Estimation Of Covariate Effects Research Articles

A comparison of various approaches to the exponential random graph model: A reanalysis of 102 student networks in school classes

This paper describes an empirical comparison of four specifications of the exponential family of random graph models (ERGM), distinguished by model specification (dyadic independence, Markov, partial conditional dependence) and, for the Markov model, by estimation method (Maximum Pseudolikelihood, Maximum Likelihood). This was done by reanalyzing 102 student networks in 57 junior high school classes. At the level of all classes combined, earlier substantive conclusions were supported by all specifications. However, the different specifications led to different conclusions for individual classes. PL produced unreliable estimates (when ML is regarded as the standard) and had more convergence problems than ML. Furthermore, the estimates of covariate effects were affected considerably by controlling for network structure, although the precise specification of the structural part (Markov or partial conditional dependence) mattered less.

Analysis of Failure Time Data with Multilevel Clustering, with Application to the Child Vitamin A Intervention Trial in Nepal

We consider the problem of estimating covariate effects in the marginal Cox proportional hazard model and multilevel associations for child mortality data collected from a vitamin A supplementation trial in Nepal, where the data are clustered within households and villages. For this purpose, a class of multivariate survival models that can be represented by a functional of marginal survival functions and accounts for hierarchical structure of clustering is exploited. Based on this class of models, an estimation strategy involving a within-cluster resampling procedure is proposed, and a model assessment approach is presented. The asymptotic theory for the proposed estimators and lack-of-fit test is established. The simulation study shows that the estimates are approximately unbiased, and the proposed test statistic is conservative under extremely heavy censoring but approaches the size otherwise. The analysis of the Nepal study data shows that the association of mortality is much greater within households than within villages.

Analytical Methods and Database Design: Implications for Transplant Researchers, 2005

Understanding how transplant data are collected is crucial to understanding how the data can be used. The collection and use of Organ Procurement and Transplantation Network/Scientific Registry of Transplant Recipients (OPTN/SRTR) data continues to evolve, leading to improvements in data quality, timeliness and scope while reducing the data collection burden. Additional ascertainment of outcomes completes and validates existing data, although caveats remain for researchers. We also consider analytical issues related to cohort choice, timing of data submission, and transplant center variations in follow-up data. All of these points should be carefully considered when choosing cohorts and data sources for analysis. The second part of the article describes some of the statistical methods for outcome analysis employed by the SRTR. Issues of cohort and follow-up period selection lead into a discussion of outcome definitions, event ascertainment, censoring and covariate adjustment. We describe methods for computing unadjusted mortality rates and survival probabilities, and estimating covariate effects through regression modeling. The article concludes with a description of simulated allocation modeling, developed by the SRTR for comparing outcomes of proposed changes to national organ allocation policies.

Quantitative genetic analysis of blood pressure reactivity to orthostatic tilt using principal components analysis

Blood pressure (BP) reactivity to orthostatic tilt may be predictive of cardiovascular disease. However, the genetic and environmental influences on BP reactivity to tilt have not been well examined. Identifying different influences on BP at rest and BP during tilt is complicated by the intercorrelation among multiple measurements. In this study, we use principal components analysis (PCA) to reduce multivariate BP data into components that are orthogonal. The objective of this study is to characterize and examine the genetic architecture of BP at rest and during head-up tilt (HUT). Specifically, we estimate the heritability of individual BP measures and three principal components (PC) derived from multiple BP measurements during HUT. Additionally, we estimate covariate effects on these traits. The study sample consisted of 444 individuals, distributed across four large families. HUT consisted of 70 degrees head-up table tilting while strapped to a tilt table. BP reactivity (deltaBP) was defined as BP during HUT minus BP while supine. Three PC extracted from the PCA were interpreted as 'general BP' (PC1), 'pulse pressure' (PC2) and 'BP reactivity' (PC3). Variance components methods were used to estimate the heritabilities of resting BP, HUT BP, deltaBP, as well as the three BP PC. Significant (P<0.05) heritabilities were found for all BP measurements, except for systolic deltaBP at 1 and 3 min, and diastolic deltaBP at 2 min. Significant genetic effects were also found for the three PC. Each of these orthogonal components is significantly influenced by somewhat different sets of covariates.

Utility of Generalized Hazards Model Incorporating Cubic B-spline Function into the Baseline Hazard Function

A generalized hazards model incorporating a cubic B-spline function into the baseline hazard function (GHMBS) was proposed as a model for estimating covariate effects in survival data analysis. The GHMBS integrated the three types of hazard models: the proportional hazards model (PHM), accelerated failure time model (AFTM), and accelerated hazards model (AHM), which enabled the likelihood principle for estimation and hypothesis testing to be applied irrespective of submodels (i.e., PHM, AFTM, and AHM). A procedure for adaptively choosing suitable knots from a set of candidate knots was proposed in order to actualize an appropriate baseline hazard function in GHMBS. The characteristic of the proposal was evaluated with bias and mean squared error of the estimation of covariate effects through a Monte-Carlo simulation experiment. A method for identifying a submodel appropriate for the data to be analyzed was also proposed based on GHMBS. The performance of the proposed model selection method was evaluated with the probability of selecting the true model through a Monte-Carlo simulation experiment based on PHM and AFTM. As a result, the proposed method achieved fairly high probabilities of identifying the true model. An application of the proposed method to actual data in a clinical trial provided a reasonable conclusion.

