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Repeated Measures
ANOVA
Quantitative Methods in HPELS
440:210
Agenda
Introduction
 The Repeated Measures ANOVA
 Hypothesis Tests with Repeated Measures
ANOVA
 Post Hoc Analysis
 Instat
 Assumptions

Introduction

Recall  There are two possible scenarios
when obtaining two sets of data for
comparison:
 Independent
samples: The data in the first
sample is completely INDEPENDENT from
the data in the second sample.
 Dependent/Related samples: The two sets of
data are DEPENDENT on one another. There
is a relationship between the two sets of data.
Introduction

Three or more data sets?


If the three or more sets of data are
independent of one another 
Analysis of Variance (ANOVA)
If the three or more sets of data are
dependent on one another 
Repeated Measures ANOVA
Agenda
Introduction
 The Repeated Measures ANOVA
 Hypothesis Tests with Repeated Measures
ANOVA
 Post Hoc Analysis
 Instat
 Assumptions

Repeated Measures ANOVA

Statistical Notation  Recall for ANOVA:
k
= number of treatment conditions (levels)
 nx = number of samples per treatment level
 N = total number of samples

N = kn if sample sizes are equal
= SX for any given treatment level
 G = ST
 MS = mean square = variance
 Tx
Repeated Measures ANOVA

Additional Statistical Notation:
P
= total score for each subject (personal
total)
 Example: If a subject was assessed three
times and had scores of 3, 4, 5  P = 12
Repeated Measures ANOVA

Formula Considerations  Recall for ANOVA:
= ST2/n – G2/N
 SSwithin = SSSinside each treatment
 SStotal = SSwithin + SSbetween
 SSbetween

SStotal = SX2 – G2/N
ANOVA

Formula Considerations:
=N–1
 dfbetween = k – 1
 dfwithin = S(n – 1)
 dftotal

dfwithin = Sdfin each treatment
ANOVA

Formula Considerations:
= s2between = SSbetween / dfbetween
 MSwithin = s2within = SSwithin / dfwithin
 F = MSbetween / MSwithin
 MSbetween
Repeated Measures ANOVA

New Formula Considerations:
 SSbetween treatments = ST2/n – G2/N
 SSbetween subjects = SP2/k – G2/N
 SSwithin  SSwithin treatments = SSSinside each treatment
 SSerror = SSwithin treatments – SSbetween subjects
 SSbetween
Repeated Measures ANOVA

New Formula Considerations:
 dfbetween treatments = k – 1
 dfwithin  dfwithin treatments = N – k
 dfbetween subjects = n – 1
 dferror = (N – k) – (n – 1)
 dfbetween
Repeated Measures ANOVA

MSbetween treatments=SSbetween treatments/dfbetween treatments
MSerror = SSerror / dferror
 F = MSbetween treatments / MSerror

Repeated Samples Designs

One-group pretest posttest (repeated
measures) design:



Perform pretest on all subjects
Administer treatments followed by posttests
Compare pretest to posttest scores and posttest to
posttest scores
O
X
O
X
O
Agenda
Introduction
 The Repeated Measures ANOVA
 Hypothesis Tests with Repeated Measures
ANOVA
 Post Hoc Analysis
 Instat
 Assumptions

Hypothesis Test: Repeated
Measuers ANOVA


Example 14.1 (p 457)
Overview:
 Researchers are interested in a behavior
modification technique on outbursts in unruly
children
 Four students (n=4) are pretested on the # of
outbursts during the course of one day
 Teachers begin using “cost-response”
technique
 Students are posttested one week later, one
month later and 6 months later
Hypothesis Test: ANOVA

Questions:
 What
is the experimental design?
 What is the independent variable/factor?
 How many levels are there?
 What is the dependent variable?
Step 1: State Hypotheses
Non-Directional
H0: µpre = µ1week = µ1month = µ6months
H1: At least one mean is different
than the others
Step 2: Set Criteria
Alpha (a) = 0.05
Critical Value:
Use F Distribution Table
Appendix B.4 (p 693)
Information Needed:
dfbetween treatments = k – 1 = 4 – 1 = 3
dferror = (N-k)-(n-1) = (16-4)-(4-1) = 9
Table B.4 (p 693)
Critical value = 3.86
Step 3: Collect Data and Calculate Statistic
Total Sum of Squares
Sum of Squares Between each Treatment
SStotal = SX2 – G2/N
SSbetween treatment = ST2/n – SG2/N
SStotal = 222 – 442/20
SSbetween treatment = 262/4+82/4+62/4+42/4 – 442/20
SStotal = 222 - 121
SSbetween treatment = (169+16+9+4) - 121
SStotal = 101
SSbetween treatment = 77
Sum of Squares Within each Treatment
Sum of Squares Error
SSwithin = SSSinside each treatment
SSerror = SSwithin treatments – SSbetween subjects
SSwithin = 11+2+9+2
SSerror = 24 - 13
SSwithin = 24
SSwithin = 11
Sum of Squares Between each Subject
SSbetween subjects = SP2/k – SG2/N
SSbetween subjects = (122/4+62/4+102/4+162/4) - 442/16
SSbetween subjects = (36+9+25+64) – 121
SSbetween subjects = 13
Raw data can be found in
Table14.3 (p 457)
Step 3: Collect Data and Calculate Statistic
F-Ratio
F = MSbetween treatment / MSerror
Mean Square Between each Treatment
F = 25.67 / 1.22
MSbetween treatment = SSbetween treatment / dfbetween treatment
F = 21.04
MSbetween treatment = 77 / 3
MSbetween = 25.67
Mean Square Error
MSerrorn = SSerror / dferror
MSerror = 11 / 9
MSwithin = 1.22
Step 4: Make Decision
Agenda
Introduction
 Repeated Measures ANOVA
 Hypothesis Tests with Repeated Measures
ANOVA
 Post Hoc Analysis
 Instat
 Assumptions

