statistical techniques to compare groups
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Statistical techniques to compare
groups

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Techniques covered in this Part
Onesample ttest
Independentsamples ttest;
Pairedsamples ttest;
Oneway analysis of variance (between groups); twoway analysis of variance (between groups);
and
nonparametric techniques.

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The different Statistical techniques to
compare groups in SPSS are:

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Assumptions
Each of the tests in this section have a number of
assumptions underlying their use. There are some
general assumptions that apply to all of the
parametric techniques discussed here (e.g. ttests,analysis of variance), and additional assumptions
associated with specific techniques.
The general assumptions are presented in thissection and the more specific assumptions are
presented in the following topics, as appropriate.

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Level of measurement
Each of these approaches assumes that the
dependent variable is measured at the interval or
ratio level, that is, using a continuous scale rather
than discrete categories.
Wherever possible when designing your study, try
to make use of continuous, rather than categorical,
measures of your dependent variable. This givesyou a wider range of possible techniques to use
when analysing your data.

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Random sampling
The techniques covered in this lecture assume that
the scores are obtained using a random sample
from the population. This is often not the case in
reallife research.

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Independence of observations
The observations that make up your data must be
independent of one another. That is, each
observation or measurement must notbe
influenced by any other observation ormeasurement. Violation of this assumption is very
serious.
There are a number of research situations that
may violate this assumption of independence. Forexample:
Studying the performance of students working inpairs or small groups. The behaviour of each
member of the group influences all other group

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Any situation where the observations or
measurements are collected in a group setting,or subjects are involved in some form of
interaction with one another, should be
considered suspect.
In designing your study you should try to ensure
that all observations are independent.
If you suspect some violation of this assumption,
Stevens (1996, p. 241) recommends that youset a more stringent alpha value (e.g. p

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Normal distributionWhat are the characteristics of a normal distribution
curve?
It is assumed that the populations from which thesamples are taken are normally distributed.
In a lot of research (particularly in the social
sciences), scores on the dependent variable are notnicely normally distributed.
Fortunately, most of the techniques are reasonably
robust or tolerant of violations of this assumption.
With large enough sample sizes (e.g. 30+), theviolation of this assumption should not cause any
major problems.
The distribution of scores for each of your groupscan be checked usin histo ramsobtained as art

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Homogeneity of variance
Techniques in this section make the assumption that
samples are obtained from populations of equal
variances.
To test this, SPSS performs the Levene test for
equality of variances as part of the ttest and analysis
of variances analyses.
If you obtain a significance value of less than .05, this
suggests that variances for the two groups are not
equal, and you have therefore violated the assumptionof homogeneity of variance.
Analysis of variance is reasonably robust to violations
of this assumption, provided the size of your groups is
reasonably similar (e.g. largest/smallest=1.5).

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Type 1 error, Type 2 error and
power
The purpose of ttests and analysis of variance is totest hypotheses. With these types of analyses there is
always the possibility of reaching the wrong
conclusion.
There are two different errors that we can make:
1. Type 1 error occurs when we think there is a
difference between our groups, but there really isnt.in other words, we may reject the null hypothesis
when it is, in fact, true .We can minimise this
possibility by selecting an appropriate alpha level.

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Type 2 error. This occurs when we fail to reject anull hypothesis when it is, in fact, false (i.e. believingthat the groups do not differ, when in fact they do).
Unfortunately these two errors are inversely related.As we try to control for a Type 1 error, we actuallyincrease the likelihood that we will commit a Type 2error.
Ideally we would like the tests that we use tocorrectly identify whether in fact there is a differencebetween our groups. This is called the power of atest.
Tests vary in terms of their power (e.g. parametrictests such as ttests, analysis of variance etc. aremore powerful than nonparametric tests).
Other factors that can influence the power of atest are:

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1. Sample Size:
When the sample size is large (e.g. 100 or more
subjects), then power is not an issue. However, whenyou have a study where the group size is small (e.g.n=20), then you need to be aware of the possibility that
a nonsignificant result may be due to insufficient
power. When small group sizes are involved it may be
necessary to adjust the alpha level to compensate.
There are tables available that will tell you how large
your sample size needs to be to achieve sufficientpower, given the effect size you wish to detect.
The higher the power, the more confident you can be
that there is no real difference between the groups.
http://localhost/var/www/apps/conversion/tmp/4SPSS%202012/Appropriate%20Sample%20Size%20in%20Survey%20Research%20+++.pdfhttp://localhost/var/www/apps/conversion/tmp/4SPSS%202012/Appropriate%20Sample%20Size%20in%20Survey%20Research%20+++.pdfhttp://localhost/var/www/apps/conversion/tmp/4SPSS%202012/Appropriate%20Sample%20Size%20in%20Survey%20Research%20+++.pdf 
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Effect size
With large samples, even very small differences
between groups can become statisticallysignificant. This does not mean that the difference
has any practical or theoretical significance.
One way that you can assess the importance of
your finding is to calculate the effect size (alsoknown as strength of association).
Effect size is a set of statistics which indicates the
relative magnitude of the differences betweenmeans. In other words, it describes the amount ofthe total variance in the dependent variable that is
predictable from knowledge of the levels of the
independent variable

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Effect size statistics, the most common of which
are:
eta squared,
Cohens d and Cohens f (see the formula)
Eta squared represents the proportion of variance
of the dependent variable that is explained by the
independent variable. Values for eta squared canrange from 0 to 1.
To interpret the strength of eta squared values the
following guidelines can be used:
.01=small effect;
.06=moderate effect; and
.14=large effect.
A number of criticisms have been levelled at eta

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Missing data
It is important that you inspect your data file formissing data. Run Descriptives and find out what
percentage of values is missing for each of your
variables.
If you find a variable with a lot of unexpected
missing data you need to ask yourself why.
You should also consider whether your missing
values are happening randomly, or whether there issome systematic pattern (e.g. lots of women failing
to answer the question about their age).

