Median Tests for Censored Survival Data; a Contingency Table Approach

Summary The median failure time is often utilized to summarize survival data because it has a more straightforward interpretation for investigators in practice than the popular hazard function. However, existing methods for comparing median failure times for censored survival data either require estimation of the probability density function or involve complicated formulas to calculate the variance of the estimates. In this article, we modify a K ‐sample median test for censored survival data ( Brookmeyer and Crowley, 1982 , Journal of the American Statistical Association 77, 433–440) through a simple contingency table approach where each cell counts the number of observations in each sample that are greater than the pooled median or vice versa. Under censoring, this approach would generate noninteger entries for the cells in the contingency table. We propose to construct a weighted asymptotic test statistic that aggregates dependent χ² ‐statistics formed at the nearest integer points to the original noninteger entries. We show that this statistic follows approximately a χ² ‐distribution with k? 1 degrees of freedom. For a small sample case, we propose a test statistic based on combined p ‐values from Fisher’s exact tests, which follows a χ² ‐distribution with 2 degrees of freedom. Simulation studies are performed to show that the proposed method provides reasonable type I error probabilities and powers. The proposed method is illustrated with two real datasets from phase III breast cancer clinical trials.     

Sample Size Determination for Designing a Strata-Matched Case-Control Study to Detect Multiple Risk Factors

Investigations of sample size for planning case-control studies have usually been limited to detecting a single factor. In this paper, we investigate sample size for multiple risk factors in strata-matched case-control studies. We construct an omnibus statistic for testing M different risk factors based on the jointly sufficient statistics of parameters associated with the risk factors. The statistic is non-iterative, and it reduces to the Cochran statistic when M = 1. The asymptotic power function of the test is a non-central chi-square with M degrees of freedom and the sample size required for a specific power can be obtained by the inverse relationship. We find that the equal sample allocation is optimum. A Monte Carlo experiment demonstrates that an approximate formula for calculating sample size is satisfactory in typical epidemiologic studies. An approximate sample size obtained using Bonferroni's method for multiple comparisons is much larger than that obtained using the omnibus test. Approximate sample size formulas investigated in this paper using the omnibus test, as well as the individual tests, can be useful in designing case-control studies for detecting multiple risk factors.     

Regression on Quantile Residual Life

Summary A time‐specific log‐linear regression method on quantile residual lifetime is proposed. Under the proposed regression model, any quantile of a time‐to‐event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring. Consistency and asymptotic normality of the regression estimator are established. An asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point. Evaluation of the test statistic does not require estimation of the variance–covariance matrix of the regression estimators, which involves the probability density function of the survival distribution with censoring. Simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic. The new regression method is applied to a breast cancer data set with long‐term follow‐up to estimate the patients' median residual lifetimes, adjusting for important prognostic factors.     

Fast approximations of pseudo-observations in the context of right censoring and interval censoring

In the context of right-censored and interval-censored data, we develop asymptotic formulas to compute pseudo-observations for the survival function and the restricted mean survival time (RMST). These formulas are based on the original estimators and do not involve computation of the jackknife estimators. For right-censored data, Von Mises expansions of the Kaplan–Meier estimator are used to derive the pseudo-observations. For interval-censored data, a general class of parametric models for the survival function is studied. An asymptotic representation of the pseudo-observations is derived involving the Hessian matrix and the score vector. Theoretical results that justify the use of pseudo-observations in regression are also derived. The formula is illustrated on the piecewise-constant-hazard model for the RMST. The proposed approximations are extremely accurate, even for small sample sizes, as illustrated by Monte Carlo simulations and real data. We also study the gain in terms of computation time, as compared to the original jackknife method, which can be substantial for a large dataset.     

Nonparametric All‐Pairs Multiple Comparisons

Nonparametric all‐pairs multiple comparisons based on pairwise rankings can be performed in the one‐way design with the Steel‐Dwass procedure. To apply this test, Wilcoxon's rank sum statistic is calculated for all pairs of groups; the maximum of the rank sums is the test statistic. We provide exact calculations of the asymptotic critical values (and P‐values, respectively) even for unbalanced designs. We recommend this asymptotic method whenever large sample sizes are present. For small sample sizes we recommend the use of the new statistic according to Baumgartner , Weiss , and Schindler (1998, Biometrics 54 , 1129–1135) instead of Wilcoxon's rank sum for the multiple comparisons. We show that the resultant procedure can be less conservative and, according to simulation results, more powerful than the original Steel‐Dwass procedure. We illustrate the methods with a practical data set.     

