exact Sig. (2-tailed)

Group 1 Male 91 .46 .50 .229 Group 2 Female 109 .54

female

Total 200 1.00

The results indicate that there is no statistically significant difference (p = 0.229). In other words, the proportion of females in this sample does not significantly differ from the hypothesized value of 50%.

Binomial testAn APA Results Section

30Index End

We hypothesized that the proportion of females is 50%. A two-tailed, binomial test was conducted to assess this research hypothesis. The observed proportion of .46 did not differ significantly from the hypothesized value of .50, two-tailed p = .23. Our results suggest that the proportion of females do not differ dramatically from males.

Chi-square goodness of fit

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A chi-square goodness of fit test allows us to test whether the observed proportions for a categorical variable differ from hypothesized proportions. For example, let's suppose that we believe that the general population consists of 10% Hispanic, 10% Asian, 10% African American and 70% White folks. We want to test whether the observed proportions from our sample differ significantly from these hypothesized proportions. Note this example employs input data (10, 10, 10, 70), in addition to A.

Menu selection:- At present the drop down menu’s cannot provide this analysis.

Syntax:- npar test /chisquare = race /expected = 10 10 10 70.

Chi-square goodness of fit

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race

Observed N

Expected N Residual

hispanic 24 20.0 4.0 asian 11 20.0 -9.0 african-amer

20 20.0 .0

white 145 140.0 5.0 Total 200

Test Statistics

race Chi-Square 5.029a df 3 Asymp. Sig.

.170

a. 0 cells (.0%) have expected frequencies less than 5. The minimum expected cell frequency is 20.0.

These results show that racial composition in our sample does not differ significantly from the hypothesized values that we supplied (chi-square with three degrees of freedom = 5.029, p = 0.170).

Index End

Two independent samples t-test

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An independent samples t-test is used when you want to compare the means of a normally distributed interval dependent variable for two independent groups. For example, using the A data file, say we wish to test whether the mean for write is the same for males and females.

Menu selection:- Analyze > Compare Means > Independent Samples T test

Syntax:- t-test groups = female(0 1) /variables = write.


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35


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Do not forget to define those “pesky” groups.

Levene's test

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In statistics, Levene's test is an inferential statistic used to assess the equality of variances in different samples. Some common statistical procedures assume that variances of the populations from which different samples are drawn are equal. Levene's test assesses this assumption. It tests the null hypothesis that the population variances are equal (called homogeneity of variance or homoscedasticity). If the resulting p-value of Levene's test is less than some critical value (typically 0.05), the obtained differences in sample variances are unlikely to have occurred based on random sampling from a population with equal variances. Thus, the null hypothesis of equal variances is rejected and it is concluded that there is a difference between the variances in the population.

Levene, Howard (1960). "Robust tests for equality of variances". In Ingram Olkin, Harold Hotelling, et al. Stanford University Press. pp. 278–292.


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Because the standard deviations for the two groups are not similar (10.3 and 8.1), we will use the "equal variances not assumed" test. This is supported by the Levene’s test p = .001).

The results indicate that there is a statistically significant difference between the mean writing score for males and females (t = -3.656, p < .0005). In other words, females have a statistically significantly higher mean score on writing (54.99) than males (50.12).

This is supported by the negative confidence interval (male - female).

Group Statistics female N Mean Std. Deviation Std. Error Mean

writing score male 91 50.1209 10.30516 1.08027

female 109 54.9908 8.13372 .77907

Independent Samples Test

Levene's Test for Equality of

Variances

F Sig.

writing score Equal variances assumed 11.133 .001

Equal variances not assumed


t-test for Equality of Means

t df Sig. (2-tailed) Mean

Difference

writing score Equal variances assumed -3.734 198 .000 -4.86995


-3.656 169.707 .000 -4.86995



Std. Error Difference

95% Confidence Interval of the Difference

Lower Upper

writing score Equal variances assumed 1.30419 -7.44183 -2.29806


1.33189 -7.49916 -2.24073


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Group Statistics female N Mean Std. Deviation Std. Error Mean

writing score male 91 50.1209 10.30516 1.08027

female 109 54.9908 8.13372 .77907


Levene's Test for Equality of

Variances

F Sig.

writing score Equal variances assumed 11.133 .001




t df Sig. (2-tailed) Mean

Difference

writing score Equal variances assumed -3.734 198 .000 -4.86995


-3.656 169.707 .000 -4.86995



Std. Error Difference

95% Confidence Interval of the Difference

Lower Upper

writing score Equal variances assumed 1.30419 -7.44183 -2.29806


1.33189 -7.49916 -2.24073

Does equality of variances matter in this case?

Two independent samples t-test - Effect Size

Statistic

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Eta square, η2 , may be computed . An η2 ranges in value from 0 to 1. It is interpreted as the proportion of variance of the test variable that is a function of the grouping variable. A value of 0 indicates that the difference in the mean scores is equal to 0, whereas a value of 1 indicates that the sample means differ, and the test scores do not differ within each group (i.e., perfect replication). You can compute η2 with the following equation:

What is a small versus a large η2 is dependent on the area of investigation. However, η2 of .01, .06, and .14 are, by convention, interpreted as small, medium, and large effect sizes, respectively.

See below.

2212

22

NNtt

Two independent samples t-test - An APA Results

Section

41Index End

An independent-samples t test was conducted to evaluate the hypothesis that the mean writing score was gender dependent. The test was significant, t (198) = -3.656, p < .0005. Male students scored lower (M = 50.12, SD = 10.31), on average, than females (M = 54.99, SD = 8.13). The 95% confidence interval for the difference in means was quite wide, ranging from -7.44 to -2.30. The eta square index indicated that 7% (η2 = .066) of the variance of the writing score was explained by gender.

It is important that you select the t test (two independent samples or paired t test) that employs most information about your data. See the example.

Wilcoxon-Mann-Whitney test

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The Wilcoxon-Mann-Whitney test is a non-parametric analog to the independent samples t-test and can be used when you do not assume that the dependent variable is a normally distributed interval variable (you only assume that the variable is at least ordinal). You will notice that the SPSS syntax for the Wilcoxon-Mann-Whitney test is almost identical to that of the independent samples t-test. We will use the same data file (the A data file) and the same variables in this example as we did in the independent t-test example above. We will not assume that write, our dependent variable, is normally distributed. Menu selection:- Analyze > Nonparametric Tests

> Legacy Dialogs > 2 Independent Samples Syntax:- npar test /m-w = write by female(0 1).

Mann-Whitney test Crichton, N. Journal Of Clinical Nursing 2000 9(4) 583-583



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The Mann-Whitney U: A Test for Assessing Whether Two Independent Samples Come from the Same Distribution Nadim NacharTutorials in Quantitative Methods for Psychology 2008 4(1) 13-20

It is often difficult, particularly when conducting research in psychology, to have access to large normally distributed samples. Fortunately, there are statistical tests to compare two independent groups that do not require large normally distributed samples. The Mann‐Whitney U is one of these tests. In the work, a summary of this test is presented. The explanation of the logic underlying this test and its application are also presented. Moreover, the forces and weaknesses of the Mann‐WhitneyU are mentioned. One major limit of the Mann‐Whitney U is that the type‑I error or alpha (α) is amplified in a situation of heteroscedasticity.

Heteroscedasticity refers to the circumstance in which the variability of a variable is unequal across the range of values of a second variable that predicts it.

http://www.tqmp.org/Content/vol04-1/p013/p013.pdf

http://www.tqmp.org/Content/vol04-1/p013/p013.pdf


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The Wilcoxon-Mann-Whitney test is sometimes used for comparing the efficacy of two treatments in trials. It is often presented as an alternative to a t test when the data are not normally distributed.

Where as a t test is a test of population means, the Mann-Whitney test is commonly regarded as a test of population medians. This is not strictly true, and treating it as such can lead to inadequate analysis of data.

Mann-Whitney test is not just a test of medians: differences in spread can be importantAnna HartBritish Medical Journal 2001 August 18; 323(7309): 391–393.

As is always the case, it is not sufficient merely to report a p value. In the case of the Mann-Whitney test, differences in spread may sometimes be as important as differences in medians, and these need to be made clear.

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1120984/



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Note that Mann-Whitney has been selected.


47

Do not forget to define those “pesky” groups.


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Ranks

female N Mean Rank

Sum of Ranks

male 91 85.63 7792.00 female 109 112.92 12308.00

writing score

Total 200

Test Statisticsa

writing score

Mann-Whitney U 3606.000 Wilcoxon W 7792.000 Z -3.329 Asymp. Sig. (2-tailed)

.001

a. Grouping Variable: female

The results suggest that there is a statistically significant difference between the underlying distributions of the write scores of males and the write scores of females (z = -3.329, p = 0.001).

Wilcoxon-Mann-Whitney test - An APA Results

Section

49Index End

A Wilcoxon test was conducted to evaluate whether writing score was affected by gender. The results indicated a significant difference, z = -3.329, p = .001. The mean of the ranks for male was 85.63, while the mean of the ranks for female was 112.92.

Chi-square test (Contingency table)

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A chi-square test is used when you want to see if there is a relationship between two categorical variables. It is equivalent to the correlation between nominal variables.

A chi-square test is a common test for nominal (categorical) data. One application of a chi-square test is a test for independence. In this case, the null hypothesis is that the occurrence of the outcomes for the two groups is equal. If your data for two groups came from the same participants (i.e. the data were paired), you should use the McNemar's test, while for k groups you should use Cochran’s Q test.

Chi-square test (Contingency table)

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In SPSS, the chisq option is used on the statistics subcommand of the crosstabs command to obtain the test statistic and its associated p‑value. Using the A data file, let's see if there is a relationship between the type of school attended (schtyp) and students' gender (female). Remember that the chi-square test assumes that the expected value for each cell is five or higher. This assumption is easily met in the examples below. However, if this assumption is not met in your data, please see the section on Fisher's exact test. Two alternate approaches are available.

EitherMenu selection:- Analyze > Tables > Custom Tables

Syntax:- crosstabs /tables = schtyp by female

/statistic = chisq phi.

Chi-square test

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Chi-square test

53Drag selected variables to the row/column boxes

Chi-square test

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Select chi-squared

Alternately

Chi-square test

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Menu selection:- Analyze > Descriptive Statistics > CrosstabsSyntax:- crosstabs /tables = schtyp by female

/statistic = chisq.

Chi-square test

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Select row and column variables.

Chi-square test

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Select Chi-square and Cramér’s V

Cramér's V (sometimes called phi, see below) is a measure of association between two nominal variables, giving a value between 0 and +1 (inclusive). It is based on Pearson's chi-squared statistic

Cramér, Harald. 1946. Mathematical Methods of Statistics. Princeton: Princeton University Press, p282. ISBN 0-691-08004-6

Chi-square test

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Case Processing Summary Cases

Valid Missing Total

N Percent N Percent N Percent type of school * female

200 100.0% 0 .0% 200 100.0%

type of school * female Crosstabulation

Count Female

Male female Total

public 77 91 168 type of school private 14 18 32 Total 91 109 200

Chi-Square Tests

Value Df

Asymp. Sig. (2-sided)

Exact Sig. (2-sided)


Pearson Chi-Square .047a 1 .828 Continuity Correctionb .001 1 .981 Likelihood Ratio .047 1 .828 Fisher's Exact Test .849 .492 Linear-by-Linear Association

.047 1 .829

N of Valid Cases 200 a. 0 cells (.0%) have expected count less than 5. The minimum expected count is 14.56. b. Computed only for a 2x2 table

These results indicate that there is no statistically significant relationship between the type of school attended and gender (chi-square with one degree of freedom = 0.047, p = 0.828).

Note 0 cells have expected count less than 5. If not use Fisher's exact test.

Chi-square testAn APA Results Section

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A two-way contingency table analysis was conducted to evaluate whether type of school exhibited a gender bias. School and gender were found to not be significantly related, Pearson χ2 (1, N = 200) = .047, p = .828, Cramér’s V = .015. The proportions of males were .85 and females .84.

Symmetric Measures

Value Approx. Sig.

Nominal by Nominal Phi .015 .828

Cramer's V .015 .828

N of Valid Cases 200

Chi-square test

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By adding “/CELLS=COUNT EXPECTED” or selecting “Cells” from the Crosstabs display you can produce the expected cells assuming independence. As required these values exceed 5.

Chi-square test

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Let's look at another example, this time looking at the relationship between gender (female) and socio-economic status (ses). The point of this example is that one (or both) variables may have more than two levels, and that the variables do not have to have the same number of levels. In this example, female has two levels (male and female) and ses has three levels (low, medium and high).

Menu selection:- Analyze > Tables > Custom TablesUsing the previous menu’s.

Syntax:- crosstabs /tables = female by ses /statistic = chisq phi.

Chi-square test

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Valid Missing Total

N Percent N Percent N Percent female * ses 200 100.0% 0 .0% 200 100.0%

female * ses Crosstabulation

Count ses

low middle high Total

male 15 47 29 91 female female 32 48 29 109

Total 47 95 58 200

Chi-Square Tests

Value df

Asymp. Sig. (2-sided)

Pearson Chi-Square 4.577a 2 .101 Likelihood Ratio 4.679 2 .096 Linear-by-Linear Association

3.110 1 .078

N of Valid Cases 200 a. 0 cells (.0%) have expected count less than 5. The minimum expected count is 21.39.

Again we find that there is no statistically significant relationship between the variables (chi-square with two degrees of freedom = 4.577, p = 0.101).

Note the absence of Fisher’s Exact Test!

Chi-square testAn APA Results Section

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A two-way contingency table analysis was conducted to evaluate whether gender was related to social economic status (SES). Gender and SES were not found to be significantly related, Pearson χ2 (2, N = 200) = 4.58, p = .101, Cramér’s V = .151. The proportions of males in low, middle and high SES were .32, .50, and .50, respectively.

Symmetric Measures

Value Approx. Sig.




Chi-square test

64

By adding “/CELLS=COUNT EXPECTED” or selecting “Cells” from the Crosstabs display you can produce the expected cells assuming independence. As required these values exceed 5.

Index End

Phi coefficient

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The measure of association, phi, is a measure which adjusts the chi square statistic by the sample size. The phi coefficient is the equivalent of the correlation between nominal variables.

It may be introduced at the same time as the Chi-square.

Select Phi

Phi coefficient

66Index End

The p values for the tests are identical.

χ2 = 4.577ϕ = 0.151n = 200

Sometimes ϕ2 is used as a measure of association.

Case Processing Summary

Cases

Valid Missing Total

N Percent N Percent N Percent

female * ses 200 100.0% 0 0.0% 200 100.0%

female * ses Crosstabulation

Count ses Total

low middle high

female male 15 47 29 91

female 32 48 29 109

Total 47 95 58 200

Chi-Square Tests

Value df Asymp. Sig. (2-

sided)

Pearson Chi-Square 4.577a 2 .101

Likelihood Ratio 4.679 2 .096

Linear-by-Linear Association 3.110 1 .078

N of Valid Cases 200 a. 0 cells (0.0%) have expected count less than 5. The minimum

expected count is 21.39.

Symmetric Measures

Value Approx. Sig.




151.0200577.42

n

Fisher's exact test

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The Fisher's exact test is used when you want to conduct a chi-square test but one or more of your cells has an expected frequency of five or less. Remember that the chi-square test assumes that each cell has an expected frequency of five or more. Fisher's exact test has no such assumption and can be used regardless of how small the expected frequency is. In SPSS you can only perform a Fisher's exact test on a 2x2 table, and these results are presented by default. Please see the results from the chi-square example above.

Analysis of a 2x2 contingency table is effectively equivalent to a test for a comparison of two proportions.

Interval estimation for the difference between independent proportions: comparison of eleven methodsRobert G. NewcombeStatistics in Medicine Volume 17, Issue 8, pages 873–890, 1998DOI: 10.1002/(SICI)1097-0258(19980430)17:8<873::AID-SIM779>3.0.CO;2-I

http://dx.doi.org/10.1002/(SICI)1097-0258(19980430)17:8%3C873::AID-SIM779%3E3.0.CO;2-I

Fisher's exact test

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As an illustration, a comparison of proportions for the data on the relationship between the type of school attended (schtyp) and students' gender (female), previously analysed using a chi-squared test p-value 0.828, Fisher’s exact test 0.849.

MTB > ptwo 109 91 91 77. Test and CI for Two Proportions

Sample X N Sample p1 91 109 0.8348622 77 91 0.846154

Difference = p (1) - p (2)Estimate for difference: -0.011291595% CI for difference: (-0.113047, 0.0904637)Test for difference = 0 (vs not = 0): Z = -0.22 P-Value = 0.828

Fisher's exact test: P-Value = 0.849

Available in SPSS see slide from Chi squared, but not as a comparison of proportions.

Fisher's exact test

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A simple web search should reveal specific tools developed for different size tables. For example

Fisher's exact test for up to 6×6 tablesFor the more adventurous

For those interested in more detail, plus a worked example see.