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The Options button in many of the SPSS statisticalprocedures offers you choices for how you want SPSSto deal with missing data.
TheExclude cases listwise option will include cases inthe analysis only if it has full data on all of the variableslisted in your variables box for that case. A case will betotally excluded from all the analyses if it is missing
even one piece of information. This can severely, andunnecessarily, limit your sample size.
The Exclude cases pairwise (sometimes shown asExclude cases analysis by analysis) option, however,
excludes the cases (persons) only if they are missingthe data required for the specific analysis. They will stillbe included in any of the analyses for which they havethe necessary information.
TheReplace with mean option, which is available insome SPSS statistical procedures, calculates the mean

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I would strongly recommend that you use
pairwise exclusionof missing data,unless you have a pressing reason to do
otherwise.
The only situation where you might need touse listwise exclusion is when you want to
refer only to a subset of cases that
provided a full set of results.

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Hypothesis testing
What is a hypothesis test
A hypothesis test uses sample data to test a
hypothesis about the population from which the
sample was taken.
When to use a hypothesis test
Use a hypothesis test to make inferences about
one or more populations when sample data areavailable.

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Why use a hypothesis test
Hypothesis testing can help answer questions such
as:
Are students achievement in science meeting orexceeding his achievement in science ?
Is the performance of teacher Abetter than theperformance of teacher B?

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Hypothesis Testing with One
Sample ttest
21

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1. Onesample ttest
What is a onesample ttest
A onesample ttest helps determine whether (thepopulation mean) is equal to a hypothesized
value (the test mean).
The test uses the standard deviation of the sample
to estimate (the population standard deviation).
If the difference between the sample mean and the
test mean is large relative to the variability of thesample mean, then is unlikely to be equal to thetest mean.

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Cont.
When to use a onesample ttest
Use a onesample ttest when continuous data
are available from a single random sample.
The test assumes the population is normallydistributed. However, it is fairly robust to
violations of this assumption for sample sizes
equal to or greater than 30, provided the
observations are collected randomly and thedata are continuous, unimodal, and
reasonably symmetric

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24
Ztest or ttest?
The shortcoming of the z test isthat it requires more information
than is usually available.
To do a ztest, we need to knowthe value of population standard
deviation to be able to computestandard error. But it is rarelyknown.

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When the population variance is unknown, we
use one samplettest
n
s
x
What if is unknown?
Cantcompute z test statistics (z score)
Z =
Population standarddeviation must be known
Can compute t statistict =
n
x
Sample standard deviation
must be known

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Hypothesis testing with a onesample
ttest
State the hypotheses Ho: = hypothesized value
H1: hypothesized value
Set the criteria for rejecting Ho
Alpha level
Critical t value

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Determining the criteria for rejecting
the Ho
If the value of texceeds some threshold or
critical valued, t, then an effect is detected
(i.e. the null hypothesis of no difference is
rejected)

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Table C.3 (p. 638 in text)

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Degrees of freedom for One Sample
ttest
degrees of freedomis the number of values
in the final calculation of a statistic that are
free to vary.
Degrees of freedom (d.f.) is computed as the
one less than the sample size (the
denominator of the standard deviation):df= n 1

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Hypothesis testing with a one
sample ttest
Compute the test statistic (tstatistic)
t =
Make statistical decision and drawconclusion
t t critical value, reject null hypothesis t < t critical value, fail to reject null hypothesis
n
s
x

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One Sample ttest Example
You are conducting an experiment to see
if a given therapy works to reduce test
anxiety in a sample of college students.
A standard measure of test anxiety isknown to produce a = 20. In the
sample you draw of 81 the mean = 18
with s= 9.
Use an alpha level of .05

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Write hypotheses
Ho: The average test anxiety in the sample ofcollege students will not be statisticallysignificantly different than 20.
Ho: = 20
H1= The average test anxiety in the sample ofcollege students will be statisticallysignificantly lower than 20.
H1: < 20

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Compute test statistic (t statistic)
t = 1820 = 2 = 2
9 / 81 1

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Compare to criteria and make
decision
tstatistic of 2 exceeds your critical value of 1.671.
Reject the null hypothesis and conclude that
average test anxiety in the sample of college
students is statistically significantly lower than
20, t = 2.0,p< .05.

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Procedures
Select Analyze/Compare Means/OneSample ttest
Notice, SPSS allows us to specify what Confidence
Interval to calculate. Leave it at 95%. Click Continue
and then Ok. The output follows.

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Notice that descriptive statistics are automatically
calculated in the onesample ttest. Does our tvalue agree
with the one in the textbook? Look at the Conf idence
Interval . Notice that i t is not th e con fidence interval of
the mean, but the confid ence interval for the dif ference
between the samp le mean and the test value we

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Hypothesis Testing with two
Sample ttest
38

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Independentsamples Ttest
An independentsamples ttest isused when you want to compare the
mean score, on some cont inuous
var iable, for two di f ferent g roups ofsubjects.

