Valerii V. Fedorov

Consequences of dichotomization

Dichotomization is the transformation of a continuous outcome (response) to a binary outcome. This approach, while somewhat common, is harmful from the viewpoint of statistical estimation and hypothesis testing. We show that this leads to loss of information, which can be large. For normally distributed data, this loss in terms of Fishers information is at least 1-2/pi (or 36%). In other words, 100 continuous observations are statistically equivalent to 158 dichotomized observations. The amount of information lost depends greatly on the prior choice of cut points, with the optimal cut point depending upon the unknown parameters. The loss of information leads to loss of power or conversely a sample size increase to maintain power. Only in certain cases, for instance, in estimating a value of the cumulative distribution function and when the assumed model is very different from the true model, can the use of dichotomized outcomes be considered a reasonable approach.

The design of multicentre trials

The analysis of data collected in multicentre trials offers challenges because the data from the individual centres must be combined in some way to give an overall evaluation of the differences between the treatments in the trial. We propose that the combined response to treatment (CRT) be used as this overall measure. The definition and estimation of the CRT can be derived from either a fixed-effects or a random-effects model. For the latter we introduce the ECRT - the expected combined response to treatment. We describe and compare both types of model and express our preference for the random-effects model. We stress that the number of patients enrolled at a centre is a random variable and show that this source of randomness inflates the variance of the estimated ECRT. Variability in enrolment rates over the centres further inflates this variance. A simple conclusion from our results is that if variability in the treatment and centre effects, in the enrolment time, in the number of patients enrolled at a centre and in the enrolment rates is not properly accounted for, then an underpowered trial may result. Using properties of estimators generated by the random-effects model we propose methods for determining the optimal number of centres and total number of patients to enrol in a trial to minimize a loss function that accounts for centre and patient costs and loss of revenue. We discuss variants of the loss function and corresponding optimization problems for different types of enrolment. We end the paper with brief generalizations of the developed techniques to the case where the response is binary.

Optimal Design of Dose Response Experiments: A Model-Oriented Approach*

We discuss optimal experimental design issues for nonlinear models arising in dose response studies. The optimization is performed with respect to various criteria which depend on the Fisher information matrix. Special attention is given to models with a variance component that depends on unknown parameters.

Optimal dose-finding designs with correlated continuous and discrete responses

In dose-finding clinical studies, it is common that multiple endpoints are of interest. For instance, in phase I/II studies, efficacy and toxicity are often the primary endpoints, which are observed simultaneously and which need to be evaluated together. Motivated by this, we confine ourselves to bivariate responses and focus on the most analytically difficult case: a mixture of continuous and categorical responses. We adopt the bivariate probit dose-response model and quantify our goal by a utility function. We study locally optimal designs, two-stage optimal designs, and fully adaptive designs under different ethical and cost constraints in the experiments. We assess the performance of two-stage designs and fully adaptive designs via simulations. Our simulations suggest that the two-stage designs are as efficient as and may be more efficient than the fully adaptive designs if there is a moderate sample size in the initial stage. In addition, two-stage designs are easier to construct and implement and thus can be a useful approach in practice.

Dose Finding Designs for Continuous Responses and Binary Utility

Often in clinical trials the observed responses are continuous but a regulatory agency will approve the drug only if the probability is sufficiently large that the efficacy measure exceeds a predefined threshold and the toxicity does not exceed another given threshold. Thus the measure of interest (utility) is based on dichotomized responses. We consider normally distributed correlated responses and build a utility function using the probit transform. Locally optimal designs are used as benchmarks for more practical designs such as composite and adaptive designs. We focus on D -criterion (i.e., all parameters of the dose-response model are of interest) and consider only two-stage designs. It is shown that the practice of reporting dichotomized responses leads to a substantial loss in the precision of estimated parameters (or in the power loss).

Comparisons of minimization and Atkinson's algorithm

Some general points regarding efficiency in clinical trials are made. Reasons as to why fitting many covariates to adjust the estimate of the treatment effect may be less problematic than commonly supposed are given. Two methods of dynamic allocation of patients based on covariates, minimization and Atkinsons approach, are compared and contrasted for the particular case where all covariates are binary. The results of Monte Carlo simulations are also presented. It is concluded that in the cases considered, Atkinsons approach is slightly more efficient than minimization although the difference is unlikely to be very important in practice. Both are more efficient than simple randomization, although it is concluded that fitting covariates may make a more valuable and instructive contribution to inferences about treatment effects than only balancing them.

Design of experiments for locally weighted regression

Abstract We consider the design of experiments when estimation is to be performed using locally weighted regression methods. We adopt criteria that consider both estimation error (variance) and error resulting from model misspecification (bias). Working with continuous designs, we use the ideas developed in convex design theory to analyze properties of the corresponding optimal designs. Numerical procedures for constructing optimal designs are developed and applied to a variety of design scenarios in one and two dimensions. Among the interesting properties of the constructed designs are the following: (1) Design points tend to be more spread throughout the design space than in the classical case. (2) The optimal designs appear to be less model and criterion dependent than their classical counterparts.(3) While the optimal designs are relatively insensitive to the specification of the design space boundaries, the allocation of supporting points is strongly governed by the points of interest and the selected weight function, if the latter is concentrated in areas significantly smaller than the design region. Some singular and unstable situations occur in the case of saturated designs. The corresponding phenomenon is discussed using a univariate linear regression example.

Design of Multicentre Clinical Trials with Random Enrolment

This chapter is devoted to the investigation of multicentre clinical trials with random enrolment, where the patients enter the centres at random according to doubly stochastic Poisson processes. We consider two-arm trials and use a random-effects model to describe treatment responses. The time needed to complete the trial (recruitment time) and the variance of the estimator of the Expected Combined Response to Treatment (ECRT) are investigated for different enrolment scenarios, and closed-form expressions and asymptotic formulae are derived. Possible delays in initiating centres and dropouts of patients are also taken into account. The developed results lead to rather simple approximate formulae which can be used to design a trial.

Optimum Design for Correlated Fields via Covariance Kernel Expansions

In this paper we consider optimal design of experiments for correlated observations. We approximate the error component of the process by an eigenvector expansion of the corresponding covariance function. Furthermore we study the limiting behavior of an additional white noise as a regularization tool. The approach is illustrated by some typical examples.

Parameter Estimation for Models with Unknown Parameters in Variance

Abstract We discuss estimation methods for multiresponse models with a variance matrix that depends on unknown parameters. An iterated estimator is proposed that is asymptotically equivalent to a maximum likelihood estimator. Numerically, this estimator is close to the iteratively reweighted least squares method. However, in the situations when the information contained in the variance component is significant, the new iterated estimator outperforms the traditional iteratively reweighted estimator. The performance of the various numerical procedures is illustrated in the simulation study.