Variance estimation for statistics computed from single recurrent event processes

This article is concerned with variance estimation for statistics that are computed from single recurrent event processes. Such statistics are important in diagnosis for each individual recurrent event process. The proposed method only assumes a semiparametric form for the first-order structure of the processes but not for the second-order (i.e., dependence) structure. The new variance estimator is shown to be consistent for the target parameter under very mild conditions. The estimator can be used in many applications in semiparametric rate regression analysis of recurrent event data such as outlier detection, residual diagnosis, as well as robust regression. A simulation study and application to two real data examples are used to demonstrate the use of the proposed method.     

A joint modeling approach for analyzing marker data in the presence of a terminal event

In many medical studies, markers are contingent on recurrent events and the cumulative markers are usually of interest. However, the recurrent event process is often interrupted by a dependent terminal event, such as death. In this article, we propose a joint modeling approach for analyzing marker data with informative recurrent and terminal events. This approach introduces a shared frailty to specify the explicit dependence structure among the markers, the recurrent, and terminal events. Estimation procedures are developed for the model parameters and the degree of dependence, and a prediction of the covariate‐specific cumulative markers is provided. The finite sample performance of the proposed estimators is examined through simulation studies. An application to a medical cost study of chronic heart failure patients from the University of Virginia Health System is illustrated.     

A parametric model for estimating prevalence,incidence, and mean bout duration from point sampling

Point or instantaneous sampling refers to the scoring of presence or absence of behavior at the end of equally spaced intervals of time and is used to estimate prevalence. The literature cited demonstrates that point sampling does not adequately estimate frequency or mean bout duration. A parametric model is developed based on exponentially distributed times of behavior and intervening nonbehavior, thus enabling estimators of mean bout length and incidence. Variance estimators are provided and a method is suggested for designing sample situations which control the variance of the prevalence estimator. The paper concludes the theoretical investigation with a thorough Monte Carlo investigation and application to a “real-life” problem. The point sample estimators compare favorably with continuous observation under appropriate choice of sampling interval and under approximately exponential assumptions.     

Semiparametric Transformation Models with Time‐Varying Coefficients for Recurrent and Terminal Events

Summary In this article, we propose a family of semiparametric transformation models with time‐varying coefficients for recurrent event data in the presence of a terminal event such as death. The new model offers great flexibility in formulating the effects of covariates on the mean functions of the recurrent events among survivors at a given time. For the inference on the proposed models, a class of estimating equations is developed and asymptotic properties of the resulting estimators are established. In addition, a lack‐of‐fit test is provided for assessing the adequacy of the model, and some tests are presented for investigating whether or not covariate effects vary with time. The finite‐sample behavior of the proposed methods is examined through Monte Carlo simulation studies, and an application to a bladder cancer study is also illustrated.     

Semiparametric Transformation Rate Model for Recurrent Event Data

In this article, we propose a class of semiparametric transformation rate models for recurrent event data subject to right censoring and potentially stopped by a terminating event (e.g., death). These transformation models include both additive rates model and proportional rates model as special cases. Respecting the property that no recurrent events can occur after the terminating event, we model the conditional recurrent event rate given survival. Weighted estimating equations are constructed to estimate the regression coefficients and baseline rate function. In particular, the baseline rate function is approximated by wavelet function. Asymptotic properties of the proposed estimators are derived and a data-dependent criterion is proposed for selecting the most suitable transformation. Simulation studies show that the proposed estimators perform well for practical sample sizes. The proposed methods are used in two real-data examples: a randomized trial of rhDNase and a community trial of vitamin A.     

Bias-corrected variance estimation and hypothesis testing for spatial point and marked point processes using subsampling

Summary We introduce novel regression extrapolation based methods to correct the often large bias in subsampling variance estimation as well as hypothesis testing for spatial point and marked point processes. For variance estimation, our proposed estimators are linear combinations of the usual subsampling variance estimator based on subblock sizes in a continuous interval. We show that they can achieve better rates in mean squared error than the usual subsampling variance estimator. In particular, for n×n observation windows, the optimal rate of n^?2 can be achieved if the data have a finite dependence range. For hypothesis testing, we apply the proposed regression extrapolation directly to the test statistics based on different subblock sizes, and therefore avoid the need to conduct bias correction for each element in the covariance matrix used to set up the test statistics. We assess the numerical performance of the proposed methods through simulation, and apply them to analyze a tropical forest data set.     

