Anderson cook

STATISTICS ROUNDTABLE

How To Choose the Appropriate Design
by Christine Anderson-Cook

hen planning an experiment, you can consider many possible sets of observations. Choosing the combinations of factors at which to collect data to make up the best design involves balancing multiple goals and objectives. You should begin by specifying a parametric model of interest to summarize thefeatures of the relationship between the response and the explanatory variables. Consider, for example, an experiment to model the connection between two variables, X 1 and X 2 , and a response, Y. If you believe the relationship is reasonably summarized by a flat plane, then the first order model, Y = β0 + β1X1 + β2X2 + ε, may be appropriate. If you believe the relationship will have some curvature,then the first

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order model with interactions, Y = β0 + β1X1 + β2X2 + β12X1X2 + ε, or the second order model, Y = β0 + β1X1 + β2X2

It’s crucial to balance good estimation of parameters and good prediction in the design space.
2 + β12X1X2 + β11X1 + β22X2 + ε, may be 2 more appropriate. If you select a model with too few parameters, then it may not capture all the features of the data. Ifyou

select a model with too many parameters, you can later perform some tests to determine whether the model can be simplified to more accurately reflect the true characteristics demonstrated by the observed data. However, if you select too large an initial model, it may reduce the precision with which you are able to estimate the parameters included in the final model.1 Balance CompetingObjectives Once an initial model has been chosen, you need to identify a criterion (or criteria) for the selection process for a best design. Typically, a good design will simultaneously balance a number of competing objectives.2 Which design is most appropriate depends on the model chosen and the specific goals of the study. Two common primary goals of conducting an experiment and then modeling theresults are: 1. To be able to estimate the parameters of the model. 2. To predict the response at any new location in the design space. Several numerical summaries for estimation and prediction can be used to make direct comparisons between potential designs. To understand the possible measures, look at a simple example involving an experiment with two factors in a square design space scaled to haveeach factor between -1 and +1 for a first order model with interaction of the form, Y = β0 + β1X1 + β2X2 + β12X1X2 + ε. In this situation, you should consider a 2 2 factorial design with two replicates at each location and one center run (design A) and a 32 factorial design (design B). Both designs are shown in Figure 1. D-Optimality D- and A-optimality are possible choices to assess the quality ofestimation of the model parameters. D-optimality focuses on desirable properties for the moment matrix of a design. It is defined as M XT X , N

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where X is a matrix that summarizes the design locations in the appropriate form for the model selected and N is the total number of observations for the design. The moment matrix of a design is important for assessing the quality of the designbecause the inverse of M, M-1 = N(XTX)-1, contains the variances and covariances of the model parameters. D-optimality seeks to maximize the determinant of the moment matrix, |M|, which is equivalent to minimizing the determinant of M-1. This minimization corresponds to finding small variances for model parameters. You should choose to scale by N because it allows you to more fairly compare designs...