A novel approach to estimation of the time to biomarker threshold: applications to HIV

In longitudinal studies of biomarkers, an outcome of interest is the time at which a biomarker reaches a particular threshold. The CD4 count is a widely used marker of human immunodeficiency virus progression. Because of the inherent variability of this marker, a single CD4 count below a relevant threshold should be interpreted with caution. Several studies have applied persistence criteria, designating the outcome as the time to the occurrence of two consecutive measurements less than the threshold. In this paper, we propose a method to estimate the time to attainment of two consecutive CD4 counts less than a meaningful threshold, which takes into account the patient‐specific trajectory and measurement error. An expression for the expected time to threshold is presented, which is a function of the fixed effects, random effects and residual variance. We present an application to human immunodeficiency virus‐positive individuals from a seroprevalent cohort in Durban, South Africa. Two thresholds are examined, and 95% bootstrap confidence intervals are presented for the estimated time to threshold. Sensitivity analysis revealed that results are robust to truncation of the series and variation in the number of visits considered for most patients. Caution should be exercised when interpreting the estimated times for patients who exhibit very slow rates of decline and patients who have less than three measurements. We also discuss the relevance of the methodology to the study of other diseases and present such applications. We demonstrate that the method proposed is computationally efficient and offers more flexibility than existing frameworks. Copyright © 2016 John Wiley & Sons, Ltd.     

Making predictions from complex longitudinal data, with application to planning monitoring intervals in a national screening programme

When biological or physiological variables change over time, we are often interested in making predictions either of future measurements or of the time taken to reach some threshold value. On the basis of longitudinal data for multiple individuals, we develop Bayesian hierarchical models for making these predictions together with their associated uncertainty. Particular aspects addressed, which include some novel components, are handling curvature in individuals' trends over time, making predictions for both underlying and measured levels, making predictions from a single baseline measurement, making predictions from a series of measurements, allowing flexibility in the error and random-effects distributions, and including covariates. In the context of data on the expansion of abdominal aortic aneurysms over time, where reaching a certain threshold leads to referral for surgery, we discuss the practical application of these models to the planning of monitoring intervals in a national screening programme. Prediction of the time to reach a threshold was too imprecise to be practically useful, and we focus instead on limiting the probability of exceeding the threshold after given time intervals. Although more complex models can be shown to fit the data better, we find that relatively simple models seem to be adequate for planning monitoring intervals.     

Variance approximation under balanced sampling

A balanced sampling design has the interesting property that Horvitz–Thompson estimators of totals for a set of balancing variables are equal to the totals we want to estimate, therefore the variance of Horvitz–Thompson estimators of variables of interest are reduced in function of their correlations with the balancing variables. Since it is hard to derive an analytic expression for the joint inclusion probabilities, we derive a general approximation of variance based on a residual technique. This approximation is useful even in the particular case of unequal probability sampling with fixed sample size. Finally, a set of numerical studies with an original methodology allows to validate this approximation.     

Asymptotic Normality in Mixtures of Power Series Distributions

Abstract.  The problem of estimating the individual probabilities of a discrete distribution is considered. The true distribution of the independent observations is a mixture of a family of power series distributions. First, we ensure identifiability of the mixing distribution assuming mild conditions. Next, the mixing distribution is estimated by non-parametric maximum likelihood and an estimator for individual probabilities is obtained from the corresponding marginal mixture density. We establish asymptotic normality for the estimator of individual probabilities by showing that, under certain conditions, the difference between this estimator and the empirical proportions is asymptotically negligible. Our framework includes Poisson, negative binomial and logarithmic series as well as binomial mixture models. Simulations highlight the benefit in achieving normality when using the proposed marginal mixture density approach instead of the empirical one, especially for small sample sizes and/or when interest is in the tail areas. A real data example is given to illustrate the use of the methodology.     

