RANDOMIZED BLOCK DESIGN//REPEATED MEASURES ANOVA

We’ll approach RM anova by way of an example:

Eight 10-year old children were shown a cartoon and asked to rate the humor in the cartoon. They were shown the same cartoon two days later and then again two days later. Each time they rated the humor of the cartoon. The experimental question is whether ratings of humor change over time.

In this example we have one within-subjects factor (time) with three levels (first, second, third). The dependent variable is the humor rating.

Here are the data:

first second third subject mean

s1 6.00 5.00 2.00 4.33

s2 5.00 5.00 4.00 4.67

s3 5.00 6.00 3.00 4.67

s4 6.00 5.00 4.00 5.00

s5 7.00 3.00 3.00 4.33

s6 4.00 2.00 1.00 2.33

s7 4.00 4.00 1.00 3.00

s8 5.00 7.00 2.00 4.67

5.25 4.625 2.50 4.125

Notes on these data: Some subjects clearly think the comic is funnier than others. You can see this by looking at the subject averages shown in the right hand column. This variability reflects individual differences in humor, and this variability in a completely randomized design would be part of the error variability in the analysis. The primary benefit of a within-subjects design (repeated measures) is that this variability can be removed from error variability, thus leaving a smaller error term and larger F obtained.

Differences between rows will be reflected in SS subjects, differences between columns will be reflected in SSwithin subjects.

In a repeated measures (or within subjects) design an individual’s score is determined by four factors: the overall (or grand) mean, the effect of condition, the individual difference effect, and the error effect.

Linear model: x_im = m + g_m + p_i + e_mi

With this linear model, the sums of squares total is partitioned into subunits: sums of squares between subjects and sums of squares within subjects. The sums of squares within subjects can be further subdivided into sums of squares between conditions and sums of squares residual.

SSTO = SSsubjects + SScondition + SSresidual

Null hypothesis of interest:

Ho: No differences between conditions

The test of this null hypothesis is an overall F test (just like the overall between subjects ANOVA F test in Chapter 14). The major difference between the within subjects F test and the between subjects F test is the choice of error term.

Testing the hypothesis of no difference between conditions:

The univariate approach to WS designs is characterized by treating “subjects” as a factor in the design. That is, a single factor within subjects design is treated as a 2 factor design, and this is called the univariate approach.

In the univariate approach, the proper error term is MSresidual.

F = SScondition/dfcondition

SSresidual/dfresidual

The degrees of freedom for condition is, as in the between subjects anova, the number of conditions minus 1.

The degrees of freedom for the residual effect is the product of the degrees of freedom for condition (p-1) and the degrees of freedom for subjects (n-1).

Thus, F = SScond/(p-1)

SSres/(p-1)(n-1)

Computation of sums of squares:

We compute a sum of squares for subjects, a sum of squares for treatment, and a sum of squares for the residual effect. However, the sums of squares for subjects is not necessary to carry out the F test for condition.

Computing sums of squares: we can use either definitional formulae or computational formulae if we are computing the sums of squares by hand. Otherwise, SPSS will return these sums of squares, but you need to know where to look on the output to find them.

Definitional formulae:

Sums of squares for treatment is computed in the usual way, taking squared differences between group means and the grand mean. This sum is then multiplied by the number of conditions.

Sums of squares residual is computed as SS(x_ij – xbar_j – xbar_i+ grand mean)² Most folk find the computational formula easier to work with.

Computational formulae for these sums of squares can be found in the text on pages 519-520.

The obtained F is simply compared to the F crit, which is easily obtained by using the F tables at the back of the book. Remember that the overall F test in ANOVA is inherently 1 tailed, so look up the critical value for your overall alpha level – no need to divide by 2.

If you F obtained is significant, then you know there is some effect of condition. If there are three or more levels of condition, then you cannot say specifically what that effect is.

A priori or post hoc comparisons are carried out in much the same way as in the completely randomized (between subject) design in Chapter 14. You can select the proper test by using the flowchart handout. The only change is that the formula for the t or q (comparison) must have MSresidual in the denominator rather than MSWG.

General comments about repeated measures ANOVA

1. The analysis of data in a repeated measures ANOVA is quite complex. We are taking the simplest, but not wisest, approach in Psych 218.

2. The complexity of repeated measures ANOVA stems largely from the fact that repeated measures ANOVA is quite sensitive to violations of one of the major assumptions of the test.

3. This assumption is called “homogeneity of treatment difference variances (HOTDV).” It is important to understand what this assumption means.

4. HOTDV means that if you were to compute sets of difference scores for all pairwise combinations, the variances of these difference scores would all be estimates of a single population of difference scores. Consider the two group case (which we analyzed as a dependent groups t test long ago). We computed one set of difference scores and used the variance of those difference scores as an estimate of the variance of the population of difference scores. Now suppose you have three groups. In this case there are actually three sets of difference scores you can compute, and the assumption of HOTDV is that the three variances you compute are all estimates of one population variance. This assumption is commonly violated.

5. Violation of HOTDV generally results in the overall test having a much higher Type I error rate than nominally set. There are several adjustments to the overall test that have been used to correct for this alpha inflation. The most common are the Greenhouse-Geisser, the Huynh-Feldt, and the Box adjustments. These show up on your SPSS output when you carry out a repeated measures ANOVA.

6. These adjustments are for the overall test only, and cannot be used for individual comparisons. The effect of violating HOTDV on individual comparisons can be inflation of Type I error rate or inflation of Type II error rate.

7. For now, in our class, we will only take the unadjusted univariate approach. I will leave it to your grad stats class to cover the various adjustments and the multivariate tests.