TABLE OF CONTENTS All the data sets considered to this point have been balanced. A balanced data set has equal replications for each treatment condition. Under the assumptions of normality and independence, all mean squares in a balanced random model are functions of chi-squared random variables. This is an important assumption if the intervals presented in the previous chapters are to maintain the stated confidence levels.
Unfortunately, conditions sometimes arise that result in unbalanced designs. For example, some measurements might be lost due to a gauge malfunction, or an operator might not be able to complete an entire experiment. When a design is unbalanced, some of the distributional properties of the mean squares are lost, and the intervals derived under the balanced model assumptions are no longer valid. In this chapter we describe an approach for constructing confidence intervals with unbalanced data using unweighted sums of squares. We present detailed results for the one-factor and two-factor models discussed previously in this book. References for unbalanced nested designs are provided in Section 8.3.9.
We begin by considering the one-factor random model presented in Chapter 2. Table 7.1 reports a modified version of the data in Table 2.1, where only one observation is collected for parts 2, 3, and 7. Recall that these data represent thermal performances (in C ) of a sample of p = 10 power modules. The measurements were all taken by a single operator.