Post Hoc Analysis

What ANOVA tells us:
 Rejection
of the H0 tells you that there is a
high PROBABILITY that AT LEAST ONE
difference exists SOMEWHERE

What ANOVA doesn’t tell us:
 Where

the differences lie
Post hoc analysis is needed to determine
which mean(s) is(are) different
Post Hoc Analysis
Post Hoc Tests: Additional hypothesis
tests performed after a significant ANOVA
test to determine where the differences lie.
 Post hoc analysis IS NOT PERFORMED
unless the initial ANOVA H0 was rejected!

Post Hoc Analysis  Type I Error






Type I error: Rejection of a true H0
Pairwise comparisons: Multiple post hoc tests
comparing the means of all “pairwise combinations”
Problem: Each post hoc hypothesis test has
chance of type I error
As multiple tests are performed, the chance of type
I error accumulates
Experimentwise alpha level: Overall probability of
type I error that accumulates over a series of
pairwise post hoc hypothesis tests
How is this accumulation of type I error controlled?
Two Methods

Bonferonni or Dunn’s Method:
 Perform
multiple t-tests of desired comparisons
or contrasts
 Make decision relative to a / # of tests
 This reduction of alpha will control for the
inflation of type I error

Specific post hoc tests:
 Note:
There are many different post hoc tests
that can be used
 Our book only covers two (Tukey and Scheffe)
Repeated Measures ANOVA

Bonferronni/Dunn’s method is appropriate
with following consideration:
 Use

related-samples T-tests
Tukey’s and Scheffe is appropriate with
following considerations:
 Replace
MSwithin with MSerror in all formulas
 Replace dfwithin with dferror in all formulas

Note: Statisticians are not in agreement
with post hoc analysis for Repeated
Measures ANOVA
Agenda
Introduction
 The Repeated Measures ANOVA
 Hypothesis Tests with Repeated Measures
ANOVA
 Post Hoc Analysis
 Instat
 Assumptions

Instat

Label three columns as follows:

Block: This groups your data by each
subject.



Treatment: This tells you which treatment
level/condition occurred for each data point.



Example: If you conducted a pretest and 2
posttests (3 total) on 5 subjects, the block
column will look like:
111222333444555
Example: If each subject (n=5) received three
treatments, the treatment column will look like:
123123123123123
Response: The data for each subject and
treatment condition
Instat

Convert the “Block” and “Treatment”
columns into “factors”:

Choose “Manage”






Choose “Column Properties”
Choose “Factor”
Select the appropriate column to be converted
Indicate the number of levels in the factor
Example: Block (5 levels, n = 5), Treatment (3
levels, k = 3)
Click OK
Instat
Choose “Statistics”


Choose “General”
Response variable:




Choose the Treatment variable
Blocking factor:



Choose the Response variable
Treatment factor:


Choose “Analysis of Variance”
Choose the Block variable
Click OK.
Interpret the p-value!!!
Instat

Post hoc analysis:

Perform multiple related samples t-Tests
with the Bonferonni/Dunn correction method
Reporting ANOVA Results

Information to include:

Value of the F statistic
 Degrees of freedom:




Between treatments: k – 1
Error: (N – k) – (n – 1)
p-value
Examples:

A significant treatment effect was observed
(F(3, 9) = 21.03, p = 0.002)
Reporting ANOVA Results

An ANOVA summary table is often
included
Source
SS
df
MS
Between
77
3
25.67
Within Treatments
24
12
Between subjects
13
3
Error
11
9
Total
101
15
1.22
F = 21.03
Agenda
Introduction
 The Analysis of Variance (ANOVA)
 Hypothesis Tests with ANOVA
 Post Hoc Analysis
 Instat
 Assumptions

Assumptions of ANOVA
Independent Observations
 Normal Distribution
 Scale of Measurement

 Interval
or ratio
Equal variances (homogeneity)
 Equal covariances (sphericity)

 If
violated a penalty is incurred
Violation of Assumptions
Nonparametric Version  Friedman Test
(Not covered)
 When to use the Friedman Test:

 Related-samples
design with three or more
groups
 Scale of measurement assumption violation:

Ordinal data
 Normality

assumption violation:
Regardless of scale of measurement
Textbook Assignment

Problems: 5, 7, 10, 23 (with post hoc)