You use repeated measures when you don't have enough subjects to make a good design or because there are other purposes (e.g., research questions) that lead to the use of this design.

Its strengths are economy of subjects and power. You can get away with fewer subjects because the real N is the number of observations, not the number of subjects.

What are Repeated Measures?

• Subjects are measured more than once. (You repeat the measure on the same people.)
• You can use fixed factors or random factors.
• Random factors: you randomly sample some levels of the independent variable from the population of levels. Thus, generalizations can be made about all of the levels in the population, even those not actually sampled. This is much like the random selection of subjects in the random sampling process.
• Fixed factors: once the levels of the independent variable are selected, subjects are randomly assigned to the levels of the independent variable. For example, you cannot assign students to "gender", and students are already assigned to their "school" or "class". Therefore, you can't generalize beyond that gender, school, or class.

Assumptions

• Assumptions are independence, normality, HOV, and a fourth one called sphericity.
• The sphericity assumption becomes a problem when you have within-subjects factors. It says that the correlations between all pairs of measurements are roughly the same. This means that measurements over several different times are not particularly robust to this assumption.
• When you have at least one between-subjects factor, then you have to satisfy yet a fifth assumption called compound symmetry.
• Compound symmetry means the variances/covariances for all pairs of levels of the fixed factor (the IV) - the between subjects factor - are the same. In other words, the groups ought to look like each other.

Fixed Effects Model

• You have a single fixed independent variable with, say, 2 levels - like 10 boys vs. 10 girls, or 10 students from Centennial and 10 students from Nederland. They are assigned to their school; you cannot change it.
• There is only one F-ratio: MSb/MSw.
• This design has more power.
• You can't generalize beyond the sample you selected.

Random Effects Model

• You have a single random independent variable with, say, 2 levels - like you randomly sample 20 fourth graders from the Boulder School District and then randomly assign them to one of two groups.
• This design has less power.
• You can generalize beyond the factor since it's randomly selected from the population. There is a tradeoff: you decrease the df for the F ratio test.

Repeated Measures: An Example with a Fixed Factor

• A one-factor repeated measures design usually means that you measure one group of subjects at several different times. Here, time (T) is a fixed factor.
• You could use this same design to measure the entire group of subjects on a set of items in a questionnaire, and treat each item as a separate independent variable.
• You have a single group of 20 subjects and you want to test all of them at three different times, sort of like pre/post test.
• The N is no longer 20 - it becomes 60 because N is the number of observations, not the number of subjects.
• df is 59: the total number of observations minus one.
• Here is your ANOVA table: T is time, S is subjects all in one group.

• Your F ratio becomes MST/MSTxS

Nested Design: An Example

• Instead of grouping all 20 subjects together, you want to split them into two groups. Each subject can be in only one group: they are nested in their group.
• You have a fixed factor A, like experiment vs. control. You have added one between group factor to the above design.
• You have J groups (Here, J=2). You randomly assign subjects to these groups.
• So you have s subjects (say 20 in all) with 10 nested in each of the two groups. This is s:A.
• As before, you want to measure all subjects at 3 different times. Time (T) is another fixed factor. It becomes a within-group factor.
• That is called a split plot design with one between-subjects factor, namely, A.
• Time (T) and any interactions with time gives rise to a set of within-group factors.
• We'll go through the ANOVA table below.

Split Plot Design

• The fixed factor (A) has 2 levels or groups (J=2), so J-1=1.
• The 20 subjects are assigned randomly to these two groups (s:A), 10 per group, so s-1=9.
• Time (T) is also a fixed factor. You have T=3 measurements of these 20 subjects over time, so T-1=2.
• Here is your ANOVA table:

• Add up the total degrees of freedom: 23 + 36 = 59 = 60 - 1. Check!

Important things to notice in the ANOVA table

• For the EMS it is not just the Error term. It is the Error term plus anything random that is either nested within the factor or crossed with the factor.
• Subjects are nested randomly within A, so you have to add the s:A to the numerator and use the s:A for the error term.
• For the T factor, you have an interaction - subjects nested in A are crossed with T. You have to add the s:AxT to both the numerators (T and AxT) and use s:AxT for the denominator.
• The degrees of freedom for T go up to 2 (because of the 3 time measurements for the T factor: 3-1=2), and 36 (s:A is 18, times 2 for the T factor).

Consequences of Violating Assumptions

• In the population, if you don't correct for non-equality of the differences in the within-subject factor pairs, you pick up more noise and have a liberal F test. You may want to run three one-way ANOVAS, but you have a tradeoff here because that will raise alpha.
• The Greenouse-Gussier correction will fix the sphericity problem. This problem is pretty common in repeated measures over time. It gives you an episilon factor. You multiply both degrees of freedom of the F-ratio by this factor by hand, then look for the critical value in the back of the book.
• For the compound symmetry assumption, you need to have the variance/covariance matrix among the groups (the between-subjects factor) to be similar to each other. If the variance/covariance matrices between groups are not similar enough to each other, you will again have a liberal F test.

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