This document provides an overview of different types of ANOVA procedures including 1-way ANOVA, ANOVA with trend analysis, 2-way ANOVA, and repeated measures ANOVA. It discusses the assumptions, calculations, and interpretations of 1-way ANOVA, including how to calculate sum of squares, mean squares, F-values, and follow up with Tukey's HSD post-hoc test. It also briefly explains 2-way ANOVA and how it is used to evaluate the combined effects of two factors on a dependent variable.

1. ASHISH DADHEECHASHISH DADHEECH
MSC NURSING IN PSYCHIATRIC
TOPIC REGARDING RESEARCH

2. ANOVA ProceduresANOVA Procedures
 ANOVA (1 way)
◦ An extension of the 2 sample t-test used to determine if there are
differences among > 2 group means
 ANOVA with trend test
◦ Test for polynomial trend in the group means
 ANOVA (2 way)
◦ Evaluate the combined effect of 2 experimental factors
 ANOVA repeated measures
◦ Extension of paired t-test for same or related subjects over time or
in differing circumstances
 ANCOVA
◦ 1 way ANOVA in which group means are adjusted by a covariate

3. 1 way ANOVA Assumptions1 way ANOVA Assumptions
Independent Samples
Normality within each group
Equal variances
◦ Within group variances are the same for each of the
groups
◦ Becomes less important if your sample sizes are
similar among your groups
 Ho: µ1= µ2= µ3=……… µk
 Ha: the population means of at least 2 groups are different

4. What is ANOVA?What is ANOVA?
 One-Way ANOVA allows us to compare the means of two or more
groups (the independent variable) on one dependent variable to
determine if the group means differ significantly from one another.
 In order to use ANOVA, we must have a categorical (or nominal) variable
that has at least two independent groups (e.g. nationality, grade level) as
the independent variable and a continuous variable (e.g., IQ score) as the
dependent variable.
 At this point, you may be thinking that ANOVA sounds very similar to
what you learned about t tests. This is actually true if we are only
comparing two groups. But when we’re looking at three or more groups,
ANOVA is much more effective in determining significant group
differences. The next slide explains why this is true.

5. Why use ANOVA vs a t-testWhy use ANOVA vs a t-test
ANOVA preserves the significance level
◦ If you took 4 independent samples from the
same population and made all possible
comparisons using t-tests (6 total
comparisons) at .05 alpha level there is a 1-
(0.95)^6= 0.26 probability that at least 1/6 of
the comparisons will result in a significant
difference
 > 25% chance we reject Ho when it is true

6. Homogeneity ofVarianceHomogeneity ofVariance
 It is important to determine whether
there are roughly an equal number of
cases in each group and whether the
amount of variance within each
group is roughly equal
• The ideal situation in ANOVA is
to have roughly equal sample sizes
in each group and a roughly equal
amount of variation (e.g., the
standard deviation) in each group
• If the sample sizes and the
standard deviations are quite
different in the various groups,
there is a problem
Group 1Group 1
Group 2Group 2

7. tt Tests vs. ANOVATests vs. ANOVA
 As you may recall, t tests allow us to decide whether the observed
difference between the means of two groups is large enough not
to be due to chance (i.e., statistically significant).
 However, each time we reach a conclusion about statistical
significance with t tests, there is a slight chance that we may be
wrong (i.e., make a Type I error—see Chapter 7). So the more t
tests we run, the greater the chances become of deciding that a t
test is significant (i.e., that the means being compared are really
different) when it really is not.
 This is why one-way ANOVA is important. ANOVA takes into
account the number of groups being compared, and provides us
with more certainty in concluding significance when we are looking
at three or more groups. Rather than finding a simple difference
between two means as in a t test, in ANOVA we are finding the
average difference between means of multiple independent groups
using the squared value of the difference between the means.

