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Lecture G89.2228 13a Power for 2 way ANOVA Random effects Expected MS Mixed models Special case: one entry per cell 1 G89.2228 Lect 13a Effect sizes and power Cohen s f (Howell s is the square ) root of the effect variance divided by within cell variance f = i =1 a 2 i a e2 f = j =1 b 2 j b 2 e f = ij2 i =1 j =1 a b ab e2 Suppose we wanted to replicate H-K&N results: fa = .10 fb= .21 fab= .02 Clearly the obtained interaction is smaller than H-K&N expected 2 G89.2228 Lect 13a Power for Interactions Example: H-K&N Suppose the expected results were .5 .5 .5 .5 .5 .5 .5 .2 .35 .5 .4 .45 fa = .10 fb= .14 fab= .14 To have 90% power for interaction sample would have had to be MUCH larger Interactions that look reasonably large may have small f values The interaction looks much smaller once adjustment for main effects takes place 3 G89.2228 Lect 13a Interaction, continued McClelland and Judd have written about the challenges of showing power for interactions Krantz and Shinn have an unpublished manuscript that suggests that tests of interaction need a sample size about 4 times as large as tests of main effects Rosenthal and Rosnow, among others, have suggested abandoning the factorial ANOVA interaction framework, and instead focus on planned contrasts of the predicted effects Note that interactions that are ordinal can sometimes disappear with a transformation of the dependent variable (and hence are less convincing) 4 G89.2228 Lect 13a Expected mean squares for the fixed effects model Model: Xijk = + i + j + ij + eijk The fact that i, j and ij sum to zero across rows and columns makes the expected mean squares very easy for balanced, fixed effects ANOVA X ij = + i + j + ij + eij X i = + i + ei X j = + j + e j X = + e The resulting expected mean squares are: 2 E ( MS A ) = bn + e2 2 E ( MS B ) = an + e2 2 E ( MS AB ) = n + e2 E ( MS E ) = e2 5 G89.2228 Lect 13a Random effects example Suppose that a memory researcher is interested in how memorable different words are across university settings She samples five universities and 40 words and tests recall using a common protocol across sites Both words and universities can be viewed as random effects Variation between the five universities suggest what variation would be like if more universities were sampled Variation between the words suggest how difficult different words can conceivably be 6 G89.2228 Lect 13a Random effects example Factor A: Words Factor B: Universities Suppose that recall is tested as follows: for each of the 5 universities and each of the 40 study words, n students are sampled and asked to recall the study word after they have been to exposed it in a crossword puzzle. This design would allow each cell of the 40x5 design to have an independent sample. We can calculate the overall recall of each of the 40 words and the level of recall for each of the five universities using means. 7 G89.2228 Lect 13a Random Effects Models The statistical model for the random effects model is as before: Yijk = + i + j + ij + ijk a b a b However, instead of assuming that = = = i =1 i j =1 j i =1 ij j =1 ij = 0, Assume that E ( i ) = E ( j ) = Ei ( ij ) = E j ( ij ) = 0. The expectation operator, E ( ) , states that the average across all hypothetical instances of is zero. It implicitly acknowledges that the average of the sample of values in this study may not be zero. The marginal and grand means become Yi = + i + + i + i Y j = + + j + j + j Y = + + + + 8 G89.2228 Lect 13a Expected MS under Random Effects The expected mean squares for the main effects of A and B include a term for the variability of ( ). 2 2 E ( MS A ) = bn + n + 2 2 2 E ( MS B ) = an + n + 2 2 E ( MS AB ) = n + 2 E ( MS E ) = 2 The F ratios used to test the main effects must use MS AB in the denominator instead of MS E Effect A: Fa 1,( a 1)(b 1) = MS A / MS AB Effect B: Fb 1,( a 1)(b 1) = MS B / MS AB The test for the interaction still uses MSE in the denominator: Effect AB: F( a 1)(b 1),ab ( n 1) = MS AB / MS E 9 G89.2228 Lect 13a Mixed Models: One Factor Fixed and one Factor Random Suppose A is fixed and B is random Yi = + i + + i + i Y j = + j + j Y = + + The Expected MS 2 2 E ( MS A ) = bn + n + 2 2 E ( MS B ) = an + 2 2 E ( MS AB ) = n + 2 E ( MS E ) = 2 The F ratios are thus: Effect A: Fb 1,ab ( n 1) = MS B / MS E Effect B: F = MS AB / MS E Effect AB: ( a 1)(b 1),ab ( n 1) 10 G89.2228 Lect 13a Fa 1,( a 1)(b 1) = MS A / MS AB Notes on Defining Effects as Random Sometimes studies are carried out in different sites or at different times E.g., Suppose a college friend and you collaborate to test a fixed effect A with 4 levels. Each of you randomly assigns 25 subjects to each level. You pool your data and plan to call study site factor B. Your total N is 200. If you treat B as fixed, the F for factor A is F(3,192). The denominator MS is MSE (within cell). If you treat B as random, the F for factor A is F(3,3). The denominator MS for the fixed effect A is MSAB. One wants to generalize across all sites, 11 but the cost in power is very high.G89.2228 Lect 13a

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