Inferential StatisticsResearch Methods

Outline• What Statistical Tests to Use?• Correlation Tests• t-Tests

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What Statistical Tests to Use?

Decision on the Statistical Tests• Depends on

The design of the research• To see the relationship of the variables?• To see if there are any changes in the

participants after certain treatment?• Etc.

Can the results be generalized?• Assumptions – conclusions – actions

Why checking assumptions?• Assumption is important

assumption conclusion action Correct assumption correct conclusion

correct action

Case: I couldn’t meet Ast today at 1.30 PM

• Assumptions 12-1 PM official lunch time in SWCU Everybody needs lunch Classes at FLL usually go from 11 AM – 1 PM then

from 2-4 PM

• Conclusions Every lecturer in SWCU will have lunch at 12-1 PM Every lecturer may teach 11 AM – 1 PM then from

2-4 PM

• Action See Ast between 1-2 PM

But…• Assumptions

Ast hates me for God knows what reasons

• Conclusions He will not see me at all

• Action That’s probably why he refuses to see

me at 1.30 PM today.

How do you know your assumptions are right?

• It’s regulation/convention But are you sure it’s regulated in SWCU and

FLL?

• It’s what usually happens in SWCU and FLL Offices are closed between 12-1 PM Lecturers are seen at campus cafes having

lunch during 12-1 PM Schedule of classes

• Where did your assumption go wrong? How can you be so sure that Ast hates you?

What has Ast to do with ResMeth?• Assumption must be correct,

otherwise the conclusion will not be correct

• What made your conclusion wrong in the case of Ast? Feelings and not what NORMALLY

happens either by regulation/convention in the POPULATION (SWCU/FLL)

• Remember NORMAL DISTRIBUTION?

Looking back at previous meetings…• The aim of doing quantitative research is to

generalize the results for the population• Assumption

Population normal distribution Sample normal distribution

• Conclusion If my sample is normally distributed, I can expect

to generalize it to the population

• Action My research recommendations can be applied in

the population

Parametric vs. Non-Parametric Tests• Some statistical tests are parametric tests

based on the normal distribution• A parametric test requires parametric data

from one of the large catalogue of distributions that statisticians have described (regulation/convention)

• Parametric data certain assumptions must be true. A parametric test for NON parametric data

inaccurate results

• very important check the assumptions before deciding which statistical test is appropriate

Correlation

Tests

• Positively related one up, the other up

• Not related at all same no matter what

• Negatively related one up, the other down

How 2 variables could be related?

Correlational Tests• Parametric Test

Pearson’s Product Moment Correlation

• Non-Parametric Spearman’s Correlation Coefficient Kendall’s tau (τ)

• To decide: Check the assumptions 1 assumption violated non-parametric

What are the underlying assumptions?1. Related pairs 2. Scale of measurements3. Normality4. Linearity5. Homoscedasticity

Testing: 1 & 2 design of the research3-5 testable using graphic & tests

Related Pairs• Data must be collected from related

pairs• 1 data from one variable, 1 data from

the other variable• E.g. Relationship between gender

and English competence Arif has data for gender “male” and for

English competence “84 points”

Scale of Measurements• Interval or ratio• Do you still remember what they are?

Continuous Not categorical

• E.g. Arif Gender nominal (categorical) Competence ratio (continuous)

• One assumption violated! Go to non-parametric (Spearman’s or

Kendall’s)

Warning!• Difference in literature

Coakes (2005) both variables must be continuous - interval

Field (2009) interval or one variable can be categorical – binary

• I’m inclined to Coakes The scatterplot when one variable is

interval and the other is binary is not homoscedasticity (I’ll show you later why this matters)

Normality• In MSExcel – (complicated!)

Histogram

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Series1Polynomial (Series1)

Normality & Linearity• In SPSS (relatively easier)

Together with descriptive statistics report & linearity

• Test by: Graphic Normality tests

Normality and Linearity• Analyze | Descriptive Statistics |

Explore Select the variable you want to test Statistics: tick• Descriptives

Plots: tick• Histogram• Normality plots with tests

Normality• From Kolmogorov-Smirnov (K-S) & Shapiro-

Wilk (S-W)

Sig. <.05 significantly different from normal distribution

competence sig. = .008 <.05 data not normal Shapiro – Wilk is more powerful (maybe K-S sig,

S-W not sig.)