8. How does ANOVA work?How does ANOVA work?
 The question that we can address using ANOVA is this: Is the average amount of
difference, or variation, between the scores of members of different samples large or
small compared to the average amount of variation within each sample, otherwise
known as random error?
 To answer this question, we have to determine three things.
 First, we have to calculate the average amount of variation within each of our
samples. This is called the mean square within (MSw) or the mean square error
(MSe).
 This is essentially the same as the standard error that we use in t tests.
 Second, we have to find the average amount of variation between the group
means. This is called the mean square between (MSb).
 This is essentially the same as the numerator in the independent samples t test.
 Third: Now that we’ve found these two statistics, we must find their ratio by
dividing the mean square between by the mean square error. This ratio provides
our F value.
 This is our old formula of dividing the statistic of interest (i.e., the average
difference between the group means) by the standard error.
 When we have our F value we can look at our family of F distributions to see if
the differences between the groups are statistically significant

9. Calculating the SSe and the SSbCalculating the SSe and the SSb
 The sum of squares error (The sum of squares error (SSe)SSe) represents the sum of the squared deviations between individual scores and theirrepresents the sum of the squared deviations between individual scores and their
respective group means on the dependent variable.respective group means on the dependent variable.
 To find theTo find the SSeSSe we:we:
 The SThe SSbSb represents the sum of the squared deviations between group means and the grand mean (the mean of allrepresents the sum of the squared deviations between group means and the grand mean (the mean of all
individual scores in all the groups combined on the dependent variable)individual scores in all the groups combined on the dependent variable)
 To find theTo find the SSbSSb we:we:
1. Subtract the grand mean from the group mean: ( – ); T indicates total, or the
mean for the total group.
2. Square each of these deviation scores: ( – )2
.
3. Multiply each squared deviation by the number of cases in the group: [n( – )2
].
Add these squared deviations from each group together: Σ[n( – )2
].
1. Subtract the group mean from each individual score in each group: (X – ).
2. Square each of these deviation scores: (X – )2
.
3. Add them all up for each group: Σ(X – )2
.
4. Then add up all of the sums of squares for all of the groups:
Σ(X1 – )2
+ Σ(X2 – )2
+ … + Σ(Xk – )2
•The only real differences between the formula for calculating the SSe and the SSb are:
1. In the SSe we subtract the group mean from the individual scores in each group, whereas in the SSb we subtract the grand mean
from each group mean.
2. In the SSb we multiply each squared deviation by the number of cases in each group. We must do this to get an approximate
deviation between the group mean and the grand mean for each case in every group.
FF == MSb/MSeMSb/MSe

10. Finding The MSe and MSbFinding The MSe and MSb
 To find the MSe and the MSb, we have to
find the sum of squares error (SSe) and the
sum squares between (SSb).
 The SSe represents the sum of the squared
deviations between individual scores and
their respective group means on the
dependent variable.
 The SSb represents the sum of the squared
deviations between group means and the
grand mean (the mean of all individual
scores in all the groups combined on the
dependent variable represented by the
symbol Xt ).
MSe = SSe / (N-K)
Where:
K=The number of groups
N=The number of cases in all the group combined.
MSb = SSb / (K-1)
Where:
K= The number of groups
 Once we have calculated the SSb and the
SSe, we can convert these numbers into
average squared deviation scores (our MSb
and MSe).
 To do this, we need to divide our SS
scores by the appropriate degrees of
freedom.
 Because we are looking at scores
between groups, our df for MSb is K-1.
 And because we are looking at individual
scores, our df for MSe is N-K.

11. Calculating anCalculating an FF -Value-Value
 Once we’ve found our MSe and MSb, calculating
our FValue is simple. We simply divide one
value by the other.
F = MSb/MSe
 After an observed F value (Fo) is determined,
you need to check in Appendix C for to find
the critical F value (Fc). Appendix C provides a
chart that lists critical values for F associated
with different alpha levels.
 Using the two degrees of freedom you have
already determined, you can see if your
observed value (Fo) is larger than the critical
value (Fc). If Fo is larger, than the value is
statistically significant and you can conclude that the
difference between group means is large enough to not
be due to chance.

