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Paired comparison data is often used to rank or order a set of items. In this paper we study a method for estimating the parameters associated with completely ordered cardinal paired comparison data. The analysis is carried out within the framework of graphical linear models but rather than using the least squares estimator, which may be difficult to analyze, we consider the average of all tree-based estimators for the connected comparison graph. The resulting estimator is a simple linear function of the sufficient statistics and has an easy to understand graph-theoretic interpretation. The statistical properties of this estimator are studied and it is shown to be unbiased, strongly consistent and asymptotically normal. Examples and numerical comparisons are provided and extensions are discussed.

Summary There are many situations where one expects an ordering among K ⩾ 2 experimental groups or treatments. Although there is a large body of literature dealing with the analysis under order restrictions, surprisingly, very little work has been done in the context of the design of experiments. Here, a principled approach to the design of experiments with ordered treatments is provided. In particular we propose two classes of designs which are optimal for testing different types of hypotheses. The theoretical findings are supplemented with thorough numerical experimentation and a concrete data example. It is shown that there is a substantial gain in power, or alternatively a reduction in the required sample size, when an experiment is both designed and analysed by using methods which account for order restrictions.

The use of historical, i.e., already existing, estimates in current studies is common in a wide variety of application areas. Nevertheless, despite their routine use, the uncertainty associated with historical estimates is rarely properly accounted for in the analysis. In this communication, we review common practices and then provide a mathematical formulation and a principled frequentist methodology for addressing the problem of drawing inferences in the presence of historical estimates. Three distinct variants are investigated in detail; the corresponding limiting distributions are found and compared. The design of future studies, given historical data, is also explored and relations with a variety of other well-studied statistical problems discussed.

In this communication we examine the relationship between maxi–min, Bayes and Nash designs for some hypothesis testing problems. In particular we consider the problem of sample allocation in the standard analysis of variance framework and show that the maxi–min design is also a Bayes solution with respect to the least favourable prior, as well as a solution to a game theoretic problem, which we refer to as a Nash design. In addition, an extension to tests for order is provided.

There are many applications in which a statistic follows, at least asymptotically, a normal distribution with a singular or nearly singular variance matrix. A classic example occurs in linear regression models under multicollinearity but there are many more such examples. There is well-developed theory for testing linear equality constraints when the alternative is two-sided and the variance matrix is either singular or nonsingular. In recent years, there is considerable, and growing, interest in developing methods for situations in which the estimated variance matrix is nearly singular. However, there is no corresponding methodology for addressing one-sided, that is, constrained or ordered alternatives. In this article, we develop a unified framework for analyzing such problems. Our approach may be viewed as the trimming or winsorizing of the eigenvalues of the corresponding variance matrix. The proposed methodology is applicable to a wide range of scientific problems and to a variety of statistical models in which inequality constraints arise. We illustrate the methodology using data from a gene expression microarray experiment obtained from the NIEHS’ Fibroid Growth Study. Supplementary materials for this article are available online.

Motivation: An important feature of microbiome count data is the presence of a large number of zeros. A common strategy to handle these excess zeros is to add a small number called pseudo-count (e.g., 1). Other strategies include using various probability models to model the excess zero counts. Although adding a pseudo-count is simple and widely used, as demonstrated in this paper, it is not ideal. On the other hand, methods that model excess zeros using a probability model often make an implicit assumption that all zeros can be explained by a common probability models. As described in this article, this is not always recommended as there are potentially three types/sources of zeros in a microbiome data. The purpose of this paper is to develop a simple methodology to identify and accomodate three different types of zeros and to test hypotheses regarding the relative abundance of taxa in two or more experimental groups. Another major contribution of this paper is to perform constrained (directional or ordered) inference when there are more than two ordered experimental groups (e.g., subjects ordered by diet or age groups or environmental exposure groups). As far as we know this is the first paper that addresses such problems in the analysis of microbiome data. Results: Using extensive simulation studies, we demonstrate that the proposed methodology not only controls the false discovery rate at a desired level of significance while competing well in terms of power with DESeq2, a popular procedure derived from RNASeq literature. As expected, the method using pseudo-counts tends to be very conservative and the classical t-test that ignores the underlying simplex structure in the data has an inflated FDR.

