Estimating correlation between alternative measures of disease progression in a longitudinal study. Modification of Diet in Renal Disease Study

A primary outcome variable in longitudinal studies is often the rate of change of a continuous measurement over time. Examples include the one-second forced expiratory volume (FEV1) in pulmonary studies or glomerular filtration rate (GFR) in renal studies. An individual patient's least-squares estimate of slope obtained from a linear regression is an imprecise measure of the true slope for that patient, and correlations involving the estimated slopes will be biased due to this measurement error. This paper presents methods for estimating the true correlation between these imprecise slope estimates, or between slope estimates and other variables measured with error. In addition to providing a simple consistent estimator of the correlation, we show how the maximum likelihood estimate of the correlation coefficient and a 100(1 - alpha) per cent confidence interval can be obtained. An example estimating the correlation between GFR and inverse serum creatinine slopes in patients with chronic renal disease is given.     

Modified robust variance estimator for generalized estimating equations with improved small-sample performance

Generalized estimating equations (GEE (Biometrika 1986; 73(1):13-22) is a general statistical method to fit marginal models for correlated or clustered responses, and it uses a robust sandwich estimator to estimate the variance-covariance matrix of the regression coefficient estimates. While this sandwich estimator is robust to the misspecification of the correlation structure of the responses, its finite sample performance deteriorates as the number of clusters or observations per cluster decreases. To address this limitation, Pan (Biometrika 2001; 88(3):901-906) and Mancl and DeRouen (Biometrics 2001; 57(1):126-134) investigated two modifications to the original sandwich variance estimator. Motivated by the ideas underlying these two modifications, we propose a novel robust variance estimator that combines the strengths of these estimators. Our theoretical and numerical results show that the proposed estimator attains better efficiency and achieves better finite sample performance compared with existing estimators. In particular, when the sample size or cluster size is small, our proposed estimator exhibits lower bias and the resulting confidence intervals for GEE estimates achieve better coverage rates performance. We illustrate the proposed method using data from a dental study.     

Estimating disease prevalence from two-phase surveys with non-response at the second phase

In this paper we compare several methods for estimating population disease prevalence from data collected by two-phase sampling when there is non-response at the second phase. The traditional weighting type estimator requires the missing completely at random assumption and may yield biased estimates if the assumption does not hold. We review two approaches and propose one new approach to adjust for non-response assuming that the non-response depends on a set of covariates collected at the first phase: an adjusted weighting type estimator using estimated response probability from a response model; a modelling type estimator using predicted disease probability from a disease model; and a regression type estimator combining the adjusted weighting type estimator and the modelling type estimator. These estimators are illustrated using data from an Alzheimer's disease study in two populations.     

Best linear inverse probability weighted estimation for two-phase designs and missing covariate regression

The inverse probability weighted estimator is often applied to two-phase designs and regression with missing covariates. Inverse probability weighted estimators typically are less efficient than likelihood-based estimators but, in general, are more robust against model misspecification. In this paper, we propose a best linear inverse probability weighted estimator for two-phase designs and missing covariate regression. Our proposed estimator is the projection of the SIPW onto the orthogonal complement of the score space based on a working regression model of the observed covariate data. The efficiency gain is from the use of the association between the outcome variable and the available covariates, which is the working regression model. One advantage of the proposed estimator is that there is no need to calculate the augmented term of the augmented weighted estimator. The estimator can be applied to general missing data problems or two-phase design studies in which the second phase data are obtained in a subcohort. The method can also be applied to secondary trait case-control genetic association studies. The asymptotic distribution is derived, and the finite sample performance of the proposed estimator is examined via extensive simulation studies. The methods are applied to a bladder cancer case-control study.     