12. Example of Dividing the Variance between an IndividualExample of Dividing the Variance between an Individual
Score and the Grand Mean into Within-Group andScore and the Grand Mean into Within-Group and
Between-Group ComponentsBetween-Group Components

13. ExampleExample
 Suppose I want to know whether
certain species of animals differ in
the number of tricks they are able
to learn. I select a random sample
of 15 tigers, 15 rabbits, and 15
pigeons for a total of 45 research
participants.
 After training and testing the
animals, I collect the data presented
in the chart to the right.
AnimalAnimal Average Number ofAverage Number of
Tricks LearnedTricks Learned
TigerTiger 12.312.3
RabbitRabbit 3.13.1
PigeonPigeon 9.79.7
SSe = 110.6 SSb = 37.3

14. Example (continued)Example (continued)
 Step 1: Degrees of Freedom:
Now that you have your SSe and your SSb the next step is to determine your two df values.
Remember, our df for MSb is K-1 and our df for MSe is N-K:
df for (MSb)= 3-1 = 2
df for (MSe) = 45-3 = 42
 Step 2: Critical F-Value:
Using these two df values, look in Appendix C to determine your critical F value. Based on the F-Value
equation, our numerator is 2 and our denominator is 42. With an alpha value of .05, our critical F
value is 3.22. Therefore, if our observed F value is larger than 3.22 we can conclude there is a
significant difference between our three groups
 Step 3: Calcuating MSb and MSe:
Using our formulas from the previous slide:MSe =SSe / (N-K) AND MSb = SSb / (K-1)
MSe = 110.6/42 = 2.63
MSb = 37.3/2 = 18.65
 Step 4: Calculating an Observed F-Value:
Finally, we can calculate our Fo using the formula Fo = MSb / MSe
Fo = 18.65 / 2.63 = 7.09

15. Interpreting Our ResultsInterpreting Our Results
 Our Fo of 7.09 is larger than our Fc of 3.22. Thus, we can conclude that
our results are statistically significanct and the difference between the
number of tricks that tigers, rabbits and pigeons can learn is not due to
chance.
 However, we do not know where this difference exists. In other words,
although we know there is a significant difference between the groups, we
do not know which groups differ significantly from one another.
 In order to answer this question we must conduct additional post-hoc
tests. Specifically, we need to conduct a Tukey HSD post-hoc test to
determine which group means are significantly different from each other.

16. Tukey HSDTukey HSD
 The Tukey test compares each group mean to every other group mean by using the familiar
formula described for t tests in Chapter 9. Specifically, it is the mean of one group minus the
mean of a second group divided by the standard error, represented by the following formula.
xs
XX
HSDTukey 21 −
= g
e
x
n
MS
s =
Where: ng = the number of cases in each group
 The Tukey HSD allows us to compares groups of the same size.
Just as with our t tests and F values, we must first determine a critical Tukey value
to compare our observed Tukey values with. Using Appendix D, locate the number
of groups we are comparing in the top row of the table, and then locate the degrees
of freedom, error (dfe) in the left-hand column (The same dfe we used to find our F
value).

17. Tukey HSD for our ExampleTukey HSD for our Example
 Now let’s useTukey HSD to determine which animals differ significantly from each other.
 First: Determine your criticalTukeyValue using Appendix D. There are three groups, and your dfe
is still 42. Thus, your critical Tukey value is approximately 3.44.
 Now, calculate observedTukey values to compare group means. Let’s look atTigers and Rabbits in
detail. Then you can calculate the other twoTukeys on your own!
xs
XX
HSDTukey 21 −
=
g
e
x
n
MS
s =
Where: ng = the number of cases in each group
Step 1: Calculate your standard error.
2.63/15 = .18
√.175 = .42
Step 2: Subtract the mean number of tricks learned by
rabbits by the mean number of tricks learned by tigers.
12.3-3.1 = 9.2
Step 3: Divide this number by the standard error.
9.2 / .42 = 21.90
Step 4: Conclude that tigers learned significantly
more tricks than rabbits.
Repeat this process for the other group comparisons

