Prediction of disease status: A regressive model approach for repeated measures

In this paper, regressive models are proposed for modeling a sequence of transitions in longitudinal data. These models are employed to predict the future status of the outcome variable of the individuals on the basis of their underlying background characteristics or risk factors. The estimation of parameters and also estimates of conditional and unconditional probabilities are shown for repeated measures. The goodness of fit tests are extended in this paper on the basis of the deviance and the Hosmer–Lemeshow procedures and generalized to repeated measures. In addition, to measure the suitability of the proposed models for predicting the disease status, we have extended the ROC curve approach to repeated measures. The procedure is shown for the conditional models for any order as well as for the unconditional model, to predict the outcome at the end of the study. The test procedures are also suggested. For testing the differences between areas under the ROC curves in subsequent follow-ups, two different test procedures are employed, one of which is based on permutation test. In this paper, an unconditional model is proposed on the basis of conditional models for the disease progression of depression among the elderly population in the USA on the basis of the Health and Retirement Survey data collected longitudinally. The illustration shows that the disease progression observed conditionally can be employed to predict the outcome and the role of selected variables and the previous outcomes can be utilized for predictive purposes. The results show that the percentage of correct predictions of a disease is quite high and the measures of sensitivity and specificity are also reasonably impressive. The extended measures of area under the ROC curve show that the models provide a reasonably good fit in terms of predicting the disease status during a long period of time. This procedure will have extensive applications in the field of longitudinal data analysis where the objective is to obtain estimates of unconditional probabilities on the basis of series of conditional transitional models.     

A review and some new results on permutation testing for multivariate problems

In recent years permutation testing methods have increased both in number of applications and in solving complex multivariate problems. When available permutation tests are essentially of an exact nonparametric nature in a conditional context, where conditioning is on the pooled observed data set which is often a set of sufficient statistics in the null hypothesis. Whereas, the reference null distribution of most parametric tests is only known asymptotically. Thus, for most sample sizes of practical interest, the possible lack of efficiency of permutation solutions may be compensated by the lack of approximation of parametric counterparts. There are many complex multivariate problems, quite common in empirical sciences, which are difficult to solve outside the conditional framework and in particular outside the method of nonparametric combination (NPC) of dependent permutation tests. In this paper we review such a method and its main properties along with some new results in experimental and observational situations (robust testing, multi-sided alternatives and testing for survival functions).     

Conditional Studentized Survival Tests for Randomly Censored Models

It is shown that in the case of heterogenous censoring distributions Studentized survival tests can be carried out as conditional permutation tests given the order statistics and their censoring status. The result is based on a conditional central limit theorem for permutation statistics. It holds for linear test statistics as well as for sup-statistics. The procedure works under one of the following general circumstances for the two-sample problem: the unbalanced sample size case, highly censored data, certain non-convergent weight functions or under alternatives. For instance, the two-sample log rank test can be carried out asymptotically as a conditional test if the relative amount of uncensored observations vanishes asymptotically as long as the number of uncensored observations becomes infinite. Similar results hold whenever the sample sizes and are unbalanced in the sense that and hold.     

Multiple Window Discrete Scan Statistics

In this article, multiple scan statistics of variable window sizes are derived for independent and identically distributed 0-1 Bernoulli trials. Both one and two dimensional, as well as, conditional and unconditional cases are treated. The advantage in using multiple scan statistics, as opposed to single fixed window scan statistics, is that they are more sensitive in detecting a change in the underlying distribution of the observed data. We show how to derive simple approximations for the significance level of these testing procedures and present numerical results to evaluate their performance.     

A toolbox of permutation tests for structural change

The sup $LM$ test for structural change is embedded into a permutation test framework for a simple location model. The resulting conditional permutation distribution is compared to the usual (unconditional) asymptotic distribution, showing that the power of the test can be clearly improved in small samples. Furthermore, the permutation test is embedded into a general framework that encompasses tools for binary and multivariate dependent variables as well as model-based permutation testing for structural change. It is also demonstrated that the methods can not only be employed for analyzing structural changes in time series data but also for recursive partitioning of cross-section data. The procedures suggested are illustrated using both artificial data and empirical applications (number of youth homicides, employment discrimination data, carbon flux in tropical forests, stock returns, and demand for economics journals).     

Randomization Inference for Outcomes with Clumping at Zero

While randomization inference is well developed for continuous and binary outcomes, there has been comparatively little work for outcomes with nonnegative support and clumping at zero. Typically, outcomes of this type have been modeled using parametric models that impose strong distributional assumptions. This article proposes new randomization inference procedures for nonnegative outcomes with clumping at zero. Instead of making distributional assumptions, we propose various assumptions about the nature of the response to treatment and use permutation inference for both testing and estimation. This approach allows for some natural goodness-of-fit tests for model assessment, as well as flexibility in selecting test statistics sensitive to different potential alternatives. We illustrate our approach using two randomized trials, where job training interventions were designed to increase earnings of participants.     

