Missing covariates in longitudinal data with informative dropouts: bias analysis and inference

We consider estimation in generalized linear mixed models (GLMM) for longitudinal data with informative dropouts. At the time a unit drops out, time-varying covariates are often unobserved in addition to the missing outcome. However, existing informative dropout models typically require covariates to be completely observed. This assumption is not realistic in the presence of time-varying covariates. In this article, we first study the asymptotic bias that would result from applying existing methods, where missing time-varying covariates are handled using naive approaches, which include: (1) using only baseline values; (2) carrying forward the last observation; and (3) assuming the missing data are ignorable. Our asymptotic bias analysis shows that these naive approaches yield inconsistent estimators of model parameters. We next propose a selection/transition model that allows covariates to be missing in addition to the outcome variable at the time of dropout. The EM algorithm is used for inference in the proposed model. Data from a longitudinal study of human immunodeficiency virus (HIV)-infected women are used to illustrate the methodology.     

Adjusting for publication bias in meta-analysis via inverse probability weighting using clinical trial registries

Publication bias is a major concern in conducting systematic reviews and meta-analyses. Various sensitivity analysis or bias-correction methods have been developed based on selection models, and they have some advantages over the widely used trim-and-fill bias-correction method. However, likelihood methods based on selection models may have difficulty in obtaining precise estimates and reasonable confidence intervals, or require a rather complicated sensitivity analysis process. Herein, we develop a simple publication bias adjustment method by utilizing the information on conducted but still unpublished trials from clinical trial registries. We introduce an estimating equation for parameter estimation in the selection function by regarding the publication bias issue as a missing data problem under the missing not at random assumption. With the estimated selection function, we introduce the inverse probability weighting (IPW) method to estimate the overall mean across studies. Furthermore, the IPW versions of heterogeneity measures such as the between-study variance and the I² measure are proposed. We propose methods to construct confidence intervals based on asymptotic normal approximation as well as on parametric bootstrap. Through numerical experiments, we observed that the estimators successfully eliminated bias, and the confidence intervals had empirical coverage probabilities close to the nominal level. On the other hand, the confidence interval based on asymptotic normal approximation is much wider in some scenarios than the bootstrap confidence interval. Therefore, the latter is recommended for practical use.     

Finding confidence limits on population growth rates: Bootstrap and analytic methods

When predicting population dynamics, the value of the prediction is not enough and should be accompanied by a confidence interval that integrates the whole chain of errors, from observations to predictions via the estimates of the parameters of the model. Matrix models are often used to predict the dynamics of age- or size-structured populations. Their parameters are vital rates. This study aims (1) at assessing the impact of the variability of observations on vital rates, and then on model’s predictions, and (2) at comparing three methods for computing confidence intervals for values predicted from the models. The first method is the bootstrap. The second method is analytic and approximates the standard error of predictions by their asymptotic variance as the sample size tends to infinity. The third method combines use of the bootstrap to estimate the standard errors of vital rates with the analytical method to then estimate the errors of predictions from the model. Computations are done for an Usher matrix models that predicts the asymptotic (as time goes to infinity) stock recovery rate for three timber species in French Guiana. Little difference is found between the hybrid and the analytic method. Their estimates of bias and standard error converge towards the bootstrap estimates when the error on vital rates becomes small enough, which corresponds in the present case to a number of observations greater than 5000 trees.     

Methods of estimation in log odds ratio regression models

McCullagh's (1984, Journal of the Royal Statistical Society, Series B 46, 250-256) approximation to the conditional maximum likelihood estimator in log odds ratio regression models is shown to have negligible asymptotic bias unless the odds ratios are large and the sample sizes in individual 2 X 2 tables are very small. In application to two sets of case-control data, it yields results virtually indistinguishable from those of the conditional analysis. A generalization of the Mantel-Haenszel estimator proposed by Davis (1985, Biometrics 41, 487-495) does not approximate the conditional results nearly as well.     

Time series modeling of neuromuscular system

The dynamic response of the human ankle joint to a bandlimited random torque perturbation superimposed on a constant bias torque is observed in normal human subjects. The applied torque input, the joint angular rotation output, and the electromyographic activity using surface electrodes from the extensor and the flexor muscles of the ankle joint were recorded. Transfer function models using time series techniques were developed for the torque — angular rotation input-output pair and for the angular rotation — electromyographic activity input-output pair. A parameter constraining technique was applied to develop more reliable models. It is shown that the asymptotic behavior of the system must be taken into account during parameter optimization to develop better predictive models.This work was supported in part by National Science Foundation grant ENG-7608754 and grants from the National Institutes of Health NS-12877 and NS-00196     

The impact of litter effects on dose-response modeling in teratology

The fitting of dose-response models to teratology data involving littermates in order to generate estimates of teratogenic risk is receiving increasing attention as a potential alternative to the "safety-factor" approach to risk estimation. In this paper, we utilize the beta-binomial distribution to introduce varying degrees of intralitter correlation, and, for purposes of illustration, consider a logistic dose-response model that describes the logit of risk as a straight-line function of ln(dose). The biases and (exact and asymptotic) variances of the maximum likelihood estimators of the intercept and slope are studied by simulation as a function of the intralitter correlation structure.     