A Monte Carlo evaluation of five interval estimators for the relative risk in sparse data

The relative risk (RR) is one of the most frequently used indices to measure the strength of association between a disease and a risk factor in etiological studies or the efficacy of an experimental treatment in clinical trials. In this paper, we concentrate attention on interval estimation of RR for sparse data, in which we have only a few patients per stratum, but a moderate or large number of strata. We consider five asymptotic interval estimators for RR, including a weighted least-squares (WLS) interval estimator with an ad hoc adjustment procedure for sparse data, an interval estimator proposed elsewhere for rare events, an interval estimator based on the Mantel-Haenszel (MH) estimator with a logarithmic transformation, an interval estimator calculated from a quadratic equation, and an interval estimator derived from the ratio estimator with a logarithmic transformation. On the basis of Monte Carlo simulations, we evaluate and compare the performance of these five interval estimators in a variety of situations. We note that, except for the cases in which the underlying common RR across strata is around 1, using the WLS interval estimator with the adjustment procedure for sparse data can be misleading. We note further that using the interval estimator suggested elsewhere for rare events tends to be conservative and hence leads to loss of efficiency. We find that the other three interval estimators can consistently perform well even when the mean number of patients for a given treatment is approximately 3 patients per stratum and the number of strata is as small as 20. Finally, we use a mortality data set comparing two chemotherapy treatments in patients with multiple myeloma to illustrate the use of the estimators discussed in this paper.     

Tests for multivariate recurrent events in the presence of a terminal event

In studies involving diseases associated with high rates of mortality, trials are frequently conducted to evaluate the effects of therapeutic interventions on recurrent event processes terminated by death. In this setting, cumulative mean functions form a natural basis for inference for questions of a health economic nature, and Ghosh and Lin (2000) recently proposed a relevant class of test statistics. Trials of patients with cancer metastatic to bone, however, involve multiple types of skeletal complications, each of which may be repeatedly experienced by patients over their lifetime. Traditionally the distinction between the various types of events is ignored and univariate analyses are conducted based on a composite recurrent event. However, when the events have different impacts on patients' quality of life, or when they incur different costs, it can be important to gain insight into the relative frequency of the specific types of events and treatment effects thereon. This may be achieved by conducting separate marginal analyses with each analysis focusing on one type of recurrent event. Global inferences regarding treatment benefit can then be achieved by carrying out multiplicity adjusted marginal tests, more formal multiple testing procedures, or by constructing global test statistics. We describe methods for testing for differences in mean functions between treatment groups which accommodate the fact that each particular event process is ultimately terminated by death. The methods are illustrated by application to a motivating study designed to examine the effect of bisphosphonate therapy on the incidence of skeletal complications among patients with breast cancer metastatic to bone. We find that there is a consistent trend towards a reduction in the cumulative mean for all four types of skeletal complications with bisphosphonate therapy; there is a significant reduction in the need for radiation therapy for the treatment of bone. The global test suggests that bisphosphonate therapy significantly reduces the overall number of skeletal complications.     

Improved confidence intervals for a difference of two cause-specific cumulative incidence functions estimated in the presence of competing risks and random censoring

A cause-specific cumulative incidence function (CIF) is the probability of failure from a specific cause as a function of time. In randomized trials, a difference of cause-specific CIFs (treatment minus control) represents a treatment effect. Cause-specific CIF in each intervention arm can be estimated based on the usual non-parametric Aalen–Johansen estimator which generalizes the Kaplan–Meier estimator of CIF in the presence of competing risks. Under random censoring, asymptotically valid Wald-type confidence intervals (CIs) for a difference of cause-specific CIFs at a specific time point can be constructed using one of the published variance estimators. Unfortunately, these intervals can suffer from substantial under-coverage when the outcome of interest is a rare event, as may be the case for example in the analysis of uncommon adverse events. We propose two new approximate interval estimators for a difference of cause-specific CIFs estimated in the presence of competing risks and random censoring. Theoretical analysis and simulations indicate that the new interval estimators are superior to the Wald CIs in the sense of avoiding substantial under-coverage with rare events, while being equivalent to the Wald CIs asymptotically. In the absence of censoring, one of the two proposed interval estimators reduces to the well-known Agresti–Caffo CI for a difference of two binomial parameters. The new methods can be easily implemented with any software package producing point and variance estimates for the Aalen–Johansen estimator, as illustrated in a real data example.     

A Class of Ratio-Cum-Product-Type Estimator

The problem of estimating the population mean using an auxiliary information has been dealt with in literature quite extensively. Ratio, product, linear regression and ratio-type estimators are well known. A class of ratio-cum-product-type estimator is proposed in this paper. Its bias and variance to the first order of approximation are obtained. For an appropriate weight ‘a’ and good range of α-values, it is found that the proposed estimator is superior than a set of estimators (i.e., sample mean, usual ratio and product estimators, SRIVASTAVA's (1967) estimator, CHAKRABARTY's (1979) estimator and a product-type estimator) which are, in fact, the particular cases of it. At optimum value of α, the proposed estimator is as efficient as linear regression estimator.     