Normality• Graphic – Histogram

not bell-shaped not normal

• Psst.. Normality line here is added as a guide. How? Try right

clicking the graphic & edit the content. Find this icon in the bar:

Normality• Is your data normally distributed?

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Linearity• How your data

for each variable falls in a linear line

• MS Excel – not possible

• SPSS – yes! See the test of

normality

Homoscedascity• How your data clustered into

certain areas when two variables are related

• To see if they have similar variance along the linear line

• Why this is important? Not wide difference between data Too wide --> not normal

Homoscedasticity• MS Excel – not possible• SPSS – yes!

Graph | Legacy Dialogs | Scatter/Dot | Simple Scatter

Choose the two variables for X axis and Y axis

• Psst.. Linear line here is added as a guide. How? Try right clicking the

graphic & edit the content. Find this icon in the bar:

HomoscedasticityGender vs.

Competence• Heteroscedasticity• Not normal

Competence vs. Graduation

• Homoscedasticity• Maybe normal

Can’t do categorical variable! Coakes wins!

Once you’ve done all of this assumption checking…• Select the correlational test the data falls

into• Our correlational tests are bivariate

correlation Between 2 variables

• We’re not dealing with partial correlation (between 2 variables plus one or more controlling variables) later when you’re more ‘grown up’ in statistics

• Pearson product-moment correlation (standardized measurement) Symbol : r or R -1 to +1 To measure size of the effect• ± 0.1 small effect• ± 0.3 medium effect• ± 0.5 large effect

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How do we measure relationships?

Pearson’s Correlation Coefficient• Using MS Excel – Data | Data Analysis |

Correlation

• Downsides Only for Pearson’s, not Spearman’s or

Kendall’s No indicator of significance of relationship Only the strength of correlation coefficient

  Competence Graduation 

Competence 1

Graduation 0.954149422 1

• Analyze | Correlate | Bivariate

• Input the variables used in Variables

• Default: Pearson• Options: Spearman

and Kendall• One- vs. two-tailed

One-tailed directional hypothesis (the more x, the more y)

Two-tailed not sure

Bivariate Correlation (Using SPSS)

• Interpretation of the result table ** significant

correlation r value Pearson

Correlation value Significant or not

Sig. <.05

• What does this numbers mean?

Pearson’s Correlation Coefficient

• Correlation result ≠ causality• Third-variable problem

Maybe there is an influence of third variable

• Direction of causality No clear indication which variable

causes the other variable to change

Warning: Causality!!!

• Non-parametric statistic Not normal data distribution, etc. Not interval data ordinal data

• Interpretation of the result table ** significant correlation rs -- Correlation coefficient value Significant or not Sig. <.05

Spearman’s Correlation Coefficient

• Non-parametric statistic Small data set which when it is ranked it

has many scores with the same rank More accurate generalization than

Spearman’s

• Interpretation of the result table ** significant correlation τ – Correlation coefficient value Significant or not Sig. <.05

Kendall’s tau (τ)

• Tell: How big Significant value

• Important Notes: No zero before the decimal point for correlation

coefficient (for example -- .87 NOT 0.87) Correlation coefficient in different letters (r, rs, or τ)

One-tailed must be reported Standard criteria for p value (probabilities)

-- .05, .01 and .001

How to Report Correlation Coefficients

• Pearson’s There is a significant correlation between X

variable and Y variable, r = .87, p (one-tailed) <.05

• Spearman’s X variable is significantly correlated with Y

variable, rs = .87 (p <.01)