A Note on Two‐Sample Tests for Comparing Intra‐Individual Genetic Sequence Diversity between Populations

Summary Gilbert, Rossini, and Shankarappa (2005 , Biometrics 61 , 106‐117) present four U‐statistic based tests to compare genetic diversity between different samples. The proposed tests improved upon previously used methods by accounting for the correlations in the data. We find, however, that the same correlations introduce an unacceptable bias in the sample estimators used for the variance and covariance of the inter‐sequence genetic distances for modest sample sizes. Here, we compute unbiased estimators for these and test the resulting improvement using simulated data. We also show that, contrary to the claims in Gilbert et al., it is not always possible to apply the Welch–Satterthwaite approximate t‐test, and we provide explicit formulas for the degrees of freedom to be used when, on the other hand, such approximation is indeed possible.     

Quantile regression models with multivariate failure time data

As an alternative to the mean regression model, the quantile regression model has been studied extensively with independent failure time data. However, due to natural or artificial clustering, it is common to encounter multivariate failure time data in biomedical research where the intracluster correlation needs to be accounted for appropriately. For right-censored correlated survival data, we investigate the quantile regression model and adapt an estimating equation approach for parameter estimation under the working independence assumption, as well as a weighted version for enhancing the efficiency. We show that the parameter estimates are consistent and asymptotically follow normal distributions. The variance estimation using asymptotic approximation involves nonparametric functional density estimation. We employ the bootstrap and perturbation resampling methods for the estimation of the variance-covariance matrix. We examine the proposed method for finite sample sizes through simulation studies, and illustrate it with data from a clinical trial on otitis media.     

Adaptive design: estimation and inference with censored data in a semiparametric model

In this article, we provide a method of estimation for the treatment effect in the adaptive design for censored survival data with or without adjusting for risk factors other than the treatment indicator. Within the semiparametric Cox proportional hazards model, we propose a bias-adjusted parameter estimator for the treatment coefficient and its asymptotic confidence interval at the end of the trial. The method for obtaining an asymptotic confidence interval and point estimator is based on a general distribution property of the final test statistic from the weighted linear rank statistics at the interims with or without considering the nuisance covariates. The computation of the estimates is straightforward. Extensive simulation studies show that the asymptotic confidence intervals have reasonable nominal probability of coverage, and the proposed point estimators are nearly unbiased with practical sample sizes.     

On Exact Unconditional Test for Linear Trend in Dose‐Response Studies

The one‐degree‐of‐freedom Cochran‐Armitage (C‐A) test statistic for linear trend has been widely applied in various dose‐response studies (e.g., anti‐ulcer medications and short‐term antibiotics, animal carcinogenicity bioassays and occupational toxicant studies). This approximate statistic relies, however, on asymptotic theory that is reliable only when the sample sizes are reasonably large and well balanced across dose levels. For small, sparse, or skewed data, the asymptotic theory is suspect and exact conditional method (based on the C‐A statistic) seems to provide a dependable alternative. Unfortunately, the exact conditional method is only practical for the linear logistic model from which the sufficient statistics for the regression coefficients can be obtained explicitly. In this article, a simple and efficient recursive polynomial multiplication algorithm for exact unconditional test (based on the C‐A statistic) for detecting a linear trend in proportions is derived. The method is applicable for all choices of the model with monotone trend including logistic, probit, arcsine, extreme value and one hit. We also show that this algorithm can be easily extended to exact unconditional power calculation for studies with up to a moderately large sample size. A real example is given to illustrate the applicability of the proposed method.     

