Essentials

176 ESSENTIALS OF STATISTICSTable 8.1, SS totalis the same for both types of ANOVA, and SS betweenis the sameas SS RM(just as the numerator of the RM t is always the same as the numeratorof an independent t test on the same data). However, SS withinis equal to 5601.16,which is the sum of SS subjectand SS subRM. You can think of the RM ANOVA startingwith SS was the basis of its error term, but then reducing that term by subtractingout SS subject, leaving only SS subRM, before creating the MS componentsof the F ratio.To be fair, we must point out that the df for the error term is also reduced inRM ANOVA, which raises the critical value. For instance, for the independentone-way ANOVA the critical F is based on df betand df w, so for our example thecritical F is F .05(2, 15) 3.68. For the RM ANOVA the critical F is based on df RMand df subRM, so for this example F .05(2, 10) 4.10, which is higher than it is forthe independent ANOVA. This is a disadvantage of the RM ANOVA, as it wasfor the RM t test. However, the F ratio for the independent ANOVA is only332.39/5601.16/15 332.39/373.41 .89, which is much smaller than the Ffor the RM ANOVA (5.0). Whereas the independent groups’ F is nowhere nearsignificance, the RM of 5.0 is larger than the critical F for this test (4.10), so theRM ANOVA is significant by the usual standards. With a reasonable consistencyof scores across participants, the reduced error term and therefore larger F of theRM ANOVA will more than compensate for the larger critical F due to reduceddegrees of freedom.AssumptionsTo be valid the RM ANOVA requires the same basic assumptions as the independentANOVA:1. The DV is measured on an interval or ratio scale.2. The DV has a normal distribution under each condition.3. Although any particular participant is measured under all conditions,different participants must be chosen randomly and independentlyfrom the population of interest.In addition, the usual homogeneity of variance assumption is modified for thisdesign. Because the error term, MS subRM, is actually an average of all the possiblepairwise interactions, it must be assumed that all of these pairwise interactions areequal in the population—this justifies averaging them into a single error term.This assumption is usually referred to as the sphericity assumption. When sphericitydoes not apply to the population, the F ratio from the one-way RM ANOVAcan exceed the critical value too easily—the rate of Type I errors can exceed the