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Summary for independentsamples t
test
Example of research question:Is there a significant difference in the mean selfesteem
scores for males and females?
What you need:
Two variables:
one categorical, independent variable (e.g.males/females); and
one continuous, dependent variable (e.g. selfesteemscores).
Assumptions:
The assumptions for this test are (continuous scale,
normality, independence of observation, homogeneity,
Cont

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Cont.
What it does:An independentsamples ttest will tell you whether
there is a statistically significant difference in the
mean scores for the two groups (that is, whether
males and females differ significantly in terms oftheir selfesteem levels).
In statistical terms, you are testing the probability
that the two sets of scores (for males and
females) came from the same population.
Nonparametric alternative:
MannWhitney Test
f

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Procedure for independentsamples
ttest
1. From the menu at the top of the screen click on:Analyze, then click on Compare means, then onIndependent Samples Ttest.
2. Move the dependent (continuous) variable (e.g. totalselfesteem) into the area labelled Test variable.
3. Move the independent variable (categorical) variable(e.g. sex) into the section labelled Grouping variable.
4. Click on Define groups and type in the numbers usedin the data set to code each group. In the current datafile 1=males, 2=females; therefore, in the Group 1 box,
type 1; and in the Group 2 box, type 2.5. Click on Continue and then OK.
The output generated from this procedure is shown below.

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I t t ti f t t f

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Interpretation of output from
independentsamples ttest
Step 1: Checking the information about thegroups
In the Group Statistics box SPSS gives you the
mean and standard deviation for each of your
groups (in this case: male/female). It also gives youthe number of people in each group (N). Always
check these values first. Do they seem right? Are
the N values for males and females correct? Or are
there a lot of missing data? If so, find out why.Perhaps you have entered the wrong code for
males and females (0 and 1, rather than 1 and 2).
Check with your codebook.

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Cont.
Step 2: Checking assumptions
Levenestest for equality of variances: This tests whether the variance (variation) of scores for
the two groups (males and females) is the same.
The outcome of this test determines which of the tvaluesthat SPSS provides is the correct one for you to use.
If your Sig. value is larger than .05 (e.g. .07, .10), youshould use the first line in the table, which refers to Equalvariances assumed.
If the significance level of Levenestest is p=.05 or less
(e.g. .01, .001), this means that the variances for the twogroups (males/females) are not the same.
Therefore your data violate the assumption of equalvariance. Dont panicSPSS provides you with analternative tvalue. You should use the information in the
second line of the ttest table, which refers to Equal
Cont

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Cont.
Step 3: Assessing differences between the
groups If the value in the Sig. (2tailed) column is equal
o r less than .05 (e.g . .03, .01, .001), then there is a
significant difference in the mean scores on your
dependent variable for each of the two groups. If the value is above .05 (e.g. .06, .10), there is no
significant difference between the two groups.
Having established that there is a significantdifference, the next step is to find out which set of
scores is higher.
C l l ti th ff t i f

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Calculating the effect size for
independentsamples ttest
Effect size statistics provide an indication of themagnitude of the differences between your
groups (not just whether the difference could
have occurred by chance).
eta squared is the most commonly used.
Eta squared can range from 0 to 1 and
represents the proportion of variance in the
dependent variable that is explained by the
independent (group) variable.
SPSS does not provide eta squared values for t
tests.

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the effect size of .006 is very small. Expressed asa percentage (multiply your eta square value by
100), only .6 per cent of the variance in self
esteem is explained by sex.
P ti th lt f

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Presenting the results for
independentsamples ttest
The results of the analysis could be presented asfollows:
An independentsamples ttest was conducted tocompare the selfesteem scores for males and
females. There was no significant difference in
scores for males (M=34.02, D=4.91) and females
[M=33.17, SD=5.71; t(434)=1.62, p=.11]. The
magnitude of the differences in the means was
very small (eta squared=.006).

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Pairedsamples Ttest
Pairedsamples ttest (also referred to as repeatedmeasures) is used when you have only one group of
people (or companies, or machines etc.) and you
collect data from them on two different occasions, or
under two different conditions. Pretest/posttest experimental designs are an example
of the type of situation where this technique is
appropriate.
It can also be used when you measure the sameperson in terms of his/her response to two different
questions.
In this case, both dimensions should be rated on the
same scale e. . from 1=not at all im ortant to 5=ver
Summary for paired samples t

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Summary for pairedsamples t
test
Example of research question: Is there a significant change in participants
fear of statistics scores following
participation in an intervention designed to
increase students confidence in their abilityto successfully complete a statistics course?
Does the intervention have an impact on
participants fear of statistics scores?

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Cont.
What you need:One set of subjects (or matched pairs). Each
person (or pair) must provide both sets of
scores.
Two variables:
one categorical independent variable (in this
case it is Time: with two different levels Time 1,
Time 2); andone continuous, dependent variable (e.g. Fear
of Statistics Test scores) measured on two
different occasions, or under different
conditions.
C t

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Cont.
What it does:A pairedsamples ttest will tell you whether there is a
statistically significant difference in the mean scores
for Time 1 and Time 2.
Assumptions: The basic assumptions for ttests.
Additional assumption: The difference between the
two scores obtained for each subject should be
normally distributed. With sample sizes of 30+,violation of this assumption is unlikely to cause any
serious problems.
Nonparametric alternative:Wilcoxon Signed Rank
Test.
Procedure for paired samples t

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Procedure for pairedsamples t
test
1. From the menu at the top of the screen click on:Analyze, then click on Compare Means, then onPaired Samples Ttest.
2. Click on the two variables that you are interestedin comparing for each subject (e.g. fost1: fear ofstats time1, fost2: fear of stats time2).
3. With both of the variables highlighted, move theminto the box labelled Paired Variables by clickingon the arrow button. Click on OK.
The output generated from this procedure isshown below

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Interpretation of output from paired

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Interpretation of output from paired
samples ttest
Step 1: Determining overall significanceIn the table labelled Paired Samples Test you need
to look in the final column, labelled Sig. (2tailed)
this is your probability value. If this value is less
than .05 (e.g. .04, .01, .001), then you can concludethat there is a significant difference between your two
scores.