The Extended Generalized Gamma Model and its Special Cases: Applications to Modeling Marriage Durations

The paper demonstrates how various parametric models for duration data such as the exponential, Weibull, and log-normal may be embedded in a single framework, and how such competing models may be assessed relative to a more comprehensive one. To illustrate the issues addressed, the survival patterns of marriages among 1203 Swedish men born 1936–1964 are studied by parametric and non-parametric survival methods. In particular, we study the sensitivity of model-choice with respect to level of aggregation of the time variable; and of covariate-effects with respect to the model chosen. In accordance with previous works our empirical results indicate that the choice of a parametric model for the duration variable is affected by the level of time aggregation. In contrast to previous results, however, our analysis shows that estimates of covariate effects are not always robust to distributional assumptions for the duration variable.

Consequences of event rate heterogeneity across non-randomized study sub-groups.

Analyses to compare non-randomized groups are more and more common in both post hoc analyses of randomized clinical trials data and in analyses of long-term observational data. In such cases, it is quite likely that there are unknown or uncollected sources of heterogeneity in event rates. Research has shown that an underlying source of heterogeneity in event rates which is not included in proportional hazards regression models leads to biased estimates for included covariate effect estimates and lower power to test them whether the source of heterogeneity is assumed to be fixed or random. We demonstrate here using several post hoc analyses of clinical trials data that a potentially common problem may be that the non-randomized groups which are to be compared have differential variability in their event rates. We then show through simulation that such underlying heterogeneity which varies across the groups, when ignored in the modelling, can lead to an attenuated regression effect estimate for comparing the two groups to each other, lower rejection rates for the effect, and Wald-based confidence intervals with potentially much lower coverage than nominal. When the groups are not significantly different, but heterogeneity differs between them, an analysis ignoring the heterogeneity can even result in a significant negative comparison.

The Additive Risk Model for Purchase Timing

This paper proposes the Additive Risk Model (ARM), first used by Aalen (1980), to explain households' interpurchase times. Unlike the Proportional Hazard Model (PHM), first proposed by Cox (1972), the ARM incorporates the effects of covariates on the individual hazard function in an additive (as opposed to multiplicative) manner. While a large number of previous studies on interpurchase timing have dealt with the question of correctly specifying the parametric distribution for interpurchase times, no study has explicitly investigated the question of correctly specifying the effects of covariates in the model. This study looks at this issue. We propose an ARM that is suitable for purchase-timing data, and compare its empirical performance to that of the PHM and the Accelerated Failure Time Model (AFTM) using scanner panel data on laundry detergents, paper towels, and toilet tissue. We find that the ARM not only estimates and validates the observed interpurchase times better than existing models, but also recovers a time-varying price elasticity and shows a high degree of robustness in the estimated covariate effects to alternative parametric specifications of the baseline hazard. The estimates of covariate parameters under the PHM, on the other hand, are highly sensitive to alternative parametric specifications of the baseline hazard.

Binary latent variable modelling and its applicationin the study of air pollution in Hong Kong.

A binary latent variable is constructed to account for the correlation between multiple binary outcomes and is treated as a dependent variable in modelling for covariate effects. This modelling method is similar to the structural equation modelling. Three models are considered: (1) all covariates affecting the binary latent variable directly; (2) some covariates affecting the binary latent variable while other affecting the manifestation of the binary outcomes; and (3) no covariates are included. Gibbs sampling, a special case of the Markov Chain Monte Carlo method, is used to estimate the parameters in the models. Simulation studies show that this method is valid and reliable in estimating covariate effects. But Model (1) fitted the data best with lowest value in the deviance information criteria. The method is illustrated by applying it to the data analysis of an environmental air pollution study. The results show that air pollution (i.e. the most versus the least polluted district) (odds ratio 1.20; 95% confidence interval 0.97-1.49; p=0.102), smoking (relative to not smoking) (2.75; 2.21-3.41; p < 0.001) and mosquito coil use (relative to non-use) (1.27; 0.99-1.62; p=0.058) had an impact on the respiratory health of male adults in Hong Kong.

