Analyzing supersaturated designs for discrete responses via generalized linear models

Abstract

A supersaturated design is a factorial design in which the number of factors to be estimated is larger than the available number of experimental runs. The cost and time required for many industrial experimentations can be reduced by using the class of supersaturated designs, since the main goal for such a design is to identify only a few of the factors under consideration that have dominant effects and to do this identification at a minimal cost. While most of the literature on supersaturated designs has focused on the construction of designs and their optimality properties, the data analysis of such designs has not been developed to a great extent. In this paper, we propose a supersaturated design analysis method, by assuming generalized linear models for discrete responses, for analyzing main effects designs and identifying simultaneously the effects that are significant. Empirical study demonstrates that this method performs well with low Type I and Type II error rates. The proposed method is therefore useful as it enables us to use supersaturated designs for analyzing data on discrete response regression models.

Appendix

For each of the 44 models presented in Table 1, 1,000 datasets were generated for each considered Scenario, and the results obtained after the application of our method are presented in Tables 6, 7, 8 and 9 for Scenario I, II, III and IV, respectively, in accordance with the threshold values examined. In the first column, the number that corresponds to each used model is given. Columns named “Type I” and ”Type II” present the average values over 1,000 simulations of the Type I and Type II error rates corresponding to every threshold value. The last line of each table presents the average Type I and Type II error values for the 44 models considered.

Comparative results for each of the following scenarios are as follows:

• Scenario I: From Table 2, we readily observe that the average Type I and Type II error values for the 44 models considered are 0.23 and 0.06, respectively, for $SU$ taken to be median$(\mathbf su )$ and $w=\frac{k}{2}$. With this choice of thresholds, the proposed method has significantly lower Type II error values, and slightly higher Type I error values (see Table 6 for comparison).

• Scenario II: From Table 3, we readily observe that the average Type I and Type II error values for the 44 models considered are 0.29 and 0.04, respectively, for $SU$ taken to be median$(\mathbf su )$ and $w=\frac{k}{2}$. With this choice of thresholds, the proposed method has significantly lower Type II error values, and slightly higher Type I error values (see Table 7 for comparison).

• Scenario III: From Table 4, we readily observe that the average Type I and Type II error values for the 44 models considered are 0.29 and 0.05, respectively, for $SU$ taken to be median$(\mathbf su )$ and $w=\frac{k}{2}$. With this choice of thresholds, the proposed method has significantly lower Type II error values, and slightly higher Type I error values (see Table 8 for comparison).

• Scenario IV: From Table 5, we readily observe that the average Type I and Type II error values for the 44 models considered are 0.30 and 0.07, respectively, for $SU$ taken to be median$(\mathbf su )$ and $w=\frac{k}{2}$. With this choice of thresholds, the proposed method has significantly lower Type II error values, and slightly higher Type I error values (see Table 9 for comparison).

Table 6 Empirical performance of the proposed algorithm for the models listed in Table 1 for Scenario I

Table 7 Empirical performance of the proposed algorithm for the models listed in Table 1 for Scenario II

Table 8 Empirical performance of the proposed algorithm for the models listed in Table 1 for Scenario III

Table 9 Empirical performance of the proposed algorithm for the models listed in Table 1 for Scenario IV

These results suggest that the proposed method for $SU$ taken to be median$(\mathbf su )$ and $w=\frac{k}{2}$ is quite robust for count response modelling. The average Type I and Type II error values are almost identical under Scenarios II, III, IV considered.

In general, we conclude that for each Scenario considered, the proposed method seems to perform efficiently after selecting $SU$ to be median$(\mathbf su )$ and $w=\frac{k}{2}$. With this choice of thresholds, the proposed method achieves the lowest Type II error values for all Scenarios considered. The fact that this choice of thresholds results in slightly higher Type I error values is not troublesome since the use of SSDs is mainly to screen the factors that should be considered for further investigation. Hence, the low Type II error rates are especially desirable, even though both Type I and Type II error rates are important and should be kept as low as possible. However, under situations of effect sparsity that holds in SSDs, Type I errors are quite likely to occur, of course.