18. Interpreting our Results. . .AgainInterpreting our Results. . .Again
 The table to the right lists the
observed Tukey values you should get
for each animal pair.
 Based on our critical Tukey value of
3.42, all of these observed values are
statistically significant.
 Now we can conclude that each of
these groups differ significantly from
one another. Another way to say this
is that tigers learned significantly
more tricks than pigeons and rabbits,
and pigeons learned significantly
more tricks than rabbits.
Animal GroupsAnimal Groups
Being ComparedBeing Compared
ObservedObserved
TukeyTukey
ValueValue
Tigers andTigers and
RabbitsRabbits 21.9021.90
Rabbits andRabbits and
PigeonsPigeons 15.7115.71
Pigeons andPigeons and
TigersTigers 6.196.19

19. 1 Way ANOVA with Trend Analysis1 Way ANOVA with Trend Analysis
Trend Analysis
◦ Groups have an order
◦ Ordinal categories or ordinal groups
◦ Testing hypothesis that means of ordered
groups change in a linear or higher order
(cubic or quadratic)

20. Factorial or 2 way ANOVA
Evaluate combined effect of 2
experimental variables (factors) on DV
 Variables are categorical or nominal
 Are factors significant separately (main effects) or
in combination (interaction effects)
Examples
 How do age and gender affect the salaries of 10
year employees?
 Investigators want to know the effects of dosage
and gender on the effectiveness of a cholesterol-
lowering drug

21. 2 way ANOVA hypothesis
Test for interaction
 Ho: there is no interaction effect
 Ha: there is an interaction effect
Test for Main Effects
◦ If there is not a significant interaction
 Ho: population means are equal across levels of
Factor A, Factor B etc
 Ha: population means are not equal across levels
of Factor A, Factor B etc
http://www.uwsp.edu/PSYCH/stat/13/anova-2w.htm

22. Factorial ANOVA in DepthFactorial ANOVA in Depth
 When dividing up the variance of a
dependent variable, such as hours of
television watched per week, into its
component parts, there are a
number of components that we can
examine:
• The main effects,
• interaction effects,
• simple effects, and
• partial and controlled effects
Main effects
Main effects
Interaction effects
Interaction effects
Simple effects
Simple effects
Partial and
Partial and
Controlled
Controlled
effectseffects

23. Main Effects and Controlled or Partial EffectsMain Effects and Controlled or Partial Effects
 When looking at the main effects, it is possible to test whether there are significant
differences between the groups of one independent variable on the dependent variable
while controlling for, or partialing out, the effects of the other independent variable(s) on
the dependent variable
• Example - Boys watch significantly more television than girls. In addition, suppose that children in
the North watch, on average, more television than children in the South.
 Now, suppose that, in my sample of children from the Northern region of the country,
there are twice as many boys as girls, whereas in my sample from the South there are twice
as many girls as boys. This could be a problem.
 Once we remove that portion of the total variance that is explained by gender, we can test
whether any additional part of the variance can be explained by knowing what region of the
country children are from.

24. When to use Factorial ANOVA…When to use Factorial ANOVA…
 Use when you have one continuous
(i.e., interval or ratio scaled)
dependent variable and two or more
categorical (i.e., nominally scaled)
independent variables
• Example - Do boys and girls differ
in the amount of television they
watch per week, on average? Do
children in different regions of the
United States (i.e., East,West,
North, and South) differ in their
average amount of television
watched per week? The average
amount of television watched per
week is the dependent variable,
and gender and region of the
country are the two independent
variables.

