Toby J. Mitchell

Bayesian Variable Selection in Linear Regression

Abstract This article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable. It is based on a Bayesian approach, intended to be as objective as possible. A probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model. The method is not fully Bayesian, however, because the ultimate choice of prior distribution from this family is affected by the data. It is assumed that the predictors represent distinct observables; the corresponding regression coefficients are assigned independent prior distributions. For each regression coefficient subject to deletion from the model, the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere, that is, a “spike and slab” distribution. The random error component is assigned a normal distribution with mean 0 and standard deviation ...

Exploratory designs for computational experiments

Abstract Recent work by Johnson et al. (J. Statist. Plann. Inference 26 (1990) 131–148) establishes equivalence of the maximin distance design criterion and an entropy criterion motivated by function prediction in a Bayesian setting. The latter criterion has been used by Currin et al. (J. Amer. Statist. Assoc. 86 (1991) 953–963) to design experiments for which the motivating application is approximation of a complex deterministic computer model. Because computer experiments often have a large number of controlled variables (inputs), maximin designs of moderate size are often concentrated in the corners of the cuboidal design region, i.e. each input is represented at only two levels. Here we will examine some maximin distance designs constructed within the class of Latin hypercube arrangements. The goal of this is to find designs which offer a compromise between the entropy/maximin criterion, and good projective properties in each dimension (as guaranteed by Latin hypercubes). A simulated annealing search algorithm is presented for constructing these designs, and patterns apparent in the optimal designs are discussed.

Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments

Abstract This article is concerned with prediction of a function y(t) over a (multidimensional) domain T, given the function values at a set of “sites” {t (1), t (2), …, t (n)} in T, and with the design, that is, with the selection of those sites. The motivating application is the design and analysis of computer experiments, where t determines the input to a computer model of a physical or behavioral system, and y(t) is a response that is part of the output or is calculated from it. Following a Bayesian formulation, prior uncertainty about the function y is expressed by means of a random function Y, which is taken here to be a Gaussian stochastic process. The mean of the posterior process can be used as the prediction function ŷ(t), and the variance can be used as a measure of uncertainty. This kind of approach has been used previously in Bayesian interpolation and is strongly related to the kriging methods used in geostatistics. Here emphasis is placed on product linear and product cubic correlation func...

Screening, predicting, and computer experiments

Many scientific phenomena are now investigated by complex computer models or codes. Given the input values, the code produces one or more outputs via a complex mathematical model. Often the code is expensive to run, and it may be necessary to build a computationally cheaper predictor to enable, for example, optimization of the inputs. If there are many input factors, an initial step in building a predictor is identifying (screening) the active factors. We model the output of the computer code as the realization of a stochastic process. This model has a number of advantages. First, it provides a statistical basis, via the likelihood, for a stepwise algorithm to determine the important factors. Second, it is very flexible, allowing nonlinear and interaction effects to emerge without explicitly modeling such effects. Third, the same data are used for screening and building the predictor, so expensive runs are efficiently used. We illustrate the methodology with two examples, both having 20 input variables. I...

Bayesian Design and Analysis of Computer Experiments: Use of Derivatives in Surface Prediction

This article is concerned with the problem of predicting a deterministic response function yo over a multidimensional domain T, given values of yo and all of its first derivatives at a set of design sites (points) in T. The intended application is to computer experiments in which yo is an output from a computer model of a physical system and each point in T represents a particular configuration of the input parameters. It is assumed that the first derivatives are already available (e.g., from a sensitivity analysis) or can be produced by the code that implements the model. A Bayesian approach in which the random function that represents prior uncertainty about yo is taken to be a stationary Gaussian stochastic process is used. The calculations needed to update the prior given observations of yo and its first derivatives at the design sites are given and are illustrated in a small example. The issue of experimental design is also discussed, in particular the criterion of maximizing the reduction in entropy...

