Max Halperin

More flexible sequential and non-sequential designs in long-term clinical trial

In this paper, we describe decision making procedures as they exist in most clinical trials,review some recently suggested approaches to monitoring and clarify how these methods allow greater flexibility in monitoring and explicit specification of data monitoring methods in the protocol.

Predictability of coronary heart disease

Abstract In the past 30 yr there has been an impressive increase in our ability to predict coronary heart disease (CHD). Some of the developments in epidemiology and statistics from which this derives are discussed. While it is not possible to specify a useful upper bound to the predictability of coronary heart disease, the paper provides guides to determining the effectiveness of various CHD risk functions, with illustrative examples from the Framingham Study, and discusses interpretative problems.

Distribution-free Confidence Intervals for Pr(X1 < X2)

SUMMARY We review the problem of obtaining a distribution-free confidence interval for the probability (p) that one random variable is less than another independent random variable based on the two-sample uncensored Wilcoxon-Mann-Whitney statistic. The pivotal quantities used by Sen (1967, Sankhya, Series A 29, 95-102) and Govindarajulu (1968, Annals of the Institute of Statistical Mathematics 20, 229-238) are modified to take into account that the variance of estimated p depends on p. A number of simulations were conducted to compare the various methods. The results suggest that the proposed modification is superior in the sense that over the entire range of p the modification generally yields values of coverage probability much closer to nominal coverage than the methods of Sen (S) and Govindarajulu (G). The coverage values for the S and G methods are very close to each other; both share the characteristics for one-sided lower confidence limits of being conservative for small values of p and nonconservative (sometimes extremely so) for large values of p. One-sided upper limits have coverages that are essentially the mirror image of lower-limit coverages. The contrasts between the modified method and the S and G methods diminish in magnitude as sample size increases but are still nontrivial for a sample size of 80. Our tentative conclusion is that the modified procedure is preferable to that of Sen and Govindarajulu and can be used when the sample size in both groups is equal to or greater than 20. Limitations of our simulations are briefly discussed. An example illustrating the use of our modified pivotal quantity and comparing it with the S and G results is presented.

Stochastic curtailing for comparison of slopes in longitudinal studies

In some clinical trials, rate of change of a physiological function is used as a surrogate for a more serious outcome. We assume an expected change linear in time for each study participant with variation in slopes and intercepts from individual to individual and repeated measures over time for each individual. We also assume that deviations of response for an individual from expected response have zero mean, constant variance, and are uncorrelated. Under these assumptions we describe ways in which stochastic curtailing as defined by Lan, Simon, and Halperin (Commun Stat Seq Anal 1:207-219, 1982) can be implemented in a two-treatment trial for one-sided comparison of slopes in the two groups. Staggered entry is taken into account, as is the possibility that some of an individuals responses are not available; this is assumed to be random. The analysis assumes the number of participants in each group is large and that most individuals have at least two measurements (including baseline value). The possibility that rate of change is not constant and its consequences are discussed.

Early stopping in the two-sample problem for bounded random variables

In a fixed sample size clinical trial, the question often asked is whether the final outcome has been determined before the data has been completely collected. A particular situation occurs when the intake of subjects is slow relative to the time required for the outcome variable to be realized. We assume that the range of values the outcome variable may take can be specified in advance. We also assume that simple randomization has been used and thus, under the null hypothesis, Students t test is a good approximation to the exact test involving the permutational distribution. In order to determine with partial data whether the outcome of the final test of hypothesis is already certain, certain nonlinear extremal problems with constraints must be solved. Analytic solutions may be expressed in the form of noniterative algorithms. Simulation studies suggest an approximation to the analytic solution. The results also provide a basis for assessing at the end of the trial, whether missing values are of consequence in the test of significance. An application based on an actual clinical trial is presented.