An analysis of neutral-alleles and variable-environment diffusion models

Three diffusion models are formulated for the evolution of a diploid population with K alleles at one locus with completely symmetric mutation and random genetic drift, a variable-environment, and all the above mechanisms. For the diallelic case, the transient behavior is studied by solving the corresponding diffusion equations by an asymptotic method valid for short time intervals. The transient behavior of the three models is compared for the case when their stationary distributions are identical. The expected amount of heterozygosity is computed using the asymptotic solution and is compared to an exact result. The asymptotic results are extended to the general case with K alleles at the locus for the symmetric mutation and variable-environment models.Research supported by the National Science Foundation under Grant MCS 79-01718     

Maximum likelihood estimators for incorrect models, with an application to ascertainment bias for continuous characters

Uncertainty about the ascertainment of human family data leads to a need for robust methods for estimating genetic and environmental effects. This in turn leads to a need for efficient techniques for estimating model parameters for data generated under one parametric model but analyzed under a second model. If the two models correspond to different ascertainment schemes for the same exponential family, simple formulas for the asymptotic means and standard errors of both conditional and unconditional MLEs can be derived. In an example for continuous sibship data, these formulas show that estimates derived from conditioning on proband value have greater asymptotic bias than two other estimators. Similarly, either conditioning on proband value or conditioning on the number of affected family members resulted in biases of up to 30% when ascertainment depended on the values of more than one affected family member.     

Dynamic sampling bias and overdispersion induced by skewed offspring distributions

Natural populations often show enhanced genetic drift consistent with a strong skew in their offspring number distribution. The skew arises because the variability of family sizes is either inherently strong or amplified by population expansions. The resulting allele-frequency fluctuations are large and, therefore, challenge standard models of population genetics, which assume sufficiently narrow offspring distributions. While the neutral dynamics backward in time can be readily analyzed using coalescent approaches, we still know little about the effect of broad offspring distributions on the forward-in-time dynamics, especially with selection. Here, we employ an asymptotic analysis combined with a scaling hypothesis to demonstrate that over-dispersed frequency trajectories emerge from the competition of conventional forces, such as selection or mutations, with an emerging time-dependent sampling bias against the minor allele. The sampling bias arises from the characteristic time-dependence of the largest sampled family size within each allelic type. Using this insight, we establish simple scaling relations for allele-frequency fluctuations, fixation probabilities, extinction times, and the site frequency spectra that arise when offspring numbers are distributed according to a power law.     

Bias in estimating association parameters for longitudinal binary responses with drop-outs

This paper considers the impact of bias in the estimation of the association parameters for longitudinal binary responses when there are drop-outs. A number of different estimating equation approaches are considered for the case where drop-out cannot be assumed to be a completely random process. In particular, standard generalized estimating equations (GEE), GEE based on conditional residuals, GEE based on multivariate normal estimating equations for the covariance matrix, and second-order estimating equations (GEE2) are examined. These different GEE estimators are compared in terms of finite sample and asymptotic bias under a variety of drop-out processes. Finally, the relationship between bias in the estimation of the association parameters and bias in the estimation of the mean parameters is explored.     

Bias analysis and the simulation‐extrapolation method for survival data with covariate measurement error under parametric proportional odds models

It has been well known that ignoring measurement error may result in substantially biased estimates in many contexts including linear and nonlinear regressions. For survival data with measurement error in covariates, there has been extensive discussion in the literature with the focus on proportional hazards (PH) models. Recently, research interest has extended to accelerated failure time (AFT) and additive hazards (AH) models. However, the impact of measurement error on other models, such as the proportional odds model, has received relatively little attention, although these models are important alternatives when PH, AFT, or AH models are not appropriate to fit data. In this paper, we investigate this important problem and study the bias induced by the naive approach of ignoring covariate measurement error. To adjust for the induced bias, we describe the simulation‐extrapolation method. The proposed method enjoys a number of appealing features. Its implementation is straightforward and can be accomplished with minor modifications of existing software. More importantly, the proposed method does not require modeling the covariate process, which is quite attractive in practice. As the precise values of error‐prone covariates are often not observable, any modeling assumption on such covariates has the risk of model misspecification, hence yielding invalid inferences if this happens. The proposed method is carefully assessed both theoretically and empirically. Theoretically, we establish the asymptotic normality for resulting estimators. Numerically, simulation studies are carried out to evaluate the performance of the estimators as well as the impact of ignoring measurement error, along with an application to a data set arising from the Busselton Health Study. Sensitivity of the proposed method to misspecification of the error model is studied as well.     

