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Montgomery & Runger

Introduction

Experimental design models with one or morÄ™ random components are found in many situations. In generaÅ‚, factor is said to be random if its levels consist of a random sample of levels selected from a large (theoretically infinite) population of levels. Conversely, a fixed factor is one in which the levels are selected nonrandomly or if the levels consist of the entire population of levels.

From these definitions, three basie types of experimental design models can be constructed: fixed-effects models, in which all factors are fixed; randorn-effects models, in which all factors are random, and mixed-effects models, in which some factors are fixed and some are random. Most basie experimental design books (Montgomery, 1991, Box, Hunter, and Hunter, 1978) discuss the fixed-effects model at length, and give brief introductions to the random and mixed models. This paper gives an introduction to the random-eflfects case, focusing on the problem of parameter estimation and hypothesis testing. Some of our comments apply to the mixed model, but it is not our intention to provide a detailed account of its treatment.

There are numerous applications of the random model. As noted by Montgomery and Runger (1993a,b), one important application is in the analysis of measurement system capability. In these studies, the experimental objective is typically to assess the total variability associated with the output performance of a measurement system, which is usually a combination of gauges or instruments, operators, parts or test units, times, and other factors. In many (if not most) cases, the factors in these experiments are viewed as random factors.

The difference between fixed and random models appears smali. For example, consider a completely randomized single-factor experiment with a factor levels and n replicates. The linear model is

Yij = p + tj + e,j, i = l,2,...,a, j = l,2,...,n    (1)

with e,j the random error term distributed as N(0,cr²). If the factor is fixed, then the x, are unknown parameters; however, if the factor is random, then the x, are random variables and we typically assume that t, is N(0,a_t²), and that all the x, and all the Ä… are statistically independent. In this particular model the similar structural appearance between the fixed and random model carries over to the analysis; specifically, hypotheses in both models are tested by the same ANOVA procedurÄ™. The model parameters are estimated diflFerently, however. The estimates of the fixed effects parameters are obtained by the method of least squares (Montgomery, 1991, provides details), while simple estimates of the variance components a² and a_x² in the random model are obtained as linear combinations of the mean squares in the ANOYA table.