There are many industrial experiments where the response variable is nonnormal. Traditionally, variance-stabilizing transformations are made on such a response in order to obtain properties needed to use ordinary least squares and analysis of variance. Generalized linear models (GLMs) offer a powerful alternative to data transformation. Specifically, the performance in response estimation and prediction for a GLM is often superior to the model built using data transformations. The confidence interval constructed around the estimate of the mean for each experimental run provides the experimenter with critical information about model quality. In generalized linear models, confidence intervals are based on asymptotic theory. As such, they are regarded as statistically valid only for large samples. Therefore, in order to use confidence intervals to compare models, it is essential to evaluate these asymptotic intervals in terms of coverage for sample sizes typically encountered in designed industrial experiments. This paper uses Monte Carlo methods to investigate the coverage of confidence intervals for the GLM for factorial experiments with 8, 16, and 32 runs.

Key Words: Design of Experiments, Generalized Linear Models, Nonconstant Variance, Nonnormality.

*By* **Sharon L. Lewis** and **Douglas C. Montgomery,
Arizona State University, Tempe, AZ 85287** and **Raymond
H. Myers, Virginia Polytechnic Institute and State University,
Blacksburg, VA 24061**

**Introduction**

DESIGNED experiments are used to build a model that describes
the relationship of a response variable of interest to one
or more independent (or design) variables. A model such as

Ordinary linear least squares is
the most widely used technique for building models based on
data from designed experiments. The method of least squares
produces best linear (i.e., minimum variance) unbiased estimators
of the regression coefficients *ß _{j}* (Montgomery
and Peck (1992), Myers (1990)). The optimality properties
of ordinary least squares depend on the assumption of constant
variance. The widespread growth in the application of designed
experiments has led to many situations where the response
is nonnormal, such as counts of defects, proportions defective,
or times to failure. These responses may follow Poisson, binomial,
and gamma distributions, respectively. In each case, the variance
is not a constant, but is rather a function of the mean. If
the fitted linear model is correct, the least squares estimators
will still be unbiased, but will no longer possess the minimum
variance property.

One option for these nonnormal response
situations is to simply ignore the problem and use normal
theory ordinary least squares with the assumptions violated.
Many researchers and practitioners assume that factorials
and fractional factorials are so robust that such an analysis
will still be useful. However, the more traditional approach
is to apply a variance-stabilizing transformation to the response
variable to *bend the data into shape*. This allows application
of classical least squares to the transformed data. A third
approach is the use of the generalized linear model (GLM).
With the GLM, normality and constant variance are no longer
required.

Myers and Montgomery (1997) give a tutorial on GLMs. In their tutorial, they show several examples comparing models built with ordinary least squares and data transformations and those built with a GLM approach. In each case, they find that a better model is possible with the GLM, where “better” is measured in terms of the model performance in response estimation and prediction. Myers and Montgomery use the length of the confidence interval on the mean response for comparing models (See also Lewis, Montgomery, and Myers (2001)).

In generalized linear models, confidence
intervals are typically based on (asymptotic) Wald inference;
therefore, these intervals are useful for a large sample size
*n* (there are exact confidence intervals available for
logistic regression, a special case of the GLM, but we do
not consider these intervals in this paper). Designed experiments
usually involve small *n*. Therefore, in order to use
confidence intervals in comparing model performance, it is
essential to evaluate these asymptotic intervals in terms
of coverage and precision for small sample applications. The
purpose of this paper is to evaluate the properties of two
different types of Wald inference based confidence intervals
for the GLM for small sample sizes such as are typically encountered
in industrial experiments. This is accomplished with a Monte
Carlo simulation study.

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