Fisher's Exact Test or Paper only

When to Use Fisher's Exact Test Keith M. Bower American Society for Quality, Six Sigma Forum Magazine, 2(4) 2003, 35-37.

http://www.physics.csbsju.edu/stats/exact_NROW_NCOLUMN_form.html

http://aoki2.si.gunma-u.ac.jp/exact/exact.html

http://www.keithbower.com/Miscellaneous/Fisher's%20Exact%20Test.htm

http://asq.org/qic/display-item/index.html?item=19534

Fisher's exact test

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For larger examples you might try (my coding)

Fisher's Exact Test

Algorithm 643

FEXACT - A Fortran Subroutine For Fisher’s Exact Test On Unordered R x C Contingency-TablesMehta, C.R. and Patel, N.R.ACM Transactions On Mathematical Software 12(2) 154-161 1986.

A Remark On Algorithm-643 - FEXACT - An Algorithm ForPerforming Fisher’s Exact Test In R x C Contingency-TablesClarkson, D.B., Fan, Y.A. and Joe, H.ACM Transactions On Mathematical Software 19(4) 484-488 1993. Index End

One-way ANOVA

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A one-way analysis of variance (ANOVA) is used when you have a categorical independent variable (with two or more categories). A normally distributed interval dependent variable. You wish to test for differences in the means of the dependent variable broken down by the levels of the independent variable. For example, using the A data file, say we wish to test whether the mean of write differs between the three program types (prog). The command for this test would be: Menu selection:- Analyze > Compare Means > One-way ANOVA

Syntax:- oneway write by prog.

Information point: Analysis of variance (ANOVA)Crichton, N.Journal Of Clinical Nursing 2000 9(3) 380-380


One-way ANOVA

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One-way ANOVA

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One-way ANOVA

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The mean of the dependent variable differs significantly among the levels of program type. However, we do not know if the difference is between only two of the levels or all three of the levels.

ANOVA writing score

Sum of Squares df

Mean Square F Sig.

Between Groups

3175.698 2 1587.849 21.275 .000

Within Groups 14703.177 197 74.635 Total 17878.875 199

One-way ANOVA

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To see the mean of write for each level of program type, Menu selection:- Analyze > Compare Means > Means

Syntax:- means tables = write by prog.

One-way ANOVA

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One-way ANOVA

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One-way ANOVA

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From this we can see that the students in the academic program have the highest mean writing score, while students in the vocational program have the lowest. For a more detailed analysis refer to Bonferroni for pairwise comparisons .


Included Excluded Total

N Percent N Percent N Percent writing score * type of program

200 100.0% 0 .0% 200 100.0%

Report

writing score type of program Mean N

Std. Deviation

general 51.3333 45 9.39778 academic 56.2571 105 7.94334 vocation 46.7600 50 9.31875 Total 52.7750 200 9.47859

One-way ANOVA

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For an effect size statistic, η2, we need to run a general linear model using an alternate approach.

Recall: Eta square, η2, ranges in value from 0 to 1. It is interpreted as the proportion of variance of the test variable that is a function of the grouping variable. A value of 0 indicates that the difference in the mean scores is equal to 0, whereas a value of 1 indicates that the sample means differ, and the test scores do not differ within each group (i.e., perfect replication).

Syntax

unianova write by prog /method=sstype(3) /intercept=include /print=etasq /criteria=alpha(.05) /design=prog.

One-way ANOVA

80

One-way ANOVA

81

One-way ANOVAAn APA Results Section

82

Tests of Between-Subjects Effects

Dependent Variable: writing score Source Type III Sum of

Squares

df Mean Square F Sig. Partial Eta

Squared

Corrected Model 3175.698a 2 1587.849 21.275 .000 .178

Intercept 460403.797 1 460403.797 6168.704 .000 .969

prog 3175.698 2 1587.849 21.275 .000 .178

Error 14703.177 197 74.635 Total 574919.000 200 Corrected Total 17878.875 199

a. R Squared = .178 (Adjusted R Squared = .169)

Index End

A one-way analysis of variance was conducted to evaluate the relationship between writing score and the type of program. The independent variable, the type of program, included three levels, general, academic and vocation. The dependent variable was the writing score. The ANOVA was significant at the .05 level, F (2, 197) = 21.28, p < .0005. The strength of relationship between the writing score and the type of program, as assessed by η2 (called partial eta squared), was strong, with the writing score factor accounting for 18% of the variance of the dependent variable.

My note!

Kruskal Wallis test

83

The Kruskal Wallis test is used when you have one independent variable with two or more levels and an ordinal dependent variable. In other words, it is the non-parametric version of ANOVA and a generalized form of the Mann-Whitney test method, since it permits two or more groups. We will use the same data file as the one way ANOVA example above (the A data file) and the same variables as in the example above, but we will not assume that write is a normally distributed interval variable. Menu selection:- Analyze > Nonparametric Tests

> Legacy Dialogs > k Independent Samples

Syntax:- npar tests /k-w = write by prog (1,3).

Kruskal Wallis test

84

Kruskal Wallis test

85

Kruskal Wallis test

86

Do not forget the range for those “pesky” groups.

Kruskal Wallis test

87

Ranks type of

program N Mean Rank

general 45 90.64 academic 105 121.56 vocation 50 65.14

writing score

Total 200

Test Statisticsa,b

writing score

Chi-Square 34.045 df 2 Asymp. Sig.

.000

a. Kruskal Wallis Test b. Grouping Variable: type of program

If some of the scores receive tied ranks, then a correction factor is used, yielding a slightly different value of chi-squared. With or without ties, the results indicate that there is a statistically significant difference (p < .0005) among the three type of programs.

Kruskal Wallis testAn APA Results Section

88Index End

A Kruskal-Wallis test was conducted to evaluate differences among the three types of program (general, academic and vocation) on median change in the writing score). The test, which was corrected for tied ranks, was significant, χ2 (2, n = 200) = 34.045, p < .001. The proportion of variability in the ranked dependent variable accounted for by the type of program variable was .17 (χ2/(n-1)), indicating a fairly strong relationship between type of program and writing score.

My note!

Paired t-test

89

A paired (samples) t-test is used when you have two related observations (i.e., two observations per subject) and you want to see if the means on these two normally distributed interval variables differ from one another. For example, using the A data file we will test whether the mean of read is equal to the mean of write.

Often used to compare before/after treatment. Menu selection:- Analyze > Compare Means > Paired-Samples T test

Syntax:- t-test pairs = read with write (paired).

Paired t-test

90

Paired t-test

91

Paired t-test

92

Paired Samples Statistics

Mean N

Std. Deviation

Std. Error Mean

reading score 52.2300 200 10.25294 .72499 Pair 1 writing score 52.7750 200 9.47859 .67024

Paired Samples Correlations

N

Correlation Sig.

Pair 1 reading score & writing score

200 .597 .000

Paired Samples Test

Paired Differences

Mean Std.

Deviation Std. Error

Mean Pair 1 reading score - writing

score -.54500 8.88667 .62838

Paired Samples Test

Paired Differences 95% Confidence Interval of

the Difference

Lower Upper t

Pair 1 reading score - writing score

-1.78414 .69414 -.867

Paired Samples Test

df

Sig. (2-tailed)

Pair 1 reading score - writing score

199 .387

These results indicate that the mean of read is not statistically significantly different from the mean of write (t = -0.867, p = 0.387).

The confidence interval includes the origin (no difference).

Paired t-testAn APA Results Section

93Index End

A paired-samples t test was conducted to evaluate whether reading and writing scores were related. The results indicated that the mean score for writing (M = 52.78, SD = 9.48) was not significantly greater than the mean score fro reading ( M = 52.23, SD = 10.25), t (199) = -.87, p = .39. The standardized effect size index, d , was .06. The 95% confidence interval for the mean difference between the two ratings was -1.78 to .69.

Recall Ntd

Wilcoxon signed rank sum test

94

The Wilcoxon signed rank sum test is the non-parametric version of a paired samples t-test. You use the Wilcoxon signed rank sum test when you do not wish to assume that the difference between the two variables is interval and normally distributed (but you do assume the difference is ordinal). We will use the same example as above, but we will not assume that the difference between read and write is interval and normally distributed. Menu selection:- Analyze > Nonparametric Tests

> Legacy Dialogs > 2 Related Samples

Syntax:- npar test /wilcoxon = write with read (paired).

Wilcoxon signed rank testCrichton, N.Journal Of Clinical Nursing 9(4) 584-584 2000



95


96

Select Wilcoxon


97

Ranks

N

Mean Rank

Sum of Ranks

Negative Ranks

97a 95.47 9261.00

Positive Ranks 88b 90.27 7944.00 Ties 15c

reading score - writing score

Total 200 a. reading score < writing score b. reading score > writing score c. reading score = writing score

Test Statisticsb

reading score

- writing score

Z -.903a Asymp. Sig. (2-tailed)

.366

a. Based on positive ranks. b. Wilcoxon Signed Ranks Test

The results suggest that there is not a statistically significant difference (p = 0.366) between read and write.

Index End

Sign test

98

If you believe the differences between read and write were not ordinal but could merely be classified as positive and negative, then you may want to consider a sign test in lieu of sign rank test. The Sign test answers the question “How Often?”, whereas other tests answer the question “How Much?”. Again, we will use the same variables in this example and assume that this difference is not ordinal. Menu selection:- Analyze > Nonparametric Tests

> Legacy Dialogs > 2 Related Samples

Syntax:- npar test /sign = read with write (paired).

Sign test

99

Sign test

100

Select Sign

For samples that are not too large also select “Exact”

Sign test

101

Frequencies N

Negative Differencesa

88

Positive Differencesb

97

Tiesc 15

writing score - reading score

Total 200 a. writing score < reading score b. writing score > reading score c. writing score = reading score

Test Statisticsa

writing score

- reading score

Z -.588 Asymp. Sig. (2-tailed)

.556

a. Sign Test

We conclude that no statistically significant difference was found (p = 0.556).

Sign testAn APA Results Section

102Index End

A Wilcoxon signed ranks test was conducted to evaluate whether reading and writing scores differed. The results indicated a non-significant difference, z = -.59, p = .56. The mean of the negative ranks were 95.47 and the positive were 90.27.

McNemar test

103

McNemar's test is a statistical test used on paired nominal data. It is applied to 2×2 contingency tables with a dichotomous trait, with matched pairs of subjects, to determine whether the row and column marginal frequencies are equal (that is, whether there is “marginal homogeneity”). For k groups use Cochran’s Q test.

You would perform McNemar's test if you were interested in the marginal frequencies of two binary outcomes. These binary outcomes may be the same outcome variable on matched pairs (like a case-control study) or two outcome variables from a single group. Continuing with the A dataset used in several above examples, let us create two binary outcomes in our dataset: himath and hiread. These outcomes can be considered in a two-way contingency table.

The null hypothesis is that the proportion of students in the himath group is the same as the proportion of students in hiread group (i.e., that the contingency table is symmetric). Menu selection:- Transform > Compute Variable

Analyze > Descriptive Statistics > Crosstabs The syntax is on the next slide.

McNemar test

104

Syntax:- COMPUTE himath=math>60.COMPUTE hiread=read>60.EXECUTE.

CROSSTABS

/TABLES=himath BY hiread /STATISTICS=MCNEMAR /CELLS=COUNT.

McNemar test

105

First the transformation

McNemar test

106Which is utilised twice, for math and read

McNemar test

107

McNemar test

108

Now the test

McNemar test

109

Select McNemar

McNemar test

110


Valid Missing Total

N Percent N Percent N Percent himath * hiread

200 100.0% 0 .0% 200 100.0%

himath * hiread Crosstabulation

Count hiread

.00 1.00 Total

.00 135 21 156 himath 1.00 18 26 44

Total 153 47 200

Chi-Square Tests

Value


McNemar Test .749a N of Valid Cases

200

a. Binomial distribution used.

McNemar's chi-square statistic suggests that there is not a statistically significant difference in the proportion of students in the himath group and the proportion of students in the hiread group.

Alternately accessing the command directly.

McNemar test

111

Menu selection:- Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples

Syntax:- NPAR TESTS/MCNEMAR=himath WITH hiread (PAIRED)/MISSING ANALYSIS/METHOD=EXACT TIMER(5).

McNemar test

112

McNemar test

113

Select McNemar and Exact

McNemar test

114

McNemar's chi-square statistic suggests that there is not a statistically significant difference in the proportion of students in the himath group and the proportion of students in the hiread group.

McNemar testAn APA Results Section

115Index End

Proportions of student scoring high in math and reading were .22 and .24, respectively. A McNemar test, which evaluates differences among related proportions, was not significant, χ2 (1, n = 200) = .10, p = .75.

Cochran’s Q test

116

In the analysis of two-way randomized block designs where the response variable can take only two possible outcomes (coded as 0 and 1), Cochran's Q test is a non-parametric statistical test to verify whether k treatments have identical effects. Your data for the k groups come from the same participants (i.e. the data are paired).

You would perform Cochran’s Q test if you were interested in the marginal frequencies of three or more binary outcomes. Continuing with the A dataset used in several above examples, let us create three binary outcomes in our dataset: himath, hiread and hiwrite. The null hypothesis is that the proportion of students in each group is the same.

Menu selection:- Transform > Compute Variable

Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples

The syntax is on the next slide.

Cochran’s Q test

117

Syntax:- COMPUTE himath=math>60.COMPUTE hiread=read>60.COMPUTE hiwrite=write>60.EXECUTE.

NPAR TESTS /COCHRAN=himath hiread hiwrite /MISSING LISTWISE /METHOD=EXACT TIMER(5).

Cochran’s Q test

118

First transform

Cochran’s Q test

119

Which is utilised three times, for math, read and write.

Now you can perform the test.

Cochran’s Q test

120

Cochran’s Q test

121

Select Friedman, Kendall’s W and Cochran’s Q also Exact

Cochran’s Q test

122

Cochran’s Q statistic (which is a chi-squared statistic) suggests that there is not a statistically significant difference in the proportion of students in the himath, hiread and hiwrite groups.

Cochran’s Q test

123

Necessary for summaryFriedman Test

Ranks

Mean Rank

himath 1.98

hiread 2.00

hiwrite 2.02

Test Statisticsa

N 200

Chi-Square .603

df 2

Asymp. Sig. .740

Exact Sig. .761

Point Probability .058

a. Friedman Test

Kendall's W Test

Ranks

Mean Rank

himath 1.98

hiread 2.00

hiwrite 2.02

Test Statistics

N 200

Kendall's Wa .002

Chi-Square .603

df 2

Asymp. Sig. .740

Exact Sig. .761

Point Probability .058

a. Kendall's Coefficient of

Concordance

Cochran’s Q testAn APA Results Section

124Index End

Proportions of student scoring high in math, reading and writing were .22, .24, and .25, respectively. A Cochran test, which evaluates differences among related proportions, was not significant, χ2 (2, n = 200) = .60, p = .74. The Kendall coefficient of concordance was .002.

Cochran’s Q test

125Index End

When you find any significant effect, you need to do a post-hoc test (as you do for ANOVA). For Cochran's Q test: run multiple McNemar's tests and adjust the p values with the Bonferroni correction (a method used to address the problem of multiple comparisons, over corrects for Type I error).

Cochran, W.G. (1950). The Comparison of Percentages in Matched Samples Biometrika, 37, 256-266.

http://www.jstor.org/stable/2332378

About the B data file

126

We have an example data set called B, which is used in Roger E. Kirk's book Experimental Design: Procedures for Behavioral Sciences (Psychology) (ISBN 0534250920). Syntax:- display dictionary

/VARIABLES s y1 y2 y3 y4.

Variable Position Measurement Level s 1 Ordinal

y1 2 Scale y2 3 Scale y3 4 Scale y4 5 Scale

About the B data file

127Index End

One-way repeated measures ANOVA

128

You would perform a one-way repeated measures analysis of variance if you had one categorical independent variable. A normally distributed interval dependent variable that was repeated at least twice for each subject. This is the equivalent of the paired samples t-test, but allows for two or more levels of the categorical variable. This tests whether the mean of the dependent variable differs by the categorical variable. In data set B, y (y1 y2 y3 y4) is the dependent variable, a is the repeated measure (a name you assign) and s is the variable that indicates the subject number. Menu selection:- Analyze > General Linear Model > Repeated Measures

Syntax:- glm y1 y2 y3 y4 /wsfactor a(4).


129


130

You chose the factor name a which you then “Add”.

You could choose something more meaningfull.


131

You chose the factor name a which you then “Add”.


132

Finally


133

Within-Subjects Factors

Measure:MEASURE_1

a Dependent Variable

1 y1 2 y2 3 y3 4 y4

Multivariate Testsb

Effect Value F Hypothesis

df Error df Sig. Pillai's Trace .754 5.114a 3.000 5.000 .055 Wilks' Lambda .246 5.114a 3.000 5.000 .055 Hotelling's Trace 3.068 5.114a 3.000 5.000 .055

a

Roy's Largest Root

3.068 5.114a 3.000 5.000 .055

a. Exact statistic b. Design: Intercept Within Subjects Design: a

Mauchly's Test of Sphericityb

Measure:MEASURE_1 Within Subjects Effect

Mauchly's W

Approx. Chi-Square df Sig.

a .339 6.187 5 .295

Mauchly's Test of Sphericityb Measure:MEASURE_1

Epsilona Within Subjects Effect

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

A .620 .834 .333 Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. a. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table.

