3. Balanced Two-Factor Crossed Random Models with Interaction
Society for Industrial and Applied Mathematics eBooks · 2005
3.1 Introduction The two-factor crossed design with interaction is the classical gauge R&R model. Typically, the two factors are referred to as “parts” and “operators.” In this chapter we consider this design for balanced experiments where both factors are random. Extensions to situations where either (i) one factor is fixed or (ii) the design is unbalanced are considered in Chapters 6 and 7, respectively. The two-factor model with no interaction is presented in Chapter 5. We begin with an example that will be used throughout this chapter. Table 3.1 reports a partial listing of a data set based on the experiment described by Houf and Berman [32]. We have modified the experiment to increase the number of operators and parts used by Houf and Berman. The response variable is the thermal performance of a module measured in C° per watt. Each response has been multiplied by 100 for convenience of scale. The data represent measurements of 20 parts recorded by six operators. Each part is measured two times by each operator. In this chapter, it is assumed that parts and operators are selected at random from larger populations. We now consider a model to represent the two-factor data shown in Table 3.1. 3.2 The Model The balanced two-factor crossed random model with interaction is Yijk = μY + Pi + Oj + (PO)ij + Eijk ,i=1,… ,p,j=1,… ,o,k=1,… ,r, 3.1 where μY is a constant and Pi, Oj, (PO)ij, Eijk are jointly independent normal random variables with means of zero and variances , , , and , respectively. The ANOVA for model (3.1) is shown in Table 3.2, and the definitions for the mean squares and means are shown in Table 3.3. Table 3.4 reports distributional properties based on the assumptions in model (3.1).