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

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

3. 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

4. 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

5. 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.

6. 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

7. 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

8. 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

9. 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

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

11. 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

12. 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;

13. 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

14. 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

15. 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)

16. 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

17. 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!

18. Fractional Factorials
Taguchi methods