• Kendall’s There was a positive relationship between X

variable and Y variable, τ = .47, p<.05

Example of Reports

t-Tests

What is it for?• Looking at the effect(s) of one

variable to another• By systematically changing some

aspect of that variable• To compare two means of the data

Comparing 2 means of data• Between-group, between-subjects or

independent design DIFFERENT participants to different

experimental manipulations

• A repeated-measures design SAME participants to different

experimental manipulations at different points in time

Comparing 2 Means Using t-Tests

Different participantsBetween groups, between subjects, or independent

design

Single Sample

From one sample compared to the

population

Test scores of a group in a semester compared to previous group’s scores

Independent or Two- Sample

Two samples with different conditions

Test scores of 2 groups with different teachers

after a semester

Same participants Repeated measures

design

Paired- or Dependent sample

From two samples of the same condition

The scores of a group before and after a

semester

Assumptions of the t-tests1. Scale of Measurement – continuous

interval2. Random sampling 3. Normality 4. Additional for Independent t-test

1. Independent of groups – inclusion into one group only, and not the other group

2. Homogeneity of variance – Levene’s test (presented in SPSS results for independent t-test)

Single Sample t-Test• Comparing the mean of

a data set with a set means of other aggregate data

• MS Excel no!• SPSS Analyze |

Compare Means | One Sample t-Test Input the Test Variable

compared Input the Test Value

(aggregate data)

Single Sample t-Test: Results & Report

• Reporting:There is no significant difference in the graduation grade between this year’s participants with previous year’s participants ( t(19) = .493, p>.05), although this year’s participants have slightly higher grade (Mean Difference = 1.4)

Significant sig. <.05t positive this data > previous aggregate data

Using MS Excel for Other t-Tests• Only for

Paired-sample T-Test Independent T-Test• Assuming equal variance• Assuming non-equal variance

Reject or accept the null hypothesis there is no difference of means in the two variables

Paired-Samples t-Test• Comparing the means of the same

group participants under two conditions

• Samples two sets of data, but paired (from the same participants)

• E.g. The pre-test vs. post-test scores of a group participants

• E.g. The scores of a group participants after being taught using picture vs. film

Paired-Sample t-Test in MSExcel• H0 = there is no difference

between the two groups• Data | Data Analysis | t-

Test: Paired two Sample for Means | Select Variable 1 & 2 | Select Output Range

• P (T<=) two-tail <t Critical two-tail = reject H0 What’s the result?

• t Stat is minus

the pre (competence) <the post (graduation)

Paired-Samples t-Test in SPSS• Analyze |

Compare Means | Paired-Samples T-Test | Input the two variables

Results• Paired-Samples Statistics• Paired-Samples Correlations

Pearson’s r and sig. (r see effect, significant <.05)

• Paired-Samples Test Mean = difference of means between groups t value = minus first variable has smaller

mean df = sample size – 1 (degree of freedom) Sig. = significant p <.05

ResultsPearson’s rsignificant sig. <.05Correlation size of

effect

significant sig. <.05t minus first variable has smaller mean

Reporting on Results

On average, the participants has significantly higher scores on variable graduation grade (M= 71.40, SE = 2.001), than on variable competence score (M= 67.95, SE = 2.328, t(19) = .00, p<.05) with large effect r = .954) Legend

• M – mean• SE – standard error• t (19) – df• r – this formula (large effect)

Independent T-test• Compare the means of two groups’

participants in two different conditions• The groups are independent of each other

MS Excel – always assume unequal variances or do  F-Test Two Sample for Variance to decide if they are equal/unequal, then choose appropriate independent t-test

SPSS -- checked using Levene’s test in the results of independent t-test

• E.g. the scores of two groups’ participants after being taught using pictures vs. film

Independent T-test using MSExcel• Data | Data Analysis |

t-Test: Two-Sample Assuming Unequal Variances | Select Variable 1 & 2 (by group) | Select Output Range

• H0 = there is no difference between the two groups

• P (T<=) two-tail <t Critical two-tail = reject H0 What’s the result?

• t Stat is minus Pictures group < film group

Independent T-test Using SPSS• Analyze |

Compare Means | Independent-Samples t-Test | Insert the test variable & grouping variable

Results • Group Statistics• Independent Samples Test

Homogeneity of Variances using Levene’s test – should be NOT significant (groups are similar) sig >.05 See sig. of equal variances assumed (otherwise See not assumed)

Mean = difference of means between groups t value = minus first group has smaller mean df = sample size – 1 (degree of freedom) Sig. = significant p <.05

Results

Sig. > .05 group is similar (good!) equal variances assumed

significant sig. >.05Mean Difference minus first group has smaller mean

Reporting on Results• On average, participants that were

taught using film had higher scores (M=72, SE=2.921), than those taught using pictures (M=70.80, SE=2. 878). This difference was not significant t(18)=-.773, p>.05.

• Legend – same as in dependent t-test

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