Optimum designs for clinical trials in personalized medicine when response variance depends on treatment.

We study optimal designs for clinical trials when the value of the response and its variance depend on treatment and covariates are included in the response model. Such designs are generalizations of Neyman allocation, commonly used in personalized medicine when external factors may have differing effects on the response depending on subgroups of patients. We develop theoretical results for D-, A-, E- and D A-optimal designs and construct semidefinite programming (SDP) formulations that support their numerical computation. D-, A-, and E-optimal designs are appropriate for efficient estimation of distinct properties of the parameters of the response models. Our formulation allows finding optimal allocation schemes for a general number of treatments and of covariates. Finally, we study frequentist sequential clinical trial allocation within contexts where response parameters and their respective variances remain unknown. We illustrate, with a simulated example and with a redesigned clinical trial on the treatment of neuro-degenerative disease, that both theoretical and SDP results, derived under the assumption of known variances, converge asymptotically to allocations obtained through the sequential scheme. Procedures to use static and sequential allocation are proposed.

Randomizing a clinical trial in neuro-degenerative disease.

The paper studies randomization rules for a sequential two-treatment, two-site clinical trial in Parkinson's disease. An important feature is that we have values of responses and five potential prognostic factors from a sample of 144 patients similar to those to be enrolled in the trial. Analysis of this sample provides a model for trial analysis. The comparison of allocation rules is made by simulation yielding measures of loss due to imbalance and of potential bias. A major novelty of the paper is the use of this sample, via a two-stage algorithm, to provide an empirical distribution of covariates for the simulation; sampling of a correlated multivariate normal distribution is followed by transformation to variables following the empirical marginal distributions. Six allocation rules are evaluated. The paper concludes with some comments on general aspects of the evaluation of such rules and provides a recommendation for two allocation rules, one for each site, depending on the target number of patients to be enrolled.

Robust Regression with Density Power Divergence: Theory, Comparisons, and Data Analysis.

Minimum density power divergence estimation provides a general framework for robust statistics, depending on a parameter α , which determines the robustness properties of the method. The usual estimation method is numerical minimization of the power divergence. The paper considers the special case of linear regression. We developed an alternative estimation procedure using the methods of S-estimation. The rho function so obtained is proportional to one minus a suitably scaled normal density raised to the power α . We used the theory of S-estimation to determine the asymptotic efficiency and breakdown point for this new form of S-estimation. Two sets of comparisons were made. In one, S power divergence is compared with other S-estimators using four distinct rho functions. Plots of efficiency against breakdown point show that the properties of S power divergence are close to those of Tukey's biweight. The second set of comparisons is between S power divergence estimation and numerical minimization. Monitoring these two procedures in terms of breakdown point shows that the numerical minimization yields a procedure with larger robust residuals and a lower empirical breakdown point, thus providing an estimate of α leading to more efficient parameter estimates.

Statistical and Proactive Analysis of an Inter-Laboratory Comparison: The Radiocarbon Dating of the Shroud of Turin.

We review the sampling and results of the radiocarbon dating of the archaeological cloth known as the Shroud of Turin, in the light of recent statistical analyses of both published and raw data. The statistical analyses highlight an inter-laboratory heterogeneity of the means and a monotone spatial variation of the ages of subsamples that suggest the presence of contaminants unevenly removed by the cleaning pretreatments. We consider the significance and overall impact of the statistical analyses on assessing the reliability of the dating results and the design of correct sampling. These analyses suggest that the 1988 radiocarbon dating does not match the current accuracy requirements. Should this be the case, it would be interesting to know the accurate age of the Shroud of Turin. Taking into account the whole body of scientific data, we discuss whether it makes sense to date the Shroud again.

A Semi-Infinite Programming based algorithm for determining T-optimum designs for model discrimination.

