A-ComVar: A Flexible Extension of Common Variance Designs

Abstract

We consider nonregular fractions of factorial experiments for a class of linear models. These models have a common general mean and main effects; however, they may have different 2-factor interactions. Here we assume for simplicity that 3-factor and higher-order interactions are negligible. In the absence of a priori knowledge about which interactions are important, it is reasonable to prefer a design that results in equal variance for the estimates of all interaction effects to aid in model discrimination. Such designs are called common variance designs and can be quite challenging to identify without performing an exhaustive search of possible designs. In this work, we introduce an extension of common variance designs called approximate common variance or A-ComVar designs. We develop a numerical approach to finding A-ComVar designs that is much more efficient than an exhaustive search. We present the types of A-ComVar designs that can be found for different number of factors, runs, and interactions. We further demonstrate the competitive performance of both common variance and A-ComVar designs using several comparisons to other popular designs in the literature.

Appendices

Appendix 1: Genetic Algorithm

In this work we used a genetic algorithm to find designs that maximize the A-ComVar objective function. This appendix provides specific details on the algorithm. Keeping with the standard genetic algorithm terminology, we use the word chromosome to describe a single candidate design. Each chromosome is comprised of the factor settings for each factor at each design point. Each of these individual factor settings is known as a gene. The population is the set of all chromosomes, i.e., all designs that we are currently considering.

We illustrate a simple version of the genetic algorithm below. In this example we search for a 6-run A-ComVar design for an experiment with three two-level factors and one interaction. We label the factors as A, B, and C. For simplicity, we assume that the population size is 3, although in real applications it will generally be larger.

Since this experiment has three two-level factors, there are 8 possible design points to pick the 6 points for our design from. The 8 points are shown in Table 7. There are ${3}\atopwithdelims (){2}$ $= 3$ possible models with all main effects and one interaction. For notational simplicity we label these models by the corresponding interaction: (AB), (AC), and (BC). Our goal is to obtain a design under which the variance of the interaction term is identical, or close to identical, under all three of these models.

Table 7 Set of possible design points for Appendix example

0. Initialization

First, each of the three chromosomes is initialized to a random start. To obtain the random start for a specific chromosome, we simply sample six of the rows in Table 7 without replacement. Our initialization procedure results in the following three chromosomes:

After initializing, we need to calculate the fitness for each of these chromosomes using the objective function in expression (2). In order to evaluate the objective function, we need to calculate $\sigma _{2i}^2$ for $i = 1, 2, 3$, which correspond to models (AB), (AC), and (BC), respectively. Then, we take the average of these three values to be $\sigma _{2}^2$ and can evaluate the objective function. These steps are illustrated below for the first chromosome.

The above procedure is repeated for each of the three chromosomes. In this case, all three designs end up having the same fitness value. We now summarize each chromosome below:

Now that we have completed the initialization process, we can begin the main loop over the algorithm.

1. Identify worst chromosome(s)

The first step is to identify the worst chromosomes. These are the chromosomes that will be replaced by new offspring. Since we only have three chromosomes in the population, we will only identify and replace the single worst chromosome. In the case of a tie (as we have here), the chromosome to be replaced is randomly chosen. In this case we have chosen chromosome 3 to be replaced.

2. Generate replacement using crossover

We next generate a replacement for the worst chromosome (3) using crossover from 2 randomly selecting remaining chromosomes. Since our example only has three chromosomes, we simply use the remaining chromosomes (1 and 2). In the crossover, a random cut point is selected, and the two chromosomes are combined using the values from the first chromosome for the factors to the left of the cut point and the values from the second chromosome for the factors to the right of the cut point. This process is illustrated below:

Note that it is possible to consider other ways of producing offspring via crossover. For example, the cut point could be different for each support point, or they could be “horizontal” instead of “vertical,” choosing certain rows from the first chromosome and the remaining rows from the second.

3. Mutation

In addition to crossover, more novelty can be introduced to the solution by randomly changing, or mutating, some of factor settings. For our purpose, the probability of each factor setting (gene) mutating is identical.

4. Replacement and Fitness Evaluation

Following Steps 3 and 4, we are now ready to replace the old chromosome with the offspring. In this step, the worst chromosome(s) is replaced by the offspring created in Steps 2–3. The fitness of this new chromosome is evaluated and stored.

Appendix 2: Tables for Example 3

Tables 8, 9, 10, 11, 12, 13, and 14 present detailed results for each of the comparisons in Example 3.

Table 8 Average percentage of correctly identified models for the common variance design $D^1$ and the Plackett–Burman design

Table 9 Average percentage of correctly identified models for A-ComVar design $D^2$ and the Plackett–Burman design

Table 10 Average percentage of correctly identified models for A-ComVar design $D^3$ and Bayes optimal design $D^4$ from Bingham and Chipman [1]

Table 11 Average percentage of correctly identified models for A-ComVar design $D^3$ and design $D^5$ from Li and Nachtsheim [14]

Table 12 Average percentage of correctly identified models for A-ComVar design $D^3$ and design $D^6$ from Ghosh and Tian [9]

Table 13 Average percentage of correctly identified models for 3-level A-ComVar design $D^2$ and CCD $D^7$

Table 14 Average percentage of correctly identified models for 3-level A-ComVar design $D^3$ and OME $D^8$