6. Balanced Two-Factor Crossed Mixed Models
Society for Industrial and Applied Mathematics eBooks · 2005
6.1 Introduction In Chapters 3 and 5 we presented two-factor models where both factors are assumed to be random effects. Although this has traditionally been the assumption in gauge R&R studies, it is often more appropriate to treat operators as fixed effects. Consider the situation described by Dolezal, Burdick, and Birch [18], in which three mechanical testers (operators) are used to monitor a process that manufactures tape drive heads (parts). A random sample of 18 heads is obtained from the process output, and the response variable “reverse overwrite” was measured for each head using the set of three testers. Reverse overwrite is a measure of residual frequency after a second frequency is placed on magnetic tape. The units of measurement are decibels and the specification limits are LSL = −41 and U SL = −33. Each tester makes three replicate measurements on each head. The total data set consists of 162 measurements. A partial listing of the data is given in Table 6.1. The testers in this situation are each attached to one of three production lines and will always be part of the testing system. Since these are the only three testers that will ever be used to monitor the process, the inference concerns only these three testers and not the population from which they were selected. Thus, the operator factor is a fixed effect. (If you are not familiar with the terms “random” and “fixed” in this context, review the material in Appendix A.) The two-factor mixed model with random parts and fixed operators is the topic of this chapter. 6.2 The Mixed Model Formulation For a set of o fixed operators, the measurements from the jth operator are represented as Yj =X+ Ej , 6.1 where Yj is a measured value of a randomly selected part, X is the true value, and Ej is the measurement error. The terms X and Ej are independent normal random variables with means μP and μEj and variances γP and γE, respectively. These assumptions imply that the o populations of operator measurements have equal variances but possibly different means.