Three way ANOVA If there was One way

Three way ANOVA If there was One way then and Two way, then you knew that there had to be……… 1

Three way Mixed Model � In a Human Factors experiment, there were two Job types, Dimensional stimulus (two or three dimensions), and five Analysts who performed ratings. � Job type and dimension may be considered fixed (repeatable) factors. � Analyst may be considered random (different in another run of the experiment). � Primary interest is consistency (or differences) between stimulus types, two or three dimensional. � Data from Hicks 1982, Fundamental Concepts in Design of Experiments (good reference). 2

Data structure (partial…) 3

Layout of Design 4

Linear Model notation 5

Linear Model 6

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ANOVA Summary 8

Detailed ANOVA with Mean Squares 9

What are “correct” F-tests? Table first… 10

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Candidate terms for coefficients For each term in the model: Then candidates for variance components: 13

EMS using Restricted Model (Cornfeld-Tukey) 14

Correct F-tests for Fixed Effects 15

Residuals by Predicted 16

Normality Plot of Residuals 17

Shapiro-Wilk Test of Normality Goodness-of-Fit Test Shapiro-Wilk W Test W 0. 984424 Prob<W 0. 8463 Note: Ho = The data is from the Normal distribution. Small p-values reject Ho. 18

Conclusions The differences among levels of the fixed effects are not greater than that which would be expected by chance over this population of Analysts. If one were to consider Analysts fixed, then one would test all terms against Experimental Error (which has lower variation), but any conclusions would only apply to these Analysts. 19

All factors fixed analysis (inference changes) 20

Dimension and Analyst 21

LS Means HSD 22

Job and Analyst 23

LS Means HSD 24