Optimal conditional error functions for the control of conditional power

Ethical considerations and the competitive environment of clinical trials usually require that any given trial have sufficient power to detect a treatment advance. If at an interim analysis the available data are used to decide whether the trial is promising enough to be continued, investigators and sponsors often wish to have a high conditional power, which is the probability to reject the null hypothesis given the interim data and the alternative of interest. Under this requirement a design with interim sample size recalculation, which keeps the overall and conditional power at a prespecified value and preserves the overall type I error rate, is a reasonable alternative to a classical group sequential design, in which the conditional power is often too small. In this article two-stage designs with control of overall and conditional power are constructed that minimize the expected sample size, either for a simple point alternative or for a random mixture of alternatives given by a prior density for the efficacy parameter. The presented optimality result applies to trials with and without an interim hypothesis test; in addition, one can account for constraints such as a minimal sample size for the second stage. The optimal designs will be illustrated with an example, and will be compared to the frequently considered method of using the conditional type I error level of a group sequential design.     

A Propensity Score-Enhanced Sequential Analytic Method for Comparative Drug Safety Surveillance

We introduce a new sequential monitoring approach to facilitate the use of observational electronic healthcare utilization databases in comparative drug safety surveillance studies comparing the safety between two approved medical products. The new approach enhances the confounder adjustment capabilities of the conditional sequential sampling procedure (CSSP), an existing group sequential method for sequentially monitoring excess risks of adverse events following the introduction of a new medical product. It applies to a prospective cohort setting where information for both treatment and comparison groups accumulates concurrently over time. CSSP adjusts for covariates through stratification and thus it may have limited capacity to control for confounding as it can only accommodate a few categorical covariates. To address this issue, we propose the propensity score (PS)-stratified CSSP, in which we construct strata based on selected percentiles of the estimated PSs. The PS is defined as the conditional probability of being treated given measured baseline covariates and is commonly used in epidemiological studies to adjust for confounding bias. The PS-stratified CSSP approach integrates this more flexible confounding adjustment, PS-stratification, with the sequential analytic approach, CSSP, thus inheriting CSSP’s attractive features: (i) it accommodates varying amounts of person follow-up time, (ii) it uses exact conditional inference, which can be important when studying rare safety outcomes, and (iii) it allows for a large number of interim tests. Further, it overcomes CSSP’s difficulty with adjusting for multiple categorical and continuous confounders.     

Estimation of a Secondary Parameter in a Group Sequential Clinical Trial

In this article, we investigate estimation of a secondary parameter in group sequential tests. We study the model in which the secondary parameter is the mean of the normal distribution in a subgroup of the subjects. The bias of the naive secondary parameter estimator is studied. It is shown that the sampling proportions of the subgroup have a crucial effect on the bias: As the sampling proportion of the subgroup at or just before the stopping time increases, the bias of the naive subgroup parameter estimator increases as well. An unbiased estimator for the subgroup parameter and an unbiased estimator for its variance are derived. Using simulations, we compare the mean squared error of the unbiased estimator to that of the naive estimator, and we show that the differences are negligible. As an example, the methods of estimation are applied to an actual group sequential clinical trial, The Beta-Blocker Heart Attack Trial.     

Conditional properties of unconditional parametric bootstrap procedures for inference in exponential families

Higher-order inference about a scalar parameter in the presenceof nuisance parameters can be achieved by bootstrapping, incircumstances where the parameter of interest is a componentof the canonical parameter in a full exponential family. Theoptimal test, which is approximated, is a conditional one basedon conditioning on the sufficient statistic for the nuisanceparameter. A bootstrap procedure that ignores the conditioningis shown to have desirable conditional properties in providingthird-order relative accuracy in approximation of p-values associatedwith the optimal test, in both continuous and discrete models.The bootstrap approach is equivalent to third-order analyticalapproaches, and is demonstrated in a number of examples to givevery accurate approximations even for very small sample sizes.     

Sample Size Reassessment in Adaptive Clinical Trials Using a Bias Corrected Estimate

Point estimation in group sequential and adaptive trials is an important issue in analysing a clinical trial. Most literature in this area is only concerned with estimation after completion of a trial. Since adaptive designs allow reassessment of sample size during the trial, reliable point estimation of the true effect when continuing the trial is additionally needed. We present a bias adjusted estimator which allows a more exact sample size determination based on the conditional power principle than the naive sample mean does.     