Asymptotic distributions of kappa statistics and their differences with many raters,many rating categories and two conditions

In clinical research and in more general classification problems, a frequent concern is the reliability of a rating system. In the absence of a gold standard, agreement may be considered as an indication of reliability. When dealing with categorical data, the well‐known kappa statistic is often used to measure agreement. The aim of this paper is to obtain a theoretical result about the asymptotic distribution of the kappa statistic with multiple items, multiple raters, multiple conditions, and multiple rating categories (more than two), based on recent work. The result settles a long lasting quest for the asymptotic variance of the kappa statistic in this situation and allows for the construction of asymptotic confidence intervals. A recent application to clinical endoscopy and to the diagnosis of inflammatory bowel diseases (IBDs) is shortly presented to complement the theoretical perspective.     

Trend tests for case-control studies of genetic markers: power,sample size and robustness

The Cochran-Armitage trend test is commonly used as a genotype-based test for candidate gene association. Corresponding to each underlying genetic model there is a particular set of scores assigned to the genotypes that maximizes its power. When the variance of the test statistic is known, the formulas for approximate power and associated sample size are readily obtained. In practice, however, the variance of the test statistic needs to be estimated. We present formulas for the required sample size to achieve a prespecified power that account for the need to estimate the variance of the test statistic. When the underlying genetic model is unknown one can incur a substantial loss of power when a test suitable for one mode of inheritance is used where another mode is the true one. Thus, tests having good power properties relative to the optimal tests for each model are useful. These tests are called efficiency robust and we study two of them: the maximin efficiency robust test is a linear combination of the standardized optimal tests that has high efficiency and the MAX test, the maximum of the standardized optimal tests. Simulation results of the robustness of these two tests indicate that the more computationally involved MAX test is preferable.     

A Two Sample CRAMÉR-VON MISES Test for Randomly Censored Data

A CRAMÉR-VON MISES type statistic is introduced for testing the equality of the underlying survival distributions of two populations when observations are subject to arbitrary right censorship. The statistic is appropriate in testing problems where a two-sided alternative is of interest. The asymptotic distribution of the statistic is found; under certain circumstances, the limiting distribution coincides with that of a one sample CRAMÉR-VON MISES type statistic for randomly censored data investigated previously. Approximations to the asymptotic distribution are discussed; an example is given.     

A non-iterative confidence interval estimating procedure for the intraclass kappa statistic with multinomial outcomes

We obtain the asymptotic sample variance of the intraclass kappa statistic for multinomial outcome data. A modified Wald type procedure based on this theory is then used for confidence interval construction. The results of a simulation study show that the proposed non-iterative approach performs very well in terms of confidence interval coverage and width for samples as small as 50. The procedure is illustrated with two examples from previously published medical studies.     

Asymptotic variances of QTL estimators with selective DNA pooling

Investigation on QTL-marker linkage usually requires a great number of observed recombinations, inferred from combined analysis of phenotypes and genotypes. To avoid costly individual genotyping, inferences on QTL position and effects can instead make use of marker allele frequencies. DNA pooling of selected samples makes allele frequency estimation feasible for studies involving large sample sizes. Linkage studies in outbred populations have traditionally exploited half-sib family designs; within the animal production context, half-sibships provide large families that are highly suitable for DNA pooling. Estimators for QTL position and effect have been proposed that make use of information from flanking markers. We present formulas derived by the delta method for the asymptotic variance of these estimators.     

Dose Response Analysis Using Robust Covariance Estimation

A dose response analysis is robustified by estimating the asymptotic covariance of the fitted model parameters by the approximate information sandwich (a sandwich statistic) under a heterogeneous variance. The robust method is described by using a nonlinear four‐parameter regression model. The usual, robust, bootstrap, and jackknife estimates of the asymptotic variance are examined for the bioassay data. Under the response of a normal distribution with changing variances over the dose levels, the performance of the usual and robust variances is investigated by Monte Carlo study. It confirms the robustness of the sandwich estimate and shows the non‐accuracy of the usual asymptotic variance estimates of fitted model parameters under the different forms of nonconstant variance structures.     