Tests of Within-Subjects Effects Measure:MEASURE_1

Source Type III Sum

of Squares df Mean

Square F Sig. Sphericity Assumed

49.000 3 16.333 11.627 .000

Greenhouse-Geisser

49.000 1.859 26.365 11.627 .001

Huynh-Feldt 49.000 2.503 19.578 11.627 .000

A

Lower-bound 49.000 1.000 49.000 11.627 .011 Sphericity Assumed

29.500 21 1.405

Greenhouse-Geisser

29.500 13.010 2.268

Huynh-Feldt 29.500 17.520 1.684

Error(a)

Lower-bound 29.500 7.000 4.214

Tests of Within-Subjects Contrasts

Measure:MEASURE_1

Source a Type III Sum

of Squares df Mean

Square F Sig. Linear 44.100 1 44.100 19.294 .003 Quadratic

4.500 1 4.500 3.150 .119 A

Cubic .400 1 .400 .800 .401 Linear 16.000 7 2.286 Quadratic

10.000 7 1.429

Error(a)

Cubic 3.500 7 .500


Measure:MEASURE_1 Transformed Variable:Average

Source Type III Sum

of Squares df Mean

Square F Sig. Intercept

578.000 1 578.000 128.444 .000

Error 31.500 7 4.500

Loads of output!!


134

Tests of Within-Subjects Effects Measure:MEASURE_1

Source Type III Sum

of Squares df Mean

Square F Sig. Sphericity Assumed

49.000 3 16.333 11.627 .000

Greenhouse-Geisser

49.000 1.859 26.365 11.627 .001

Huynh-Feldt 49.000 2.503 19.578 11.627 .000

A

Lower-bound 49.000 1.000 49.000 11.627 .011 Sphericity Assumed

29.500 21 1.405

Greenhouse-Geisser

29.500 13.010 2.268

Huynh-Feldt 29.500 17.520 1.684

Error(a)

Lower-bound 29.500 7.000 4.214

You will notice that this output gives four different p-values. The output labelled “sphericity assumed” is the p-value (<0.0005), that you would get if you assumed compound symmetry in the variance-covariance matrix. Because that assumption is often not valid, the three other p-values offer various corrections (the Huynh-Feldt, H-F, Greenhouse-Geisser, G-G and Lower-bound). No matter which p-value you use, our results indicate that we have a statistically significant effect of a at the .05 level.


An APA Results Section

135Index End

A one-way within-subjects ANOVA was conducted with the factor being the number of years married and the dependent variable being y. The means and standard deviations for scores are presented above . The results for the ANOVA indicated a significant time effect, Wilks’s λ = .25, F (3, 21) = 11.63, p < .0005, multivariate η2 = .75 (1-λ).

My note!

Bonferroni for pairwise comparisons

136

This is a minor extension of the previous analysis.

Menu selection:- Analyze

> General Linear Model > Repeated Measures

Syntax:-

GLM y1 y2 y3 y4 /WSFACTOR=a 4 Polynomial /METHOD=SSTYPE(3)

Only the additional outputs are presented.


137

This table simply provides important descriptive statistics for the analysis as shown below.

Descriptive Statistics

Mean Std. Deviation N

3.0000 1.51186 8

3.5000 .92582 8

4.2500 1.03510 8

6.2500 2.12132 8


138

Using post hoc tests to examine whether estimated marginal means differ for levels of specific factors in the model.

Estimated Marginal Means a

Estimates

Measure:MEASURE_1

a Mean Std. Error

95% Confidence Interval

Lower Bound Upper Bound

1 3.000 .535 1.736 4.264

2 3.500 .327 2.726 4.274

3 4.250 .366 3.385 5.115

4 6.250 .750 4.477 8.023


139

The results presented in the previous Tests of Within-Subjects Effects table, the Huynh-Feldt (p < .0005) informed us that we have an overall significant difference in means, but we do not know where those differences occurred.

This table presents the results of the Bonferroni post-hoc test, which allows us to discover which specific means differed.

Remember, if your overall ANOVA result was not significant, you should not examine the Pairwise Comparisons table.

Pairwise Comparisons

Measure:MEASURE_1

(I) a (J) a

Mean Difference

(I-J) Std. Error Sig.a

95% Confidence Interval for

Differencea


1 2 -.500 .327 1.000 -1.690 .690

3 -1.250 .491 .230 -3.035 .535

4 -3.250* .726 .017 -5.889 -.611

2 1 .500 .327 1.000 -.690 1.690

3 -.750 .412 .668 -2.248 .748

4 -2.750 .773 .056 -5.562 .062

3 1 1.250 .491 .230 -.535 3.035

2 .750 .412 .668 -.748 2.248

4 -2.000 .681 .131 -4.477 .477

4 1 3.250* .726 .017 .611 5.889

2 2.750 .773 .056 -.062 5.562

3 2.000 .681 .131 -.477 4.477

Based on estimated marginal means

a. Adjustment for multiple comparisons: Bonferroni.

*. The mean difference is significant at the .05 level.


140

We can see that there was a significant difference between 1 and 4 (p = 0.017), while 2 and 4 merit further consideration.

In true SPSS style the results are duplicated.

Pairwise Comparisons

Measure:MEASURE_1

(I) a (J) a

Mean Difference

(I-J) Std. Error Sig.a

95% Confidence Interval for

Differencea


1 2 -.500 .327 1.000 -1.690 .690

3 -1.250 .491 .230 -3.035 .535

4 -3.250* .726 .017 -5.889 -.611

2 1 .500 .327 1.000 -.690 1.690

3 -.750 .412 .668 -2.248 .748

4 -2.750 .773 .056 -5.562 .062

3 1 1.250 .491 .230 -.535 3.035

2 .750 .412 .668 -.748 2.248

4 -2.000 .681 .131 -4.477 .477

4 1 3.250* .726 .017 .611 5.889

2 2.750 .773 .056 -.062 5.562

3 2.000 .681 .131 -.477 4.477

Based on estimated marginal means

a. Adjustment for multiple comparisons: Bonferroni.

*. The mean difference is significant at the .05 level.


141

The table provides four variants of the F test. Wilks' lambda is the most commonly reported. Usually the same substantive conclusion emerges from any variant. For these data, we conclude that none of effects are significant (p = 0.055). See next slide.

Multivariate Tests

Value F Hypothesis df Error df Sig.

Partial Eta

Squared

Pillai's trace .754 5.114a 3.000 5.000 .055 .754

Wilks' lambda .246 5.114a 3.000 5.000 .055 .754

Hotelling's trace 3.068 5.114a 3.000 5.000 .055 .754

Roy's largest root 3.068 5.114a 3.000 5.000 .055 .754

Each F tests the multivariate effect of a. These tests are based on the linearly independent pairwise comparisons

among the estimated marginal means.

a. Exact statistic


142

Wilks lambda is the easiest to understand and therefore the most frequently used. It has a good balance between power and assumptions. Wilks lambda can be interpreted as the multivariate counterpart of a univariate R-squared, that is, it indicates the proportion of generalized variance in the dependent variables that is accounted for by the predictors.

Correct Use of Repeated Measures Analysis of Variance E. Park, M. Cho and C.-S. Ki. Korean J. Lab. Med. 2009 29 1-9

Wilks' lambda performs, in the multivariate setting, with a combination ofdependent variables, the same role as the F-test performs in one-way analysis of variance. Wilks' lambda is a direct measure of the proportion of variance in the combination of dependent variables that is unaccounted for by the independent variable (the grouping variable or factor). If a large proportion of the variance is accounted for by the independent variable then it suggests that there is an effect from the grouping variable and that the groups have different mean values.

Information Point: Wilks' lambda Nicola Crichton Journal of Clinical Nursing, 9, 381-381, 2000.

Index End

http://synapse.koreamed.org/Synapse/Data/PDFData/0039KJLM/kjlm-29-1.pdf


About the C data file

143

The C data set contains 3 pulse measurements from each of 30 people assigned to 2 different diet regiments and 3 different exercise regiments. Syntax:- display dictionary

/VARIABLES id diet exertype pulse time highpulse. Variable Position

id 1 diet 2

exertype 3 pulse 4 time 5

highpulse 6

About the C data file

144Index End

Repeated measures logistic regression

145

If you have a binary outcome measured repeatedly for each subject and you wish to run a logistic regression that accounts for the effect of multiple measures from single subjects, you can perform a repeated measures logistic regression. In SPSS, this can be done using the GENLIN command and indicating binomial as the probability distribution and logit as the link function to be used in the model. In C, if we define a "high" pulse as being over 100, we can then predict the probability of a high pulse using diet regime. Menu selection:- Analyze > Generalized Estimating Equations

However see the next slide.


146

While the drop down menu’s can be employed to set the arguments it is simpler to employ the syntax window.

Syntax:- GENLIN highpulse (REFERENCE=LAST) BY diet (order=DESCENDING) /MODEL diet DISTRIBUTION=BINOMIAL LINK=LOGIT /REPEATED SUBJECT=id CORRTYPE=EXCHANGEABLE.

For completeness the drop down menu saga is shown, some 9 slides!


147


148


149


150


151


152


153


154


155


156

Model Information Dependent Variable highpulsea Probability Distribution Binomial Link Function Logit Subject Effect

1 id

Working Correlation Matrix Structure Exchangeable a. The procedure models .00 as the response, treating 1.00 as the reference category.


N Percent Included 90 100.0% Excluded

0 .0%

Total 90 100.0%

Correlated Data Summary

Number of Levels Subject Effect

id 30

Number of Subjects 30 Minimum 3 Number of

Measurements per Subject

Maximum 3

Correlation Matrix Dimension 3

Categorical Variable Information

N Percent .00 63 70.0% 1.00 27 30.0%

Dependent Variable

highpulse

Total 90 100.0% 2.00 45 50.0% 1.00 45 50.0%

Factor diet

Total 90 100.0%

Goodness of Fitb Value

Quasi Likelihood under Independence Model Criterion (QIC)a

113.986

Corrected Quasi Likelihood under Independence Model Criterion (QICC)a

111.340

Dependent Variable: highpulse

Model: (Intercept), diet a. Computed using the full log quasi-

likelihood function. b. Information criteria are in small-is-better form.

Tests of Model Effects

Type III

Source Wald Chi-

Square df Sig.

(Intercept) 8.437 1 .004

diet 1.562 1 .211

Dependent Variable: highpulse Model: (Intercept), diet

Parameter Estimates

95% Wald Confidence Interval

Parameter B Std. Error Lower Upper

(Intercept) 1.253 .4328 .404 2.101

[diet=2.00] -.754 .6031 -1.936 .428

[diet=1.00] 0a . . .

(Scale) 1

Loads of output!!


157Index End

Parameter Estimates Hypothesis Test

Parameter

Wald Chi-

Square df Sig. (Intercept) 8.377 1 .004 [diet=2.00] 1.562 1 .211 [diet=1.00] . . . (Scale) Dependent Variable: highpulse Model: (Intercept), diet a. Set to zero because this parameter is redundant.

These results indicate that diet is not statistically significant (Wald Chi-Square = 1.562, p = 0.211).

Factorial ANOVA

158

A factorial ANOVA has two or more categorical independent variables (either with or without the interactions) and a single normally distributed interval dependent variable. For example, using the A data file we will look at writing scores (write) as the dependent variable and gender (female) and socio-economic status (ses) as independent variables, and we will include an interaction of female by ses. Note that in SPSS, you do not need to have the interaction term(s) in your data set. Rather, you can have SPSS create it/them temporarily by placing an asterisk between the variables that will make up the interaction term(s). For the approach adopted here, this step is automatic. However, see the syntax example below. Menu selection:- Analyze > General Linear Model > Univariate

Syntax:- glm write by female ses.

Factorial ANOVA

159

AlternateSyntax:- UNIANOVA write BY female ses

/METHOD=SSTYPE(3) /INTERCEPT=INCLUDE /CRITERIA=ALPHA(0.05) /DESIGN=female ses female*ses.

Note the interaction term, female*ses.

Factorial ANOVA

160

Factorial ANOVA

161

Factorial ANOVA

162

These results indicate that the overall model is statistically significant (F = 5.666, p < 0.0005). The variables female and ses are also statistically significant (F = 16.595, p < 0.0005 and F = 6.611, p = 0.002, respectively). However, note that interaction between female and ses is not statistically significant (F = 0.133, p = 0.875).

Index End

Between-Subjects Factors Value Label N

female .00 male 91

1.00 female 109

ses 1.00 low 47

2.00 middle 95

3.00 high 58

Tests of Between-Subjects Effects Dependent Variable:writing score

Source

Type III Sum of

Squares df Mean Square F Sig.

Corrected Model 2278.244a 5 455.649 5.666 .000

Intercept 473967.467 1 473967.467 5893.972 .000

female 1334.493 1 1334.493 16.595 .000

ses 1063.253 2 531.626 6.611 .002

female * ses 21.431 2 10.715 .133 .875

Error 15600.631 194 80.416

Total 574919.000 200

Corrected Total 17878.875 199

a. R Squared = 0127 (Adjusted R Squared = 0105)

Friedman test

163

You perform a Friedman test when you have one within-subjects independent variable with two or more levels and a dependent variable that is not interval and normally distributed (but at least ordinal). We will use this test to determine if there is a difference in the reading, writing and math scores. The null hypothesis in this test is that the distribution of the ranks of each type of score (i.e., reading, writing and math) are the same. To conduct a Friedman test, the data need to be in a long format (see the next topic). Menu selection:- Analyze > Nonparametric Tests

> Legacy Dialogs > K Related Samples

Syntax:- npar tests /friedman = read write math.

Friedman test

164

Friedman test

165

Friedman test

166Index End

Ranks

Mean Rank

reading score

1.96

writing score 2.04 math score 2.01

Test Statisticsa

N 200 Chi-Square

.645

df 2 Asymp. Sig.

.724

a. Friedman Test

Friedman's chi-square has a value of 0.645 and a p-value of 0.724 and is not statistically significant. Hence, there is no evidence that the distributions of the three types of scores are different.

Reshaping data

167

This example illustrates a wide data file and reshapes it into long form. Consider the data containing the kids and their heights at one year of age (ht1) and at two years of age (ht2).

This is called a wide format since the heights are wide. We may want the data to be long, where each height is in a separate observation.

FAMID BIRTH HT1 HT2 1.00 1.00 2.80 3.40 1.00 2.00 2.90 3.80 1.00 3.00 2.20 2.90 2.00 1.00 2.00 3.20 2.00 2.00 1.80 2.80 2.00 3.00 1.90 2.40 3.00 1.00 2.20 3.30 3.00 2.00 2.30 3.40 3.00 3.00 2.10 2.90 Number of cases read: 9 Number of cases listed: 9

Reshaping data

168

We may want the data to be long, where each height is in a separate observation.

Data may be restructured using the point and click function in SPSS, or pre-processing with Excel.

Index End

FAMID BIRTH AGE HT 1.00 1.00 1.00 2.80 1.00 1.00 2.00 3.40 1.00 2.00 1.00 2.90 1.00 2.00 2.00 3.80 1.00 3.00 1.00 2.20 1.00 3.00 2.00 2.90 2.00 1.00 1.00 2.00 2.00 1.00 2.00 3.20 2.00 2.00 1.00 1.80 2.00 2.00 2.00 2.80 2.00 3.00 1.00 1.90 2.00 3.00 2.00 2.40 3.00 1.00 1.00 2.20 3.00 1.00 2.00 3.30 3.00 2.00 1.00 2.30 3.00 2.00 2.00 3.40 3.00 3.00 1.00 2.10 3.00 3.00 2.00 2.90 Number of cases read: 18 Number of cases listed: 18

Ordered logistic regression

169

Ordered logistic regression is used when the dependent variable is ordered, but not continuous. For example, using the A data file we will create an ordered variable called write3, to use in our logistic regression. This variable will have the values 1, 2 and 3, indicating a low, medium or high writing score. We do not generally recommend categorizing a continuous variable in this way; we are simply creating a variable to use for this example. Menu selection:- Transform > Recode into Different Variables

Syntax:- if write ge 30 and write le 48 write3 = 1.if write ge 49 and write le 57 write3 = 2.if write ge 58 and write le 70 write3 = 3.execute.


170

First compute the new variable.


171

Choose a sensible name.


172

“Add” to create rules and finally “Change”


173

finally “continue”


174

use “change” to execute


175

We will use gender (female), reading score (read) and social studies score (socst) as predictor variables in this model, against our new variable (write3). We will use a logit link and on the print subcommand we have requested the parameter estimates, the (model) summary statistics and the test of the parallel lines assumption. Menu selection:- Analyze > Regression > Ordinal

Syntax:- plum write3 with female read socst /link = logit /print = parameter summary tparallel.