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Cont.
Step 2: Comparing mean valuesHaving established that there is a significant
difference, the next step is to find out which set of
scores is higher (Time 1 or Time 2). To do this, look
in the first printout box, labelled Paired SamplesStatistics. This box gives you the Mean scores for
each of the two sets of scores.
Calculating the effect size for paired

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Calculating the effect size for paired
samples ttest
Given our eta squared value of .50, we can
conclude that there was a large effect, with a
substantial difference in the Fear of Statistics scores

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Caution
Although we obtained a significant difference inthe scores before/after the intervention, we
cannot say that the intervention caused the drop
in Fear of Statistics Test scores. Research is
never that simple, unfortunately! There are manyother factors that may have also influenced the
decrease in fear scores.
Wherever possible, the researcher should try to
anticipate these confounding factors and eithercontrol for them or incorporate them into the
research design.
Presenting the results for paired

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Presenting the results for paired
samples ttest
A pairedsamples ttest was conducted toevaluate the impact of the intervention on
students scores on the Fear of Statistics Test(FOST). There was a statistically significant
decrease in FOST scores from Time 1 (M=40.17,
SD=5.16) to Time 2 [M=37.5, SD=5.15,
t(29)=5.39, p

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Hypothesis Testing
Analysis Of Variance
61

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Oneway analysis of variance
In many research situations, however, we areinterested in comparing the mean scores of more thantwo groups. In this situation we would use analysis ofvariance (ANOVA).
Oneway analysis of variance involves oneindependent variable (referred to as a factor), whichhas a number of different levels. These levelscorrespond to the different groups or conditions.
For example, in comparing the effectiveness of three
different teaching styles on students Maths scores,you would have one factor (teaching style) with threelevels (e.g. whole class, small group activities, selfpaced computer activities).
The dependent variable is a continuous variable (in
Cont.

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Analysis of variance is so called because it compares
the variance (variability in scores) between the different
groups (believed to be due to the independent variable)with the variability within each of the groups (believed to
be due to chance).
An F ratio is calculated which represents the variance
between the groups, divided by the variance within the
groups.
A large F ratio indicates that there is more variability
between the groups (caused by the independent
variable) than there is within each group (referred to as
the error term).
A significant F test indicates that we can reject the null
hypothesis, which states that the population means are
Cont

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Cont.
There are two different types of oneway ANOVA :
betweengroups analysis of variance, which isused when you have different subjects or cases in
each of your groups (this is referred to as an
independent groups design); and
repeatedmeasures analysis of variance, which is
used when you are measuring the same subjects
under different conditions (or measured at
different points in time) (this is also referred to asa withinsubjects design).

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Planned comparisons andPosthoccomparisons

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Planned comparisons
Planned comparisons (also know as a priori) are used
when you wish to test specific hypotheses (usuallydrawn from theory or past research) concerning the
differences between a subset of your groups (e.g. do
Groups 1 and 3 differ significantly?).
Planned comparisons do not control for the increasedrisks of Type 1 errors.
If there are a large number of differences that you wish
to explore, it may be safer to use the alternative
approach (posthoc comparisons), which is designed toprotect against Type 1 errors.
The other alternative is to apply what is known as a
Bonferroni adjustment to the alpha level that you will use
to ud e statistical si nificance. This involves settin a
P t h i

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Posthoccomparisons Posthoc comparisons (also known as a posteriori) are
used when you want to conduct a whole set ofcomparisons, exploring the differences between each of
the groups or conditions in your study.
Posthoc comparisons are designed to guard against the
possibility of an increased Type 1 error due to the largenumber of different comparisons being made.
With small samples this can be a problem, as it can be
very hard to find a significant result, even when the
apparent difference in scores between the groups isquite large.
There are a number of different posthoc tests that you
can use, and these vary in terms of their nature and
strictness. The assumptions underlying the posthoc tests
P t H t t th t l i

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Multiple Comparison Tests
AND Range TestsRange Tests Only
Multiple Comparison Tests
Only
Tukeys HSD (honestlysignificant difference) test
Tukeys b (AKA, TukeysWSD (Wholly Significant
Difference))
Bonferroni (don't use with 5
groups or greater)
Hochbergs GT2 SNK (StudentNewmanKeuls)
Sidak
Gabriel Duncan
Dunnett (compares a
control group to the other
groups without comparing
the other groups to eachother)
Scheffe (confidence
intervals that are fairly
wide)
REGW F (RyanEinot
GabrielWelsch F test)
LSD (least significant
difference)
REGW Q (RyanEinot
GabrielWelsch range test)
Post Hoc tests that assume equal variance

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Post Hoc tests
Fisher's LSD (Least Significant Different)This test is the most liberal of all Post Hoc tests and its critical t forsignificance is not affected by the number of groups. This test isappropriate when you have 3 means to compare. It is notappropriate for additional means.
Bonferroni (AKA, Dunns Bonferroni)
This test does not require the overall ANOVA to be significant. It isappropriate when the number of comparisons (c = number ofcomparisons = k(k1))/2) exceeds the number of degrees offreedom (df) between groups (df = k1). This test is veryconservative and its power quickly declines as the c increases. Agood rule of thumb is that the number of comparisons (c) be nolarger than the degrees of freedom (df).
NewmanKeuls
If there is more than one true null hypothesis in a set of means, thistest will overestimate they familywise error rate. It is appropriate touse this test when the number of comparisons exceeds the
number of degrees of freedom (df) between groups (df = k1) and
Cont