On restricted mean time in favor of treatment

The restricted mean time in favor (RMT-IF) of treatment is a nonparametric effect size for complex life history data. It is defined as the net average time the treated spend in a more favorable state than the untreated over a prespecified time window. It generalizes the familiar restricted mean survival time (RMST) from the two-state life–death model to account for intermediate stages in disease progression. The overall estimand can be additively decomposed into stage-wise effects, with the standard RMST as a component. Alternate expressions of the overall and stage-wise estimands as integrals of the marginal survival functions for a sequence of landmark transitioning events allow them to be easily estimated by plug-in Kaplan–Meier estimators. The dynamic profile of the estimated treatment effects as a function of follow-up time can be visualized using a multilayer, cone-shaped “bouquet plot.” Simulation studies under realistic settings show that the RMT-IF meaningfully and accurately quantifies the treatment effect and outperforms traditional tests on time to the first event in statistical efficiency thanks to its fuller utilization of patient data. The new methods are illustrated on a colon cancer trial with relapse and death as outcomes and a cardiovascular trial with recurrent hospitalizations and death as outcomes. The R-package rmt implements the proposed methodology and is publicly available from the Comprehensive R Archive Network (CRAN).     

Nonparametric inference of general while-alive estimands for recurrent events

Measuring the treatment effect on recurrent events like hospitalization in the presence of death has long challenged statisticians and clinicians alike. Traditional inference on the cumulative frequency unjustly penalizes survivorship as longer survivors also tend to experience more adverse events. Expanding a recently suggested idea of the “while-alive” event rate, we consider a general class of such estimands that adjust for the length of survival without losing causal interpretation. Given a user-specified loss function that allows for arbitrary weighting, we define as estimand the average loss experienced per unit time alive within a target period and use the ratio of this loss rate to measure the effect size. Scaling the loss rate by the width of the corresponding time window gives us an alternative, and sometimes more photogenic, way of showing the data. To make inferences, we construct a nonparametric estimator for the loss rate through the cumulative loss and the restricted mean survival time and derive its influence function in closed form for variance estimation and testing. As simulations and analysis of real data from a heart failure trial both show, the while-alive approach corrects for the false attenuation of treatment effect due to patients living longer under treatment, with increased statistical power as a result. The proposed methods are implemented in the R-package WA , which is publicly available from the Comprehensive R Archive Network (CRAN).     

Comparative evaluation of a new effective population size estimator based on approximate bayesian computation

We describe and evaluate a new estimator of the effective population size (N(e)), a critical parameter in evolutionary and conservation biology. This new "SummStat" N(e) estimator is based upon the use of summary statistics in an approximate Bayesian computation framework to infer N(e). Simulations of a Wright-Fisher population with known N(e) show that the SummStat estimator is useful across a realistic range of individuals and loci sampled, generations between samples, and N(e) values. We also address the paucity of information about the relative performance of N(e) estimators by comparing the SummStat estimator to two recently developed likelihood-based estimators and a traditional moment-based estimator. The SummStat estimator is the least biased of the four estimators compared. In 32 of 36 parameter combinations investigated using initial allele frequencies drawn from a Dirichlet distribution, it has the lowest bias. The relative mean square error (RMSE) of the SummStat estimator was generally intermediate to the others. All of the estimators had RMSE > 1 when small samples (n = 20, five loci) were collected a generation apart. In contrast, when samples were separated by three or more generations and N(e) < or = 50, the SummStat and likelihood-based estimators all had greatly reduced RMSE. Under the conditions simulated, SummStat confidence intervals were more conservative than the likelihood-based estimators and more likely to include true N(e). The greatest strength of the SummStat estimator is its flexible structure. This flexibility allows it to incorporate any potentially informative summary statistic from population genetic data.     

Joint Models of Longitudinal Data and Recurrent Events with Informative Terminal Event

This article presents semiparametric joint models to analyze longitudinal data with recurrent events (e.g. multiple tumors, repeated hospital admissions) and a terminal event such as death. A broad class of transformation models for the cumulative intensity of the recurrent events and the cumulative hazard of the terminal event is considered, which includes the proportional hazards model and the proportional odds model as special cases. We propose to estimate all the parameters using the nonparametric maximum likelihood estimators (NPMLE). We provide the simple and efficient EM algorithms to implement the proposed inference procedure. Asymptotic properties of the estimators are shown to be asymptotically normal and semiparametrically efficient. Finally, we evaluate the performance of the method through extensive simulation studies and a real-data application.     