Modelling Low Pay Transition Probabilities, Accounting for Panel Attrition, Non-Response, and Initial Conditions

We model annual low pay transition probabilities taking account of three potentially endogenous selections: two sample drop-out mechanisms (panel attrition, non-employment) and ‘initial conditions’ (base-year low pay status). This model, and variants that ignore one or more of these selection mechanisms, are fitted to data for men from the British Household Panel Survey. Tests of the ignorability of the endogenous selection mechanisms suggest that ‘economic’ selection mechanisms such as initial conditions and retention of employment are more important than the ‘survey’ selection mechanism (attrition). However, consistent with related US research, relatively simple models provide estimates of covariate effects that differ little from the estimates from the complicated models.

Discrete proportional hazards models for mismeasured outcomes.

Outcome mismeasurement can lead to biased estimation in several contexts. Magder and Hughes (1997, American Journal of Epidemiology 146, 195-203) showed that failure to adjust for imperfect outcome measures in logistic regression analysis can conservatively bias estimation of covariate effects, even when the mismeasurement rate is the same across levels of the covariate. Other authors have addressed the need to account for mismeasurement in survival analysis in selected cases (Snapinn, 1998, Biometrics 54, 209-218; Gelfand and Wang, 2000, Statistics in Medicine 19, 1865-1879; Balasubramanian and Lagakos, 2001, Biometrics 57, 1048-1058, 2003, Biometrika 90, 171-182). We provide a general, more widely applicable, adjusted proportional hazards (APH) method for estimation of cumulative survival and hazard ratios in discrete time when the outcome is measured with error. We show that mismeasured failure status in a standard proportional hazards (PH) model can conservatively bias estimation of hazard ratios and that inference, in most practical situations, is more severely affected by poor specificity than by poor sensitivity. However, in simulations over a wide range of conditions, the APH method with correctly specified mismeasurement rates performs very well.

Cancer aggregation and complex segregation analysis of families with female non-smoking lung cancer probands in Taiwan

Previous studies have found that having a first-degree blood relative with lung cancer was a possible predictor of lung cancer risk, but some studies have indicated that the association is non-significant or only significant for a subset of the studied population. To determine the familial aggregation and whether there is any evidence for a gene controlling the susceptibility to developing lung cancer in female non-smokers, multiple logistic regression methods for estimating covariate effects and maximum likelihood segregation analyses were performed using data from 216 female non-smoking lung cancer probands (2328 individuals) in a population-based case–control study. Having a family history of lung cancer was found to be a significant predictor of lung cancer for non-smoking females (Adjusted Odds Ratio (OR)=5.7, 95% Confidence Interval (CI)=1.9–16.9). Having a female relative with lung cancer (adjusted OR=14.4, 95% CI=2.7–75.5) was more strongly associated with the lung cancer risk than was having a male relative with lung cancer. This association was stronger for probands aged less than 60 years at onset (adjusted OR=11.2, 95% CI=2.2–56.9). All of the Mendelian models fitted the data significantly better than the sporadic (no major type) model or the environmental model (P<0.00l). The Mendelian codominant models provided the best fit of the data for the early onset probands and showed a stronger effect for a major susceptibility locus for non-smoking lung cancer probands. The results of this study provide evidence that a rare autosomal codominant gene may influence the risk lung cancer in non-smoker and is responsible for the familial aggregation observed in non-smoking lung cancer patients.

Bayesian Geoadditive Seemingly Unrelated Regression

Parametric seemingly unrelated regression (SUR) models are a common tool for multivariate regression analysis when error variables are reasonably correlated, so that separate univariate analysis may result in inefficient estimates of covariate effects. A weakness of parametric models is that they require strong assumptions on the functional form of possibly nonlinear effects of metrical covariates. In this paper, we develop a Bayesian semiparametric SUR model, where the usual linear predictors are replaced by more flexible additive predictors allowing for simultaneous nonparametric estimation of such covariate effects and of spatial effects. The approach is based on appropriate smoothness priors which allow different forms and degrees of smoothness in a general framework. Inference is fully Bayesian and uses recent Markov chain Monte Carlo techniques.

Nonparametric identification of daily activity durations using kernel density estimators

Modeling of activity duration has become an important aspect in characterizing activity and travel patterns. The standard approach for analyzing activity duration is to use hazard-based models to account for the duration dependence within an activity while estimating covariate effects and heterogeneity. The effects of covariates and heterogeneity on duration have been modeled using parametric and nonparametric regression techniques. In this paper, we present a nonparametric pattern recognition approach that can precede hazard-based duration modeling to identify heterogeneity patterns that may undermine duration modeling if not detected. The technique utilizes a kernel estimate of the probability density function (pdf) of various activity durations and allows for statistical comparison of distributions to evaluate differences between groups of individuals. Preliminary testing on the first wave of a panel survey indicates that the approach is insightful in evaluating covariate effects while allowing visual inspection of heterogeneity in the data.