T-optimum designs for model discrimination are notoriously difficult to find because of the computational difficulty involved in solving an optimization problem that involves two layers of optimization. Only a handful of analytical T-optimal designs are available for the simplest problems; the rest in the literature are found using specialized numerical procedures for a specific problem. We propose a potentially more systematic and general way for finding T-optimal designs using a Semi-Infinite Programming (SIP) approach. The strategy requires that we first reformulate the original minimax or maximin optimization problem into an equivalent semi-infinite program and solve it using an exchange-based method where lower and upper bounds produced by solving the outer and the inner programs, are iterated to convergence. A global Nonlinear Programming (NLP) solver is used to handle the subproblems, thus finding the optimal design and the least favorable parametric configuration that minimizes the residual sum of squares from the alternative or test models. We also use a nonlinear program to check the global optimality of the SIP-generated design and automate the construction of globally optimal designs. The algorithm is successfully used to produce results that coincide with several T-optimal designs reported in the literature for various types of model discrimination problems with normally distributed errors. However, our method is more general, merely requiring that the parameters of the model be estimated by a numerical optimization.

Bias and loss: the two sides of a biased coin.

The paper assesses biased-coin designs for sequential treatment allocation in clinical trials. Comparisons emphasise the importance of considering randomness, as well as treatment balance, which are calculated as bias and loss. In the numerical examples, the responses are assumed normally distributed, perhaps after transformation, and balance is required over a set of covariates. The effect of covariate distribution on the properties of five allocation rules is investigated, with an emphasis on methods of comparison, which also apply to other forms of response. The concept of admissibility shows that the widely used minimisation rule is outperformed by Atkinson's rule derived from the theory of optimum experimental design. We present a simplified form of this rule. For this rule, the ability to guess the next treatment allocation decreases with study size. For the other rules, it is constant.

Optimum design of experiments for enzyme inhibition kinetic models.

We find closed-form expressions for the D-optimum designs for three- and four-parameter nonlinear models arising in kinetic models for enzyme inhibition. We calculate the efficiency of designs over a range of parameter values and make recommendations for design when the parameter values are not well known. In a three-parameter experimental example, a standard design has an efficiency of 18.2% of the D-optimum design. Experimental results from a standard design with 120 trials and a D-optimum design with 21 trials give parameter estimates that are in close agreement. The estimated standard errors of these parameter estimates confirm our theoretical results on efficiency and thus on the serious savings that can be made by the use of D-optimum designs.

Potential application of D-optimal designs in the efficient investigation of cytochrome P450 inhibition kinetic models.

Correctly chosen d-optimal designs provide efficient experimental schemes when the aim of the investigation is to obtain precise estimates of parameters. In the current work, estimates of parameters refer to the enzyme kinetic parameters V(max) and K(m), but they also refer to the inhibition constant K(i). In general, this experimental approach is performed on a grid of values of the design variables. However, this approach may not be very efficient, in the sense that the parameter estimates (V(max), K(m), and K(i)) have unnecessarily high variances. For good estimates of parameters, the most efficient designs consist of clusters of replicates of a few sets of experimental conditions. The current study compares the application of such d-optimal designs with that of a conventional approach in assessing the competitive inhibitory potency of fluconazole and sertraline toward CYP2C9 and 2D6, respectively. In each instance, the parameter estimates, namely V(max), K(m), and K(i), were predicted well using the d-optimal design compared with those measured using the rich data sets, for both inhibitors. We show that d optimality can provide more efficient designs for estimating the model parameters, including K(i). We also show that real cost savings can be made by carefully planning studies that use the theory of optimal experimental design.

Optimal experimental design in chromatography.

Many experimental programs in chromatography involve the estimation of a second-order response surface. The paper focuses on the use of the methods of optimal experimental design to find efficient response surface designs. We give tables of the properties of designs with two, three and four factors. References are given to tables of designs and to SAS programs for their construction.

Bayesian adaptive biased-coin designs for clinical trials with normal responses.

Adaptive designs are used in phase III clinical trials for skewing the allocation pattern toward the better treatments. We use optimum design theory to derive a skewed Bayesian biased-coin procedure for sequential designs with continuous responses. The skewed designs are used to provide adaptive designs, the performance of which is studied numerically and theoretically. Important properties are loss and the proportion of allocation to the better treatment.

The distribution of loss in two-treatment biased-coin designs.

The paper compares randomized rules of the biased-coin type for the sequential allocation of treatments in a clinical trial. An important characteristic is the loss, which measures the increase in the variance of parameter estimates due to the imbalance caused by randomization. Simulations are used to find the small-sample distribution of loss. For some rules a simple chi-squared approximation to the asymptotic distribution holds well down to very small sample sizes.