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Cont. Tukey's HSD (Honestly Significant Difference)
This test is perhaps the most popular post hoc. It reducesType I error at the expense of Power. It is appropriate to usethis test when one desires all the possible comparisonsbetween a large set of means (6 or more means).
Tukey's b (AKA, TukeysWSD (Wholly Significant
Difference))This test strikes a balance between the NewmanKeuls andTukey's more conservative HSD regarding Type I error andPower. Tukey's b is appropriate to use when one is makingmore than k1 comparisons, yet fewer than (k(k1))/2comparisons, and needs more control of Type I error thanNewmanKuels.
Scheffe
This test is the most conservative of all post hoc tests.Compared to Tukey's HSD, Scheffe has less Power whenmaking pairwise (simple) comparisons, but more Powerwhen making complex comparisons. It is appropriate to use
Oneway betweengroups ANOVA

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One way between groups ANOVA
with posthoc tests
Oneway betweengroups analysis of variance isused when you have one independent (grouping)
variable with three or more levels (groups) and one
dependent continuous variable.
The oneway part of the title indicates there is onlyone independent variable, and betweengroupsmeans that you have different subjects or cases in
each of the groups.
Summary for oneway between

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Summary for one way between
groups ANOVA with posthoc tests
What you need: Two variables: one categorical independent variable with three
or more distinct categories. This can also be a
continuous variable that has been recoded to
give three equal groups (e.g. age group: subjectsdivided into 3 age categories, 29 and younger,
between 30 and 44, 45 or above). For
instructions on how to do this see Chapter 8; and
one continuous dependent variable (e.g.
optimism).

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Cont.
What it does:Oneway ANOVA will tell you whether there are
significant differences in the mean scores on the
dependent variable across the three groups.
Posthoc tests can then be used to find outwhere these differences lie.
Nonparametric alternative: KruskalWallis Test

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Population distribution of response variable ineach group is normal
Standard deviations of population distributions
for the groups are equal
Independent randomsamples
(In practice, in a lot of cases, these arent
strictly met, but we do ANOVA anyway)
Assumptions

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Null hypothesis:
H0: 1= 2= 3
Alternative or research hypothesis:
Ha: 1 2or 1 3or 3 3
Hypotheses

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Probability of making error in decision to reject
null hypothesis For this test choose = 0.05
Level of significance

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Test statistic
gNWSS
gBSSF
1
estimateWithin
estimateBetween
11 gdf
gNdf 2

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Between estimate of variance
Between estimate calculations
1
2
2
g
yyns
ii
Group N Mean Group Mean  Difference Times
Grand Mean Squared N
Walk 46 10.20 3.900 15.210 699.660
Drive 228 14.43 0.330 0.109 24.829
Bus 17 20.35 6.250 39.063 664.063
Total 291 14.10 1388.552
Divided by g1 (between estimate) 694.276
Grand
meanBS
S

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Within estimate of variance
Within estimate calculations
gN
sns
ii
2
21
Group N Variance Variance
Times
n i  1
Walk 46 46.608 2143.965
Drive 228 68.857 15699.351
Bus 17 170.877 2904.912Total 291 20748.228
Divided by Ng (within estimate) 72.042
WS
S

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Calculating the Fstatistic
Fstatistic calculation
637.9042.72
276.694
estimateWithin
estimateBetweenF

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df1(degrees of freedom in numerator) Number of samples/groups  1
= 3  1 = 2
df2(degrees of freedom in denominator)
Total number of cases  number of groups =2916  3 = 288
Degrees of freedom

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Separate tables for each probability df1(degrees of freedom in numerator) across
top
df2(degrees of freedom in denominator) down
side Values of Fin table
For degrees of freedom not given, use nextlower value
Table of Fdistribution

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Find Fvalues for degrees of freedom (2, 313)= 0.05, F= 3.07 (2, 120)= 0.01, F= 4.79 (2, 120)= 0.001, F= 7.31 (2, 120)
F= 9.637 > F= 7.31 for = 0.001
pvalue < 0.001
pvalue

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pvalue < 0.001 is less than = 0.05 Reject null hypothesis that all means are
equal
Conclude that at least one of the means is
different from the others
Conclusion

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Example
Treatment 1 Treatment 2 Treatment 3 Treatment 460 inches 50 48 47
67 52 49 6742 43 50 5467 67 55 6756 67 56 6862 59 61 6564 67 61 6559 64 60 5672 63 59 6071 65 64 65

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Example
Treatment 1 Treatment 2 Treatment 3 Treatment 460 inches 50 48 47
67 52 49 6742 43 50 5467 67 55 6756 67 56 6862 59 61 6564 67 61 6559 64 60 5672 63 59 6071 65 64 65
Step 1) calculate the sum of
squares between groups:
Mean for group 1 = 62.0
Mean for group 2 = 59.7
Mean for group 3 = 56.3
Mean for group 4 = 61.4
Grand mean= 59.85
SSB = [(6259.85)2+ (59.759.85)2+ (56.359.85)2+ (61.459.85)2 ]xn per
group= 19.65x10= 196.5

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Example
Treatment 1 Treatment 2 Treatment 3 Treatment 460 inches 50 48 47
67 52 49 6742 43 50 5467 67 55 6756 67 56 6862 59 61 6564 67 61 6559 64 60 5672 63 59 6071 65 64 65
Step 2) calculate the sum of
squares within groups:
(6062)2+(6762)2+(4262)
2+(6762)2+(5662)2+(62
62)2+(6462)2+(5962)2+
(7262)2+(7162)2+(50
59.7)2+(5259.7)2+(43
59.7)2
+6759.7)2
+(6759.7)
2+(6959.7)2+.(sum of40 squared deviations) =
2060.6

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Step 3) Fill in the ANOVA table
3 196.5 65.5 1.14 .344
36 2060.6 57.2
Source of variation d.f. Sum of squares Mean Sum of
Squares
Fstatistic pvalue
Between
Within
Total 39 2257.1
Procedure for oneway between

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y
groups ANOVA with posthoc tests1. From the menu at the top of the screen click on:
Analyze, then click on Compare Means, then on OnewayANOVA.
2. Click on your dependent (continuous) variable (e.g. Totaloptimism). Move this into the box marked Dependent Listby clicking on the arrow button.
3. Click on your independent, categorical variable (e.g.agegp3). Move this into the box labelled Factor.
4. Click the Options button and click on Descriptive,Homogeneity of variance test, BrownForsythe, Welshand Means Plot.
5. For Missing values, make sure there is a dot in theoption marked Excludecases analysis by analysis. If not,click on this option once. Click on Continue.
6. Click on the button marked Post Hoc. Click on Tukey.
7. Click on Continue and then OK.
The output is shown below.