Secondary outcome analysis for data from an outcome‐dependent sampling design

Outcome‐dependent sampling (ODS) scheme is a cost‐effective way to conduct a study. For a study with continuous primary outcome, an ODS scheme can be implemented where the expensive exposure is only measured on a simple random sample and supplemental samples selected from 2 tails of the primary outcome variable. With the tremendous cost invested in collecting the primary exposure information, investigators often would like to use the available data to study the relationship between a secondary outcome and the obtained exposure variable. This is referred as secondary analysis. Secondary analysis in ODS designs can be tricky, as the ODS sample is not a random sample from the general population. In this article, we use the inverse probability weighted and augmented inverse probability weighted estimating equations to analyze the secondary outcome for data obtained from the ODS design. We do not make any parametric assumptions on the primary and secondary outcome and only specify the form of the regression mean models, thus allow an arbitrary error distribution. Our approach is robust to second‐ and higher‐order moment misspecification. It also leads to more precise estimates of the parameters by effectively using all the available participants. Through simulation studies, we show that the proposed estimator is consistent and asymptotically normal. Data from the Collaborative Perinatal Project are analyzed to illustrate our method.     

Cluster effects and simultaneity in multilevel models

For small group sizes, the GLS estimator in multilevel models is biased and inconsistent when the random cluster effects are correlated with the regressors. A fixed effects approach, conditioning on the cluster effects, provides consistent estimates for the slope parameters. The two estimators are equivalent when group sizes are large. The same results obtain for two-stage estimation procedures that allow for some of the regressors to be simultaneously determined with the dependent variable. The GLS and fixed effects estimators are applied to data on acute care hospital utilization in the UK, allowing for health authority district effects. © 1997 John Wiley & Sons, Ltd.     

Estimation of the time of a linear trend in monitoring survival time

Change point estimation is recognized as an essential tool of root cause analyses within quality control programs as it enables clinical experts to search for potential causes of change in hospital outcomes more effectively. In this paper, we consider estimation of the time when a linear trend disturbance has occurred in survival time following an in-control clinical intervention in the presence of variable patient mix. To model the process and change point, a linear trend in the survival time of patients who underwent cardiac surgery is formulated using hierarchical models in a Bayesian framework. The data are right censored since the monitoring is conducted over a limited follow-up period. We capture the effect of risk factors prior to the surgery using a Weibull accelerated failure time regression model. We use Markov Chain Monte Carlo to obtain posterior distributions of the change point parameters including the location and the slope size of the trend and also corresponding probabilistic intervals and inferences. The performance of the Bayesian estimator is investigated through simulations and the result shows that precise estimates can be obtained when they are used in conjunction with the risk-adjusted survival time cumulative sum control chart (CUSUM) control charts for different trend scenarios. In comparison with the alternatives, step change point model and built-in CUSUM estimator, more accurate and precise estimates are obtained by the proposed Bayesian estimator over linear trends. These superiorities are enhanced when probability quantification, flexibility and generalizability of the Bayesian change point detection model are also considered.     

Comparison of adaptive treatment strategies based on longitudinal outcomes in sequential multiple assignment randomized trials

In sequential multiple assignment randomized trials, longitudinal outcomes may be the most important outcomes of interest because this type of trials is usually conducted in areas of chronic diseases or conditions. We propose to use a weighted generalized estimating equation (GEE) approach to analyzing data from such type of trials for comparing two adaptive treatment strategies based on generalized linear models. Although the randomization probabilities are known, we consider estimated weights in which the randomization probabilities are replaced by their empirical estimates and prove that the resulting weighted GEE estimator is more efficient than the estimators with true weights. The variance of the weighted GEE estimator is estimated by an empirical sandwich estimator. The time variable in the model can be linear, piecewise linear, or more complicated forms. This provides more flexibility that is important because, in the adaptive treatment setting, the treatment changes over time and, hence, a single linear trend over the whole period of study may not be practical. Simulation results show that the weighted GEE estimators of regression coefficients are consistent regardless of the specification of the correlation structure of the longitudinal outcomes. The weighted GEE method is then applied in analyzing data from the Clinical Antipsychotic Trials of Intervention Effectiveness. Copyright © 2016 John Wiley & Sons, Ltd.     