Density estimation in live-trapping studies

Unbiased estimation of population density is a major and unsolved problem in animal trapping studies. This paper describes a new and general method for estimating density from closed-population capture–recapture data. Many estimators exist for the size (N) and mean capture probability ( p ) of a closed population. These statistics suffer from an unknown bias due to edge effect that varies with trap layout and home range size. The mean distance between successive captures of an individual (     ) provides information on the scale of individual movements, but is itself a function of trap spacing and grid size. Our aim is to define and estimate parameters that do not depend on the trap layout. In the new method, simulation and inverse prediction are used to estimate jointly the population density (D) and two parameters of individual capture probability, magnitude (g₀) and spatial scale (σ), from the information in     , p and     . The method uses any configuration of traps (e.g. grid, web or line) and any choice of closed-population estimator. It is assumed that home ranges have a stationary distribution in two dimensions, and that capture events may be simulated as the outcome of competing Poisson processes in time. The method is applied to simulated and field data. The estimator appears unusually robust and free from bias.     

Properties and unbiased estimation of F- and D-statistics in samples containing related and inbred individuals

The Patterson F- and D-statistics are commonly used measures for quantifying population relationships and for testing hypotheses about demographic history. These statistics make use of allele frequency information across populations to infer different aspects of population history, such as population structure and introgression events. Inclusion of related or inbred individuals can bias such statistics, which may often lead to the filtering of such individuals. Here, we derive statistical properties of the F- and D-statistics, including their biases due to the inclusion of related or inbred individuals, their variances, and their corresponding mean squared errors. Moreover, for those statistics that are biased, we develop unbiased estimators and evaluate the variances of these new quantities. Comparisons of the new unbiased statistics to the originals demonstrates that our newly derived statistics often have lower error across a wide population parameter space. Furthermore, we apply these unbiased estimators using several global human populations with the inclusion of related individuals to highlight their application on an empirical dataset. Finally, we implement these unbiased estimators in open-source software package funbiased for easy application by the scientific community.     

On estimation in relative survival

Estimation of relative survival has become the first and the most basic step when reporting cancer survival statistics. Standard estimators are in routine use by all cancer registries. However, it has been recently noted that these estimators do not provide information on cancer mortality that is independent of the national general population mortality. Thus they are not suitable for comparison between countries. Furthermore, the commonly used interpretation of the relative survival curve is vague and misleading. The present article attempts to remedy these basic problems. The population quantities of the traditional estimators are carefully described and their interpretation discussed. We then propose a new estimator of net survival probability that enables the desired comparability between countries. The new estimator requires no modeling and is accompanied with a straightforward variance estimate. The methods are described on real as well as simulated data.     

Mean and variance of ratio estimators used in fluorescence ratio imaging

BACKGROUND: The ratio of two measured fluorescence signals (called x and y) is used in different applications in fluorescence microscopy. Multiple instances of both signals can be combined in different ways to construct different ratio estimators. METHODS: The mean and variance of three estimators for the ratio between two random variables, x and y, are discussed. Given n samples of x and y, we can intuitively construct two different estimators: the mean of the ratio of each x and y and the ratio between the mean of x and the mean of y. The former is biased and the latter is only asymptotically unbiased. Using the statistical characteristics of this estimator, a third, unbiased estimator can be constructed. RESULTS: We tested the three estimators on simulated data, real-world fluorescence test images, and comparative genome hybridization (CGH) data. The results on the simulated and real-world test images confirm the presented theory. The CGH experiments show that our new estimator performs better than the existing estimators. CONCLUSIONS: We have derived an unbiased ratio estimator that outperforms intuitive ratio estimators.     

An Efficient Regression Type Estimator in Survey Sampling

For the estimation of population mean in simple random sampling, an efficient regression-type estimator is proposed which is more efficient than the conventional regression estimator and hence than mean per unit estimator, ratio and product estimators and many other estimators proposed by various authors. Some numerical examples are included for illustration.     

Estimating pairwise relatedness from dominant genetic markers

Knowledge of the genetic relatedness between a pair of individuals is important in many research areas of quantitative genetics, conservation genetics, evolution and ecology. Many estimators have been developed to estimate such pairwise relatedness (r) using codominant markers, such as microsatellites and enzymes. In contrast, only two estimators are proposed to use dominant markers, such as random amplified polymorphic DNAs (RAPDs) and amplified fragment length polymorphisms (AFLPs), in relatedness inference. They are both biased estimators, and their statistical properties and robustness to the sampling errors in allele frequency have not been investigated. In this short paper, I propose two new pairwise relatedness estimators for dominant markers, and compare them in precision, accuracy and robustness to sampling with the two previous estimators using simulations. It was found that the new estimator based on the least squares approach is unbiased when allele frequencies are known or estimated from a sample without correcting for sampling effects. It has, however, a low precision and as a result, an intermediate overall performance among the four estimators in terms of the mean squared deviation (MSD) of estimates from actual values of r. The new estimator based on a similarity index is slightly biased but has generally the lowest MSD among the four estimators compared, regardless of the number of loci, type of actual relationships, allele frequencies known or estimated from samples. Simulations also show that the confidence intervals estimated by bootstrapping are appropriate for different estimators provided that the number of loci used in the estimation is not small.