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betweengroups ANOVA with posthoc

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tests
DescriptivesThis table gives you information about each group (number ineach group, means, standard deviation, minimum andmaximum, etc.) Always check this table first. Are the Ns foreach group correct?
Test of homogeneity of varianceso The homogeneity of variance option gives you Levenestest for
homogeneity of variances, which tests whether the variance inscores is the same for each of the three groups.
o Check the significance value (Sig.) for Levenestest. If thisnumber is greater than .05 (e.g. .08, .12, .28), then you havenot violated the assumption of homogeneity of variance.
o If you have found that you violated this assumption you willneed to consult the table in the output headed Robust Tests ofEquality of Means. The two tests shown there (Welsh and
BrownForsythe) are preferable when the assumption of the
Cont.

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ANOVA
This table gives both betweengroups and withingroups
sums of squares, degrees of freedom etc. The main thing
you are interested in is the column marked Sig. If the Sig.
value is less than or equal to .05 (e.g. .03, .01, .001),
then there is a significant difference somewhere among
the mean scores on your dependent variable for the threegroups.
Multiple comparisons
You should look at this table only if you found asignificant difference in your overall ANOVA. That is, if
the Sig. value was equal to or less than .05. The
posthoc tests in this table will tell you exactly where
the differences among the groups
Cont.

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Means plotsThis plot provides an easy way to compare the
mean scores for the different groups.
Warning: these plots can be misleading.
Depending on the scale used on the Y axis (inthis case representing Optimism scores), even
small differences can look dramatic.

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Calculating effect sizeThe information you need to calculate eta squared,one of the most
common effect size statistics, is provided in theANOVA table (a calculator would be useful here). Theformula is:
Cohen classifies .01 as a small effect, .06 as a
medium effect and .14 as a large effect.

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Warning
In this example we obtained a statisticallysignificant result, but the actual difference in themean scores of the groups was very small (21.36,
22.10, 22.96). This is evident in the small effectsize obtained (eta squared=.02). With a largeenough sample (in this case N=435), quite smalldifferences can become statistically significant,
even if the difference between the groups is oflittle practical importance. Always interpret yourresults carefully, taking into account all theinformation you have available. Dont rely too
heavily on statistical significancemany other

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Presenting the results
A oneway betweengroups analysis of variance wasconducted to explore the impact of age on levels of
optimism, as measured by the Life Orientation test (LOT).
Subjects were divided into three groups according to their
age (Group 1: 29 or less; Group 2: 30 to 44; Group 3: 45and above). There was a statistically significant difference
at thep

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Twoway betweengroupsANOVA
Twoway betweengroups
ANOVA

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ANOVA
Twoway means that there are two independentvariables, and betweengroups indicates that
different people are in each of the groups. This
technique allows us to look at the individual and
joint effect of two independent variables on onedependent variable.
The advantage of using a twoway design is that
we can test the main effect for each independentvariable and also explore the possibility of an
interaction effect.
An interaction effect occurs when the effect of one
independent variable on the dependent variable
S f t ANOVA

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Summary for twoway ANOVA
Example of research question:What is the impact of age and gender on
optimism? Does gender moderate the
relationship between age and optimism?
What you need: Three variables:
two categorical independent variables (e.g. Sex:
males/females; Age group: young, middle, old);
andone continuous dependent variable (e.g. total
optimism).
Assumptions: the assumptions underlying ANOVA.

C t

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Cont.
What it does: Twoway ANOVA allows you to simultaneously
test for the effect of each of your independent
variables on the dependent variable and also
identifies any interaction effect.
For example, it allows you to test for:
sex differences in optimism;
differences in optimism for young, middle andold subjects; and
the interaction of these two variablesis therea difference in the effect of age on optimism for
males and females?
v uanalysis

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y
P d f t ANOVA

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Procedure for twoway ANOVA
1. From the menu at the top of the screen click on:Analyze, then click on General Linear Model, thenon Univariate.
2. Click on your dependent, continuous variable (e.g.total optimism) and move it into the box labelledDependent variable.
3. Click on your two independent, categoricalvariables (sex, agegp3: this is age grouped into threecategories) and move these into the box labelled Fixed
Factors.4. Click on the Options button.
Click on Descriptive Statistics, Estimates of effectsize and Homogeneity tests.
Click on Continue.
Cont.5 Click on the Post Hoc button

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5. Click on the Post Hoc button.
From the Factors listed on the lefthand side choose the
independent variable(s) you are interested in (this variableshould have three or more levels or groups: e.g. agegp3).
Click on the arrow button to move it into the Post Hoc Tests forsection.
Choose the test you wish to use (in this case Tukey).
Click on Continue.
6. Click on the Plots button.
In the Horizontal box put the independent variable that has themost groups (e.g. agegp3).
In the box labelled Separate Lines put the other independentvariable (e.g. sex).
Click on Add. In the section labelled Plots you should now see your two
variables listed (e.g. agegp3*sex).
The output

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Interpretation of output from twoway
ANOVA

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ANOVA Descriptive statistics.
These provide the mean scores, standard deviations
and N for each subgroup. Check that these values are
correct.
LevenesTest of Equality of Error Variances. This test provides a test of one of the assumptions
underlying analysis of variance. The value you are most
interested in is the Sig. level. You want this to be greater
than .05, and therefore not significant. A significant result
(Sig. value less than .05) suggests that the variance of your
dependent variable across the groups is not equal.
If you find this to be the case in your study it is
recommended that you set a more stringent significance
level e. . .01 for evaluatin the results of our twowa
Cont.