On evaluating how well a biomarker can predict treatment response with survival data

One of the objectives of personalized medicine is to take treatment decisions based on a biomarker measurement. Therefore, it is often interesting to evaluate how well a biomarker can predict the response to a treatment. To do so, a popular methodology consists of using a regression model and testing for an interaction between treatment assignment and biomarker. However, the existence of an interaction is not sufficient for a biomarker to be predictive. It is only necessary. Hence, the use of the marker‐by‐treatment predictiveness curve has been recommended. In addition to evaluate how well a single continuous biomarker predicts treatment response, it can further help to define an optimal threshold. This curve displays the risk of a binary outcome as a function of the quantiles of the biomarker, for each treatment group. Methods that assume a binary outcome or rely on a proportional hazard model for a time‐to‐event outcome have been proposed to estimate this curve. In this work, we propose some extensions for censored data. They rely on a time‐dependent logistic model, and we propose to estimate this model via inverse probability of censoring weighting. We present simulations results and three applications to prostate cancer, liver cirrhosis, and lung cancer data. They suggest that a large number of events need to be observed to define a threshold with sufficient accuracy for clinical usefulness. They also illustrate that when the treatment effect varies with the time horizon which defines the outcome, then the optimal threshold also depends on this time horizon.     

Bayesian disclosure risk assessment: predicting small frequencies in contingency tables

Summary.  We propose an approach for assessing the risk of individual identification in the release of categorical data. This requires the accurate calculation of predictive probabilities for those cells in a contingency table which have small sample frequencies, making the problem somewhat different from usual contingency table estimation, where interest is generally focused on regions of high probability. Our approach is Bayesian and provides posterior predictive probabilities of identification risk. By incorporating model uncertainty in our analysis, we can provide more realistic estimates of disclosure risk for individual cell counts than are provided by methods which ignore the multivariate structure of the data set.     

Toward the evaluation of P(X(t) > Y(t)) when both X(t) and Y(t) are inactivity times of two systems

The inactivity time, also known as reversed residual life, has been a topic of increasing interest in the literature. In this investigation, based on the comparison of inactivity times of two devices, we introduce and study a new estimate of the probability of the inactivity time of one device exceeding that of another device. The problem studied in this paper is important for engineers and system designers. It would enable them to compare the inactivity times of the products and, hence to design better products. Several properties of this probability are established. Connections between the target probability and the reversed hazard rates of the two devices are established. In addition, some of the reliability properties of the new concept are investigated extending the well-known probability ordering. Finally, to illustrate the introduced concepts, many examples and applications in the context of reliability theory are included.     

Flexible semi-parametric regression of state occupational probabilities in a multistate model with right-censored data

Inference for the state occupation probabilities, given a set of baseline covariates, is an important problem in survival analysis and time to event multistate data. We introduce an inverse censoring probability re-weighted semi-parametric single index model based approach to estimate conditional state occupation probabilities of a given individual in a multistate model under right-censoring. Besides obtaining a temporal regression function, we also test the potential time varying effect of a baseline covariate on future state occupation. We show that the proposed technique has desirable finite sample performances and its performance is competitive when compared with three other existing approaches. We illustrate the proposed methodology using two different data sets. First, we re-examine a well-known data set dealing with leukemia patients undergoing bone marrow transplant with various state transitions. Our second illustration is based on data from a study involving functional status of a set of spinal cord injured patients undergoing a rehabilitation program.     

A frequentist understanding of sets of measures

We present a mathematical theory of objective, frequentist chance phenomena that uses as a model a set of probability measures. In this work, sets of measures are not viewed as a statistical compound hypothesis or as a tool for modeling imprecise subjective behavior. Instead we use sets of measures to model stable (although not stationary in the traditional stochastic sense) physical sources of finite time series data that have highly irregular behavior. Such models give a coarse-grained picture of the phenomena, keeping track of the range of the possible probabilities of the events. We present methods to simulate finite data sequences coming from a source modeled by a set of probability measures, and to estimate the model from finite time series data. The estimation of the set of probability measures is based on the analysis of a set of relative frequencies of events taken along subsequences selected by a collection of rules. In particular, we provide a universal methodology for finding a family of subsequence selection rules that can estimate any set of probability measures with high probability.     

