1 G Lect 13b G Lecture 13b Mixed models Special case: one entry per cell Equal vs. unequal cell n's

2 G Lect 13b Mixed Models: One Factor Random, One Factor Fixed Suppose we replicate a simple experiment at a sample of different universities –Experimental factors: fixed –Universities: considered random Suppose we measure attention span of 30 children at five developmental times –Times: fixed –Individual children: random Suppose we manipulate three learning motivation protocols and study their effects with a five samples of study words –Learning protocols: fixed –Study words: random

3 G Lect 13b Mixed Models: Expected Mean Squares Model: Y ijk = µ +  i +  j +  ij + e ijk Define A as fixed and B as random The Expected MS The F ratios are thus: –Effect A: –Effect B: –Effect AB:

4 G Lect 13b 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 MS E (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 MS AB. One wants to generalize across all sites, but the cost in power is very high.

5 G Lect 13b Special Case: Single entry per cell Suppose that all forty words (a=40) were tested with the same set of b subjects from a single university. Factor A is Words (random) and Factor B is subjects (random). No replications within cell are available, and so it is not possible to estimate MS E. Under the random effects model, MS AB is the appropriate error term, and it can be estimated. Under a mixed model (if words were specific fixed conditions), the appropriate test of the subjects effect is not possible. By convention, we assume that there is no interaction and test the random effect with MS AB.

6 G Lect 13b Example: Data on Stem Completion: CA _ _ _

7 G Lect 13b Example: Analysis

8 G Lect 13b New issue: Unequal cell n's in ANOVA designs When the sample size is not the same in the cells of a factorial ANOVA, and when the cell sizes are not proportional, then we say we have a nonorthogonal design. Nonorthogonal = “correlated”, and in a nonorthogonal design the indicators of Factor A and of Factor B are correlated. The lack of balance implies that cells may have different impacts on main effects of A and of B as a function of the n i ’s when the main effects are computed as simple averages of all available data.

9 G Lect 13b Approaches to nonorthogonal designs Unweighted Means solution –Discussed by Howell –Called Unique effects in SPSS –Called Type III effects in SAS –Equivalent to “adjusting” all effects for the presence of all others –Sums of squares from analysis do not add up “Experimental” Approach of SPSS –Called Type II effects in SAS –Main effects are adjusted for each other but not for interaction –Two way interactions adjusted for each other, but not for higher order interactions –Sums of squares do not add up “Hierarchical” Approach of SPSS –Called Type I effects in SAS –Hierarchical regression approach used –First factor is not adjusted for anything –Second factor is adjusted for first only –Third term is adjusted for first and second (and so on) –Sums of squares do add up.

10 G Lect 13b ANOVA in Experiments and ANOVA in surveys Experimental data are usually balanced or almost balanced. When n i ’s are close, the various approaches lead to same results In surveys, sample sizes in factor level are often determined by the population structure. If you expect to work with unbalanced designs, although ANOVA can be treated as a special case of regression, use of the regression approach directly will provide you with more options (e.g. the hierarchical approach)