Experimental Design & Analysis Random and Fixed Factors; Fractional Factorials April 10, 2007 DOCTORAL SEMINAR, SPRING SEMESTER 2007

Outline Fixed and random factors Fractional factorial designs Fixed factor model Random factor model Mixed factor model Examples SAS Fractional factorial designs

Fixed Factors Most ANOVA designs are fixed effects models, meaning that data are collected on all relevant categories of the independent variables Relevance of levels are dictated by some theoretical underpinning E.g. comparing dosages of a drug (5 mg, 10 mg, 20 mg) are of theoretical interest, perhaps representing low, medium, high levels Comparing sexes with respect to some response variable, the factor sex is fully represented with 2 levels

Random Factors In random effects models, in contrast, data are collected only for a sample of categories For instance, a researcher may study the effect of item order in a questionnaire. Six items could be ordered 720 ways. However, the researcher may limit herself to the study of a sample of six of these 720 ways The random effect model in this case would test the null hypothesis that the effects of ordering = 0 Studies examining whether facial symmetry is important in mate selection may include a variety of symmetric faces Within-subjects designs include a random effect

Fixed and Random Factors Random effects are factors which meet 2 criteria: Replaceability: The levels (categories) of the factor (independent variable) are randomly or arbitrarily selected, and could be replaced by other, equally acceptable levels. Generalization: The researcher wishes to generalize findings beyond the particular, randomly or arbitrarily selected levels in the study.

Mixed Effects Model Treatment by replication design is a common mixed effects model A fixed factor, such as male and female faces A random factor, or replication factor, representing variety of bilateral symmetry, such as distance of facial features

Fixed and Random Factors Effects shown in the ANOVA table for a random factor design are interpreted a bit differently from standard, within-groups designs Main effect of the fixed treatment variable is the average effect of the treatment across the randomly sampled or arbitrarily selected categories of the random variable The effect of the fixed by random (e.g. treatment by replication) interaction indicates the variance of the treatment effect across the categories of the random variable Main effect of the random effect variable (e.g. the replication effect) is of no theoretical interest as its levels are arbitrary cases from a large population of equally acceptable cases

Random and Fixed Factors For one-way ANOVA, computation of F is the same for fixed and random effects, but computation differs when there are two or more independent variables Resulting ANOVA table still gives similar sums of squares and F-ratios for the main and interaction effects, and is read similarly Assumptions: normality, homogeneity of variances, and sphericity, but robust to violations of these assumptions

Testing for Significance Test for significance of: A and B are fixed factors A and B are random factors A is a fixed factor and B is a random factor A main effect MSA/MSE MSA/MSAB B main effect MSB/MSE MSB/MSAB AB interaction MSAB/MSE

Testing for Significance What is impact of assuming a random factor is a fixed factor Type I error? Type II error?

Random and Fixed Factors Treating a random factor as a fixed factor will inflate Type I error If a random factor is treated as a fixed factor, research findings may pertain only to the particular arbitrary cases studied and findings and inferences may be different with alternative cases

SAS Procedures For random effects models or mixed effects models PROC GLM DATA = mydata; CLASS factor1 factor2; MODEL depvar = factor1|factor2; RANDOM factor1 / TEST; RUN;

Fractional Factorials Even if the number of factors, k, in a design is small, the 2k runs specified for a full factorial can quickly become very large E.g. 26 = 64 runs is for a two-level, full factorial design with six factors

Fractional Factorials The solution to this problem is to use only a fraction of the runs specified by the full factorial design Which runs to make and which to leave out is the subject of interest here We use various strategies that ensure an appropriate choice of runs Properly chosen fractional factorial designs for 2-level experiments have the desirable properties of being both balanced and orthogonal Assume that higher-order interactions are not of interest

Fractional Factorials Imagine testing the effects of attention variables on processing of negation Duration of exposure (10 sec vs. 20 sec vs. 40 sec) Cognitive load (7 digit vs. 3 digit) Self relevance (self vs. other)

Fractional Factorials Two-Level Factors Three-Level Factor B C X -1 x1 +1 x2 x3 3 x 2 x 2 design = 12 conditions! Are all cells and interactions of comparable value? Consider running fractional factorial Yoke cells

Fractional Factorials Two-Level Factors Three-Level Factor B C X 3 digit self 10 sec 7 digit 20 sec other 40 sec 4 conditions!

Fractional Factorials Taguchi methods