There are many situations in which a researcher would like to analyse data from a two-way layout. Often, the assumptions of linearity and normality may not hold. To address such situations, we introduce a semiparametric model. The model extends the well-known density ratio model from the one-way to the two-way layout and provides a useful framework for semiparametric analysis of variance type problems under order restrictions. In particular, the likelihood ratio order is emphasized. The model enables highly efficient inference without resorting to fully parametric assumptions or the use of transformations. Estimation and testing procedures under order restrictions are developed and investigated in detail. It is shown that the model is robust to misspecification, and several simulations suggest that it performs well in practice. The methodology is illustrated using two data examples; in the first, the response variable is discrete, whereas in the second, it is continuous.

Summary The comparison of two or more ordered experimental groups based on multivariate data is common in a variety of applications such as toxicology, clinical trials and drug development, to name just a few. In this article, we develop a nonparametric methodology for analyzing such data. In particular we propose a global K sample nonparametric test for order among vector valued outcomes. The testing procedure can also be used in a post-hoc fashion to answer questions about the ordering of subgroups and/or single outcomes within any subset of experimental groups. Such a methodology does not currently exist. In contrast with standard methodology such as multivariate analysis of variance (MANOVA), and its nonparametric analogues, we do not assume that the groups differ only by a location parameter or that the components of the response vector have the same marginal distributions between and across groups, that is, we allow for the shape of the distribution to change across groups. We emphasize that our test compares the outcome distributions, not just their mean tendencies, and explicitly incorporates and exploits the order constraints. Consequently it is more powerful than the existing unordered tests. The methodology is illustrated using genotoxicity data where the effect of hydrogen peroxide exposure on damage to DNA is evaluated using a comet assay.

This paper is motivated by the recent interest in the analysis of high-dimensional microbiome data. A key feature of these data is the presence of "structural zeros" which are microbes missing from an observation vector due to an underlying biological process and not due to error in measurement. Typical notions of missingness are unable to model these structural zeros. We define a general framework which allows for structural zeros in the model and propose methods of estimating sparse high-dimensional covariance and precision matrices under this setup. We establish error bounds in the spectral and Frobenius norms for the proposed estimators and empirically verify them with a simulation study. The proposed methodology is illustrated by applying it to the global gut microbiome data of Yatsunenko and others (2012. Human gut microbiome viewed across age and geography. Nature 486, 222-227). Using our methodology we classify subjects according to the geographical location on the basis of their gut microbiome.

In this article, we compare four nonparametric estimators of a distribution function (DF), estimated under a stochastic order restriction. The estimators are compared by simulation using four criteria: (1) the estimation of cumulative DFs; (2) the estimation of quantiles; (3) the estimation of moments and other functionals; and (4) as tools for testing for stochastic order. Our simulation study shows that estimators based on the pointwise maximum-likelihood estimator (p-MLE) outperform all other estimators when the underlying distributions are ‘close’ to each other. The gain in efficiency may be as high as 25%. If the DFs are far apart then the p-MLE may not be the best. However, the efficiency loss using the p-MLE relative to the best estimator in each case is generally low (about 5%). We also find that the test based on the p-MLE is the most powerful in the majority of cases although the gain in power relative to other tests is generally small.

A family of simple two sample rank tests for testing for stochastic order is introduced. The proposed tests are motivated by the stochastic ordering of the ranks and methods for combining p -values which capitalize on this ordering. We show that the new tests coincide with some well known rank tests. Their large sample distributions are derived under various assumptions on the sample sizes. Some optimality properties are discussed. Simulations show that the new tests perform well, often better than specialized tests designed for testing for order among two populations.

Summary In a variety of applications researchers collect data that are sampled under ordered experimental conditions. In such situations it is reasonable to assume that outcomes are ordered stochastically by the level of the treatment. For example a toxicologist may want to assess the effect of a toxin on the reproductive system by studying the number of offspring produced by an animal as a function of the dose that it received. A variety of scientific questions arise in such a context. The most basic is whether a dose–response relationship is present. If so, determining the minimal toxic dose is of great interest. The strength of the ordering between groups i

The estimator of Hsieh and Turnbull (1996) for the binormal receiver operating characteristic (ROC) curve is extended from grouped to ungrouped data. The new estimator is shown to be consistent and asymptotically normally distributed, and simulation results show that it outperforms Hsieh and Turnbull's original estimator.

