Within-Subjects

(also known as "repeated measures")

ANOVA

As you should be becoming more sophisticated, I am going to be less explicit with my illustrations. I want you to begin to pull yourself up by your bootstraps.

This method is used when we have several measures on the same group of subjects. It allows for increased power and reduced error because each of the subjects are compared to themselves (eliminating the error that arises from differences among subjects). For instance, I may want to take three equivalent (although perhaps alternate forms to avoid carryover) achievement measures for a group of ten students at 1) the first class day, 2) midterm, and 3) final exam day.

If you understood and can perform a factorial ANOVA, you will be able to do a one-way repeated measures design, because it is essentially the same thing. The two factors are conditions (A) and subjects (S). The table below shows our measures for the ten students at three different times.

This is essentially an A x S factorial. The only difference is that computation of the final F statistics for the difference between A levels (the different test times) is accomplished by using the AxS interaction term in the denominator of the F ratio (the numerator is calculated in the same way as the between groups MS_BG was calculated). Of course, it is the differences between subjects that we are trying to eliminate, so we do not calculate the main effect for S.

F = MS_A / MS_AxS

Now, I want to introduce you to a new form of summation notation which is common in ANOVA texts. It uses periods (or dots) to indicate which means are used. You can compare the SS formulas to the factorial formulas if you have trouble understanding this notation at first. The way it works is this. Normally, authors use the little i to denote rows, and the little j to denote columns. When you see X_ij, it refers to the value in row i and column j . When you see the notation X_.j , it refers to the average of all rows in column j. When you see X_i. , it means the average of all columns in row i . When you see X_.. it means the average of all cells (the grand mean). With a little practice, you will pick up this notation very easily. The formulas for the within subjects ANOVA sums of squares are given below. The degrees of freedom are calculated just as they were for the values in factorial ANOVA.

For your assignment, I want you to test the omnibus null of no difference between scores on the achievement test at the three different testing occasions. Post hoc tests are different for this type of ANOVA, so I will post a good review on that on Thursday.

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