A weighted logistic regression model for estimation of recurrence of adenomas

In a colorectal polyp prevention trial, some participants might have their follow-up colonoscopy conducted before the scheduled time (i.e. at the end of the trial). This results in variable follow-up lengths for participants and the data of recurrence status at the end of the trial can be considered as current status data. In this paper, we use a weighted logistic regression model to estimate recurrence rate of adenoma data at the end of the trial. The weights are used to adjust for variable follow-up. We show that logistic regression tends to underestimate recurrence rate. In a simulation study, we show that Kaplan-Meier estimator derived from the right endpoint of the current status data tends to overestimate recurrence rate in contrast to logistic regression and the weighted logistic regression method can produce reasonable estimates of recurrence rate even under a high non-compliance rate compared to conventional logistic regression and Kaplan-Meier estimator. The method described here is illustrated with an example from a colon cancer study.     

Double robust estimator of average causal treatment effect for censored medical cost data

In observational studies, estimation of average causal treatment effect on a patient's response should adjust for confounders that are associated with both treatment exposure and response. In addition, the response, such as medical cost, may have incomplete follow‐up. In this article, a double robust estimator is proposed for average causal treatment effect for right censored medical cost data. The estimator is double robust in the sense that it remains consistent when either the model for the treatment assignment or the regression model for the response is correctly specified. Double robust estimators increase the likelihood the results will represent a valid inference. Asymptotic normality is obtained for the proposed estimator, and an estimator for the asymptotic variance is also derived. Simulation studies show good finite sample performance of the proposed estimator and a real data analysis using the proposed method is provided as illustration. Copyright © 2016 John Wiley & Sons, Ltd.     

ESTIMATING THE PROPORTION OF TREATMENT EFFECT EXPLAINED BY A SURROGATE MARKER

In this paper, we measure the extent to which a biological marker is a surrogate endpoint for a clinical event by the proportional reduction in the regression coefficient for the treatment indicator due to the inclusion of the marker in the Cox regression model. We estimate this proportion by applying the partial likelihood function to two Cox models postulated on the same failure time variable. We show that the resultant estimator is asymptotically normal with a simple variance estimator. One can construct confidence intervals for the proportion by using the direct normal approximation to the point estimator or by using Fieller's theorem. Extensive simulation studies demonstrate that the proposed methods are appropriate for practical use. We provide applications to HIV/AIDS clinical trials. © 1997 John Wiley & Sons, Ltd.     

A semi‐parametric threshold regression analysis of sexually transmitted infections in adolescent women

Time‐to‐event analysis of sexually transmitted infection data is often complicated by the existence of nonproportional hazards and nonlinear independent variable effects. Methods without the proportional hazards assumption, such as threshold regression models, have been successfully used in many applications. This paper seeks to extend the existing threshold regression models to accommodate the nonlinear independent variable effects. Specifically, we incorporated penalized and regression splines to the threshold regression models for added modeling flexibility. Cross validation methods were used for the selection of the number of knots and for the determination of smoothing parameters. Variance estimates were proposed for inference purposes. Simulation results showed that the proposed methods were able to achieve nonparametric function and parametric coefficient estimates that are close to their true values. Simulation also demonstrated satisfactory performance of variance estimates. Using the proposed methods, we analyzed time from sexual debut to the first infection with Chlamydia trachomatis infection in a group of young women. Analysis shows that the lifetime number of sexual partners has a nonlinear effect on the risk of C. trachomatis infection and the infection risks were differential by ethnicity and age of sexual debut. Copyright © 2009 John Wiley & Sons, Ltd.     