The dose-response relationship for ethyl methanesulfonate-induced mutations at the hypoxanthine-guanine phosphoribosyl transferase locus in Chinese hamster ovary cells

The frequency of ethyl methanesulfonate (EMS)-induced mutations to 6-thioguanine resistance in a Chinese hamster ovary cells done K1-BH4 was studied at many EMS doses including the minimally lethal range (0–100 μg/ml) as well as the exponential killing portion (100–800 μg/ml) of the survival curve. The mutation frequency increases approximately in proportion with increasing EMS concentration at a fixed treatment time. The pooled data for the observed mutation frequency, f(X), as a function of EMS dose X, is adequately described by a linear function f(X)=10−6(8.73+3.45 X), where 0≤X≤800 μg/ml. One interpretation of the linear dose-response is that, as a result of EMS treatment, ethylation of cellular constituents occurs, which is directly responsible for the mutation. Biochemical analyses demonstrate that most of the randomly isolated 6-thioguanine-resistant variants possess a highly reduced or undetectable level of HGPRT activity suggesting that the EMS-induced mutations to 6-thioguanine resistance affect primarily, if not exclusively, the HGPRT locus.

Computer Construction of “D-Optimal” First-Order Designs

This paper presents the results of a study in which the computer algorithm DETMAX was used for the purpose of constructing n-run “D-optimal” designs over a cubic region of interest for the first-order model E(y) = β0 + β0 x 1 + … + β p x p . These results suggest some general “rules” (actually conjectures) for the construction of such designs. For p ≤ 9, all but 12 combinations of n and p are covered by these “rules;” the 12 exceptions are discussed separately. The resolution IV designs obtained by folding over these “D-optimal” first-order designs are also discussed and are shown to compare favorably with designs previously published.

The dose-response relationship for ultraviolet-light-induced mutations at the hypoxanthine-guanine phosphoribosyltransferase locus in Chinese hamster ovary cells

Exposure of Chinese hamster ovary (CHO) cells clone K1BE4 to ultraviolet (UV) light at doses up to 86 ergs/mm2 did not significantly reduce cell survival, but UV doses of 86–648 ergs/mm2 produced an exponential cell killing. Observed mutation frequency to 6-thioguannine resistance induced by UV increases approximately in proportion to increasing doses up to 260 ergs/mm2 in a range of 5–648 ergs/mm2 examined. The pooled data of mutation frequency f(X) as a function of dose X from 0–260 ergs/mm2 is adequately described by f(X)=10−6 (13.6+2.04 X). That the UVinduced mutations to 6-thioguanine resistance affects the hypoxanthineguanine phosphoribosyltransferase (HGPRT) locus is supported by the observation that all randomly isolated drugresistant colonies contained highly reduced or undetectable HGPRT activity.

Nonparametric estimation of the distribution of time to onset for specific diseases in survival/sacrifice experiments

This paper concerns the analysis of an animal survival/sacrifice experiment designed to investigate the incidence of a particular disease of interest. The disease is assumed to be irreversible, and detectable only at death, for example by a necropsy. Each observation can be of one of three types: (i) death caused by the disease, (ii) death from a competing cause such as sacrifice, with the disease present, or (iii) death with the disease absent. A two-dimensional EM algorithm is proposed for the nonparametric maximum likelihood estimation of the distributions of the time to onset and of the time to death from the disease. These can be compared with nonparametric estimators recently proposed by Kodell , Shaw and Johnson (1982, Biometrics 38, 43-58) and by Dinse and Lagakos (1982, Biometrics 38, 921-932). A slight modification of the algorithm permits the construction of likelihood-based interval estimates of quantiles of the distributions. Some extensions and generalizations are indicated.

D-optimal fractions of three-level factorial designs

D-optimal fractions of three-level factorial designs for p factors are constructed for factorial effects models (2 ≤ p ≤ 4) and quadratic response surface models (2 ≤ p ≤ 5). These designs are generated using an exchange algorithm for maximizing |X′X| and an algorithm which produces D-optimal balanced array designs. The design properties for the DETMAX designs and the balanced array designs are tabulated. An example is given to illustrate the use of such designs.