25. Results of a Factorial ANOVAResults of a Factorial ANOVA
 Two main effects, one for my
comparison of boys and girls and
one for my comparison of
children from different regions of
the country
• Definition: Main effects are
differences between the group
means on the dependent
variable for any independent
variable in the ANOVA
model.
 An interaction effect, or simply an
interaction
• Definition: An interaction is
present when the differences
between the group means on
the dependent variable for
one independent variable
varies according to the level
of a second independent
• Interaction effects are
also known as
moderator effects
#=x
#=x
#=x#=x
Main EffectMain Effect
InteractionInteraction

26. InteractionsInteractions
 Factorial ANOVA allows researchers to
test whether there are any statistical
interactions present
 The level of possible interactions
increases as the number of independent
variables increases
• When there are two independent variables
in the analysis, there are two possible main
effects and one possible two-way
interaction effect
# variables# variables
# variables# variables
# of possible# of possible
interactionsinteractions
# of possible# of possible
interactionsinteractions

27. Example of an InteractionExample of an Interaction
 The relationship between gender and
amount of television watched
depends on the region of the country
• We appear to have a two-way
interaction here
Mean Amounts of Television Viewed by Gender and Region.Mean Amounts of Television Viewed by Gender and Region.
NorthNorth EastEast WestWest SouthSouth Overall Averages by GenderOverall Averages by Gender
GirlsGirls
2020 1515 1515 1010 1515
BoysBoys 2525 2020 2020 2525 22.522.5
OverallOverall
AveragesAverages 22.522.5 17.517.5 17.517.5 17.517.5
• We can see is that there is aWe can see is that there is a
consistent pattern for theconsistent pattern for the
relationship between genderrelationship between gender
and amount of televisionand amount of television
viewed in three of the regionsviewed in three of the regions
(North, East, and West), but in(North, East, and West), but in
the fourth region (South) thethe fourth region (South) the
pattern changes somewhatpattern changes somewhat

28. Graph of an InteractionGraph of an Interaction
0
5
10
15
20
25
30
North East West South
Region
Hoursperweekoft.v.watched
Boys
Girls

29. Example of a Factorial ANOVAExample of a Factorial ANOVA
 We conducted a study to see
whether high school boys and girls
differed in their self-efficacy, whether
students with relatively high GPAs
differed from those with relatively
low GPAs in their self-efficacy, and
whether there was an interaction
between gender and GPA on self-
efficacy
 Students’ self-efficacy was the
dependent variable. Self-efficacy
means how confident students are in
their ability to do their schoolwork
successfully
 The results are presented on the
next 2 slides

30. Example (continued)Example (continued)
SPSS results for gender by GPA factorial ANOVA.SPSS results for gender by GPA factorial ANOVA.
GenderGender GPAGPA MeanMean Std. Dev.Std. Dev. NN
GirlGirl 1.001.00 3.66673.6667 .7758.7758 121121
Girl 2.002.00 4.00504.0050 .7599.7599 133133
Girl TotalTotal 3.84383.8438 .7845.7845 254254
BoyBoy 1.001.00 3.93093.9309 .8494.8494 111111
Boy 2.002.00 4.08094.0809 .8485.8485 103103
Boy TotalTotal 4.00314.0031 .8503.8503 214214
TotalTotal 1.001.00 3.79313.7931 .8208.8208 232232
Total 2.002.00 4.03814.0381 .7989.7989 236236
Total TotalTotal 3.91673.9167 .8182.8182 468468
These descriptive statistics reveal that high-achievers (group 2 in the GPA column)
have higher self-efficacy than low achievers (group 1) and that the difference
between high and low achievers appears to be larger among girls than among boys

31. Example (continued)Example (continued)
ANOVA ResultsANOVA Results
SourceSource
Type IIIType III
Sum ofSum of
SquaresSquares dfdf Mean SquareMean Square FF Sig.Sig.
EtaEta
SquareSquare
Corrected ModelCorrected Model 11.40211.402 33 3.8013.801 5.8545.854 .001.001 .036.036
InterceptIntercept 7129.4357129.435 11 7129.4357129.435 10981.56610981.566 .000.000 .959.959
GenderGender 3.3543.354 11 3.3543.354 5.1665.166 .023.023 .011.011
GPAGPA 6.9126.912 11 6.9126.912 10.64610.646 .001.001 .022.022
Gender * GPAGender * GPA 1.0281.028 11 1.0281.028 1.5841.584 .209.209 .003.003
ErrorError 301.237301.237 464464 .649.649
TotalTotal 7491.8897491.889 468468
Corrected TotalCorrected Total 312.639312.639 467467
These ANOVA statistics reveal that there is a statistically significant main effect for gender,
another significant main effect for GPA group, but no significant gender by GPA group
interaction. Combined with the descriptive statistics on the previous slide we can conclude
that boys have higher self-efficacy than girls, high GPA students have higher self-efficacy
than low GPA students, and there is no interaction between gender and GPA on self-
efficacy. Looking at the last column of this table we can also see that the effect sizes are
quite small (eta squared = .011 for the gender effect and .022 for the GPA effect).