176


177


178


179Index End


N Marginal

Percentage write3 1.00 61 30.5%

2.00 61 30.5% 3.00 78 39.0%

Valid 200 100.0% Missing 0 Total 200

Model Fitting Information

Model -2 Log

Likelihood Chi-Square df Sig. Intercept Only 376.226 Final 252.151 124.075 3 .000 Link function: Logit.

Pseudo R-Square

Cox and Snell .462 Nagelkerke .521 McFadden .284 Link function: Logit.

Parameter Estimates

Estimate Std. Error Wald df Sig. Threshold [write3 = 1.00] 9.704 1.203 65.109 1 .000

[write3 = 2.00] 11.800 1.312 80.868 1 .000 Location female 1.285 .322 15.887 1 .000

read .118 .022 29.867 1 .000 socst .080 .019 17.781 1 .000

Parameter Estimates

95% Confidence Interval

Lower Bound Upper Bound Threshold [write3 = 1.00] 7.347 12.061

[write3 = 2.00] 9.228 14.372 Location female .653 1.918

read .076 .160 socst .043 .117

Link function: Logit.

Test of Parallel Linesa

Model -2 Log

Likelihood Chi-Square df Sig. Null Hypothesis 252.151 General 250.104 2.047 3 .563 The null hypothesis states that the location parameters (slope coefficients) are the same across response categories. a. Link function: Logit.

The results indicate that the overall model is statistically significant (p < .0005), as are each of the predictor variables (p < .0005). There are two thresholds for this model because there are three levels of the outcome variable. We also see that the test of the proportional odds assumption is non-significant (p = 0.563). One of the assumptions underlying ordinal logistic (and ordinal probit) regression is that the relationship between each pair of outcome groups is the same. In other words, ordinal logistic regression assumes that the coefficients that describe the relationship between, say, the lowest versus all higher categories of the response variable are the same as those that describe the relationship between the next lowest category and all higher categories, etc. This is called the proportional odds assumption or the parallel regression assumption. Because the relationship between all pairs of groups is the same, there is only one set of coefficients (only one model). If this was not the case, we would need different models (such as a generalized ordered logit model) to describe the relationship between each pair of outcome groups.

c

c

Factorial logistic regression

180

A factorial logistic regression is used when you have two or more categorical independent variables but a dichotomous dependent variable. For example, using the A data file we will use female as our dependent variable, because it is the only dichotomous variable in our data set; certainly not because it is common practice to use gender as an outcome variable. We will use type of program (prog) and school type (schtyp) as our predictor variables. Because prog is a categorical variable (it has three levels), we need to create dummy codes for it. SPSS will do this for you by making dummy codes for all variables listed after the keyword with. SPSS will also create the interaction term; simply list the two variables that will make up the interaction separated by the keyword by.

Menu selection:- Analyze > Regression > Binary Logistic

Simplest to realise via the syntax window. Syntax:- logistic regression female with prog schtyp prog by schtyp /contrast(prog) = indicator(1).


181


182

Note that the identification of prog as the categorical variable is made below.


183

Use Ctrl with left mouse key to select two variables then >a*b> for the product term.


184

Define categorical variables


185

Indicator(1) identifies value 1 as the (first) reference category


186


Unweighted Casesa N Percent

Selected Cases Included in Analysis 200 100.0

Missing Cases 0 .0

Total 200 100.0 Unselected Cases 0 .0

Total 200 100.0

a. If weight is in effect, see classification table for the total number of cases.

Dependent Variable Encoding

Original Value Internal Value

male 0 female 1

Categorical Variables Codings

Frequency

Parameter coding

(1) (2)

type of program general 45 .000 .000

academic 105 1.000 .000

vocation 50 .000 1.000 Block 0: Beginning Block

Classification Tablea,b

Observed

Predicted female Percentage

Correct male female

Step 0 female Male 0 91 .0

Female 0 109 100.0

Overall Percentage 54.5

a. Constant is included in the model.

b. The cut value is .500

Variables in the Equation B S.E. Wald df Sig. Exp(B)

Step 0 Constant .180 .142 1.616 1 .204 1.198

Variables not in the Equation Score df Sig.

Step 0 Variables Prog .053 2 .974

prog(1) .049 1 .826

prog(2) .007 1 .935

Schtyp .047 1 .828

prog * schtyp .031 2 .985

prog(1) by schtyp .004 1 .950

prog(2) by schtyp .011 1 .917

Overall Statistics 2.923 5 .712 Block 1: Method = Enter

Omnibus Tests of Model Coefficients Chi-square df Sig.

Step 1 Step 3.147 5 .677

Block 3.147 5 .677

Model 3.147 5 .677

Model Summary

Step -2 Log

likelihood Cox & Snell R

Square Nagelkerke R

Square

1 272.490a .016 .021

a. Estimation terminated at iteration number 4 because parameter estimates changed by less than .001.

Classification Tablea

Observed


Correct male female

Step 1 female Male 32 59 35.2

Female 31 78 71.6


a. The cut value is .500

Loads of output!!


187Index End


Step 1 Step 3.147 5 .677

Block 3.147 5 .677

Model 3.147 5 .677

The results indicate that the overall model is not statistically significant (Likelihood ratio Chi2 = 3.147, p = 0.677). Furthermore, none of the coefficients are statistically significant either. This shows that the overall effect of prog is not significant.

Correlation

188

A correlation (Pearson correlation) is useful when you want to see the relationship between two (or more) normally distributed interval variables. For example, using the A data file we can run a correlation between two continuous variables, read and write. Menu selection:- Analyze > Correlate > Bivariate

Syntax:- correlations /variables = read write.

Understanding and Interpreting Correlations - an Interactive Visualization

http://rpsychologist.com/d3/correlation/

http://rpsychologist.com/d3/correlation/

Correlation

189

Correlation

190

Select Pearson

Correlation

191

Correlations reading score writing score

reading score Pearson Correlation 1 .597

Sig. (2-tailed) .000

N 200 200

writing score Pearson Correlation .597 1


N 200 200

In the first example above, we see that the correlation between read and write is 0.597. By squaring the correlation and then multiplying by 100, you can determine what percentage of the variability is shared, 0.597 when squared is .356409, multiplied by 100 would be 36%. Hence read shares about 36% of its variability with write.

No need for diagonal entries, nor to duplicate the results!

Correlation

192

.00-.19 “very weak” .20-.39 “weak” .40-.59 “moderate” .60-.79 “strong” .80-1.0 “very strong”

Which is based on the coefficient of determination (r2). Which indicates the proportion of variance in each of two correlated variables which is shared by both.

An index of the degree of lack of relationship is also available. It is the square root of the proportion of unexplained variance and is called the coefficient of alienation (1-r2)½. This in turn leads to an estimate of error reduction 1-(1-r2)½.

As a rule of thumb use the following guide for the absolute value of correlation (r):

A Graphic and Tabular Aid To Interpreting Correlation Coefficients J.F. Voorhees Monthly Weather Review 54 423 1926.

Correlation

193

In the second example, we will run a correlation between a dichotomous variable, female, and a continuous variable, write. Although it is assumed that the variables are interval and normally distributed, we can include dummy variables when performing correlations. Menu selection:- Analyze > Correlate > Bivariate

Syntax:- correlations /variables = female write.

Correlation

194

In the output for the second example, we can see the correlation between write and female is 0.256. Squaring this number yields .065536, meaning that female shares approximately 6.5% of its variability with write.

Correlations female writing score

female Pearson Correlation 1 .256


N 200 200

writing score Pearson Correlation .256 1


N 200 200

CorrelationAn APA Results Section

For A Table Of Correlations

195Index End

Correlation coefficients were computed among the five self-concept scales. Using the Bonferroni approach to control for Type I error across the 10 correlations, a p value of less than .005 (.05/10 = .005) was required for significance. The results of the correlational analyses presented in the table show that 7 out of the 10 correlations were statistically significant and were greater than or equal to .35 (the critical value at p = .005 and N-2 degrees of freedom).

My note!Five variables, 5×4 = 20, but all pairs are duplicated, so 10 correlations

Simple linear regression

196

Simple linear regression allows us to look at the linear relationship between one normally distributed interval predictor and one normally distributed interval outcome variable. For example, using the A data file, say we wish to look at the relationship between writing scores (write) and reading scores (read); in other words, predicting write from read. Menu selection:- Analyze > Regression > Linear Regression

Syntax:- regression /missing listwise /statistics coeff outs ci(95) r anova /criteria=pin(.05) pout(.10) /noorigin /dependent write /method=enter read.

Regression analysis Crichton, N. Journal Of Clinical Nursing 10(4) 462-462 2001

Information point: coefficient of determination, R2 Crichton, N. Journal Of Clinical Nursing 8(4) 379-279 1999





197


198


199


200

Variables Entered/Removedb

Model Variables Entered

Variables Removed Method

1 reading score . Enter

a. All requested variables entered.

b. Dependent Variable: writing score

Model Summary

Model R R Square Adjusted R

Square Std. Error of the

Estimate

1 .597a .356 .353 7.62487

a. Predictors: (Constant), reading score

ANOVAb

Model Sum of Squares df Mean Square F Sig.

1 Regression 6367.421 1 6367.421 109.521 .000a

Residual 11511.454 198 58.139

Total 17878.875 199

a. Predictors: (Constant), reading score b. Dependent Variable: writing score

Coefficientsa

Model

Unstandardized Coefficients Standardized Coefficients

t Sig. B Std. Error Beta

1 (Constant) 23.959 2.806 8.539 .000

reading score .552 .053 .597 10.465 .000

a. Dependent Variable: writing score

We see that the relationship between write and read is positive (.552) and based on the t-value (10.47) and p-value (<0.0005), we would conclude this relationship is statistically significant. Hence, we would say there is a statistically significant positive linear relationship between reading and writing.

Take care with in/dependent assumptions.

201

Simple linear regression -Plot

Graphs > Legacy Dialogs > Scatter/Dot

Simple scatterSyntax

GRAPH /SCATTERPLOT(BIVAR)=read WITH write /MISSING=LISTWISE.

Note that the fitted line must be added interactively.

202


203


204


To fit a line 1. Open the output file

2. Double click on the graph (the chart editor will open)

3. Click on the reference line icon 4. Click apply and close

205



An APA Results Section

206

Index End

A linear regression analysis was conducted to evaluate the prediction of writing score from the reading score. The scatterplot for the two variables, is shown, indicates that the two variables are linearly related such that as reading score increases the writing score increases. The regression equation for predicting the writing score is

Writing score = .552 Reading score + 23.959

The 95% confidence interval for the slope, .448 to .656 does not contain the value of zero, and therefore overall strength is significantly related to the overall injury index. The correlation between the reading and writing scores was .60. Approximately 36% of the variance of the writing score was accounted for by its linear relationship with the reading score.

Coefficientsa

Model 95.0% Confidence Interval for B


1 (Constant) 18.426 29.492

reading score .448 .656


Non-parametric correlation

207

A Spearman correlation is used when one or both of the variables are not assumed to be normally distributed and interval (but are assumed to be ordinal). The values of the variables are converted to ranks and then correlated. In our example, we will look for a relationship between read and write. We will not assume that both of these variables are normal and interval. Menu selection:- Analyze > Correlate > Bivariate

Syntax:- nonpar corr /variables = read write /print = spearman.


208


209

Select Spearman


210

Correlations reading score writing score

Spearman's rho reading score Correlation Coefficient 1.000 .617

Sig. (2-tailed) . .000

N 200 200

writing score Correlation Coefficient .617 1.000

Sig. (2-tailed) .000 .

N 200 200

The results suggest that the relationship between read and write ( = 0.617, p < 0.0005) is statistically significant.


211Index End

Spearman’s correlation works by calculating Pearson’s correlation on the ranked values of this data. Ranking (from low to high) is obtained by assigning a rank of 1 to the lowest value, 2 to the next lowest and so on. Thus the p value is only “correct” if there are no ties in the data. In the event that ties occur an exact calculation should be employed. SPSS does not do this. However the estimated value is usually reliable enough.

Comparison Of Values Of Pearson’s And Spearman’s Correlation Coefficients On The Same Sets Of Data Jan Hauke and Tomasz KossowskiQuaestiones Geographicae 2011 30(2) 87-93

http://geoinfo.amu.edu.pl/qg/archives/2011/QG302_087-093.pdf

http://geoinfo.amu.edu.pl/qg/archives/2011/QG302_087-093.pdf

Simple logistic regression

212

Logistic regression assumes that the outcome variable is binary (i.e., coded as 0 and 1). We have only one variable in the A data file that is coded 0 and 1, and that is female. We understand that female is a silly outcome variable (it would make more sense to use it as a predictor variable). But we can use female as the outcome variable to illustrate how the code for this command is structured and how to interpret the output. The first variable listed after the logistic command is the outcome (or dependent) variable, and all of the rest of the variables are predictor (or independent) variables. In our example, female will be the outcome variable, and read will be the predictor variable. As with ordinary least squares regression, the predictor variables must be either dichotomous or continuous; they cannot be categorical. Menu selection:- Analyze > Regression > Binary Logistic

Syntax:- logistic regression female with read.


213


214


215



Selected Cases Included in Analysis 200 100.0

Missing Cases 0 .0

Total 200 100.0 Unselected Cases 0 .0

Total 200 100.0




male 0 female 1

Block 0: Beginning Block

Classification Tablea,b

Observed


Correct male female

Step 0 female male 0 91 .0

female 0 109 100.0


a. Constant is included in the model.

b. The cut value is .500


Step 0 Constant .180 .142 1.616 1 .204 1.198

Variables not in the Equation Score df Sig.

Step 0 Variables read .564 1 .453

Overall Statistics .564 1 .453

Block 1: Method = Enter


Step 1 Step .564 1 .453

Block .564 1 .453

Model .564 1 .453

Model Summary

Step -2 Log


Square Nagelkerke R

Square

1 275.073a .003 .004



Observed


Correct male female

Step 1 female male 4 87 4.4

female 5 104 95.4




Step 1a read -.010 .014 .562 1 .453 .990

Constant .726 .742 .958 1 .328 2.067

a. Variable(s) entered on step 1: read.

Loads of output!!


216Index End



Step 1 Step .564 1 .453

Block .564 1 .453

Model .564 1 .453

Model Summary

Step -2 Log


Square Nagelkerke R

Square

1 275.073a .003 .004



Observed


Correct male female

Step 1 female male 4 87 4.4

female 5 104 95.4




Step 1a read -.010 .014 .562 1 .453 .990

Constant .726 .742 .958 1 .328 2.067

a. Variable(s) entered on step 1: read.

The results indicate that reading score (read) is not a statistically significant predictor of gender (i.e., being female), Wald = 0.562, p = 0.453. Likewise, the test of the overall model is not statistically significant, likelihood ratio Chi-squared = 0.564, p = 0.453.

Multiple regression

217

Multiple regression is very similar to simple regression, except that in multiple regression you have more than one predictor variable in the equation. For example, using the A data file we will predict writing score (write) from gender (female), reading (read), math, science and social studies (socst) scores. Menu selection:- Analyze > Regression > Linear Regression

Syntax:- regression variable = write female read math science socst

/dependent = write /method = enter.

Multiple regression

218

Multiple regression

219

Note additional independent variables within box

Multiple regression

220

The results indicate that the overall model is statistically significant (F = 58.60, p < 0.0005). Furthermore, all of the predictor variables are statistically significant except for read.

Variables Entered/Removedb

Model Variables Entered

Variables Removed Method

1 social studies score, female, science score, math score, reading score

. Enter

a. All requested variables entered. b. Dependent Variable: writing score

Model Summary

Model R R Square Adjusted R

Square Std. Error of the

Estimate

1 .776a .602 .591 6.05897

a. Predictors: (Constant), social studies score, female, science score, math score, reading score

ANOVAb

Model Sum of Squares df Mean Square F Sig.

Regression 10756.924 5 2151.385 58.603 .000a

Residual 7121.951 194 36.711 1

Total 17878.875 199

a. Predictors: (Constant), social studies score, female, science score, math score, reading score b. Dependent Variable: writing score

Coefficientsa

Unstandardized Coefficients Standardized Coefficients

Model B Std. Error Beta t Sig.

(Constant) 6.139 2.808 2.186 .030

female 5.493 .875 .289 6.274 .000

reading score .125 .065 .136 1.931 .055

math score .238 .067 .235 3.547 .000

science score .242 .061 .253 3.986 .000

1

social studies score .229 .053 .260 4.339 .000


Multiple regression An APA Results Section

221

A multiple regression analysis was conducted to evaluate how well gender, reading and math scores predicted writing score. The predictors were the three indices, while the criterion variable was the writing score. The linear combination of measures was significantly related to the score, F (5, 194) = 58.6, p <.0005. The sample multiple correlation coefficient was .78, indicating that approximately 60% (.7762 = .602) of the variance of the writing score in the sample can be accounted for by the linear combination of the other measures. The table (coefficients table above) presents indices to indicate the relative strength of the variables.

My notes!

Multiple regression - Alternatives

222

There are problems with stepwise model selection procedures, these notes are a health warning.

