Supersaturated design including an orthogonal base

Recently, many supersaturated designs have been proposed. A supersaturated design is a fractional factorial design in which the number of factors is greater than the number of experimental runs. The main thrust of the previous studies has been to generate more columns while avoiding large values of squared inner products among all design columns. These designs would be appropriate if the probability for each factor being active is uniformly distributed. When factors can be partitioned into two groups, namely, with high and low probabilities of each factor being active, it is desirable to maintain orthogonality among columns to be assigned to the factors in the high-probability group. We discuss a supersaturated design including an orthogonal base which is suitable for this common situation. Mathematical results on the existence of the supersaturated designs are shown, and the construction of supersaturated designs is presented. We next discuss some properties of the proposed supersaturated designs based on the squared inner products.     

Characterizing projected designs: repeat and mirror-image runs

Two-level designs are useful to examine a large number of factors in an efficient manner. It is typically anticipated that only a few factors will be identified as important ones. The results can then be reanalyzed using a projection of the original design, projected into the space of the factors that matter. An interesting question is how many intrinsically different type of projections are possible from an initial given design. We examine this question here for the Plackett and Burman screening series with N= 12, 20 and 24 runs and projected dimensions k≤5. As a characterization criterion, we look at the number of repeat and mirror-image runs in the projections. The idea can be applied toany two-level design projected into fewer dimensions.     

Some bivariate uniform distributions

Bivariate uniform distributions with dependent components are readily derived by distribution function transformations of the components of non-uniform dependent continuous bivariate random variables (X,Y). Contour plots of joint density functions show the various, and varying, forms of dependence which can arise from different distributional forms for (X,Y) and aids the choice of bivariate uniform distributions as empirical models.     

Numerical computation of rectangular bivariate and trivariate normal and t probabilities

Algorithms for the computation of bivariate and trivariate normal and t probabilities for rectangles are reviewed. The algorithms use numerical integration to approximate transformed probability distribution integrals. A generalization of Plackett's formula is derived for bivariate and trivariate t probabilities. New methods are described for the numerical computation of bivariate and trivariate t probabilities. Test results are provided, along with recommendations for the most efficient algorithms for single and double precision computations.     

A strategy for constructing multivariate distributions

Given a random vector (X₁,…, X_n) for which the univariate and bivariate marginal distributions belong to some specified families of distributions, we present a procedure for constructing families of multivariate distributions with the specified univariate and bivariate margins. Some general properties of the resulting families of multivariate distributions are reviewed. This procedure is illustrated by generalizing the bivariate Plackett (1965) and Clayton (1978) distributions to three dimensions. In addition to providing rich families of models for data analysis, this method of construction provides a convenient way of simulating observations from multivariate distributions with specific types of univariate and bivariate marginal distributions. A general algorithm for simulating random observations from these families of multivariate distributions is presented     

The validation of surrogate end points by using data from randomized clinical trials: a case-study in advanced colorectal cancer

Summary.  In many therapeutic areas, the identification and validation of surrogate end points is of prime interest to reduce the duration and/or size of clinical trials. Buyse and co-workers and Burzykowski and co-workers have proposed a validation strategy for end points that are either normally distributed or (possibly censored) failure times. In this paper, we address the problem of validating an ordinal categorical or binary end point as a surrogate for a failure time true end point. In particular, we investigate the validity of tumour response as a surrogate for survival time in evaluating fluoropyrimidine-based experimental therapies for advanced colorectal cancer. Our analysis is performed on data from 28 randomized trials in advanced colorectal cancer, which are available through the Meta-Analysis Group in Cancer.