Small-sample estimation of negative binomial dispersion, with applications to SAGE data

We derive a quantile-adjusted conditional maximum likelihood estimator for the dispersion parameter of the negative binomial distribution and compare its performance, in terms of bias, to various other methods. Our estimation scheme outperforms all other methods in very small samples, typical of those from serial analysis of gene expression studies, the motivating data for this study. The impact of dispersion estimation on hypothesis testing is studied. We derive an "exact" test that outperforms the standard approximate asymptotic tests.     

Some properties of an estimator for the basic reproduction number of the general epidemic model.

In this paper some properties of a convenient estimator, derived from a martingale estimating function, for the basic reproduction number of the general epidemic model are given for both finite and large samples. These properties give some guidelines for using this convenient estimator. It is shown that it underestimates the parameter and that the bias tends to zero when the population size and the initial number of infectives are increased simultaneously. The bias cannot be removed for a fixed number of introductory infectives. However, the estimator is asymptotically unbiased, conditional on a major outbreak. A simulation study shows that the central limit theorem applies for moderate population sizes.     

Insights into latent class analysis of diagnostic test performance

Latent class analysis is used to assess diagnostic test accuracy when a gold standard assessment of disease is not available but results of multiple imperfect tests are. We consider the simplest setting, where 3 tests are observed and conditional independence (CI) is assumed. Closed-form expressions for maximum likelihood parameter estimates are derived. They show explicitly how observed 2- and 3-way associations between test results are used to infer disease prevalence and test true- and false-positive rates. Although interesting and reasonable under CI, the estimators clearly have no basis when it fails. Intuition for bias induced by conditional dependence follows from the analytic expressions. Further intuition derives from an Expectation Maximization (EM) approach to calculating the estimates. We discuss implications of our results and related work for settings where more than 3 tests are available. We conclude that careful justification of assumptions about the dependence between tests in diseased and nondiseased subjects is necessary in order to ensure unbiased estimates of prevalence and test operating characteristics and to provide these estimates clinical interpretations. Such justification must be based in part on a clear clinical definition of disease and biological knowledge about mechanisms giving rise to test results.     

Exact confidence bounds following adaptive group sequential tests

Summary .  We provide a method for obtaining confidence intervals, point estimates, and p-values for the primary effect size parameter at the end of a two-arm group sequential clinical trial in which adaptive changes have been implemented along the way. The method is based on applying the adaptive hypothesis testing procedure of Müller and Schäfer (2001, Biometrics 57, 886–891) to a sequence of dual tests derived from the stage-wise adjusted confidence interval of Tsiatis, Rosner, and Mehta (1984, Biometrics 40, 797–803). In the nonadaptive setting this confidence interval is known to provide exact coverage. In the adaptive setting exact coverage is guaranteed provided the adaptation takes place at the penultimate stage. In general, however, all that can be claimed theoretically is that the coverage is guaranteed to be conservative. Nevertheless, extensive simulation experiments, supported by an empirical characterization of the conditional error function, demonstrate convincingly that for all practical purposes the coverage is exact and the point estimate is median unbiased. No procedure has previously been available for producing confidence intervals and point estimates with these desirable properties in an adaptive group sequential setting. The methodology is illustrated by an application to a clinical trial of deep brain stimulation for Parkinson's disease.     

Comparison of properties of tests for assessing tumor clonality

SUMMARY: In a recent article Begg et al. (2007, Biometrics 63, 522-530) proposed a statistical test to determine whether or not a diagnosed second primary tumor is biologically independent of the original primary tumor, by comparing patterns of allelic losses at candidate genetic loci. The proposed concordant mutations test is a conditional test, an adaptation of Fisher's exact test, that requires no knowledge of the marginal mutation probabilities. The test was shown to have generally good properties, but is susceptible to anticonservative bias if there is wide variation in mutation probabilities between loci, or if the individual mutation probabilities of the parental alleles for individual patients differ substantially from each other. In this article, a likelihood ratio test is derived in an effort to address these validity issues. This test requires prespecification of the marginal mutation probabilities at each locus, parameters for which some information will typically be available in the literature. In simulations this test is shown to be valid, but to be considerably less efficient than the concordant mutations test for sample sizes (numbers of informative loci) typical of this problem. Much of the efficiency deficit can be recovered, however, by restricting the allelic imbalance parameter estimate to a prespecified range, assuming that this parameter is in the prespecified range.     