Semiparametric estimation of the transformation model by leveraging external aggregate data in the presence of population heterogeneity

Leveraging information in aggregate data from external sources to improve estimation efficiency and prediction accuracy with smaller scale studies has drawn a great deal of attention in recent years. Yet, conventional methods often either ignore uncertainty in the external information or fail to account for the heterogeneity between internal and external studies. This article proposes an empirical likelihood-based framework to improve the estimation of the semiparametric transformation models by incorporating information about the t-year subgroup survival probability from external sources. The proposed estimation procedure incorporates an additional likelihood component to account for uncertainty in the external information and employs a density ratio model to characterize population heterogeneity. We establish the consistency and asymptotic normality of the proposed estimator and show that it is more efficient than the conventional pseudopartial likelihood estimator without combining information. Simulation studies show that the proposed estimator yields little bias and outperforms the conventional approach even in the presence of information uncertainty and heterogeneity. The proposed methodologies are illustrated with an analysis of a pancreatic cancer study.     

Empirical Likelihood Semiparametric Regression Analysis for Longitudinal Data

A semiparametric regression model for longitudinal data is considered.The empirical likelihood method is used to estimate the regressioncoefficients and the baseline function, and to construct confidenceregions and intervals. It is proved that the maximum empiricallikelihood estimator of the regression coefficients achievesasymptotic efficiency and the estimator of the baseline functionattains asymptotic normality when a bias correction is made.Two calibrated empirical likelihood approaches to inferencefor the baseline function are developed. We propose a groupwiseempirical likelihood procedure to handle the inter-series dependencefor the longitudinal semiparametric regression model, and employbias correction to construct the empirical likelihood ratiofunctions for the parameters of interest. This leads us to provea nonparametric version of Wilks' theorem. Compared with methodsbased on normal approximations, the empirical likelihood doesnot require consistent estimators for the asymptotic varianceand bias. A simulation compares the empirical likelihood andnormal-based methods in terms of coverage accuracies and averageareas/lengths of confidence regions/intervals.     

Inverse probability weighting estimation of the volume under the ROC surface in the presence of verification bias

In diagnostic medicine, the volume under the receiver operating characteristic (ROC) surface (VUS) is a commonly used index to quantify the ability of a continuous diagnostic test to discriminate between three disease states. In practice, verification of the true disease status may be performed only for a subset of subjects under study since the verification procedure is invasive, risky, or expensive. The selection for disease examination might depend on the results of the diagnostic test and other clinical characteristics of the patients, which in turn can cause bias in estimates of the VUS. This bias is referred to as verification bias. Existing verification bias correction in three‐way ROC analysis focuses on ordinal tests. We propose verification bias‐correction methods to construct ROC surface and estimate the VUS for a continuous diagnostic test, based on inverse probability weighting. By applying U‐statistics theory, we develop asymptotic properties for the estimator. A Jackknife estimator of variance is also derived. Extensive simulation studies are performed to evaluate the performance of the new estimators in terms of bias correction and variance. The proposed methods are used to assess the ability of a biomarker to accurately identify stages of Alzheimer's disease.     

Incorporating an Asymptotic Parameter into the Weibull Model to Describe Plant Disease Progress

The Weibull model is a flexible growth model that describes both general population growth and plant disease progress. However, lack of an asymptotic parameter has limited its wider application. In the present study, an asymptotic parameter K was introduced into the original Weibull model, written as; y = K {1 − exp [− ( t − a ) ^c ]}, in which a , b , c and K are location, scale, shape, and asymptotic parameters, respectively, y is the proportion of disease and t is time. A wide range of simulated disease progress data sets were generated using logistic, Gompertz and monomolecular models by specifying different parameter values, and fitted to both original and modified Weibull models. The modified model provided statistically better fits for all data than the original model. The modified model can thus improve the curve-fitting ability of the original model which often failed to converge, especially when the asymptote is less than 1.0. Actual disease progress data on wheat leaf rust and tomato root rot with different asymptotic values were also used to compare the original and modified Weibull models. The modified model provided a statistically better fit than the original model, and model estimates of asymptotic parameter K were nearly identical to the actual disease maxima reflecting the characteristics of the host-pathosystem. Comparison of logistic, Gompertz, and Weibull models including parameter K by fitting to the observed data on wheat leaf rust and tomato root rot revealed the applicability of the modified Weibull model, which in a majority of cases provided a statistically superior fit.     