Mapping quantitative trait loci using multiple phenotypes in general pedigrees

The use of correlated phenotypes may dramatically increase the power to detect the underlying quantitative trait loci (QTLs). Current approaches for multiple phenotypes include regression-based methods, the multivariate variance of components method, factor analysis and structural equations. Issues with these methods include: 1) They are computation intensive and are subject to problems of optimization algorithms; 2) Existing claims on the asymptotic distribution of the likelihood ratio statistic for the multivariate variance of components method are contradictory and erroneous; 3) The dimension reduction of the parameter space under the null hypothesis, a phenomenon that is unique to multivariate analyses, makes the asymptotic distribution of the likelihood ratio statistic more complicated than expected. In this article, three cases of varying complexity are considered. For each case, the efficient score statistic, which is asympotically equivalent to the likelihood ratio statistic, is derived, so is its asymptotic distribution [correction]. These methods are straightforward to calculate. Finite-sample properties of these score statistics are studied through extensive simulations. These score statistics are for use with general pedigrees.     

Statistical inference for self-designing clinical trials with a one-sided hypothesis

In the process of monitoring clinical trials, it seems appealing to use the interim findings to determine whether the sample size originally planned will provide adequate power when the alternative hypothesis is true, and to adjust the sample size if necessary. In the present paper, we propose a flexible sequential monitoring method following the work of Fisher (1998), in which the maximum sample size does not have to be specified in advance. The final test statistic is constructed based on a weighted average of the sequentially collected data, where the weight function at each stage is determined by the observed data prior to that stage. Such a weight function is used to maintain the integrity of the variance of the final test statistic so that the overall type I error rate is preserved. Moreover, the weight function plays an implicit role in termination of a trial when a treatment difference exists. Finally, the design allows the trial to be stopped early when the efficacy result is sufficiently negative. Simulation studies confirm the performance of the method.     

Sample sizes based on the log-rank statistic in complex clinical trials

The log-rank test is frequently used to compare survival curves. While sample size estimation for comparison of binomial proportions has been adapted to typical clinical trial conditions such as noncompliance, lag time, and staggered entry, the estimation of sample size when the log-rank statistic is to be used has not been generalized to these types of clinical trial conditions. This paper presents a method of estimating sample sizes for the comparison of survival curves by the log-rank statistic in the presence of unrestricted rates of noncompliance, lag time, and so forth. The method applies to stratified trials in which the above conditions may vary across the different strata, and does not assume proportional hazards. Power and duration, as well as sample sizes, can be estimated. The method also produces estimates for binomial proportions and the Tarone-Ware class of statistics.     

Steady state of homozygosity and Gst for the island model

We examine homozygosity and G(st) for a subdivided population governed by the finite island model. Assuming an infinite allele model and strong mutation we show that the steady state distributions of G(st) and homozygosity have asymptotic expansions in the mutation rate. We use this observation to derive asymptotic expansions for various moments of homozygosity and to derive rigorous formulas for the mean and variance of G(st). We show that G(st) approximately 1/(1+2Nm), similarly to the well known formula of Wright for the infinite island model, and that the variance of G(st) goes to zero as mutation increases.     

Confidence interval for rate ratio in a 2 x 2 table with structural zero: an application in assessing false-negative rate ratio when combining two diagnostic tests

In this article, we consider problems with correlated data that can be summarized in a 2 x 2 table with structural zero in one of the off-diagonal cells. Data of this kind sometimes appear in infectious disease studies and two-step procedure studies. Lui (1998, Biometrics54, 706-711) considered confidence interval estimation of rate ratio based on Fieller-type, Wald-type, and logarithmic transformation statistics. We reexamine the same problem under the context of confidence interval construction on false-negative rate ratio in diagnostic performance when combining two diagnostic tests. We propose a score statistic for testing the null hypothesis of nonunity false-negative rate ratio. Score test-based confidence interval construction for false-negative rate ratio will also be discussed. Simulation studies are conducted to compare the performance of the new derived score test statistic and existing statistics for small to moderate sample sizes. In terms of confidence interval construction, our asymptotic score test-based confidence interval estimator possesses significantly shorter expected width with coverage probability being close to the anticipated confidence level. In terms of hypothesis testing, our asymptotic score test procedure has actual type I error rate close to the pre-assigned nominal level. We illustrate our methodologies with real examples from a clinical laboratory study and a cancer study.