Various algorithms have been developed for aiding in model selection. Many of them are “automatic”, in the sense that they have a “stopping rule” (which it might be possible for the researcher to set or change from a default value) based on criteria such as value of a t-statistic or an F-statistic. Others might be better termed “semi-automatic,” in the sense that they automatically list various options and values of measures that might be used to help evaluate them. Caution: Different regression software may use the same name (e.g., “Forward Selection” or “Backward Elimination”) to designate different algorithms. Be sure to read the documentation to know find out just what the algorithm does in the software you are using - in particular, whether it has a stopping rule or is of the “semi-automatic” variety.


223

The reasons for not using a stepwise procedure are as follows.

There is a great deal of arbitrariness in the procedures. Forwards and backwards stepwise methods will in general give different “best models”. There are differing criteria for accepting or rejecting a variable at any stage and also for when to stop and declare the current model “best”. The process gives a false impression of statistical sophistication. Often a complex stepwise analysis is presented, when no proper thought has been given to the real issues involved.


224

Stepwise regressions are nevertheless important for three reasons. First, to emphasise that there is a considerable problem in choosing a model out of so many, so considerable that a variety of automated procedures have been devised to “help”. Second to show that while purely statistical methods of choice can be constructed, they are unsatisfactory. And third, because they are fairly popular ways of avoiding constructive thinking about model selection, you may well come across them. You should know that they exist and roughly how they work.

Stepwise regressions probably do have a useful role to play, when there are large numbers of x‑variables, when all prior information is taken carefully into account in inclusion/exclusion of variables, and when the results are used as a preliminary sifting of the many x-variables. It would be rare for a stepwise regression to produce convincing evidence for or against a scientific hypothesis.


225

“... perhaps the most serious source of error lies in letting statistical procedures make decisions for you.”

Good P.I. and Hardin J.W., Common Errors in Statistics (and How to Avoid Them), 4th Edition, Wiley, 2012, p. 3.

“Don't be too quick to turn on the computer. By passing the brain to compute by reflex is a sure recipe for disaster.”

Good P.I. and Hardin J.W., Common Errors in Statistics (and How to Avoid Them), 4th Edition, Wiley, 2012, p. 152.


226

“We do not recommend such stopping rules for routine use since they can reject perfectly reasonable sub-models from further consideration. Stepwise procedures are easy to explain, inexpensive to compute, and widely used. The comparative simplicity of the results from stepwise regression with model selection rules appeals to many analysts. But, such algorithmic model selection methods must be used with caution.”

Cook R.D. and Weisberg S., Applied Regression Including Computing and Graphics, Wiley, 1999, p. 280.


227

Stopping stepwise: Why stepwise and similar selection methods are bad, and what you should use

A common problem in regression analysis is that of variable selection. Often, you have a large number of potential independent variables, and wish to select among them, perhaps to create a ‘best’ model. One common method of dealing with this problem is some form of automated procedure, such as forward, backward, or stepwise selection. We show that these methods are not to be recommended, and present better alternatives using PROC GLMSELECT and other methods.

Contains useful references, is based on SAS.

http://www.nesug.org/proceedings/nesug07/sa/sa07.pdf

http://www.nesug.org/proceedings/nesug07/sa/sa07.pdf


228

In a large world where parameters need to be estimated from small or unreliable samples, the function between predictive accuracy and the flexibility of a model (e.g., number of free parameters) is typically inversely U shaped. Both too few and too many parameters can hurt performance. Competing models of strategies should be tested for their predictive ability, not their ability to fit already known data.

Pitt M.A., Myung I.J. and Zhang S. 2002. “Toward a method for selecting among computational models for cognition” Psychol. Rev. 109 472–491 DOI: 10.1037/0033-295X.109.3.472

http://dx.doi.org/10.1037/0033-295X.109.3.472


229

What strategies might we adopt?

Heuristics are a subset of strategies; strategies also include complex regression or Bayesian models. The part of the information that is ignored is covered by Shah and Oppenheimer’s (2008) list of five aspects (see below). The goal of making judgments more quickly and frugally is consistent with the goal of effort reduction, where “frugal” is often measured by the number of cues that a heuristic searches.


230

Many definitions of heuristics exist, Shah and Oppenheimer (2008) proposed that all heuristics rely on effort reduction by one or more of the following:

(a) examining fewer cues, (b) reducing the effort of retrieving cue values, (c) simplifying the weighting of cues, (d ) integrating less information, (e) examining fewer alternatives.

Shah, A.K. and Oppenheimer, D.M. 2008 “Heuristics Made Easy: An Effort-Reduction Framework” Psychological Bulletin 134(2) 207-222 DOI: 10.1037/0033-2909.134.2.207

http://dx.doi.org/10.1037/0033-2909.134.2.207


231

Two alternative techniques, dominance analysis (Budescu 1993) and relative weight analysis (Johnson 2000), have been developed that permit more accurate partitioning of variance among correlated predictors. Simulation research clearly shows that these measures of importance perform quite well across a variety of conditions and are recommended for this purpose (LeBreton et al. 2004). Despite calls advocating for the wider use of these indices (Tonidandel and LeBreton 2011), researchers seem reluctant to do so.Budescu, D. V. 1993 “Dominance analysis: A new approach to the problem of relative importance of predictors in multiple regression” Psychological Bulletin, 114, 542–551 DOI: 10.1037/0033-2909.114.3.542.Johnson, J. W. 2000 “A heuristic method for estimating the relative weight of predictor variables in multiple

regression” Multivariate Behavioral Research, 35, 1–19 DOI: 10.1207/S15327906MBR3501_1.LeBreton, J. M., Ployhart, R. E. and Ladd, R. T. 2004 “A Monte Carlo comparison of relative importance methodologies” Organizational Research Methods, 7, 258–282 DOI: 10.1177/1094428104266017.Tonidandel, S., and LeBreton, J. M. 2011 “Relative importance analyses: A useful supplement to multiple regression analyses” Journal of Business and Psychology, 26, 1–9 DOI: 10.1007/s10869-010-9204-3.

Index End

http://dx.doi.org/10.1037/0033-2909.114.3.542

http://dx.doi.org/10.1207/S15327906MBR3501_1

http://dx.doi.org/10.1177/1094428104266017

http://dx.doi.org/10.1007/s10869-010-9204-3

Analysis of covariance

232

Analysis of covariance is like ANOVA, except in addition to the categorical predictors you also have continuous predictors as well. For example, the one way ANOVA example used write as the dependent variable and prog as the independent variable. Let's add read as a continuous variable to this model, as shown below. Menu selection:- Analyze > General Linear Model > Univariate

Syntax:- glm write with read by prog.


233


234

To help understand all these applications, concentrate on the introduction of the variables etc.


235Index End


1.00 general 45

2.00 academic 105

type of program

3.00 vocation 50


Dependent Variable:writing score

Source Type III Sum of

Squares df Mean Square F Sig.

Corrected Model 7017.681a 3 2339.227 42.213 .000 Intercept 4867.964 1 4867.964 87.847 .000 read 3841.983 1 3841.983 69.332 .000 prog 650.260 2 325.130 5.867 .003 Error 10861.194 196 55.414

Total 574919.000 200

Corrected Total 17878.875 199

a. R Squared = 0393 (Adjusted R Squared = 0383)

The results indicate that even after adjusting for the reading score (read), the writing scores still significantly differ by program type (prog), F = 5.867, p = 0.003.

Multiple logistic regression

236

Multiple logistic regression is like simple logistic regression, except that there are two or more predictors. The predictors can be interval variables or dummy variables, but cannot be categorical variables. If you have categorical predictors, they should be coded into one or more dummy variables. We have only one variable in our data set that is coded 0 and 1, and that is female. We understand that female is a silly outcome variable (it would make more sense to use it as a predictor variable). But we can use female as the outcome variable to illustrate how the code for this command is structured and how to interpret the output. The first variable listed after the logistic regression command is the outcome (or dependent) variable, and all of the rest of the variables are predictor (or independent) variables (listed after the keyword with). In our example, female will be the outcome variable, and read and write will be the predictor variables. Menu selection:- Analyze > Regression > Binary Logistic

Syntax:- logistic regression female with read write.


237


238


239



Included in Analysis 200 100.0

Missing Cases 0 .0

Selected Cases

Total 200 100.0

Unselected Cases 0 .0 Total 200 100.0




male 0

female 1 Block 0: Beginning Block

Classification Tablea,b Predicted female Observed male female

Percentage

Correct

male 0 91 .0 female

female 0 109 100.0

Step 0


a. Constant is included in the model. b. The cut value is .500

Variables in the Equation

B S.E. Wald df Sig. Exp(B)

Step 0 Constant .180 .142 1.616 1 .204 1.198

Variables not in the Equation

Score df Sig.

read .564 1 .453 Variables

write 13.158 1 .000

Step 0

Overall Statistics 26.359 2 .000



Step 27.819 2 .000

Block 27.819 2 .000

Step 1

Model 27.819 2 .000

Model Summary

Step -2 Log


Square Nagelkerke R

Square

1 247.818a .130 .174



Predicted female Observed male female

Percentage Correct

male 54 37 59.3 female

female 30 79 72.5

Step 1





read -.071 .020 13.125 1 .000 .931

write .106 .022 23.075 1 .000 1.112

Step 1a

Constant -1.706 .923 3.414 1 .065 .182

a. Variable(s) entered on step 1: read, write.

Loads of output!!


240Index End



read -.071 .020 13.125 1 .000 .931

write .106 .022 23.075 1 .000 1.112

Step 1a

Constant -1.706 .923 3.414 1 .065 .182

a. Variable(s) entered on step 1: read, write.

These results show that both read and write are significant predictors of female.

Discriminant analysis

241

Discriminant analysis is used when you have one or more normally distributed interval independent variable(s) and a categorical dependent variable. It is a multivariate technique that considers the latent dimensions in the independent variables for predicting group membership in the categorical dependent variable. For example, using the A data file, say we wish to use read, write and math scores to predict the type of program a student belongs to (prog). Menu selection:- Analyze > Classify > Discriminant

Syntax:- Discriminant groups = prog(1, 3) /variables = read write math.


242


243

Do not forget to define the range for Prog.


244

Analysis Case Processing Summary

Unweighted Cases N Percent

Valid 200 100.0 Missing or out-of-range group codes

0 .0

At least one missing discriminating variable

0 .0

Both missing or out-of-range group codes and at least one missing discriminating variable

0 .0

Excluded

Total 0 .0 Total 200 100.0

Group Statistics

Valid N (listwise) type of program Unweighted Weighted

reading score 45 45.000

writing score 45 45.000

general

math score 45 45.000



academic




vocation




Total

math score 200 200.000 Analysis 1 Summary of Canonical Discriminant Functions

Eigenvalues

Function Eigenvalue % of Variance Cumulative % Canonical

Correlation

1 .356a 98.7 98.7 .513 2 .005a 1.3 100.0 .067

a. First 2 canonical discriminant functions were used in the analysis.

Wilks' Lambda

Test of Function(s) Wilks' Lambda Chi-square df Sig.

1 through 2 .734 60.619 6 .000 2 .995 .888 2 .641

Standardized Canonical Discriminant

Function Coefficients

Function 1 2

reading score .273 -.410 writing score .331 1.183

math score .582 -.656

Structure Matrix

Function 1 2

math score .913* -.272

reading score .778* -.184 writing score .775* .630

Pooled within-groups correlations between discriminating variables and

standardized canonical discriminant functions Variables ordered by absolute size of

correlation within function.

*. Largest absolute correlation between each variable and any discriminant

function

Functions at Group Centroids

Function

type of program 1 2

general -.312 .119

academic .536 -.020 vocation -.844 -.066

Unstandardized canonical discriminant

functions evaluated at group means

Loads of output!!


245Index End

Functions at Group Centroids

Function type of program 1 2

general -.312 .119 academic .536 -.020 vocation -.844 -.066

Unstandardized canonical discriminant functions evaluated at group means

Clearly, the SPSS output for this procedure is quite lengthy, and it is beyond the scope of this page to explain all of it. However, the main point is that two canonical variables are identified by the analysis, the first of which seems to be more related to program type than the second.

One-way MANOVA

246

MANOVA (multivariate analysis of variance) is like ANOVA, except that there are two or more dependent variables. In a one-way MANOVA, there is one categorical independent variable and two or more dependent variables. For example, using the A data file, say we wish to examine the differences in read, write and math broken down by program type (prog). Menu selection:- Analyse > General Linear Model > Multivariate

Syntax:- glm read write math by prog.

One-way MANOVA

247

One-way MANOVA

248

One-way MANOVA

249


1.00 general 45 2.00 academic 105

type of program

3.00 vocation 50

Multivariate Testsc Effect Value F Hypothesis df Error df Sig.

Pillai's Trace .978 2883.051a 3.000 195.000 .000 Wilks' Lambda .022 2883.051a 3.000 195.000 .000 Hotelling's Trace 44.355 2883.051a 3.000 195.000 .000

Intercept

Roy's Largest Root 44.355 2883.051a 3.000 195.000 .000 Pillai's Trace .267 10.075 6.000 392.000 .000 Wilks' Lambda .734 10.870a 6.000 390.000 .000 Hotelling's Trace .361 11.667 6.000 388.000 .000

prog

Roy's Largest Root .356 23.277b 3.000 196.000 .000 a. Exact statistic b. The statistic is an upper bound on F that yields a lower bound on the significance level. c. Design: Intercept + prog


Source Dependent Variable Type III Sum of

Squares df Mean Square reading score 3716.861a 2 1858.431 writing score 3175.698b 2 1587.849

Corrected Model

math score 4002.104c 2 2001.052 reading score 447178.672 1 447178.672 writing score 460403.797 1 460403.797

Intercept

math score 453421.258 1 453421.258 reading score 3716.861 2 1858.431 writing score 3175.698 2 1587.849

prog

math score 4002.104 2 2001.052 reading score 17202.559 197 87.323 writing score 14703.177 197 74.635

Error

math score 13463.691 197 68.344 reading score 566514.000 200 writing score 574919.000 200

Total

math score 571765.000 200 reading score 20919.420 199 writing score 17878.875 199

Corrected Total

math score 17465.795 199

Concentrate on the third table

One-way MANOVA

250Index End

Concluding output table.

The students in the different programs differ in their joint distribution of read, write and math.

Multivariate multiple regression

251

Multivariate multiple regression is used when you have two or more dependent variables that are to be predicted from two or more independent variables. In our example, we will predict write and read from female, math, science and social studies (socst) scores. Menu selection:- Analyse > General Linear Model > Multivariate

Syntax:- glm write read with female math science socst.


252


253


254

Multivariate Testsb

Effect Value F Hypothesis df Error df Sig.

Pillai's Trace .030 3.019a 2.000 194.000 .051

Wilks' Lambda .970 3.019a 2.000 194.000 .051

Hotelling's Trace .031 3.019a 2.000 194.000 .051

Intercept

Roy's Largest Root .031 3.019a 2.000 194.000 .051

Pillai's Trace .170 19.851a 2.000 194.000 .000

Wilks' Lambda .830 19.851a 2.000 194.000 .000


female


Pillai's Trace .160 18.467a 2.000 194.000 .000

Wilks' Lambda .840 18.467a 2.000 194.000 .000


math


Pillai's Trace .166 19.366a 2.000 194.000 .000

Wilks' Lambda .834 19.366a 2.000 194.000 .000


science


Pillai's Trace .221 27.466a 2.000 194.000 .000

Wilks' Lambda .779 27.466a 2.000 194.000 .000


socst


a. Exact statistic b. Design: Intercept + female + math + science + socst


Source Dependent Variable Type III Sum of

Squares df Mean Square

writing score 10620.092a 4 2655.023 Corrected Model

reading score 12219.658b 4 3054.915

writing score 202.117 1 202.117 Intercept

reading score 55.107 1 55.107

writing score 1413.528 1 1413.528 female

reading score 12.605 1 12.605

writing score 714.867 1 714.867 math

reading score 1025.673 1 1025.673

writing score 857.882 1 857.882 science

reading score 946.955 1 946.955

writing score 1105.653 1 1105.653 socst

reading score 1475.810 1 1475.810

writing score 7258.783 195 37.225 Error

reading score 8699.762 195 44.614

writing score 574919.000 200 Total

reading score 566514.000 200

writing score 17878.875 199 Corrected Total

reading score 20919.420 199

Concentrate on the this table


255Index End


Source Dependent Variable F Sig.

writing score 71.325 .000 Corrected Model

reading score 68.474 .000

writing score 5.430 .021 Intercept


writing score 37.973 .000 female

reading score .283 .596

writing score 19.204 .000 math


writing score 23.046 .000 science


writing score 29.702 .000 socst


writing score Error

reading score

writing score Total

reading score

writing score Corrected Total

reading score

a. R Squared = 0594 (Adjusted R Squared = 0586) b. R Squared = 0584 (Adjusted R Squared = 0576)

Concluding table.

These results show that all of the variables in the model have a statistically significant relationship with the joint distribution of write and read.

Canonical correlation

256

Canonical correlation is a multivariate technique used to examine the relationship between two groups of variables. For each set of variables, it creates latent variables and looks at the relationships among the latent variables. It assumes that all variables in the model are interval and normally distributed. SPSS requires that each of the two groups of variables be separated by the keyword with. There need not be an equal number of variables in the two groups (before and after the with). In this case {read, write} with {math, science}.