Consistent Estimation in Mendelian Randomization with Some Invalid Instruments Using a Weighted Median Estimator

Developments in genome‐wide association studies and the increasing availability of summary genetic association data have made application of Mendelian randomization relatively straightforward. However, obtaining reliable results from a Mendelian randomization investigation remains problematic, as the conventional inverse‐variance weighted method only gives consistent estimates if all of the genetic variants in the analysis are valid instrumental variables. We present a novel weighted median estimator for combining data on multiple genetic variants into a single causal estimate. This estimator is consistent even when up to 50% of the information comes from invalid instrumental variables. In a simulation analysis, it is shown to have better finite‐sample Type 1 error rates than the inverse‐variance weighted method, and is complementary to the recently proposed MR‐Egger (Mendelian randomization‐Egger) regression method. In analyses of the causal effects of low‐density lipoprotein cholesterol and high‐density lipoprotein cholesterol on coronary artery disease risk, the inverse‐variance weighted method suggests a causal effect of both lipid fractions, whereas the weighted median and MR‐Egger regression methods suggest a null effect of high‐density lipoprotein cholesterol that corresponds with the experimental evidence. Both median‐based and MR‐Egger regression methods should be considered as sensitivity analyses for Mendelian randomization investigations with multiple genetic variants.     

Instrumental variable approach to estimating the scalar-on-function regression model with measurement error with application to energy expenditure assessment in childhood obesity

Wearable device technology allows continuous monitoring of biological markers and thereby enables study of time-dependent relationships. For example, in this paper, we are interested in the impact of daily energy expenditure over a period of time on subsequent progression toward obesity among children. Data from these devices appear as either sparsely or densely observed functional data and methods of functional regression are often used for their statistical analyses. We study the scalar-on-function regression model with imprecisely measured values of the predictor function. In this setting, we have a scalar-valued response and a function-valued covariate that are both collected at a single time period. We propose a generalized method of moments-based approach for estimation, while an instrumental variable belonging in the same time space as the imprecisely measured covariate is used for model identification. Additionally, no distributional assumptions regarding the measurement errors are assumed, while complex covariance structures are allowed for the measurement errors in the implementation of our proposed methods. We demonstrate that our proposed estimator is L₂ consistent and enjoys the optimal rate of convergence for univariate nonparametric functions. In a simulation study, we illustrate that ignoring measurement error leads to biased estimations of the functional coefficient. The simulation studies also confirm our ability to consistently estimate the function-valued coefficient when compared to approaches that ignore potential measurement errors. Our proposed methods are applied to our motivating example to assess the impact of baseline levels of energy expenditure on body mass index among elementary school–aged children.     

Analysis of non-ignorable missing and left-censored longitudinal data using a weighted random effects tobit model

In a longitudinal study with response data collected during a hospital stay, observations may be missing because of the subject's discharge from the hospital prior to completion of the study or the death of the subject, resulting in non-ignorable missing data. In addition to non-ignorable missingness, there is left-censoring in the response measurements because of the inherent limit of detection. For analyzing non-ignorable missing and left-censored longitudinal data, we have proposed to extend the theory of random effects tobit regression model to weighted random effects tobit regression model. The weights are computed on the basis of inverse probability weighted augmented methodology. An extensive simulation study was performed to compare the performance of the proposed model with a number of competitive models. The simulation study shows that the estimates are consistent and that the root mean square errors of the estimates are minimal for the use of augmented inverse probability weights in the random effects tobit model. The proposed method is also applied to the non-ignorable missing and left-censored interleukin-6 biomarker data obtained from the Genetic and Inflammatory Markers of Sepsis study.     

A new estimation with minimum trace of asymptotic covariance matrix for incomplete longitudinal data with a surrogate process

Missing data is a very common problem in medical and social studies, especially when data are collected longitudinally. It is a challenging problem to utilize observed data effectively. Many papers on missing data problems can be found in statistical literature. It is well known that the inverse weighted estimation is neither efficient nor robust. On the other hand, the doubly robust (DR) method can improve the efficiency and robustness. As is known, the DR estimation requires a missing data model (i.e., a model for the probability that data are observed) and a working regression model (i.e., a model for the outcome variable given covariates and surrogate variables). Because the DR estimating function has mean zero for any parameters in the working regression model when the missing data model is correctly specified, in this paper, we derive a formula for the estimator of the parameters of the working regression model that yields the optimally efficient estimator of the marginal mean model (the parameters of interest) when the missing data model is correctly specified. Furthermore, the proposed method also inherits the DR property. Simulation studies demonstrate the greater efficiency of the proposed method compared with the standard DR method. A longitudinal dementia data set is used for illustration. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimating the relation between dietary intake obtained from a food frequency questionnaire and true average intake