32. Example: Burger (1986)Example: Burger (1986)
 Examined the effects of choice and public versus private evaluation
on college students’ performance on an anagram-solving task
 One dependent and two independent variables
• Dependent variable: the number of anagrams solved by
participants in a 2-minute period
• Independent variables: choice or no choice; public or private
Mean number of anagrams solved for fourMean number of anagrams solved for four
treatment groups.treatment groups.
PublicPublic PrivatePrivate
ChoiceChoice
NoNo
ChoiceChoice
ChoiceChoice
NoNo
ChoiceChoice
Number ofNumber of
anagrams solvedanagrams solved
19.5019.50 14.8614.86 14.9214.92 15.3615.36

33. Burger (1986) ConcludedBurger (1986) Concluded
 Found a main effect for choice, with students in the two choice groups
combined solving more anagrams, on average, than students in the two no-
choice groups combined
 Found a main effect for public over private performance
 Found an interaction between choice and public/private. Note the
difference between the public and private performance groups in the
Choice condition.
0
5
10
15
20
25
Choice No Choice
Numberofanagramscorrectlysolved
Public
Private

34. Repeated-Measures ANOVARepeated-Measures ANOVA
versus Pairedversus Paired tt TestTest
 Similar to a paired, or dependent samples t test, repeated-
measures ANOVA allows you to test whether there are significant
differences between the scores of a single sample on a single
variable measured at more than one time.
 Unlike paired t tests, repeated-measures ANOVA lets you
◦ Examine change on a variable measured across more than two time
points;
◦ Include a covariate in the model;
◦ Include categorical (i.e., between-subjects) independent variables in the
model;
◦ Examine interactions between within-subject and between-subject
independent variables on the dependent variable

35. Different Types of Repeated MeasuresDifferent Types of Repeated Measures
ANOVAs (and when to use each type)ANOVAs (and when to use each type)
 The most basic model
One sample
One dependent variable
measured on an interval or
ratio scale
The dependent variable is
measured at least two different
times
 Example: Measuring the
reaction time of a sample of
people before they drink two
beers and after they drink two
beers
Time 1: Reaction time
with no drinks
Time 2: Reaction time
after two beers
Time,
or trial

36. Different Types of RepeatedDifferent Types of Repeated
Measures ANOVAs (and when to useMeasures ANOVAs (and when to use
each type)each type) Adding a between-subjectsAdding a between-subjects
independent variable to the basicindependent variable to the basic
modelmodel
 One sample with multiple categories (e.g.,One sample with multiple categories (e.g.,
boys and girls; American, French, andboys and girls; American, French, and
Greek), or multiple samplesGreek), or multiple samples
 One dependent variable measured on anOne dependent variable measured on an
interval or ratio scaleinterval or ratio scale
 The dependent variable is measured at leastThe dependent variable is measured at least
two different times. (Time, or trial, is thetwo different times. (Time, or trial, is the
independent, within-subjects variable)independent, within-subjects variable)
 Example: Measuring the reaction timeExample: Measuring the reaction time
of men and women before they drinkof men and women before they drink
two beers and after they drink twotwo beers and after they drink two
beersbeers
Time 1, Group 1:
Reaction time of
men with no
drinks
Time 2, Group 1:
Reaction time
of men
after two beers
Time, or
trial
Time 1, Group 2:
Reaction time of
women
after two beers
Time 2, Group 2:
Reaction time
of women
after two beers