In many applications researchers collect multivariate binary response data under two or more, naturally ordered, experimental conditions. In such situations one is often interested in using all binary outcomes simultaneously to detect an ordering among the experimental conditions. To make such comparisons we develop a general methodology for testing for the multivariate stochastic order between K ≥ 2 multivariate binary distributions. The proposed test uses order restricted estimators which, according to our simulation study, are more efficient than the unrestricted estimators in terms of mean squared error. The power of the proposed test was compared with several alternative tests. These included procedures which combine individual univariate tests for order, such as union intersection tests and a Bonferroni based test. We also compared the proposed test with unrestricted Hotelling's T(2) type test. Our simulations suggest that the proposed method competes well with these alternatives. The gain in power is often substantial. The proposed methodology is illustrated by applying it to a two-year rodent cancer bioassay data obtained from the US National Toxicology Program (NTP). Supplemental materials are available online.

In medical studies, endpoints are often measured for each patient longitudinally. The mixed-effects model has been a useful tool for the analysis of such data. There are situations in which the parameters of the model are subject to some restrictions or constraints. For example, in hearing loss studies, we expect hearing to deteriorate with time. This means that hearing thresholds which reflect hearing acuity will, on average, increase over time. Therefore, the regression coefficients associated with the mean effect of time on hearing ability will be constrained. Such constraints should be accounted for in the analysis. We propose maximum likelihood estimation procedures, based on the expectation-conditional maximization either algorithm, to estimate the parameters of the model while accounting for the constraints on them. The proposed methods improve, in terms of mean square error, on the unconstrained estimators. In some settings, the improvement may be substantial. Hypotheses testing procedures that incorporate the constraints are developed. Specifically, likelihood ratio, Wald, and score tests are proposed and investigated. Their empirical significance levels and power are studied using simulations. It is shown that incorporating the constraints improves the mean squared error of the estimates and the power of the tests. These improvements may be substantial. The methodology is used to analyze a hearing loss study.

We present a novel semi-parametric model for two-sample survival data, and an estimation method with a simple, closed-form solution. We study analytically the asymptotics of the estimators, and conduct a small simulation study.

It is shown that if a binary regression function is increasing then retrospective sampling induces a stochastic ordering of the covariate distributions among the responders, which we call cases, and the non-responders, which we call controls. We also show that if the covariate distributions are stochastically ordered then the regression function must be increasing. This means that testing whether the regression function is monotone is equivalent to testing whether the covariate distributions are stochastically ordered. Capitalizing on these new probabilistic observations we proceed to develop two new non-parametric tests for stochastic order. The new tests are based on either the maximally selected, or integrated, chi-bar statistic of order one. The tests are easy to compute and interpret and their large sampling distributions are easily found. Numerical comparisons show that they compare favorably with existing methods in both small and large samples. We emphasize that the new tests are applicable to any testing problem involving two stochastically ordered distributions.

It is shown that retrospective sampling induces stochastic order relations in case-control studies. More specifically if the regression function is increasing and the covariates are positively dependent, then the covariates for cases are larger, with respect to some multivariate stochastic order, than the covariates of the controls. Strong dependence concepts yield strong multivariate stochastic orders. Conversely, different multivariate stochastic orders imply different monotonicity properties on the regression function. The results carry over to marginal models, transformed models and to problems involving confounders. The results set forth a new theoretical foundation for the analysis of case-control studies.

Identifiability, existence and uniqueness of the non-parametric maximum likelihood estimator (NPMLE) in biased sampling models are thoroughly investigated. The conditions for existence and uniqueness are considered from a different perspective and proofs are simplified. An extension to semi-parametric models is provided.

We provide a method for finding the optimal double sampling plan for estimating the mean value of a continuous outcome. It is assumed that the fallible and true outcome data are related by a multivariate linear regression model where only some of the explanatory variables are sampled. Conditions under which double sampling is preferred over standard sampling plans are determined. An application of the method to a well-known data set on air pollution is presented.