Knowledge of the regression relation between dietary intake reported on a food frequency questionnaire and true average intake is useful in interpreting results from nutritional epidemiologic studies and in planning such studies. Studies which validate a questionnaire against a food record may be used to estimate this regression relation provided the food record is completed by each subject on at least two occasions. Using data collected from women aged 45-69 years during 1985-1986 in the pilot study of the Women's Health Trial, the authors show how variation in diet over time and intraindividual correlation between a questionnaire and food record obtained close together in time affects the estimation of the regression. The authors' method provides estimates of the regression slope and the questionnaire "bias" that are corrected for these effects, together with standard errors. A computer program in the SAS language, for carrying out the analysis, is provided.     

Weighted quantile regression for analyzing health care cost data with missing covariates

Analysis of health care cost data is often complicated by a high level of skewness, heteroscedastic variances and the presence of missing data. Most of the existing literature on cost data analysis have been focused on modeling the conditional mean. In this paper, we study a weighted quantile regression approach for estimating the conditional quantiles health care cost data with missing covariates. The weighted quantile regression estimator is consistent, unlike the naive estimator, and asymptotically normal. Furthermore, we propose a modified BIC for variable selection in quantile regression when the covariates are missing at random. The quantile regression framework allows us to obtain a more complete picture of the effects of the covariates on the health care cost and is naturally adapted to the skewness and heterogeneity of the cost data. The method is semiparametric in the sense that it does not require to specify the likelihood function for the random error or the covariates. We investigate the weighted quantile regression procedure and the modified BIC via extensive simulations. We illustrate the application by analyzing a real data set from a health care cost study. Copyright © 2013 John Wiley & Sons, Ltd.     

Testing and Correcting for Non-Random Selection Bias due to Censoring: An Application to Medical Costs

Censoring is a common problem with medical cost data. Methods from traditional survival analysis are not directly applicable to estimate medical costs since patients accumulate costs with different rate functions over time, leading to negatively biased estimates. Heckman's two-step estimator results in large variances when identical explanatory variables that influence selection are included in the structural equation, i.e. when there are no exclusion restrictions. This paper provides a systematic treatment of the correction for nonrandom sample selection bias of medical cost data where the selection rule is described by a censored regression model. The proposed method first uses the duration of time a patient is tracked for the selection, rather than a binary variable, namely whether or not the duration is censored. Second, using Tobit residuals instead of the inverse Mills ratio in the structural equation allows us to decrease large variances introduced by the Heckman model when there are no exclusion restrictions. We show that the resulting estimators are consistent and asymptotically normal. Simulation studies confirmed our results. Moreover, we derive a simple test to determine possible sample selection bias due to censoring. Data from a study on the medical cost of breast, prostate, colon, and lung cancer is used as an application of the method.     

Identification of subgroups with differential treatment effects for longitudinal and multiresponse variables

We describe and evaluate a regression tree algorithm for finding subgroups with differential treatments effects in randomized trials with multivariate outcomes. The data may contain missing values in the outcomes and covariates, and the treatment variable is not limited to two levels. Simulation results show that the regression tree models have unbiased variable selection and the estimates of subgroup treatment effects are approximately unbiased. A bootstrap calibration technique is proposed for constructing confidence intervals for the treatment effects. The method is illustrated with data from a longitudinal study comparing two diabetes drugs and a mammography screening trial comparing two treatments and a control. Copyright © 2016 John Wiley & Sons, Ltd.