Discretizing a compound distribution with application to categorical modelling

Many probability distributions can be represented as compound distributions. Consider some parameter vector as random. The compound distribution is the expected distribution of the variable of interest given the random parameters. Our idea is to define a partition of the domain of definition of the random parameters, so that we can represent the expected density of the variable of interest as a finite mixture of conditional densities. We then model the mixture probabilities of the conditional densities using information on population categories, thus modifying the original overall model. We thus obtain specific models for sub-populations that stem from the overall model. The distribution of a sub-population of interest is thus completely specified in terms of mixing probabilities. All characteristics of interest can be derived from this distribution and the comparison between sub-populations easily proceeds from the comparison of the mixing probabilities. A real example based on EU-SILC data is given. Then the methodology is investigated through simulation.     

Estimating reference ranges from stratified two-stage samples

We present methodology for estimating age-specific reference ranges by using data from two-stage samples. On the basis of the information obtained in the first stage, the initial sample is stratified and random subsamples are drawn from each stratum, where the selection probabilities in this second-stage sampling may be different across strata in the population. The variable for which the reference ranges are to be established is measured at the second phase. The approach involves maximum likelihood estimation of the parameters of the age-specific distributions and separate estimation of the population stratum probabilities. These are combined to yield estimates of the quantiles of interest. The issue of variance estimation for the estimated quantiles is also addressed. The methodology is applied to the estimation of reference ranges for a cognitive test score in a study of non-demented older Japanese-Americans.     

Dynamic Probabilities of Restrictions in State Space Models: An Application to the Phillips Curve

Using the Savage–Dickey density ratio and an alternative approach that uses more relaxed assumptions, we develop methods to calculate the probability that a restriction holds at a point in time without assuming that the restriction holds at any other points in time. Both approaches use MCMC output only from the unrestricted model to compute the time-varying posterior probabilities for all models of interest. Using U.S. data, we find the probability that the long-run Phillips curve is vertical to be fairly high, but decreases over time. The probability that the NAIRU is not identified fluctuates over time, but increases after 1990.     

Statistical modelling and saddle-point approximation of tail probabilities for accumulated splice loss in fibre-optic networks

Tail probabilities are calculated by saddle-point approximation in a probabilistic-statistical model for the accumulated splice loss that results from a number of fusion splices in the installation of fibre-optic networks. When these probabilities, representing the risk of exceeding a specified total loss, can be controlled and kept low, the requirements on the individual losses can be substantially relaxed from their customary settings. As a consequence, it should be possible to save considerable installation time and cost. The probabilistic model, which can be theoretically motivated, states that the individual loss is basically exponentially distributed, but with a Gaussian contribution added and truncated at a set value, and that the loss is additive over splices. An extensive set of installation data fitted well with this model, except for occasional high losses. Therefore, the model described was extended to allow for a frequency of unspecified high losses of this sort. It is also indicated how the model parameters can be estimated from data.     

Proportional hazards regression with interval censored data using an inverse probability weight

The prevalence of interval censored data is increasing in medical studies due to the growing use of biomarkers to define a disease progression endpoint. Interval censoring results from periodic monitoring of the progression status. For example, disease progression is established in the interval between the clinic visit where progression is recorded and the prior clinic visit where there was no evidence of disease progression. A methodology is proposed for estimation and inference on the regression coefficients in the Cox proportional hazards model with interval censored data. The methodology is based on estimating equations and uses an inverse probability weight to select event time pairs where the ordering is unambiguous. Simulations are performed to examine the finite sample properties of the estimate and a colon cancer data set is used to demonstrate its performance relative to the conventional partial likelihood estimate that ignores the interval censoring.     