Distribution-free confidence intervals for a parameter of Wilcoxon-Mann-Whitney type for ordered categories and progressive censoring

Halperin, Gilbert, and Lachin (1987, Biometrics 43, 71-80) obtain confidence intervals for Pr(X less than Y) based on the two-sample Wilcoxon statistic for continuous data. Their approach is applied here to ordered categorical data and right-censored continuous data, using the generalization zeta = Pr(X less than Y) + 1/2Pr(X = Y) to account for ties. Deviations from nominal coverage probability for various sample sizes and values of zeta are obtained via simulation of either three or six ordered categories based on underlying Poisson or exponential distributions. The simulation results indicate that the proposed method performs quite well, and it is apparently superior to the approach of Hochberg (1981, Communications in Statistics--Theory and Methods A10, 1719-1732) for values of zeta far from 1/2.

ANALYSES OF BIRTH-RANK DATA1

The birth-rank problem is one in which it is required to determine, retrospectively, if the incidence of a condition, say a disease condition, in individuals is affected by the birth order of the individual. For each individual one knows both his birth rank and the total number of siblings in his family. To avoid biases in this kind of study it may be important that data be restricted to completed families in which all children have passed through the age period during which the disease occurs. The study is retrospective in that one considers the birth ranks of affected (or unaffected) children rather than the incidence rates of disease at varying ranks of birth. A number of papers reporting the results of birth-rank investigations or providing methods of statistical analysis for such investigations have appeared (references in text: also Penrose [1934 a, b] and Malzberg [1938]). The publication by Keeping [1952], in particular, describes a number of these methods. In the present report some of these methods are reviewed, simplified, and extended. The problem of maternal age effects, covered in some of the reports, will not be considered here. For purposes of simplicity, in what follows we will consider that the unit of sampling is the diseased individual, not the family containing a diseased individual. In some of the existing publications data reported are in the form of the birth ranks of the diseased and of the non-diseased members in each family; the analysis made, however, is frequently the same as would apply in the case we wish to consider here3. Problems

Conditional Distribution-Free Tests for the Two-Sample Problem in the Presence of Right Censoring

Abstract Two-sample rank tests for survival data in the presence of arbitrary right censoring are considered. We distinguish between administrative censoring, arising because survival study participants do not enter as a cohort, and censoring due to “loss to follow-up.” We show how conditionally distribution-free tests can be constructed in certain situations. Conditional versions of the generalized Wilcoxon and Mantel statistics are shown to be asymptotically normal in the conditional reference set, but with modified means and variances. Efficiency of these tests relative to asymptotically distribution-free competitors is unity, providing the censoring distributions are discrete, the same for both samples, and providing loss-to-follow-up (LFU) distributions are the same for the two samples. When these assumptions do not hold, efficiency can deteriorate considerably, being poorest, other things being equal, when the censoring distribution is continuous.

Grouping and linear regression

With a large number of observations, the method of grouping is often employed to provide simpler graphs or tables. When one investigates the relationship between two variables, one usually groups based on the magnitude of the independent variable, and then plots the dependent variable averages against independent variable averages to get a clearer graph. If grouping is based on the magnitude of the dependent variable, the plot of group means as indicated above does not appropriately describe the relationship of the dependent variable to the independent variable. These results are demonstrated theoretically for the special case of bivariate normality (and thus linear regression), but would be expected to be similar for other distribution assumptions. An example is given from an epidemiological study.

Design and Sensitivity Evaluation of Follow-Up Studies for Risk Factor Assessment

In follow-up studies of samples from human populations, a major goal is frequently the assessment of association of risk factors with some disease outcome. For a completed study it seems of interest to quantify the sensitivity of the study, that is, to prescribe the intensity of association which the study would be able to detect with high probability. On the other hand, it is of interest to determine the size of the cohort and the length of follow-up needed to give a high probability of detecting an association of a given intensity. A basis for answering these questions is provided under the assumptions of underlying joint normality of distribution of risk factors, a Hotellings T2 test of equality of risk factors among cases and noncases, and a probit plane in the risk factors as a risk-function model.