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Tests Of BetweenSubjects Effects.
This gives you a number of pieces of information, notnecessarily in the order in which you need to check
them.
Interaction effects
The first thing you need to do is to check for the
possibility of an interaction effect (e.g. that the
influence of age on optimism levels depends on
whether you are a male or a female). In the SPSS output the line we need to look at is
labeled AGEGP3*SEX. To find out whether the
interaction is significant, check the Sig. column for
that line. If the value is less than or equal to .05
Cont.

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Main effects
In the lefthand column, find the variable you are interested in(e.g. AGEGP3) To determine whether there is a main effect
for each independent variable, check in the column marked
Sig. next to each variable.
Effect sizeThe effect size for the agegp3 variable is provided in the
column labelled Partial Eta Squared (.018).
Using Cohens (1988) criterion, this can be classified as
small (see introduction to Part Five). So, although this effectreaches statistical significance, the actual difference in the
mean values is very small. From the Descriptives table we
can see that the mean scores for the three age groups
(collapsed for sex) are 21.36, 22.10, 22.96. The difference
between the rou s a ears to be of little ractical
Cont

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Cont.
PosthocPosthoc tests are relevant only if you have
more than two levels (groups) to your
independent variable. These tests
systematically compare each of your pairs ofgroups, and indicate whether there is a
significant difference in the means of each.
SPSS provides these posthoc tests as part
of the ANOVA output. tests
Cont

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Cont.
Multiple comparisonsThe results of the posthoc tests are provided in
the table labelled Multiple Comparisons. We
have requested the Tukey Honestly Significant
Difference test, as this is one of the morecommonly used tests. Look down the column
labelled Sig. for any values less than .05.
Significant results are also indicated by a little
asterisk in the column labelled Mean Difference.
Cont

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Cont.
PlotsYou will see at the end of your SPSS output a
plot of the optimism scores for males and
females, across the three age groups. This plot
is very useful for allowing you to visually inspectthe relationship among your variables.
Additional analyses if you obtain a
significant interaction effect

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significant interaction effect
Conduct an analysis of simple effects. This meansthat you will look at the results for each of the
subgroups separately. This involves splitting the
sample into groups according to one of your
independent variables and running separate oneway ANOVAs to explore the effect of the other
variable.
Use the SPSS Split File option. This option allows
you to split your sample according to one
categorical variable and to repeat analyses
separately for each group.
Procedure for splitting the sample

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Procedure for splitting the sample1. From the menu at the top of the screen click on: Data,
then click on Split File.2. Click on Organize output by groups.
3. Move the grouping variable (sex) into the box markedGroups based on.
4. This will split the sample by sex and repeat anyanalyses that follow for these two groups separately.
5. Click on OK.
After splitting the file you then perform a oneway ANOVA
Important: When you have finished these analysesfor the separate groups you must turn the Spl i tF ile op t ion off
Presenting the results from twoway
ANOVA

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ANOVA
A twoway betweengroups analysis of variance wasconducted to explore the impact of sex and age on levels of
optimism, as measured by the Life Orientation test (LOT).
Subjects were divided into three groups according to their
age (Group 1: 1829 years; Group 2: 3044 years; Group3: 45 years and above). There was a statistically significantmain effect for age [F(2, 429)=3.91, p=.02]; however, the
effect size was small (partial eta squared=.02). Posthoc
comparisons using the Tukey HSD test indicated that the
mean score for the 1829 age group (M=21.36, SD=4.55)was significantly different from the 45 + group (M=22.96,SD=4.49). The 3044 age group (M=22.10, SD=4.15) did
not differ significantly from either of the other groups. The
main effect for sex [F(1, 429)=.30,p=.59] and the
= =

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Analysis of covariance(ANCOVA)
Analysis of covariance (ANCOVA)

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Analysis of covariance is an extension of analysis of
variance that allows you to explore differences between
groups while statistically controlling for an additional
(continuous) variable. This additional variable (called a
covariate) is a variable that you suspect may be
influencing scores on the dependent variable.
SPSS uses regression procedures to remove the
variation in the dependent variable that is due to the
covariate/s, and then performs the normal analysis of
variance techniques on the corrected or adjustedscores.
By removing the influence of these additional variables
ANCOVA can increase the power or sensitivity of the F
test.
Uses of ANCOVA

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ANCOVA can be used when you have a twogroup pre
test/posttest design (e.g. comparing the impact of twodifferent interventions, taking before and after
measures for each group). The scores on the pretest
are treated as a covariate to control for preexisting
differences between the groups. This makes ANCOVAvery useful in situations when you have quite small
sample sizes, and only small or medium effect sizes
(see discussion on effect sizes in the introduction to
Part. Five). Under these circumstances (which are very
common in social science research), Stevens (1996)
recommends the use of two or three carefully chosen
covariates to reduce the error variance and increase
your chances of detecting a significant difference
CONT.
ANCOVA is also handy when you have been unable to

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ANCOVA is also handy when you have been unable to
randomly assign your subjects to the different groups, but
instead have had to use existing groups (e.g. classes ofstudents). As these groups may differ on a number of
different attributes (not just the one you are interested in),
ANCOVA can be used in an attempt to reduce some of
these differences. The use of wellchosen covariates canhelp reduce the confounding influence of group
differences. This is certainly not an ideal situation, as it is
not possible to control for all possible differences; however,
it does help reduce this systematic bias. The use ofANCOVA with intact or existing groups is somewhat of a
contentious one among writers in the field. It would be a
good idea to read more widely if you find yourself in this
situation Some of these issues are summarised in Stevens
1 N 30 O l t t t
Comparing means