Asymptotic distribution of stage-grouped population models

Matrix models are often used to predict the dynamics of size-structured or age-structured populations. The asymptotic behaviour of such models is defined by their malthusian growth rate lambda, and by their stationary distribution w that gives the asymptotic proportion of individuals in each stage. As the coefficients of the transition matrix are estimated from a sample of observations, lambda and w can be considered as random variables whose law depends on the distribution of the observations. The goal of this study is to specify the asymptotic law of lambda and w when using the maximum likelihood estimators of the coefficients of the transition matrix. We prove that lambda and w are asymptotically normal, and the expressions of the asymptotic variance of lambda and of the asymptotic covariance matrix of w are given. The convergence speed of lambda and w towards their asymptotic law is studied using simulations. The results are applied to a real case study that consists of a Usher model for a tropical rain forest in French Guiana. They permit to assess the number of trees to measure to get a given precision on the estimated asymptotic diameter distribution, which is an important information on tropical forest management.     

The second Southern African Bird Atlas Project: Causes and consequences of geographical sampling bias

Using the Southern African Bird Atlas Project (SABAP2) as a case study, we examine the possible determinants of spatial bias in volunteer sampling effort and how well such biased data represent environmental gradients across the area covered by the atlas. For each province in South Africa, we used generalized linear mixed models to determine the combination of variables that explain spatial variation in sampling effort (number of visits per 5′ × 5′ grid cell, or “pentad”). The explanatory variables were distance to major road and exceptional birding locations or “sampling hubs,” percentage cover of protected, urban, and cultivated area, and the climate variables mean annual precipitation, winter temperatures, and summer temperatures. Further, we used the climate variables and plant biomes to define subsets of pentads representing environmental zones across South Africa, Lesotho, and Swaziland. For each environmental zone, we quantified sampling intensity, and we assessed sampling completeness with species accumulation curves fitted to the asymptotic Lomolino model. Sampling effort was highest close to sampling hubs, major roads, urban areas, and protected areas. Cultivated area and the climate variables were less important. Further, environmental zones were not evenly represented by current data and the zones varied in the amount of sampling required representing the species that are present. SABAP2 volunteers' preferences in birding locations cause spatial bias in the dataset that should be taken into account when analyzing these data. Large parts of South Africa remain underrepresented, which may restrict the kind of ecological questions that may be addressed. However, sampling bias may be improved by directing volunteers toward undersampled regions while taking into account volunteer preferences.     

Evolution of eusociality in diploid species

Haplodiploid species are naturally biased by their genetic structure toward the evolution of sterile worker castes, as shown by W. D. Hamilton (1964. J. Theoret. Biol., 7, 1–16, 17–52). Diploid species do not have this intrinsic genetic bias toward eusociality. Nonetheless, true sociality has evolved in the diploid ancestors of the modern termites, and varying degrees of quasisociality are not uncommon in diploid species, including mammals. A genetic bias toward investment in relatives rather than offspring can arise in a diploid species as a result of inbreeding. The consequences of several regular incestuous breeding systems are analyzed in detail. It is shown that, under certain conditions, there is a natural bias toward an alternation of inbred and outbred generations. As this alternation proceeds, the genetic bias toward eusociality rapidly approaches an asymptotic value of 4(1 + 2f₀)/3(1 + 3f₀), where f₀ is the average coefficient of relationship for the outbreeding pairs. For f₀ close to zero, the genetic bias toward eusociality is close to 1.33, which is even larger than the genetic bias of 1.25 in haplodiploid species. Under other conditions there may be repeated incestuous matings between successive outbreeding generations. In this case the bias toward eusociality can be as large as 2.     

Integral equation models for endemic infectious diseases

Summary Endemic infectious diseases for which infection confers permanent immunity are described by a system of nonlinear Volterra integral equations of convolution type. These constant-parameter models include vital dynamics (births and deaths), immunization and distributed infectious period. The models are shown to be well posed, the threshold criteria are determined and the asymptotic behavior is analysed. It is concluded that distributed delays do not change the thresholds and the asymptotic behaviors of the models.This work was partially supported by NIH Grant AI 13233.     

Modelling publication bias and p-hacking

Publication bias and p-hacking are two well-known phenomena that strongly affect the scientific literature and cause severe problems in meta-analyses. Due to these phenomena, the assumptions of meta-analyses are seriously violated and the results of the studies cannot be trusted. While publication bias is very often captured well by the weighting function selection model, p-hacking is much harder to model and no definitive solution has been found yet. In this paper, we advocate the selection model approach to model publication bias and propose a mixture model for p-hacking. We derive some properties for these models, and we compare them formally and through simulations. Finally, two real data examples are used to show how the models work in practice.