Canonical correlation are the correlations of two canonical (latent) variables, one representing a set of independent variables, the other a set of dependent variables. There may be more than one such linear correlation relating the two sets of variables, with each correlation representing a different dimension by which the independent set of variables is related to the dependent set. The purpose of the method is to explain the relation of the two sets of variables, not to model the individual variables.


257

Canonical correlation analysis is the study of the linear relations between two sets of variables. It is the multivariate extension of correlation analysis.

Suppose you have given a group of students two tests of ten questions each and wish to determine the overall correlation between these two tests. Canonical correlation finds a weighted average of the questions from the first test and correlates this with a weighted average of the questions from the second test. The weights are constructed to maximize the correlation between these two averages. This correlation is called the first canonical correlation coefficient.

You can create another set of weighted averages unrelated to the first and calculate their correlation. This correlation is the second canonical correlation coefficient. This process continues until the number of canonical correlations equals the number of variables in the smallest group.


258

In statistics, canonical-correlation analysis is a way of making sense of cross-covariance matrices. If we have two vectors X = (X1, ..., Xn) and Y = (Y1, ..., Ym) of random variables, and there are correlations among the variables, then canonical-correlation analysis will find linear combinations of the Xi and Yj which have maximum correlation with each other (Härdle and Léopold 2007). T. R. Knapp notes “virtually all of the commonly encountered parametric tests of significance can be treated as special cases of canonical-correlation analysis, which is the general procedure for investigating the relationships between two sets of variables.” The method was first introduced by Harold Hotelling in 1936.

Härdle, Wolfgang and Simar, Léopold (2007). “Canonical Correlation Analysis”. Applied Multivariate Statistical Analysis. pp. 321–330. Canonical Correlation Analysis - Springer ISBN 978-3-540-72243-4. Knapp, T. R. (1978). “Canonical correlation analysis: A general parametric significance-testing system”. Psychological Bulletin 85(2): 410–416 DOI: 10.1037/0033-2909.85.2.410 .Hotelling, H. (1936). “Relations Between Two Sets of Variates”. Biometrika 28 (3–4): 321–377 DOI: 10.1093/biomet/28.3-4.321 .

http://link.springer.com/chapter/10.1007%2F978-3-540-72244-1_14

http://dx.doi.org/10.1037/0033-2909.85.2.410

http://dx.doi.org/10.1093/biomet/28.3-4.321


259

The manova command, for canonical correlation, is one of the SPSS commands that can only be accessed via syntax; there is not a sequence of pull-down menus or point-and-clicks that could arrive at this analysis.

Syntax:- manova read write with math science /discrim all alpha(1) /print=sig(eigen dim).


260

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - * * * * * * * * * * * * * * * * * A n a l y s i s o f V a r i a n c e -- Design 1 * * * * * * * * * * * * * * EFFECT .. WITHIN CELLS Regression Multivariate Tests of Significance (S = 2, M = -1/2, N = 97 ) Test Name Value Approx. F Hypoth. DF Error DF Sig. of F Pillais .59783 41.99694 4.00 394.00 .000 Hotellings 1.48369 72.32964 4.00 390.00 .000 Wilks .40249 56.47060 4.00 392.00 .000 Roys .59728 Note.. F statistic for WILKS' Lambda is exact. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Eigenvalues and Canonical Correlations Root No. Eigenvalue Pct. Cum. Pct. Canon Cor. Sq. Cor 1 1.48313 99.96283 99.96283 .77284 .59728 2 .00055 .03717 100.00000 .02348 .00055 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Dimension Reduction Analysis Roots Wilks L. F Hypoth. DF Error DF Sig. of F 1 TO 2 .40249 56.47060 4.00 392.00 .000 2 TO 2 .99945 .10865 1.00 197.00 .742 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - EFFECT .. WITHIN CELLS Regression (Cont.) Univariate F-tests with (2,197) D. F. Variable Sq. Mul. R Adj. R-sq. Hypoth. MS Error MS F Sig. of F read .51356 .50862 5371.66966 51.65523 103.99081 .000 write .43565 .42992 3894.42594 51.21839 76.03569 .000 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

The output shows the linear combinations corresponding to the first canonical correlation. At the bottom of the output are the two canonical correlations. These results indicate that the first canonical correlation is .7728.


261Index End

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - * * * * * * * * * * * * * * * * * A n a l y s i s o f V a r i a n c e -- Design 1 * * * * * * * * * * * * * * EFFECT .. WITHIN CELLS Regression Multivariate Tests of Significance (S = 2, M = -1/2, N = 97 ) Test Name Value Approx. F Hypoth. DF Error DF Sig. of F Pillais .59783 41.99694 4.00 394.00 .000 Hotellings 1.48369 72.32964 4.00 390.00 .000 Wilks .40249 56.47060 4.00 392.00 .000 Roys .59728 Note.. F statistic for WILKS' Lambda is exact. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Eigenvalues and Canonical Correlations Root No. Eigenvalue Pct. Cum. Pct. Canon Cor. Sq. Cor 1 1.48313 99.96283 99.96283 .77284 .59728 2 .00055 .03717 100.00000 .02348 .00055 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Dimension Reduction Analysis Roots Wilks L. F Hypoth. DF Error DF Sig. of F 1 TO 2 .40249 56.47060 4.00 392.00 .000 2 TO 2 .99945 .10865 1.00 197.00 .742 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - EFFECT .. WITHIN CELLS Regression (Cont.) Univariate F-tests with (2,197) D. F. Variable Sq. Mul. R Adj. R-sq. Hypoth. MS Error MS F Sig. of F read .51356 .50862 5371.66966 51.65523 103.99081 .000 write .43565 .42992 3894.42594 51.21839 76.03569 .000 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

The F-test in this output tests the hypothesis that the first canonical correlation is not equal to zero. Clearly, F = 56.4706 is statistically significant. However, the second canonical correlation of .0235 is not statistically significantly different from zero (F = 0.1087, p = 0.742).

Factor analysis

262

Factor analysis is a form of exploratory multivariate analysis that is used to either reduce the number of variables in a model or to detect relationships among variables. All variables involved in the factor analysis need to be interval and are assumed to be normally distributed. The goal of the analysis is to try to identify factors which underlie the variables. There may be fewer factors than variables, but there may not be more factors than variables. For our example, let's suppose that we think that there are some common factors underlying the various test scores. We will include subcommands for varimax rotation and a plot of the eigenvalues. We will use a principal components extraction and will retain two factors. Menu selection:- Analyze > Dimension Reduction > Factor Syntax:- factor

/variables read write math science socst /criteria factors(2) /extraction pc /rotation varimax /plot eigen.

Factor analysis Crichton, N. Journal Of Clinical Nursing 10(4) 562-562 2001


Factor analysis

263

Factor analysis

264

Factor analysis

265

Factor analysis

266

Factor analysis

267

Communalities

Initial Extraction

reading score 1.000 .736 writing score 1.000 .704 math score 1.000 .750 science score 1.000 .849 social studies score 1.000 .900

Extraction Method: Principal Component Analysis.

Total Variance Explained

Initial Eigenvalues

Extraction Sums of Squared Loadings

Component Total % of Variance Cumulative % Total

1 3.381 67.616 67.616 3.381 2 .557 11.148 78.764 .557 3 .407 8.136 86.900

4 .356 7.123 94.023

5 .299 5.977 100.000

Total Variance Explained

Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings

Component % of Variance Cumulative % Total % of Variance Cumulative %

1 67.616 67.616 2.113 42.267 42.267 2 11.148 78.764 1.825 36.497 78.764 3

4

5

Extraction Method: Principal Component Analysis.

Communality (which is the opposite of uniqueness) is the proportion of variance of the variable (i.e., read) that is accounted for by all of the factors taken together, and a very low communality can indicate that a variable may not belong with any of the factors.

Factor analysis

268

The scree plot may be useful in determining how many factors to retain.

Factor analysis

269Index End

Component Matrixa

Component 1 2

reading score .858 -.020 writing score .824 .155 math score .844 -.195 science score .801 -.456 social studies score .783 .536

Extraction Method: Principal Component Analysis. a. 2 components extracted.

Rotated Component Matrixa

Component 1 2

reading score .650 .559 writing score .508 .667 math score .757 .421 science score .900 .198 social studies score .222 .922

Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. a. Rotation converged in 3 iterations.

Component Transformation Matrix

Component 1 2

1 .742 .670 2 -.670 .742

Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.

From the component matrix table, we can see that all five of the test scores load onto the first factor, while all five tend to load not so heavily on the second factor. The purpose of rotating the factors is to get the variables to load either very high or very low on each factor. In this example, because all of the variables loaded onto factor 1 and not on factor 2, the rotation did not aid in the interpretation. Instead, it made the results even more difficult to interpret.

270

Normal probabilityMany statistical methods require that the numeric variables we are working with have an approximate normal distribution. For example, t-tests, F-tests, and regression analyses all require in some sense that the numeric variables are approximately normally distributed.

271

Normal probabilityFollowing a thorough review, it is suggested “a compelling rationalization of the widely noted robustness of the fixed-effects ANOVA to non-normality”.“It has often been reported that violation of the normality assumption should be of little concern.”

Glass, G.V., P.D. Peckham, and J.R. Sanders. 1972. Consequences of failure to meet assumptions underlying fixed effects analyses of variance and covariance. Rev. Educ. Res. 42: 237-288. DOI: 10.3102/00346543042003237

http://dx.doi.org/10.3102/00346543042003237

272

Normal probabilityAnother review which also suggests alternate tests “non-normality appeared to have little effect on Type I error performance but could have more serious implications for statistical power”.

Lix, L.M., J.C. Keselman, and H.J. Keselman. 1996. Consequences of assumption violations revisited: A quantitative review of alternatives to the one-way analysis of variance F test. Rev. Educ. Res. 66: 579-619. DOI: 10.3102/00346543066004579

http://dx.doi.org/10.3102/00346543066004579

273

Normal probability plotTools for Assessing Normality include

Histogram and BoxplotNormal Quantile Plot (also called Normal

Probability Plot)

Goodness of Fit Tests such asAnderson-Darling TestKolmogorov-Smirnov TestLillefor’s Test

Shapiro-Wilk Test

Problem: they don’t always agree!

274

Normal probability plotYou could produce conventional descriptive statistics, a histogram with a superimposed normal curve, and a normal scores plot also called a normal probability plot.

The pulse data from data set C is employed.

275

Normal probability plotAnalyze

> Descriptive Statistics > Explore

Under plots select histogram, also normality plots with tests, descriptive statistics and boxplots are default options.

276

Normal probability plot

277


278


These tests are considered in the next section.

279


280


If the data is “normal” the non-linear vertical axis in a probability plot should result in an approximately linear scatter plot representing the raw data.

281

Normal probability plotDetrended normal P-P plots depict the actual deviations of data points from the straight horizontal line at zero. No specific pattern in a detrended plot indicates normality of the variable.

282


283

Normal probability plotGraphs

> Legacy Dialogs

> Histogram

Tick – display normal curve

284


285


286

Normal probability plotGraphs

> Legacy Dialogs

> Line

Select Simple and Groups of Cases the use Define to choose the variable and select “cum %”.

287


288


289


If you wish to superimpose a normal curve, it is probably simpler in Excel! This approach underlies the Kolmogorov-Smirnov test.

290

Normal probability plotYou are seeking to assess the normality of the data. The pulse data from data set C is employed.

The P-P plot is a normal probability plot with the data on the horizontal axis and the expected z-scores if our data was normal on the vertical axis. When our data is approximately normal the spacing of the two will agree resulting in a plot with observations lying on the reference line in the normal probability plot.

291


1501351201059075

20

15

10

5

0

pulse

Freq

uenc

y

Mean 99.7StDev 14.86N 90

Histogram of pulseNormal

15014013012011010090807060

100

80

60

40

20

0

pulse

Perc

ent

Mean 99.7StDev 14.86N 90

Empirical CDF of pulseNormal

1501401301201101009080pulse

Boxplot of pulse

1501251007550

99.9

99

959080706050403020105

1

0.1

pulse

Perc

ent

Mean 99.7StDev 14.86N 90AD 4.635P-Value <0.005

Probability Plot of pulseNormal - 95% CI

Index End

292

SkewnessThe skewness is the third centralised normalised moment.

If skewness is positive, the data are positively skewed or skewed right, meaning that the right tail of the distribution is longer than the left. If skewness is negative, the data are negatively skewed or skewed left, meaning that the left tail is longer.

If skewness = 0, the data are perfectly symmetrical. But a skewness of exactly zero is quite unlikely for real-world data, so how can you interpret the skewness number? Bulmer (1979) suggests this rule of thumb:

If skewness is less than −1 or greater than +1, the distribution is highly skewed. If skewness is between −1 and −½ or between +½ and +1, the distribution is

moderately skewed. If skewness is between −½ and +½, the distribution is approximately symmetric.

Bulmer, M. G. 1979. Principles of Statistics. Dover.

293

SkewnessBut what do I mean by “too much for random chance to be the explanation”? To answer that, you need to divide the sample skewness G1 by the standard error of skewness (SES) to get the test statistic, which measures how many standard errors separate the sample skewness from zero: test statistic: ZG1 = G1/SES where This formula is adapted from page 85 of Cramér (1997). The critical value of ZG1 is approximately 2. (This is a two-tailed test of skewness ≠ 0 at roughly the 0.05 significance level.) If ZG1 < −2, the population is very likely skewed negatively (though you don’t know by how much).If ZG1 is between −2 and +2, you can’t reach any conclusion about the skewness of

the population: it might be symmetric, or it might be skewed in either direction.If ZG1 > 2, the population is very likely skewed positively (though you don’t know by how much).

Cramér, Duncan 1997 Basic Statistics for Social Research, Routledge.

)3)(1)(2()1(6

nnn

nnSES

294

SkewnessIt is available via Analyze > Descriptive Statistics > Descriptives

The syntax is

DESCRIPTIVES VARIABLES=x3 x2 x rt_x ln_x x_1 x_2 x_3 /STATISTICS=SKEWNESS.

295

Skewness

Index End

Descriptive StatisticsN SkewnessStatistic Statistic Std. Error Ratio

x3 90 2.208 0.254 8.69x2 90 1.885 0.254 7.42x 90 1.550 0.254 6.10rt_x 90 1.380 0.254 5.43ln_x 90 1.208 0.254 4.76x_1 90 -0.864 0.254 -3.40x_2 90 -0.526 0.254 -2.07x_3 90 -0.203 0.254 -0.80

296

KurtosisThe kurtosis is the fourth centralised normalised moment.

The question is similar to the question about skewness, and the answers are similar too. You divide the sample excess kurtosis by the standard error of kurtosis (SEK) to get the test statistic, which tells you how many standard errors the sample excess kurtosis is from zero:

test statistic: ZG2 = G2 / SEK where

The formula is adapted from page 89 of Cramér (1979).

Cramér, Duncan 1997 Basic Statistics for Social Research, Routledge.

)5)(3(12

2

nn

nSESSEK

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KurtosisThe critical value of ZG2 is approximately 2. (This is a two-tailed test of excess kurtosis ≠ 0 at approximately the 0.05 significance level.)

If ZG2 < −2, the population very likely has negative excess kurtosis (kurtosis <3, platykurtic), though you don’t know how much.

If ZG2 is between −2 and +2, you can’t reach any conclusion about the

kurtosis: excess kurtosis might be positive, negative, or zero. If ZG2 > +2, the population very likely has positive excess kurtosis

(kurtosis >3, leptokurtic), though you don’t know how much.

Cramér, Duncan. 1997.Basic Statistics for Social Research. Routledge.

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Kurtosis!The rules for determining the type of distribution based on skewness and kurtosis may however vary among statisticians. Evans (2007) for instance, suggested that distribution with skewness value of greater than 1 or less than -1 could be considered as highly skewed. Those with skewness value of between 0.5 and 1 or between -1 and -0.5 is said to have moderately skewed distribution whereas a value between 0.5 and -0.5 indicates relative symmetry. Brown (1997) on the other hand, proposed to the practitioners to make use of the standard error of skewness (SES) and standard error of kurtosis (SEK) in deciding whether the tested data could be assumed to come from a normal distribution. He suggested that the data could be assumed as normally distributed if the skewness and kurtosis values lie within the range of ±2×SES and ±2×SEK, respectively. Some practitioners favour one and some favour the others. Nonetheless, skewness and kurtosis do not provide conclusive information about normality. Hence, it is always a good practice to supplement the skewness and kurtosis coefficients with other methods of testing normality such as the graphical methods and formal tests of normality.

J.R. Evans, “Statistics, data analysis and decision making”, 3rd edition, Prentice Hall, pp. 60, 2007.J.D. Brown, “Skewness and kurtosis,” Shiken: JALT Testing & Evaluation SIG Newsletter, vol. 1, pp. 18 – 20, 1997.

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KurtosisIt is available via Analyze > Descriptive Statistics > Descriptives

The syntax is

DESCRIPTIVES VARIABLES=x3 x2 x rt_x ln_x x_1 x_2 x_3 /STATISTICS=KURTOSIS.