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1 group N 30 Onesample ttestN< 30 Normally distributed Onesample ttest
Not normal Sign test2 groups Independen
tN 30 ttestN< 30 Normally distributed ttest
Not normal MannWhitney Uor Wilcoxonsignedrank test
Paired N 30 paired ttestN< 30 Normally distributed paired ttest
Not normal Wilcoxon signedrank test3 or more
groupsIndependen
tNormally
distributed1 factor One way anova
2 factors two or other anovaNot normal KruskalWallis oneway
analysis of varianceby ranksDependent Normally
distributedRepeated measures anova
Not normal Friedman twoway analysis ofvarianceby ranks
THE END

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Test Statistic for Testing a SinglePopulation Mean ()

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Population Mean ()
n
s
Xt
XSE
Xt
oo
or)(
~ tdistribuion
with df = n1.
In general the basic form of a test statistic is given by:
)()()(
estimateSEvalueedhypothesizestimatet
post hoc

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post hoc
Once you have determined that differences existamong the means, post hoc range tests and
pairwise multiple comparisons can determine
which means differ. Range tests identify
homogeneous subsets of means that are notdifferent from each other. Pairwise multiple
comparisons test the difference between each
pair of means, and yield a matrix where asterisks
indicate significantly different group means at an
alpha level of 0.05 (SPSS, Inc.).
Singlestep tests
f

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TukeyKramer: The TukeyKramer test is an extension of the Tukey test to
unbalanced designs. Unlike Tukey test for balanced designs, it is not exact.
The FWE of the TukeyKramer test may be less than ALPHA. It is lessconservative for only slightly unbalanced designs and more conservative
when differences among samples sizes are bigger.
Hochbergs GF2: The GF2 test is similar to Tukey, but the critical
values are based on the studentized maximum modulus distribution instead
of the studentized range. For balanced or unbalanced oneway anova, itsFWE does not exceed ALPHA. It is usually more conservative than the
TukeyKramer test for unbalanced designs and it is always more
conservative than the Tukey test for balanced designs.
Gabriel: Like the GF2 test, the Gabriel test is based on studentized
maximum modulus. It is equivalent to the GF2 test for balanced onewayanova. For unbalanced oneway anova, it is less conservative than GF2, but
its FWE may exceed ALPHA in highly unbalanced designs.
Dunnett: The Dunnettstest is a test to use when the only pariwisecomparisons of interest are comparisons with a control. It is an exact test,
that is, its FWE is exactly equal to ALPHA, forbalanced as well as
Singlestep tests

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Below are short descriptions of these tests.
LSD: The LSD (Least Significant Difference) test is a twostep
test. First the ANOVA F test is performed. If it is significant at
level ALPHA, then all pairwise ttests are carried out, each at
level ALPHA. If the F test is not significant, then the
procedure terminates. The LSD test does not control the
FWE. Bonferroni: The Bonferroni multiple comparison test is a
conservative test, that is, the FWE is not exactly equal
to ALPHA, but is less than ALPHA in most situations. It is
easy to apply and can be used for any set of comparisons.Even though the Bonferroni test controls the FEW rate, in
many situations it may be too conservative and not have
enough power to detect significant differences.
Singlestep tests Sidak : Sidak adjusted pvalues are also easy to compute.

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Sidak: Sidak adjusted p values are also easy to compute.
The Sidak test gives slightly smaller adjusted pvalues than
Bonferroni, but it guarantees the strict control of FWE onlywhen the comparisons are independent .
Scheffe: The Scheffe test is used in ANOVA analysis
(balanced, unbalanced, with covariates). It controls for the
FWE for all possible contrasts, not only pairwise comparisons
and is too conservative in cases when pairwise comparisons
are the only comparisons of interest.
Tukey: The Tukey test is based on the studentized range
distribution (standardized maximum difference between the
means). For oneway balanced anova, the FWE of the Tukeytest is exactly equal the assumed value of ALPHA. The Tukey
test is also exact for oneway balanced anova with correlated
errors when the type of correlation structure is compound
symmetry
Appendix

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Appendix
Effect size statistics, the most common of which are:
eta squared,

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q ,
Cohens d and Cohens f
ANOVA Table

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ANOVA Table
Between
(k groups)
k1 SSB(sum of squared
deviations of group
means from grand
mean)
SSB/k1 Go to
Fk1,nkk
chart
Total
variation
nk1 TSS
(sum of squared deviations of
observations from grand mean)
Source of
variation d.f.
Sum of
squares
Mean Sum
of Squares
Fstatistic pvalue
Within(n individuals per
group)
nkkSSW(sum of squared
deviations of
observations from
their group mean)
s2=SSW/nkk
knk
SSWk
SSB
1
TSS=SSB + SSW
Character ist ics o f a Normal Distr ibut ion1) Continuous Random Variable.
2) Bellshaped curve.

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2) Bell shaped curve.
3) The normal curve extends indefinitely in both directions,
approaching, but never touching, the horizontal axis as it does so.4) Unimodal
5) Mean = Median = Mode
6) Symmetrical with respect to the mean. That is, 50% of the area
(data) under the curve lies to the left of the mean and 50% of the
area (data) under the curve lies to the right of the mean.
7) (a) 68% of the area (data) under the curve is within one
standard deviation of the mean
(b) 95% of the area (data) under the curve is within two standard
deviations of the mean(c) 99.7% of the area (data) under the curve is within three
t d d d i ti f th