On the Weak Consistency of Permutation Tests

Consistency of some nonparametric tests with real variables has been studied by several authors under the assumption that population variance is finite and/or in the presence of some violations of the data exchangeability between samples. Since main inferential conclusions of permutation tests concern the actual dataset, where sample sizes are held fixed, we consider the notion of consistency in the weak version (in probability). Here, we characterize weak consistency of permutation tests assuming population mean is finite and without assuming existence of population variance. Moreover, since permutation test statistics do not require to be standardized, we do not assume that data are homoscedastic in the alternative. Several application examples to mostly used test statistics are discussed. A simulation study and some hints for robust testing procedures are also presented.     

Exact conditional inference for two uniform populations

We discuss the nature of ancillary information in the context of the continuous uniform distribution. In the one-sample problem, the existence of sufficient statistics mitigates conditioning on the ancillary configuration. In the two-sample problem, additional ancillary information becomes available when the ratio of scale parameters is known. We give exact results for conditional inferences about the common scale parameter and for the difference in location parameters of two uniform distributions. The ancillary information affects the precision of the latter through a comparison of the sample value of the ratio of scale parameters with the known population value. A limited conditional simulation compares the Type I errors and power of these exact results with approximate results using the robust pooled t-statistic.     

Comparing two independent incidence rates using conditional and unconditional exact tests

Several unconditional exact tests, which are constructed to control the Type I error rate at the nominal level, for comparing two independent Poisson rates are proposed and compared to the conditional exact test using a binomial distribution. The unconditional exact test using binomial p-value, likelihood ratio, or efficient score as the test statistic improves the power in general, and are therefore recommended. Unconditional exact tests using Wald statistics, whether on the original or square-root scale, may be substantially less powerful than the conditional exact test, and is not recommended. An example is provided from a cardiovascular trial.     

The evaluation of DNA evidence in pedigrees requiring population inference

Summary. The evaluation of nuclear DNA evidence for identification purposes is performed here taking account of the uncertainty about population parameters. Graphical models are used to detail the hypotheses being debated in a trial with the aim of obtaining a directed acyclic graph. Graphs also clarify the set of evidence that contributes to population inferences and they also describe the conditional independence structure of DNA evidence. Numerical illustrations are provided by re-examining three case-studies taken from the literature. Our calculations of the weight of evidence differ from those given by the authors of case-studies in that they reveal more conservative values.     

On asymptotic non-normality of null distributions of mrpp statistics

Severe departures from normality occur frequently for null distributions of statistics associated with applications of mulLi-response permutation procedures (MRPP) for either small or large finite populations. This paper describes the commonly encountered situation associated with asymptotic non-normality for null distributions of MRPP statistics which does not depend on the underlying multivariate distribution. In addition, this paper establishes the existence of a non-degenerate underlying distribution for which the null distributions of MRPP statistics are asymptotically non-normal for essentially all size structure configurations. It is known that MRPP statistics are symmetric versions of a broader class of statistics, most of which are asymmetric. Because of the non-normality associated with null distributions of MRPP statistics, this paper includes necessary results for inferences based on the exact first three moments of anv statistic in this broader class (analogous to existing results for MRPP statistics).     

Exact unconditional testing procedures for comparing two independent Poisson rates

We consider seven exact unconditional testing procedures for comparing adjusted incidence rates between two groups from a Poisson process. Exact tests are always preferable due to the guarantee of test size in small to medium sample settings. Han [Comparing two independent incidence rates using conditional and unconditional exact tests. Pharm Stat. 2008;7(3):195–201] compared the performance of partial maximization p-values based on the Wald test statistic, the likelihood ratio test statistic, the score test statistic, and the conditional p-value. These four testing procedures do not perform consistently, as the results depend on the choice of test statistics for general alternatives. We consider the approach based on estimation and partial maximization, and compare these to the ones studied by Han (2008) for testing superiority. The procedures are compared with regard to the actual type I error rate and power under various conditions. An example from a biomedical research study is provided to illustrate the testing procedures. The approach based on partial maximization using the score test is recommended due to the comparable performance and computational advantage in large sample settings. Additionally, the approach based on estimation and partial maximization performs consistently for all the three test statistics, and is also recommended for use in practice.     

Local Box–Cox transformation on time-varying parametric models for smoothing estimation of conditional CDF with longitudinal data

Nonparametric estimation and inferences of conditional distribution functions with longitudinal data have important applications in biomedical studies, such as epidemiological studies and longitudinal clinical trials. Estimation approaches without any structural assumptions may lead to inadequate and numerically unstable estimators in practice. We propose in this paper a nonparametric approach based on time-varying parametric models for estimating the conditional distribution functions with a longitudinal sample. Our model assumes that the conditional distribution of the outcome variable at each given time point can be approximated by a parametric model after local Box–Cox transformation. Our estimation is based on a two-step smoothing method, in which we first obtain the raw estimators of the conditional distribution functions at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. Applications of our two-step estimation method have been demonstrated through a large epidemiological study of childhood growth and blood pressure. Finite sample properties of our procedures are investigated through a simulation study. Application and simulation results show that smoothing estimation from time-variant parametric models outperforms the existing kernel smoothing estimator by producing narrower pointwise bootstrap confidence band and smaller root mean squared error.     