It is well known that for any sample size the estimator for the slope in the simple linear measurement error model does not possess even a first moment. Addressing this issue constrained maximum likelihood estimators (MLEs) are developed using three different approaches, under a number of identifiability assumptions. The constrained MLEs have finite moments of all orders. They are asymptotically equivalent to the MLE, which is known to be consistent and asymptotically normal. Numerical investigation shows that the constrained MLEs perform well under a wide variety of experimental settings.

Objective: The extent to which noncompletion of a clinical trial relates to outcomes has implications for choosing the most appropriate method for contending with missing data due to dropout. We examined whether dropout relates to outcome in clinical trials of antipsychotic medication. Methods: Data from 5 large clinical trials of schizophrenia (n = 3483) were examined separately. Patients were aggregated into groups based on their final study visit. Group mean Positive and Negative Syndrome Scale (PANSS) total scores for each visit were computed and graphed. Change from baseline to end point for each group was computed and examined using ANCOVA. Cox regression modeling was used to examine baseline PANSS total and change as predictors of time to dropout. Results: In all 5 trials there was a statistically significantly relationship between time in trial and improvement. The longer the patients remained in the trial the more that they improved, with trial completers showing the most improvement at each time point. Higher baseline PANSS scores and symptom deterioration indicated by increased PANSS preceding the final study visit prior to dropout corresponded significantly with a greater likelihood of dropout. Conclusions: Dropout in clinical trials of antipsychotic medications corresponds with efficacy outcomes, the dynamics of symptom change and baseline symptom severity. Therefore, methods for statistical analysis should examine both efficacy and dropout and cannot assume that missing data due to dropout are completely at random.

We study the conditions under which the sample mean is self-consistent, and therefore an optimal predictor, for an arbitrary observation in the sample.

This paper discusses the estimation of the sojourn time distribution in a latent state using observational data. We assume that observations are available only if an event has occurred. This type of sampling is referred to as retrospective ascertainment. Our focus is the information content of various constructions of the likelihood function.

Motivated by a statistical model for the structural line segment relationship developed for computer vision applications we derive an estimator for the slope of a regression line in univariate measurement error models. We show that under the typical side conditions, this estimator coincides, in most cases, with the maximum likelihood estimator for the normal structural model. Its large sample properties are derived.

Luo & Tsai (2012, Biometrika) introduced the proportional likelihood ratio model. They proposed an iterative algorithm for the estimation of the baseline distribution function but did not establish its convergence. Here we provide a proof of convergence. Copyright 2013, Oxford University Press.

SUMMARY Overdiagnosis refers to the situation where a screening exam detects a disease that would have otherwise been undetected in a person’s lifetime. The disease would have not have been diagnosed because the individual would have died of other causes prior to its clinical onset. Although the probability of overdiagnosis is an important quantity for understanding early detection programs it has not been rigorously studied. We analyze an idealized early detection program and derive the mathematical expression for the probability of overdiagnosis. The results are studied numerically for prostate cancer and applied to a variety of screening schedules. Our investigation indicates that the probability of overdiagnosis is remarkably high.

SUMMARY Although case-control studies are widely used for evaluating the benefit of early detection programs, the theoretical basis underlying this application has not been well developed. In this paper the properties of chronic disease case-control studies for evaluating early detection programs are investigated. An idealized case-control study is analyzed and the theoretical expression for the odds ratio associated with the benefit of screening is derived. The odds ratio is related to the natural history of disease and the screening program. Our results indicate that case-control studies result in odds ratios that are surprisingly close to unity and consequently have low power.

Summary We study the effect of misclassification of a binary covariate on the parameters of a logistic regression model. In particular we consider 2 � 2 � 2 tables. We assume that a binary covariate is subject to misclassification that may depend on the observed outcome. This type of misclassification is known as (outcome dependent) differential misclassification. We examine the resulting asymptotic bias on the parameters of the model and derive formulas for the biases and their approximations as a function of the odds and misclassification probabilities. Conditions for unbiased estimation are also discussed. The implications are illustrated numerically using a case control study. For completeness we briefly examine the effect of covariate dependent misclassification of exposures and of outcomes.