37. Different Types of RepeatedDifferent Types of Repeated
Measures ANOVAs (and when to useMeasures ANOVAs (and when to use
each type)each type) Adding a covariate to the between-Adding a covariate to the between-
subjects independent variable to thesubjects independent variable to the
basic modelbasic model
 One sample with multiple categories (e.g., boys andOne sample with multiple categories (e.g., boys and
girls; American, French, and Greek), or multiplegirls; American, French, and Greek), or multiple
samplessamples
 One dependent variable measured on an interval orOne dependent variable measured on an interval or
ratio scaleratio scale
 One covariate measured on an interval/ratio scale orOne covariate measured on an interval/ratio scale or
dichotomouslydichotomously
 The dependent variable is measured at least twoThe dependent variable is measured at least two
different times. (Time, or trial, is the independent,different times. (Time, or trial, is the independent,
within-subjects variable)within-subjects variable)
 Example: Measuring the reaction time of menExample: Measuring the reaction time of men
and women before they drink two beers andand women before they drink two beers and
after they drink two beers, controlling forafter they drink two beers, controlling for
weight of the participantsweight of the participants
Time 1, Group 1:
Reaction time of
men with no
drinks
Time 2, Group 1:
Reaction time
of men
after two beers
Time, or
trial
Time 1, Group 2:
Reaction time of
women
after two beers
Time 2, Group 2:
Reaction time
of women
after two beers
Covariate

38. Types of Variance in Repeated-Types of Variance in Repeated-
Measures ANOVAMeasures ANOVA
 Within-subject
◦ Variance in the dependent variable attributable to change, or difference,
over time or across trials (e.g., changes in reaction time within the
sample from the first test to the second test)
 Between-subject
◦ Variance in the dependent variable attributable to differences between
groups (e.g., men and women).
 Interaction
◦ Variance in the dependent across time, or trials, that differs by levels of
the between-subjects variable (i.e., groups). For example, if women’s
reaction time slows after drinking two beers but men’s reaction time
does not.
 Covariate
◦ Variance in the dependent variable that is attributable to the covariate.
Covariates are included to see whether the independent variables are
related to the dependent variable after controlling for the covariate. For
example, does the reaction time of men and women change after
drinking two beers once we control for the weight of the individuals?

39. Repeated Measures SummaryRepeated Measures Summary
 Repeated-measures ANOVA should be used when you have multiple
measures of the same interval/ratio scaled dependent variable over
multiple times or trials.
 You can use it to partition the variance in the dependent variable into
multiple components, including
◦ Within-subjects (across time or trials)
◦ Between-subjects (across multiple groups, or categories of an independent
variable)
◦ Covariate
◦ Interaction between the within-subjects and between-subjects independent
predictors
 As with all ANOVA procedures, repeated-measures ANOVA assumes
that the data are normally distributed and that there is homogeneity of
variance across trials and groups.

40. ANCOVA
1 way ANOVA with a twist
 Means not compared directly
 Means adjusted by a covariate
 Covariate is not controlled by researcher
 It is intrinsic to the subject observed

41. Analysis ofAnalysis of CovarianceCovariance
 In ANOVA, the idea is to test whether there are
differences between groups on a dependent variable after
controlling for the effects of a different variable, or set of
variables
 We have already discussed how we can examine the
effects of one independent variable on the dependent
variable after controlling for the effects of another
independent variable
 We can also control for the effects of a covariate. Unlike
independent variables in ANOVA, covariates do not have
to be categorical, or nominal, variables

42. ANCOVA examples
A car dealership wants to know if placing
trucks, SUV’s, or sports cars in the
display area impacts the number of
customers who enter the showroom.
Other factors may also impact the
number such as the outside weather
conditions. Thus outside weather
becomes the covariate.

43. Assumptions
Covariate must be quantitative an linearly
related to the outcome measure in each
group
 Regression line relating covariate to the response
variable
 Slopes of regression lines are equal
 Ho: regression lines for each group are parallel
 Ha: at least two of the regression lines are not parallel
 Ho: all group means adjusted by the covariate are equal
 Ha: at least 2 means adjusted by the covariate are not
equal