Nonparametric empirical Bayes method for comparison of treatment effects with application to stress urinary incontinence data

Hierarchical models are widely used in medical research to structure complicated models and produce statistical inferences. In a hierarchical model, observations are sampled conditional on some parameters and these parameters are sampled from a common prior distribution. Bayes and empirical Bayes (EB) methods have been effectively applied in analyzing these models. Despite many successes, parametric Bayes and EB methods may be sensitive to misspecification of prior distributions. In this paper, without specific restriction on the form of the prior distribution, we propose a nonparametric EB method to estimate the treatment effect of each group and develop a testing procedure to compare between-group differences. Simulation studies demonstrate that the proposed EB method was more efficient than some standard procedures. An illustrative example is provided with data from a clinical trial evaluating a new treatment for patients with stress urinary incontinence.     

A Bayesian nonparametric causal model

Typically, in the practice of causal inference from observational studies, a parametric model is assumed for the joint population density of potential outcomes and treatment assignments, and possibly this is accompanied by the assumption of no hidden bias. However, both assumptions are questionable for real data, the accuracy of causal inference is compromised when the data violates either assumption, and the parametric assumption precludes capturing a more general range of density shapes (e.g., heavier tail behavior and possible multi-modalities). We introduce a flexible, Bayesian nonparametric causal model to provide more accurate causal inferences. The model makes use of a stick-breaking prior, which has the flexibility to capture any multi-modalities, skewness and heavier tail behavior in this joint population density, while accounting for hidden bias. We prove the asymptotic consistency of the posterior distribution of the model, and illustrate our causal model through the analysis of small and large observational data sets.     

Block permutation principles for the change analysis of dependent data

We study an AMOC model with an abrupt change in the mean and dependent errors that form a linear process. Different kinds of statistics are considered, such as maximum-type statistics (particularly different CUSUM procedures) or sum-type statistics. Approximations of the critical values for change-point tests are obtained through permutation methods. The theoretical results show that the original test statistics and their corresponding block permutation counterparts follow the same distributional asymptotics. The main step in the proof is to obtain limit theorems for the corresponding rank statistics and then use laws of large numbers to obtain the permutation asymptotics conditionally on the given data.     

Exact Non-Parametric Confidence, Prediction and Tolerance Intervals with Progressive Type-II Censoring

Abstract.  This article extends recent results [Scand. J. Statist. 28 (2001) 699] about exact non-parametric inferences based on order statistics with progressive type-II censoring. The extension lies in that non-parametric inferences are now covered where the dependence between involved order statistics cannot be circumvented. These inferences include: (a) tolerance intervals containing at least a specified proportion of the parent distribution, (b) prediction intervals containing at least a specified number of observations in a future sample, and (c) outer and/or inner confidence intervals for a quantile interval of the parent distribution. The inferences are valid for any parent distribution with continuous distribution function. The key result shows how the probability of an event involving k dependent order statistics that are observable/uncensored with progressive type-II censoring can be represented as a mixture with known weights of corresponding probabilities involving k dependent ordinary order statistics. Further applications/developments concerning exact Kolmogorov-type confidence regions are indicated.     

A note on simultaneous inference with rerandomization tests

Valid simultaneous confidence intervals based on rerandomization are provided for the first time. They are derived from joint confidence regions which are constructed by testing for all possible parametric values. A simple exampe illustrates these confidence intervals and compares inferences from them with other methods.     

Conditional and unconditional tests with historical controls

Score statistics utilizing historical control data have been proposed to test for increasing trend in tumour occurrence rates in laboratory carcinogenicity studies. Novel invariance arguments are used to confirm, under slightly weaker conditions, previously established asymptotic distributions (mixtures of normal distributions) of tests unconditional on the tumor response rate in the concurrent control group. Conditioning on the control response rate, an ancillary statistic, leads to a new conditional limit theorem in which the test statistic converges to an unknown random variable. Because of this, a subasymptotic approximation to the conditional limiting distribution is also considered. The adequacy of these large-sample approximations in finite samples is evaluated using computer simulation. Bootstrap methods for use in finite samples are also proposed. The application of the conditional and unconditional tests is illustrated using bioassay data taken from the literature. The results presented in this paper are used to formulate recommendations for the use of tests for trend with historical controls in practice.     

Resampling schemes with low resampling intensity and their applications in testing hypotheses

The paper explores statistical features of different resampling schemes under low resampling intensity. The original sample is considered in a very general framework of triangular arrays, without independence or equally distributed assumptions, although improvements under such conditions are also provided. We show that low resampling schemes have very interesting and flexible properties, providing new insights into the performance of widely used resampling methods, including subsampling, two-sample unbalanced permutation statistics or wild bootstrap. It is shown that, under regularity assumptions, resampling tests with critical values derived by the appertaining low resampling procedures are asymptotically valid and there is no loss of power compared with the power function of an ideal (but unfeasible) parametric family of tests. Moreover we show that in several contexts, including regression models, they may act as a filter for the normal part of a limit distribution, turning down the influence of outliers.