We compare three methods which can be used to analyse the influence of birth order and other factors on health outcomes in multiple birth data. We consider marginal models based on generalized estimating equations (GEE) and two kinds of conditional models; conditional logistic regression (CLR) and mixed effects models (MEM). Although the models may be written similarly, there are differences in both the interpretation and the numerical values assigned to the parameters. Our main conclusion is that GEE and MEM are preferable to CLR since they provide more flexibility in dealing with missing values and covariates. The choice between GEE and MEM is less obvious and depends on the data, the parameter of interest and statistical power.

SUMMARY Recently genetic epidemiologists have begun using case–control family study designs to investigate the role of genetic and environmental risk factors in disease etiology. The objective of these studies is to assess the association of environmental factors with the disease trait; to characterize the disease genes using segregation analysis; and to quantify the residual familial aggregation after controlling for environmental and genetic factors. Typically these objectives are achieved by conducting separate studies and analysis. This paper describes an estimating equation based approach for a combined association, segregation and aggregation analysis on data from case–control family studies. Simulations indicate that the method performs well in a variety of settings. The method is illustrated using simulated family history data made available to participants in a recent Genetic Analysis Workshop.

We provide a method for estimating the sample mean of a continuous outcome in a stratified population using a double sampling scheme. The stratified sample mean is a weighted average of stratum specific means. It is assumed that the fallible and true outcome data are related by a simple linear regression model in each stratum. The optimal stratified double sampling plan, i.e. , the double sampling plan that minimizes the cost of sampling for fixed variances, or alternatively, minimizes the variance for fixed costs, is found and compared to a standard sampling plan. The design parameters are the total sample size and the number of doubly sampled units in each stratum. We show that the optimal double sampling plan is a function of the between-strata and within-strata cost and variance ratios. The efficiency gains, relative to standard sampling plans, under broad set of conditions, are considerable.

The exponential distribution is a fundamental model in parametric survival analysis. In this paper the exponential regression model with a canonical parameterization, λ=β t x , is explored. Large sample and exact significance tests are developed to test the relationship between survival and one covariate. The standard unconditional likelihood approach is compared with two new tests, which are based on the conditional distribution of the sufficient statistics. The various testing procedures are compared numerically. Finally, the relationship with the semiparametric proportional hazard model is investigated.

We provide a method for finding the optimal double sampling plan for estimating the mean value of a continuous outcome. It is assumed that the fallible and true outcome data are related by a simple linear regression model. The design parameters are the total sample size, N, and the number of doubly sampled units n. We show that under certain conditions the efficiency gains, relative to standard sampling plans are considerable.

There is growing interest in the design and implementation of cancer prevention trials. The key idea is to have agents which interfere with carcinogenesis and/or the preclinical stage. In this article we develop multi-stage stochastic models for the planning of cancer prevention trials. For known inputs it is possible to calculate the incidence of disease for the control and intervention groups. Consequently we find designs that balance the required sample size and follow-up time while guaranteeing prespecified error probabilities. Moreover such models can incorporate the mode of action of the intervention as well as compliance. The model has been applied to breast cancer to determine the implications for planning breast cancer intervention trials. Although the model addresses issues in cancer prevention, it is quite general and may be suitable for other chronic diseases.

There is a growing interest in planning and implementing broad-scale clinical trials with a focus on prevention and screening. Often, the data-generating mechanism for such experiments can be viewed as a semi-Markov process. In this communication, we develop general expressions for the steady-state probabilities for regenerative semi-Markov processes. Hence, the probability of being in a certain state at the time of recruitment to a clinical trial can be calculated. An application to breast cancer prevention is demonstrated.

This paper investigates the urn sampling analogue for the score statistic relating survival to covariates assuming a proportional hazard model. The exact permutation distribution can be calculated as well as the exact low order moments for arbitrary censoring patterns. The asymptotic distribution of the score statistic is an easy consequence. The method is naturally extended to deal with the multivariate case, time varying covariates and interval censoring. Finally the relationship between the censoring process, the survival times and covariates are studied considering different reference sets for the distribution of the score statistic. Some assumptions about the censoring process are investigated and as a consequence the effect of censoring is clarified.