Psy 633 Repeated-Measures ANOVA (GW 13)

  I.  The Logic

F = variance between treatments / variance expected by chance or error

Because individual differences can be eliminated or removed from the error in a repeated-measures study, the structure of the final F-ratio can be modified as follows: 

F = variance between treatments (no individual differences possible)/variance expected by chance (with individual differences removed)

The partitioning of variability for a repeated-measures design:

 

 

II. Notation & Formulas

 

 

 

The first stage of the repeated-measures ANOVA uses the same notation and formulas as the between-subjects ANOVA. In the first stage, total variability is divided into variability between treatments and variability within treatments.

The second stage is new--it removes individual differences from the within treatment variability, making for a smaller, more precise estimate of error . The remaining variability in the denominator is called residual variance or error variance because it measures how much variance is expected just by chance after the individual differences have been removed.

k = number of treatments

n = number of scores in each treatment

N = total number of scores in the entire study

G = grand total of all scores in the experiment

T = the sum of the scores in each treatment condition

P = the total of scores for each participant (participant totals)--THIS IS NEW--only possible in a repeated (or paired samples) design

III. Examples

 

Example #1

Number of Formatting Commands used During Each Test Session

         Test 1  Test 2  Test 3  Test 4  Test 5        

      P1          5         6       14        19        22

      P2         2         7        11        18        24

      P3         1         3         8         25        22

      P4         6         5         9         18        22

      P5         6         8       15         22        29

 

 

Example #2

Table 14-3  (p. 457)
The effect of response-cost treatment on the number of outbursts in class after different periods of time.

 

Table 14-4  (p. 460)
Analysis of variance summary for Example 14.1.

Questions to think about:

What two general types of information are reported in an APA results section? Which type of information is generally reported first?

How can the results of a study be depicted graphically?

 

How is eta squared computed for a single-factor within-subjects design?
Remember that our goal is to calculate the proportion of total variability that has not been explained by other factors. So we can represent the denominator in two ways--they will both get us to the same result. 

  OR    

 

Post-Hoc Tests to follow up on a significant overall ANOVA:

For repeated measures and a small number of planned comparisons, use a Dunn test adustment to evaluate follow-up t-tests.

When the overall ANOVA is significant:

We will conduct pairwise t-tests

Instead of setting Type I error at .05 for each comparison, we will decrease our per comparison Type I error by:

Dunn test = alpha per comparison / number of comparisons  

This is a way of managing our overall experimentwise Type I error that accumulates over the entire set of analyses. 







How can we use Tukey's HSD for post-hoc tests?

Recall that in a between-subjects design,

 

In a repeated-measures design, and use degrees of freedom for MSError when looking up q

************************************************************Not required material

How can we use the Scheffe test with a repeated-measures design? 

As before, the numerator of the F-ratio is the MSbetween treatments that uses the SS for only the two treatments we’re interested in and divides by k-1 from the entire experiment.

The denominator is the error term (MSerror ) that was used for the overall ANOVA.

Looking up the critical value: As in chapter 13, df for the numerator is k from entire experiment – 1. And since we’re using the error term from the overall ANOVA, we end up with the same df in the denominator as the overall ANOVA. Bottom line: We use the same critical value that was used to evaluate the overall ANOVA.