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Kurtosis

Index End

Descriptive StatisticsN KurtosisStatistic Statistic Std. Error Ratio

x3 90 4.776 0.503 9.50x2 90 3.356 0.503 6.67x 90 2.162 0.503 4.30rt_x 90 1.651 0.503 3.28ln_x 90 1.201 0.503 2.39x_1 90 0.480 0.503 0.95x_2 90 -0.001 0.503 0.00x_3 90 -0.256 0.503 -0.51

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Does It Really Matter?“Students t test and more generally the ANOVA F test are robust to non-normality” (Fayers 2011).

However

“Thus a clearer statement is that t tests and ANOVA are ‘robust against type-I errors’. This of course accords with the enthusiasm that many researchers have inobtaining ‘‘significant’’ p values.The aim of the article (see next slide) is to show that type-II errors can be substantially increased if non-normality is ignored.” (Fayers 2011).

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Does It Really Matter?Alphas, betas and skewy distributions: two ways of getting the wrong answer Peter Fayers Adv. Health Sci. Educ. Theory Pract. 2011 16(3) 291–296 DOI: 10.1007/s10459-011-9283-6

Introduction to Robust Estimation and Hypothesis Testing (2nd ed.). Wilcox, R. R., 2005, Burlington MA:Elsevier Academic Press. ISBN 978-0-12-751542-7.

Robustness to Non-Normality of Common Tests for the Many-Sample Location Problem Khan A. and Rayner G.D.Journal Of Applied Mathematics And Decision Sciences, 2003, 7(4), 187:206 DOI: 10.1155/S1173912603000178

Index End



http://dx.doi.org/10.1007/s10459-011-9283-6

http://dx.doi.org/10.1155/S1173912603000178

Tukey's ladder of powers

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Tukey has designed a family of power transformations (close cousin to the Box-Cox transformations, but with a visual aspect useful to find the appropriate transformation to promote symmetry and linearity relationships.

These transformations preserve order, preserve proximities and are smooth functions (not producing jumps or peaks). y1 is the untransformed (raw) variable, y0 is replaced by the logarithm that provides the appropriate transformation between the square root and the reciprocal.

You can also use lower and higher powers as listed, as well intermediate ones, i.e. y2.5 will be stronger than y² but less than y³.

Tukey, J. W. (1977) Exploratory Data Analysis. Addison-Wesley, Reading, MA.

3 y3

2 y2

1 y1

½ 0 ln(y)-1 y-1

-2 y-2

-3 y-3

y

Cartoon

http://i.imgur.com/2cFlr3Q.jpg


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A transformation is simply a means of representing the data in a different coordinate system. In addition to restoring normality, the transformation often reduces heteroscedasticity. (Non-constancy of the variance of a measure over the levels of the factor under study.) This is important, because constant variance is often an assumption of parametric tests. Subsequent statistical analyses are performed on the transformed data; the results are interpreted with respect to the original scale of measurement. Achieving an appropriate transformation is a trial-and-error process. A particular transformation is applied and the new data distribution tested for normality; if the data are still non-normal, the process is repeated. Nevertheless, there are certain generalities that can be used to direct your efforts, as certain types of data typically respond to particular transformations. For example Square-root transforms are often appropriate for count data, which tend to follow Poisson distributions. Arcsine (sin-1) transforms are used for data that are percentages or proportions, and tend to fit binomial distributions. Log and square-root transforms are part of a larger class of transforms known as the ladder of powers.


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Transform > Compute Variable

See normal probability plot section for graphical options.


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SORT CASES BY pulse(D).COMPUTE ID=$casenum.EXECUTE.

Compute work = $sysmis.if id=1 work=pulse.Execute.

RMV /pmax=SMEAN(work).

COMPUTE pulse1=1 + pulse/pmax.EXECUTE.COMPUTE y3=pulse1 ** 3.EXECUTE.COMPUTE y2=pulse1 ** 2.EXECUTE.COMPUTE y=pulse1 .EXECUTE.COMPUTE rt_y=SQRT(pulse1).EXECUTE.COMPUTE ln_y=LN(pulse1).EXECUTE.COMPUTE y_1=pulse1 ** -1.EXECUTE.COMPUTE y_2=pulse1 ** -2.EXECUTE.COMPUTE y_3=pulse1 ** -3.EXECUTE.EXAMINE VARIABLES=y2 y rt_y y_1 y_2 y_3 /COMPARE VARIABLE /PLOT=BOXPLOT /STATISTICS=NONE /NOTOTAL /MISSING=LISTWISE.

The pulse data from data set C is employed.

It is scaled by these first compute statement to aid interpretation of the plots.

This step is non-essential, only aiding the graphical presentation.

The “x” variables have not been scaled, being simply powers of “pulse”.


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Normal

Which appears most “normal”?x-2 x-3


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Normal - 95% CI

Which appears most “normal”?x-2 x-3


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x3 x2 x1 x0 x x-1 x-2 x-3Mean 1061245 10158 99.7 4.592 9.96 0.01022 0.000106 1.12E-06StDev 567251 3310 14.86 0.137 0.711 0.001296 2.52E-05 3.75E-07N 90 90 90 90 90 90 90 90AD 8.615 6.523 4.635 3.048 3.799 1.824 0.988 0.529p-value <0.005 <0.005 <0.005 <0.005 <0.005 <0.005 0.013 0.172

In general if the normal distribution fits the data, then the plotted points will roughly form a straight line. In addition the plotted points will fall close to the fitted line. Also the Anderson-Darling statistic will be small, and the associated p-value will be larger than the chosen α-level (usually 0.05). So the test rejects the hypothesis of normality when the p-value is less than or equal to α.


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To test for normality in SPSS you can perform a Kolmogorov-Smirnov Test,

Analyze > Nonparametric tests

> Legacy Dialogs> 1-Sample

Kolmogorov-Smirnov Test


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The Asymp. Sig. (2 tailed) value is also known as the p-value. This tells you the probability of getting the results if the null hypothesis were actually true (i.e. it is the probability you would be in error if you rejected the null hypothesis).

One-Sample Kolmogorov-Smirnov Test x3 x2 x rt_x ln_x x-1 x-2 x-3N 90 90 90 90 90 90 90 90Normal Parametersa,b Mean 1061244.5667 10158.4111 99.7000 9.9599 4.5924 .0102 .0001 .0000

Std. Deviation 567251.11996 3309.53301 14.85847 .71097 .13698 .00130 .00003 .00000Most Extreme Differences

Absolute .255 .221 .192 .178 .163 .133 .105 .079Positive .255 .221 .192 .178 .163 .063 .060 .054Negative -.173 -.139 -.108 -.094 -.080 -.133 -.105 -.079

Kolmogorov-Smirnov Z 2.422 2.099 1.821 1.684 1.544 1.263 .993 .745Asymp. Sig. (2-tailed) .000 .000 .003 .007 .017 .082 .278 .635a. Test distribution is Normal.

b. Calculated from data.


314

Despite the scaling the log. transform spoils the final plot.


315

You are seeking the most normal data visually.

Probably one of y_1, y_2 or y_3 transform.

316

Tukey's ladder of powersMany statistical methods require that the numeric variables you are working with have an approximately normal distribution. Reality is that this is often times not the case. One of the most common departures from normality is skewness, in particular, right skewness.

When the data is plotted vs. the expected z-scores the normal probability plot shows right skewness by a downward bending curve.

When the data is plotted vs. the expected z-scores the normal probability plot shows left skewness by an upward bending curve.

317

Tukey’s Ladder of PowersTukey (1977) describes an orderly way of re-expressing variables using a power transformation. If a transformation for x of the type xλ, results in an effectively linear probability plot, then we should consider changing our measurement scale for the rest of the statistical analysis. There is no constraint on values of λ that we may consider. Obviously choosing λ=1 leaves the data unchanged. Negative values of λ are also reasonable. Tukey (1977) suggests that it is convenient to simply define the transformation when λ=0 to be the logarithmic function rather than the constant 1.

318

Tukey’s Ladder of PowersIn general if the normal distribution fits the data, then the plotted points will roughly form a straight line. In addition the plotted points will fall close to the fitted line. Also the Anderson-Darling statistic will be small, and the associated p-value will be larger than the chosen α-level (usually 0.05). So the test rejects the hypothesis of normality when the p-value is less than or equal to α.

319

Tukey’s Ladder of PowersTo test for normality is SPSS you can perform a Kolmogorov-Smirnov Test

Analyze > Nonparametric tests > 1-Sample Kolmogorov-Smirnov Test

The Asymp. Sig. (2 tailed) value is also known as the p-value. This tells you the probability of getting the results if the null hypothesis were actually true (i.e. it is the probability you would be in error if you rejected the null hypothesis).

320

Tukey’s Ladder of PowersThe hypothesis are H0 the distribution of x is normalH1 the distribution of x is not normal If the p-value is less than .05, you reject the normality assumption, and if the p-value is greater than .05, there is insufficient evidence to suggest the distribution is not normal (meaning that you can proceed with the assumption of normality.) In summary if the test is significant (lower than or equal to 0.05) implies the data is not normally distributed.

321

Tukey’s Ladder of PowersTo read more about Normality tests.

Comparisons Of Tests For Normality With A Cautionary NoteDyer, A.R.Biometrika 61(1) 185-189 1974 DOI: 10.1093/biomet/61.1.185

A Comparison Of Various Tests Of NormalityYazici, Berna and Yolacan, SenayJournal Of Statistical Computation And Simulation 77(2) 175-183 2007 DOI: 10.1080/10629360600678310

Comparisons of various types of normality testsB. W. Yap and C. H. Sim Journal of Statistical Computation and Simulation 81(12) 2141-2155 2011 DOI: 10.1080/00949655.2010.520163

Index End

http://dx.doi.org/10.1093/biomet/61.1.185

http://dx.doi.org/10.1080/10629360600678310

http://dx.doi.org/10.1080/00949655.2010.520163

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Median SplitThere is quite a literature to suggest that, even though it is nice and convenient to sort people into 2 groups and then use a t test to compare group means, you lose considerable power. Cohen (1983) has said that breaking subjects into two groups leads to the loss of 1/5 to 2/3 of the variance accounted for by the original variables. The loss in power is equivalent to tossing out 1/3 to 2/3 of the sample.

The Cost Of Dichotomization or paperCohen, J. Applied Psychological Measurement 7(3) 249-253 1983

Index End

http://dx.doi.org/10.1177/014662168300700301

http://conservancy.umn.edu/handle/107497

323

Likert ScaleThese are not exhaustive notes, rather some thoughts on preparing a Likert Scale. Some 11,000 papers employ Likert scales, many debate the issues sketched below.

Statements are often rated on a five point Likert (1932) scale. Weijters et al. (2010) report people completing surveys with 7-point scales are more susceptible to extreme responding (i.e. picking one or other of the endpoints). They are also more likely to make mistakes when just the end-points are labelled. These authors recommend using 5-point scales with each item on the scale fully labelled (e.g. strongly agree, agree, neutral, disagree, strongly disagree).

Likert, R. (1932). A technique for the measurement of attitudes. Archives of Psychology, 22(140) 1-55.

Weijters, Bert; Cabooter, Elke; Schillewaert, Niels (2010) The effect of rating scale format on response styles: The number of response categories and response category labels. International Journal Of Research In Marketing 27(3) 236-247 DOI: 10.1016/j.ijresmar.2010.02.004 (see final slide in this section).

http://www.voteview.com/pdf/Likert_1932.pdf

http://dx.doi.org/10.1016/j.ijresmar.2010.02.004

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Likert ScaleNegative coding, for a Likert scale from 1 to n. If the response was x and it need negative coding, simply form n+1-x with Excel (say). So 1→n and n→1.

Test it for all possible values in your favourite scale.

B C B C2 n 5 2 n 53 x 2 3 x 2

4 negative coding 4

4 negative

coding =C2+1-C3

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Likert ScaleHowever, a 7-point Likert scale is recommended to maximise the sensitivity of the scale (Allen and Seaman 2007, Cummins and Gullone 2000), while Leung (2011) covers the full spectrum of choices.

Allen, I.E., and Seaman, C.A. (2007). Likert scales and data analyses. Quality Progress, 40(7), 64-65.

Cummins, R.A., and Gullone, E. (2000). Why we should not use 5-point Likert scales: The case for subjective quality of life measurement. In Proceedings, second international conference on quality of life in cities (p74-93). Singapore: National University of Singapore.

Leung, S-O. (2011). A Comparison of Psychometric Properties and Normality in 4-, 5-, 6-, and 11-Point Likert Scales, Journal of Social Service Research 37(4) 412-421 DOI: 10.1080/01488376.2011.580697 an odd place for a technical paper.

http://asq.org/quality-progress/2007/07/statistics/likert-scales-and-data-analyses.html

https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=1&ved=0CCEQFjAA&url=https%3A%2F%2Fwww.deakin.edu.au%2Fresearch%2Facqol%2Fiwbg%2Fcorrespondence%2F2002%2FCummins_Gullone_2000_Likert_Scales.doc&ei=4UPnVNbdNcvmUvzZgPAL&usg=AFQjCNG1Xiwktxe8YxjgLOxJqB8HRKiCWg&sig2=5CWgaErWRWbqb_ud8BJyoA&bvm=bv.86475890,d.d24

https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=1&ved=0CCEQFjAA&url=https%3A%2F%2Fwww.deakin.edu.au%2Fresearch%2Facqol%2Fiwbg%2Fcorrespondence%2F2002%2FCummins_Gullone_2000_Likert_Scales.doc&ei=4UPnVNbdNcvmUvzZgPAL&usg=AFQjCNG1Xiwktxe8YxjgLOxJqB8HRKiCWg&sig2=5CWgaErWRWbqb_ud8BJyoA&bvm=bv.86475890,d.d24

http://dx.doi.org/10.1080/01488376.2011.580697

326

Likert ScaleYou can even cope with mixed scales. It is unclear why different scales may have been used in the same experiment, but it has been shown that data is still comparable when it has been re-scaled (Dawes 2008).

Dawes J. (2008) Do data characteristics change according to the number of scale points used? An experiment using 5 point, 7 point and 10 point scales International Journal of Market Research. 51 (1) 61-77.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.417.9488&rep=rep1&type=pdf



327

Likert ScaleWhat about a mid-point? It may be better if the scale contained a neutral midpoint (Tsang 2012). This decision (an odd/even scale) depends whether respondents are being forced to exclude the neutral position with an even scale.

Tsang K.K (2012) The use of midpoint on Likert Scale: The implications for educational research Hong Kong Teachers’ Centre Journal 11 121-130.

An odd number of points allow people to select a middle option. An even number forces respondents to take sides. An even number is appropriate when you want to know what direction the people in the middle are leaning. However, forcing people to choose a side, without a middle point, may frustrate some respondents (Wong et al. 1993).

Wong, C.-S., Tam, K.-C., Fung, M.-Y., and Wan, K. (1993). Differences between odd and even number of response scale: Some empirical evidence. Chinese Journal of Psychology, 35, 75-86.

http://edb.org.hk/HKTC/download/journal/j11/HKTCJv11_11-B02.pdf

http://edb.org.hk/HKTC/download/journal/j11/HKTCJv11_11-B02.pdf

http://psycnet.apa.org/psycinfo/1995-11886-001

http://psycnet.apa.org/psycinfo/1995-11886-001

328

Likert ScaleSince they have no neutral point, even-numbered Likert scales force the respondent to commit to a certain position (Brown, 2006) even if the respondent may not have a definite opinion.

There are some researchers who prefer scales with 7 items or with an even number of response items (Cohen, Manion, and Morrison, 2000).

Brown, J.D. (2000). What issues affect Likert-scale questionnaire formats? JALT Testing and Evaluation SIG, 4, 27-30.

Cohen, L., Manion, L. and Morrison, K. (2000). Research methods in education (5th ed.). London: Routledge Falmer.

http://jalt.org/test/bro_7.htm

329

Likert ScaleThe change of response order in a Likert-type scale altered participant responses and scale characteristics. Response order is the order in which options of a Likert-type scale are offered (Weng 2000).

How many scale divisions or categories should be used (1 to 10; 1 to 7; -3 to +3)? Should there be an odd or even number of divisions? (Odd gives neutral centre value; even forces respondents to take a non-neutral position.) What should the nature and descriptiveness of the scale labels be? What should the physical form or layout of the scale be? (graphic, simple linear, vertical, horizontal) Should a response be forced or be left optional?

Li-Jen Weng 2000 Effects of Response Order on Likert-Type Scales, Educational and Psychological Measurement 60(6) 908-924 DOI: 10.1177/00131640021970989

http://dx.doi.org/10.1177/00131640021970989

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Likert Scale

Index EndWeijters et al. (2010) DOI: 10.1016/j.ijresmar.2010.02.004

http://dx.doi.org/10.1016/j.ijresmar.2010.02.004

331

WinsorizeWinsorising or Winsorization is a transformation by limiting extreme values in the data to reduce the effect of possibly spurious outliers. It is named after the engineer-turned-biostatistician Charles P. Winsor (1895-1951).