A Global Sensitivity Test for Evaluating Statistical Hypotheses with Nonidentifiable Models

Summary We consider the problem of evaluating a statistical hypothesis when some model characteristics are nonidentifiable from observed data. Such a scenario is common in meta‐analysis for assessing publication bias and in longitudinal studies for evaluating a covariate effect when dropouts are likely to be nonignorable. One possible approach to this problem is to fix a minimal set of sensitivity parameters conditional upon which hypothesized parameters are identifiable. Here, we extend this idea and show how to evaluate the hypothesis of interest using an infimum statistic over the whole support of the sensitivity parameter. We characterize the limiting distribution of the statistic as a process in the sensitivity parameter, which involves a careful theoretical analysis of its behavior under model misspecification. In practice, we suggest a nonparametric bootstrap procedure to implement this infimum test as well as to construct confidence bands for simultaneous pointwise tests across all values of the sensitivity parameter, adjusting for multiple testing. The methodology's practical utility is illustrated in an analysis of a longitudinal psychiatric study.     

The effect of conditional dependence on the evaluation of diagnostic tests

The accuracy of a new diagnostic test is often determined by comparison with a reference test which also has unknown error rates. Maximum likelihood estimation of the error rates of both tests is possible if they are simultaneously applied to two populations with different disease prevalences. The estimation procedure assumes that the two tests are independent, conditional on a subject's true diagnostic status. If the tests are conditionally dependent, error rates for both tests can be substantially underestimated. Estimators for the prevalence rates in the two populations can be positively or negatively biased, depending on the relative magnitude of the two conditional covariances and the value of the prevalence parameter.     

Conditional Point Estimation in Adaptive Group Sequential Test Designs

It is well known that point estimates in group sequential designs are biased. This also applies to adaptive designs that enable, e.g., data driven reassessments of group sample sizes. For triangular designs, Whitehead (1986) (Biometrika 73 , 573–581) proposed a bias adjusted estimate. But this estimate is not feasible in adaptive designs although it is in group sequential designs. Furthermore, there is a waste of information because it does not use the information at which stage the trial was stopped. We present a modification which does use this information and which is applicable to adaptive designs. The modified estimate achieves an improvement in group sequential designs and shows similar results in adaptive designs.     

Conditional Rejection Probabilities of Student's t‐test and Design Adaptations

For clinical trials with interim analyses conditional rejection probabilities play an important role when stochastic curtailment or design adaptations are performed. The conditional rejection probability gives the conditional probability to finally reject the null hypothesis given the interim data. It is computed either under the null or the alternative hypothesis. We investigate the properties of the conditional rejection probability for the one sided, one sample t‐test and show that it can be non monotone in the interim mean of the data and non monotone in the non‐centrality parameter for the alternative. We give several proposals how to implement design adaptations (that are based on the conditional rejection probability) for the t‐test and give a numerical example. Additionally, the conditional rejection probability given the interim t‐statistic is investigated. It does not depend on the unknown σ and can be used in stochastic curtailment procedures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regressive models for risk prediction of repeated multinomial outcomes: An illustration using Health and Retirement Study data

Life expectancy is increasing in many countries and this may lead to a higher frequency of adverse health outcomes. Therefore, there is a growing demand for predicting the risk of a sequence of events based on specified factors from repeated outcomes. We proposed regressive models and a framework to predict the joint probabilities of a sequence of events for multinomial outcomes from longitudinal studies. The Markov chain is used to link marginal and sequence of conditional probabilities to predict the joint probability. Marginal and sequence of conditional probabilities are estimated using marginal and regressive models. An application is shown using the Health and Retirement Study data. The bias of parameter estimates for all models from all bootstrap simulation is less than 1% in most of the cases. The estimated mean squared error is also very low. Results from the simulation study show negligible bias and the usefulness of the proposed model. The proposed model and framework would be useful to solve real-life problems from various fields and big data analysis.     

