Analysis of Unbalanced Data from an Experiment with Random Block Effects and Unequally Spaced Factor Levels

Abstract

We examine a 3² factorial experiment that was run on each of 12 days with extra replications of the middle point. Usually, the literature on the analysis of factorial experiments performed in blocks treats the blocks as a fixed effect. Data collected from this experiment, however, were analyzed by treating the block effect as random. The combination of quantitative factor levels and random block effects provides an opportunity to discuss issues that are not commonly encountered in the literature on mixed model analyses of data. In particular, we: (a) discuss modeling the interaction between blocks and regression coefficients; (b) discuss the effect of coding on the estimation of degrees of freedom (df) of the fixed terms; (c) present an oddity with estimating the intercept df; and (d) discuss inference on the mean of the responses. How the experiment may have been run and its consequences are also discussed.

Key Words:

• Estimation of the mean in mixed models
• Fixed or random orthogonal blocks
• PROC MIXED
• Type II and Type III sums of squares