The computation of many statistics can be heavily influenced by extreme values. One approach to providing a more robust computation of the statistic is to Winsorize the data before computing the statistic.

Apart from confusion about the correct spelling. There is the ambiguity about where the precise percentile sits.

332

WinsorizeTo Winsorize the data, tail values are set equal to some specified percentile of the data. For example, for a 90% Winsorization, the bottom 5% of the values are set equal to the value corresponding to the 5th percentile while the upper 5% of the values are set equal to the value corresponding to the 95th percentile.

Just because a method existsdoes not necessarily mean its a great idea!!

333

WinsorizeThe pulse data from data set C is employed.

Analyze > Descriptive Statistics > Frequencies

334

Winsorize

Select statistics

335

Winsorize

Add desired percentiles, 5 then 95

336

Winsorize

For brevity do not display frequency tables

337

Winsorize

Note the percentiles and enter them into the next slide.

338

WinsorizeTransform > Compute Variable

339

Winsorize

Select Old and New Values

Choose asensible newname

340

Winsorize

Then Add

341

Winsorize

Then Add

342

Winsorize

Then Add

Retain allother values

343

Winsorize

Finally, continue then OK

344

WinsorizeTo check your results

Analyze > Descriptive Statistics > Descriptives

345

Winsorize

OK

346

Winsorize

As desired.

347

WinsorizeSyntax:- freq var pulse /format = notable /percentiles = 5 95.

compute winsor = pulse.if pulse <= 83 winsor = 83.if pulse >= 137.25 winsor = 137.25.

descriptives variables=pulse winsor /statistics=mean stddev min max.

348

WinsorizeTrimming and Winsorization: A reviewW. J. Dixon and K. K. Yuen Statistische Hefte June 1974, Volume 15, Issue 2-3, pp 157-170

This paper provides a literature review for robust statistical procedures trimming and Winsorization that were first proposed for estimating location, but were later extended to other estimation and testing problems. Performance of these techniques under normal and long-tailed distributions are discussed.

http://link.springer.com/article/10.1007/BF02922904

349

WinsorizeWinsorisation for estimates of changeDaniel LewisPapers presented at the ICES-III, June 18-21, 2007, Montreal, Quebec, Canada paper

Outliers are a common problem in business surveys which, if left untreated, can have a large impact on survey estimates. For business surveys in the UK Office for National Statistics (ONS), outliers are often treated by modifying their values using a treatment known as Winsorisation. The method involves identifying a cut-off for outliers. Any values lying above the cut-offs are reduced towards the cut-off. The cut-offs are derived in a way that approximately minimises the Mean Square Error of level estimates. However, for many surveys estimates of change are more important. This paper looks at a variety of methods for Winsorising specifically for estimates of change. The measure of change investigated is the difference between two consecutive estimates of total. The first step is to derive potential methods for Winsorising this type of change. Some of these methods prove more practical than others. The methods are then evaluated, using change estimates derived by taking the difference between two regular Winsorised level estimates as a comparison.

http://www.amstat.org/meetings/ices/2007/proceedings/ICES2007-000113.PDF

350

WinsorizeSpeaking Stata: Trimming to tasteCox, N.J.Stata Journal 2013 13(3) 640-666

Trimmed means are means calculated after setting aside zero or more values in each tail of a sample distribution. Here we focus on trimming equal numbers in each tail. Such trimmed means define a family or function with mean and median as extreme members and are attractive as simple and easily understood summaries of the general level (location, central tendency) of a variable. This article provides a tutorial review of trimmed means, emphasizing the scope for trimming to varying degrees in describing and exploring data. Detailed remarks are included on the idea's history, plotting of results, and confidence interval procedures.

Note uses Stata! Index End

http://www.stata-journal.com/article.html?article=st0313

351

General Linear ModelsGenerally, the various statistical analyses are taught independently from each other. This makes it difficult to learn new statistical analyses, in contexts that differ. The paper gives a short technical introduction to the general linear model (GLM), in which it is shown that ANOVA (one-way, factorial, repeated measure and analysis of covariance) is simply a multiple correlation/regression analysis (MCRA). Generalizations to other cases, such as multivariate and nonlinear analysis, are also discussed. It can easily be shown that every popular linear analysis can be derived from understanding MCRA.

They present the identities shown on the next two slides.

General Linear Models: An Integrated Approach to Statistics Sylvain Chartier and Andrew FaulknerTutorials in Quantitative Methods for Psychology 2008 4(2) 65-78

352

General Linear Models

353

General Linear Models

Index End

354

Centre DataSometimes to facilitate analysis it is necessary to centre a data set, for instance to give it a mean of zero. In this case consider the reading score in data set A.

355

Centre DataWhy center? Different authors have made different recommendations regarding the centring of independent variables. Some have recommended mean-centring (i.e., subtracting the mean from the value of the original variable so that it has a mean of 0); others z-standardization (which does the same, and then divides by the standard deviation, so that it has a mean of 0 and a standard deviation of 1); others suggest leaving the variables in their raw form. In truth, with the exception of cases of extreme multi-collinearity (which may arise in multiple-regression/correlation), the decision does not make any major difference. For instance the p value for an interaction term any subsequent interaction plot should be identical whichever way it is done (Dalal and Zickar 2012; Kromrey and Foster-Johnson 1998).Dalal, D. K. and Zickar, M. J. 2012 “Some common myths about centering predictor variables in moderated multiple regression and polynomial regression” Organizational Research

Methods, 15, 339–362 DOI: 10.1177/1094428111430540Kromrey, J. D. and Foster-Johnson, L. 1998 “Mean centering in moderated multiple regression: Much ado about nothing” Educational and Psychological Measurement, 58, 42–67 DOI: 10.1177/0013164498058001005

356

Centre DataTo check we have achieved our goal we generate descriptive statistics.

357

Centre DataTo check we have achieved our goal we generate descriptive statistics.

Syntax

EXAMINE VARIABLES=read /PLOT BOXPLOT STEMLEAF /COMPARE GROUPS /STATISTICS DESCRIPTIVES /CINTERVAL 95 /MISSING LISTWISE /NOTOTAL.

358

Centre DataNote, the mean is 53.23.

359

Centre DataTo create a column of means.

360

Centre DataTo create a column of means.

Syntax

AGGREGATE /OUTFILE=* MODE=ADDVARIABLES /BREAK= /read_mean=MEAN(read).

361

Centre DataFinally compute the desired variable.

362

Centre DataFinally compute the desired variable.

Syntax

COMPUTE read_centre=read - read_mean.EXECUTE.

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Centre DataRepeat descriptive statistics on the new variable. Now zero mean.

Index End

364

Correlation - ComparisonIf simply comparing two correlations, a simple web search should reveal numerous analytic tools. Alternately see, for example,

Kanji, G.K. 1999 “100 Statistical Tests” London: SAGE.

Test 14 which includes a worked example.

1

11 1

1ln21

rrz

31

11 n

2

22 1

1ln21

rrz

31

22 n

22

21

21 zzz

365

Correlation - ComparisonSeveral procedures that use summary data to test hypotheses about Pearson correlations and ordinary least squares regression coefficients have been described in various books and articles. No single resource describes all of the most common tests. Furthermore, many of these tests have not yet been implemented in popular statistical software packages such as SPSS. The article (next slide) describes all of the most common tests and provide SPSS programs to perform them. When they are applicable, the code also computes 100 × (1 − α)% confidence intervals corresponding to the tests. For testing hypotheses about independent regression coefficients, they demonstrate one method that uses summary data and another that uses raw data (i.e., Potthoff analysis). When the raw data are available, the latter method is preferred, because use of summary data entails some loss of precision due to rounding.

366

Correlation - ComparisonSPSS and SAS programs for comparing Pearson correlations and OLS regression coefficients

Bruce Weaver and Karl L. Wuensch

Behavior Research Methods September 2013, Volume 45, Issue 3, pp 880-895

DOI: 10.3758/s13428-012-0289-7

For the code access. Not the earlier version on the journal web site.

367

Correlation - ParadoxValidation of selection tests or other selection variables is most frequently accomplished through correlation and regression. It is quite common to note that both sexes or two ethnic groups were included in the sample. Sometimes in the combined sex or ethnic groups the correlation of the predictor and criterion may be moderate or large, but within each group the correlation of predictor and criterion is low or zero. This seeming paradox is not a psychological phenomenon, but a consequence of the mathematics of regression and correlation. The problem would seem to require a Within and Between Analysis, but having only two groups prohibits use of the methodology. A general hierarchical linear models analysis is proposed and demonstrated by the authors.

M.J. Ree , T.R. Carretta and J.A. Earles 1999 “In Validation Sometimes Two Sexes Are One Too Many: A Tutorial, Human Performance”, 12(1), 79-88, DOI: 10.1207/s15327043hup1201_4 Index End

368

Sobel TestThe Sobel test will assess whether a mediator (see more extensive notes on Mediation on the main web page) carries the influence of an independent variable to a dependent variable.

The Sobel test works well only in large samples. It is recommended using this test only if the user has no access to raw data. If you have the raw data, bootstrapping offers a much better alternative that imposes no distributional assumptions. Consult Preacher and Hayes (2004, 2008) for details and easy-to-use macros that run the necessary regression analyses for you:Preacher, K. J., & Hayes, A. F. (2008) “Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models” Behavior Research Methods, 40, 879-891 DOI: 10.3758/BRM.40.3.879.Preacher, K. J., & Hayes, A. F. (2004) “SPSS and SAS procedures for estimating indirect effects in simple Mediation models”Behavior Research Methods, Instruments, & Computers, 36, 717-731 DOI: 10.3758/BF03206553.

See How can I perform a Sobel test on a single mediation effect in SPSS? But its not simple!

Index End

369

Structural Equation ModellingThis tutorial begins with an overview of structural equation modeling (SEM) that includes the purpose and goals of the statistical analysis as well as terminology unique to this technique. I will focus on confirmatory factor analysis (CFA), a special type of SEM. After a general introduction, CFA is differentiated from exploratory factor analysis (EFA), and the advantages of CFA techniques are discussed. Following a brief overview, the process of modeling will be discussed and illustrated with an example using data from a HIV risk behavior evaluation of homeless adults (Stein & Nyamathi, 2000). Techniques for analysis of nonnormally distributed data as well as strategies for model modification are shown. The empirical example examines the structure of drug and alcohol use problem scales. Although these scales are not specific personality constructs, the concepts illustrated in this article directly correspond to those found when analyzing personality scales and inventories. Computer program syntax and output for the empirical example from a popular SEM program (EQS 6.1; Bentler, 2001) are included.

Structural equation modeling: Reviewing the basics and moving forwardUllman, Jodie B.Journal Of Personality Assessment 87(1) 35-50 2006DOI: 10.1207/s15327752jpa8701_03

370

Structural Equation ModellingStein, J. A., & Nyamathi, A. M. (2000). Gender differences in behavioural and psychosocial predictors of HIV testing and return for test results in a high-risk population. AIDS Care, 12, 343–356. DOI: 10.1080/09540120050043007

Bentler, P.M. (2001). EQS 6 structural equations program manual. Encino, CA: Multivariate Software.

Download Eqs 6 software

This page serves as a gateway to a tutorial on structural equation modeling or SEM.

Index End

371

QuartilesAre quartiles well defined?

Hyndman and Fan (1996) investigated nine different methods implemented in the statistical software to calculate sample quantiles. Sample quantiles providing nonparametric estimators of their population counterparts are based on one or two order statistics from the sample x

1,

…,xn. Adopting notation used in Hyndman and Fan (1996) let us define

the p100%-th sample quantile given by the ith method as follows

for some constants m∈R and 0 ≤ γ ≤ 1 chosen appropriately for each method. The value of γ is a function of j=⌊np+m⌋ and g=np+m−j. Values of these parameters corresponding to particular methods are given in the table. For more details on those methods we refer the reader to Hyndman and Fan (1996).

Hyndman R.J., Fan, Y. (1996), Sample quantiles in statistical packages, The American Statistician 50, 361-365, DOI: 10.2307/2684934

njnji xxpQ :1:)1()(

nmjp

nmj 1

372

QuartilesMethod 1 m = 0

otherwise1

0g if0

Method 2 m = 0

otherwise1

0g if5.0

Method 3 m = −0.5

otherwise1

even is j and 0g if0

Method 4 m = 0 γ = g

Method 5 m = 0.5 γ = g

Method 6 m = p γ = g

Method 7 m = 1 − p γ = g

Method 8 m = ⅓ (p + 1) γ = g

Method 9 m = ¼ p + ⅜ γ = g

Method 6 corresponds to Excel (quartile.exc(), also SPSS and Minitab.Method 7 corresponds to Excel (quartile() and quartile.inc())

373

QuartilesWhat ever method we choose, the first and the third quartiles given by the ith method are given by

where k=⌊0.75n+m⌋ and l=⌊0.25n+m⌋. Consequently, the interquartile range based on a sample of size n, produced by the ith method (i=1,…,9), has the following form

nknki xxQ :1:)1()75.0(

nlnli xxQ :1:)1()25.0(

)())(1( :1:1:: nlnknlnkin xxxxIQR

374

QuartilesGiven the following random sorted data0.008228533 0.012108943 0.043745942 0.054314571 0.214140779 0.251766741 0.289577348 0.290472163 0.342535936 0.384891271 0.452616365 0.463141946 0.496310054 0.603820413 0.645915266 0.739309969 0.786093766 0.813916662 0.832916098 0.951772594

Method 1 2 3 4 5 6 7 8 9n 20 20 20 20 20 20 20 20 20m 0 0 -0.5 0 0.5 0.25 0.75 0.42 0.4375p 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25j 5 5 4 5 5 5 5 5 5g 0.00 0.00 0.50 0.00 0.50 0.25 0.75 0.42 0.44γ 0 0.5 1 0.00 0.50 0.25 0.75 0.42 0.44l 5 5 4 5 5 5 5 5 5

x1 0.2141 0.2330 0.2141 0.2141 0.2330 0.2235 0.2424 0.2298 0.2306

Lower Quartile

375

Quartiles

Index End

Given the following random sorted data0.008228533 0.012108943 0.043745942 0.054314571 0.214140779 0.251766741 0.289577348 0.290472163 0.342535936 0.384891271 0.452616365 0.463141946 0.496310054 0.603820413 0.645915266 0.739309969 0.786093766 0.813916662 0.832916098 0.951772594

Method 1 2 3 4 5 6 7 8 9n 20 20 20 20 20 20 20 20 20m 0 0 -0.5 0 0.5 0.75 0.25 0.58 0.5625p 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75j 15 15 14 15 15 15 15 15 15g 0.00 0.00 0.50 0.00 0.50 0.75 0.25 0.58 0.56γ 0 0.5 1 0.00 0.50 0.75 0.25 0.58 0.56k 15 15 14 15 15 15 15 15 15

x3 0.6459 0.6926 0.6459 0.6459 0.6926 0.7160 0.6693 0.7004 0.6984

Upper Quartile

376

Does It Always Matter?Scientists think in terms of confidence intervals – they are inclined to accept a hypothesis if the probability that it is true exceeds 95 per cent. However within the law “beyond reasonable doubt” appears to be a claim that there is a high probability that the hypothesis – the defendant’s guilt – is true.

A Story Can Be More Useful Than MathsJohn KayFinancial Times 26 February 2013

377

Does It Always Matter?…we slavishly lean on the crutch of significance testing because, if we didn’t, much of psychology would simply fall apart. If he was right, then significance testing is tantamount to psychology’s “dirty little secret.”

Significance tests as sorcery: Science is empirical - significance tests are notCharles LambdinTheory and Psychology 22(1) 67–90 2012

378

Does It Always Matter?The first rule of performing a project

1 The supervisor is always right

The second rule of performing a project

2 If the supervisor is wrong, rule 1 applies

379

Does It Always Matter? Probably!

Estimation based on effect sizes, confidence intervals, and meta-analysis usually provides a more informative analysis of empirical results than does statistical significance testing, which has long been the conventional choice in psychology. The sixth edition of the American Psychological Association Publication Manual now recommends that psychologists should, wherever possible, use estimation and base their interpretation of research results on point and interval estimates.

The statistical recommendations of the American Psychological Association Publication Manual: Effect sizes, confidence intervals, and meta-analysisGeoff Cumming, Fiona Fidler, Pav Kalinowski and Jerry LaiAustralian Journal of Psychology 2012 64 138–146

380

Does It Always Matter? Probably!

The debate is ongoing!

Cumming, G. 2014 “The new statistics: Why and how. Psychological Science”, 25, 7-29 DOI: 10.1177/0956797613504966

Savalei V. and Dunn E. 2015 “Is the call to abandon p-values the redherring of the replicability crisis?” Frontiers in Psychology 6:245.DOI: 10.3389/fpsyg.2015.00245

Index End

381

SPSS Tips

Now you should go and try for yourself.

Each week a cluster is booked to follow this session. This will enable you to come and go as you please.

Obviously other timetabled sessions for this module take precedence.

Index

what statistical analysis should i use?

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