Application of a non-homogeneous Markov chain with seasonal transition probabilities to ozone data

In this work, we assume that the sequence recording whether or not an ozone exceedance of an environmental threshold has occurred in a given day is ruled by a non-homogeneous Markov chain of order one. In order to account for the possible presence of cycles in the empirical transition probabilities, a parametric form incorporating seasonal components is considered. Results show that even though some covariates (namely, relative humidity and temperature) are not included explicitly in the model, their influence is captured in the behavior of the transition probabilities. Parameters are estimated using the Bayesian point of view via Markov chain Monte Carlo algorithms. The model is applied to ozone data obtained from the monitoring network of Mexico City, Mexico. An analysis of how the methodology could be used as an aid in the decision-making is also given.     

Continuous threshold models with two-way interactions in survival analysis

Proportional hazards model with the biomarker–treatment interaction plays an important role in the survival analysis of the subset treatment effect. A threshold parameter for a continuous biomarker variable defines the subset of patients who can benefit or lose from a certain new treatment. In this article, we focus on a continuous threshold effect using the rectified linear unit and propose a gradient descent method to obtain the maximum likelihood estimation of the regression coefficients and the threshold parameter simultaneously. Under certain regularity conditions, we prove the consistency, asymptotic normality and provide a robust estimate of the covariance matrix when the model is misspecified. To illustrate the finite sample properties of the proposed methods, we simulate data to evaluate the empirical biases, the standard errors and the coverage probabilities for both the correctly specified models and misspecified models. The proposed continuous threshold model is applied to a prostate cancer data with serum prostatic acid phosphatase as a biomarker.     

Gaussian models for degradation processes-part I: Methods for the analysis of biomarker data

We present two stochastic models that describe the relationship between biomarker process values at random time points, event times, and a vector of covariates. In both models the biomarker processes are degradation processes that represent the decay of systems over time. In the first model the biomarker process is a Wiener process whose drift is a function of the covariate vector. In the second model the biomarker process is taken to be the difference between a stationary Gaussian process and a time drift whose drift parameter is a function of the covariates. For both models we present statistical methods for estimation of the regression coefficients. The first model is useful for predicting the residual time from study entry to the time a critical boundary is reached while the second model is useful for predicting the latency time from the infection until the time the presence of the infection is detected. We present our methods principally in the context of conducting inference in a population of HIV infected individuals.     

Why a time effect often has a limited impact on capture‐recapture estimates in closed populations

The author is concerned with log‐linear estimators of the size N of a population in a capture‐recapture experiment featuring heterogeneity in the individual capture probabilities and a time effect. He also considers models where the first capture influences the probability of subsequent captures. He derives several results from a new inequality associated with a dispersive ordering for discrete random variables. He shows that in a log‐linear model with inter‐individual heterogeneity, the estimator N is an increasing function of the heterogeneity parameter. He also shows that the inclusion of a time effect in the capture probabilities decreases N in models without heterogeneity. He further argues that a model featuring heterogeneity can accommodate a time effect through a small change in the heterogeneity parameter. He demonstrates these results using an inequality for the estimators of the heterogeneity parameters and illustrates them in a Monte Carlo experiment     

A Bernoulli autoregressive moving average model applied to rainfall occurrence

Abstract

In this article, we propose a new model for binary time series involving an autoregressive moving average structure. The proposed model, which is an extension of the GARMA model, can be used for calculating the forecast probability of an occurrence of an event of interest in cases where these probabilities are dependent on previous observations in the near term. The proposed model is used to analyze a real dataset involving a series that contains only data 0 and 1, indicating the absence or presence of rain in a city located in the central region of São Paulo state, Brazil.     

The use of multi-state capture-recapture models to address questions in evolutionary ecology

Multi-state capture-recapture models can be used to estimate survival rates in populations that are stratified by location or by state variables associated with individual animals. In populations stratified by location, movement probabilities can be estimated and used to test hypotheses relevant to population genetics and evolutionary ecology. When the interest is in state variables, these models permit estimation and testing of hypotheses about state-specific survival probabilities. If the state variable of interest is reproductive activity or success, then the multi-state modeling approach can be used to test hypotheses about life history trade-offs and a possible cost of reproduction.