Dopamine agonists and antagonists can produce an attenuation of response bias in a temporal discrimination task depending on discriminability of target duration

The current study examined the effects of the D2 agonist (quinpirole) and D2 antagonist (eticlopride) on temporal discrimination performance in a conditional discrimination task (Experiment I) and a delayed conditional discrimination task (Experiment II). In both experiments rats discriminated between a scheduled stimulus duration of 3 s versus 9 s. Consistent with previous reports, overall discrimination performance decreased in a dose-dependent manner with both drugs. Changes in response bias (the tendency to choose-short or choose-long irrespective of actual stimulus duration), however, were best characterized in terms of both drugs tending to attenuate the bias effects normally observed during baseline drug-free performance. Specifically, the 'choose-short' bias observed in Experiment I and at a relatively short, 0.1 s, delay in Experiment II became less extreme with increasing doses. In addition, the 'choose-long' bias observed at a relatively long, 6 s, delay in Experiment II also became less extreme with increasing doses. Thus, whether there was an apparent shift from a short response bias to long, or vice versa, was the product of the delay interval between stimulus presentation and choice rather than whether the drug in question was a D2 agonist or antagonist. Such an attenuation of bias may have arisen because of subjects confounding the delay interval with the actual discriminative stimulus duration.     

Bias-reduced estimators and confidence intervals for odds ratios in genome-wide association studies

Genome-wide association studies (GWAS) provide an important approach to identifying common genetic variants that predispose to human disease. A typical GWAS may genotype hundreds of thousands of single nucleotide polymorphisms (SNPs) located throughout the human genome in a set of cases and controls. Logistic regression is often used to test for association between a SNP genotype and case versus control status, with corresponding odds ratios (ORs) typically reported only for those SNPs meeting selection criteria. However, when these estimates are based on the original data used to detect the variant, the results are affected by a selection bias sometimes referred to the "winner's curse" (Capen and others, 1971). The actual genetic association is typically overestimated. We show that such selection bias may be severe in the sense that the conditional expectation of the standard OR estimator may be quite far away from the underlying parameter. Also standard confidence intervals (CIs) may have far from the desired coverage rate for the selected ORs. We propose and evaluate 3 bias-reduced estimators, and also corresponding weighted estimators that combine corrected and uncorrected estimators, to reduce selection bias. Their corresponding CIs are also proposed. We study the performance of these estimators using simulated data sets and show that they reduce the bias and give CI coverage close to the desired level under various scenarios, even for associations having only small statistical power.     

Conditional Markov Chain Design for Accrual Clinical Trials

Randomization in a comparative experiment has, as one aim, the control of bias in the initial selection of experimental units. When the experiment is a clinical trial employing the accrual of patients, two additional aims are the control of admission bias and control of chronologic bias. This can be accomplished by using a method of randomization, such as the “biased coin design” of Efron, which sequentially forces balance. As an extension of Efron's design, this paper develops a class of conditional Markov chain designs. The detailed randomization employed utilizes the sequential imbalances in the treatment allocation as states in a Markov process. Through the use of appropriate transition probabilities, a range of possible designs can be attained. An additional objective of physical randomization is to provide a model for data analysis. Such a randomization theoretic analysis is presented for the current designs. In addition, Monte Carlo sampling results are given to support the proposed normal theory approximation to the exact randomization distribution.     

Small-sample confidence regions in exponential families

This article presents an algorithm for small-sample conditional confidence regions for two or more parameters for any discrete regression model in the generalized linear interactive model family. Regions are constructed by careful inversion of conditional hypothesis tests. This method presupposes the use of approximate or exact techniques for enumerating the sample space for some components of the vector of sufficient statistics conditional on other components. Such enumeration may be performed exactly or by exact or approximate Monte Carlo, including the algorithms of Kolassa and Tanner (1994, Journal of the American Statistical Association 89, 697-702; 1999, Biometrics 55, 246-251). This method also assumes that one can compute certain conditional probabilities for a fixed value of the parameter vector. Because of a property of exponential families, one can use this set of conditional probabilities to directly compute the conditional probabilities associated with any other value of the vector of the parameters of interest. This observation dramatically reduces the computational effort required to invert the hypothesis test to obtain the confidence region. To construct a region with confidence level 1 - alpha, the algorithm begins with a grid of values for the parameters of interest. For each parameter vector on the grid (corresponding to the current null hypothesis), one transforms the initial set of conditional probabilities using exponential tilting and then calculates the p value for this current null hypothesis. The confidence region is the set of parameter values for which the p value is at least alpha.