Sampling Adjusted Analysis of Dynamic Additive Regression Models for Longitudinal Data

We consider a modelling approach to longitudinal data that aims at estimating flexible covariate effects in a model where the sampling probabilities are modelled explicitly. The joint modelling yields simple estimators that are easy to compute and analyse, even if the sampling of the longitudinal responses interacts with the response level. An incorrect model for the sampling probabilities results in biased estimates. Non‐representative sampling occurs, for example, if patients with an extreme development (based on extreme values of the response) are called in for additional examinations and measurements. We allow covariate effects to be time‐varying or time‐constant. Estimates of covariate effects are obtained by solving martingale equations locally for the cumulative regression functions. Using Aalen's additive model for the sampling probabilities, we obtain simple expressions for the estimators and their asymptotic variances. The asymptotic distributions for the estimators of the non‐parametric components as well as the parametric components of the model are derived drawing on general martingale results. Two applications are presented. We consider the growth of cystic fibrosis patients and the prothrombin index for liver cirrhosis patients. The conclusion about the growth of the cystic fibrosis patients is not altered when adjusting for a possible non‐representativeness in the sampling, whereas we reach substantively different conclusions about the treatment effect for the liver cirrhosis patients.

Assessing human fertility using several markers of ovulation.

In modelling human fertility one ideally accounts for timing of intercourse relative to ovulation. Measurement error in identifying the day of ovulation can bias estimates of fecundability parameters and attenuate estimates of covariate effects. In the absence of a single perfect marker of ovulation, several error prone markers are sometimes obtained. In this paper we propose a semi-parametric mixture model that uses multiple independent markers of ovulation to account for measurement error. The model assigns each method of assessing ovulation a distinct non-parametric error distribution, and corrects bias in estimates of day-specific fecundability. We use a Monte Carlo EM algorithm for joint estimation of (i) the error distribution for the markers, (ii) the error-corrected fertility parameters, and (iii) the couple-specific random effects. We apply the methods to data from a North Carolina fertility study to assess the magnitude of error in measures of ovulation based on urinary luteinizing hormone and metabolites of ovarian hormones, and estimate the corrected day-specific probabilities of clinical pregnancy. Published in 2001 by John Wiley & Sons, Ltd.

Comments on the Papers by Gelman et al. and Wakefield and Salway

Lianne Sheppard (University of Washington, Seattle) As both the papers by Gelman et al. and Wakefield and Salway emphasize, a deeper understanding of ecologic studies is vital to their application. There is a fundamental misalignment in many ecologic studies because the level of analysis, the group, does not match the level of inference, the individual. In their stimulating papers, Gelman and his colleagues and Wakefield and Salway address this misalignment by concentrating on features of ecologic regressions under fundamentally different frameworks. Gelman and his colleagues restrict their attention to binary or categorical covariates and cast their models in terms that are familiar in the political science literature. Within this structure they suggest some workable approaches to identifying violations of model assumptions. They rightly highlight the fundamental inability to evaluate all the assumptions that are necessary in an ecologic regression with the data at hand. Wakefield and Salway focus on the more general challenge of coping with continuous covariates in a group level setting. They present their framework for ecologic studies in the context of environmental epidemiologic applications. In all applications, group level studies, unlike multilevel studies, cannot separately estimate covariate effects that operate differently through groups and individuals. Since the additional effects of group membership (often called contextual effects (Firebaugh, 1978)) are much more likely to derive from the social rather than the physical environment, the intended application is crucial to the potential for valid inference from group level studies. It is natural to assume that environmental epidemiologic applications will require fewer untestable assumptions than ecologic studies in the social and behavioural sciences. Wakefield and Salway emphasize the importance of the within-area joint distribution of the exposure variables. Of particular value, they clarify the variance of an aggregate outcome (conditional on the exposure) induced under a range of assumptions about the exposure. They restrict their detailed treatment to cases where the within-area outcomes are independent. In many applications this assumption will be too strong and consequently there will be additional dependences in the variance models. A second feature of ignoring the within-area exposure distribution is its influence on the bias of the relative risk parameter estimates for all but the additive risk model. For instance, in their epidemiologic example, Gelman and his colleagues substitute average covariate measurements for individual values in a non-linear model. This is common practice in epidemiology (see for example Guthrie and Sheppard's (2000) discussion of Greenland and Robins's (1994) treatment of a similar radon model). The fitted ecologic model is clearly misspecified when inference is directed at the individual. The implications of this misspecification depend on the full within-area exposure distribution, not just the mean, features which are not included in the ecologic model. Gelman and his colleagues describe ramifications that are specific to radon exposures, whereas Wakefield and Salway give a more general perspective. These papers, as well as others in this issue, give fresh insights into many important considerations for ecologic studies.

Retrospective Ascertainment of Recurrent Events: An Application to Time to Pregnancy

Retrospectively ascertained data are common in many areas, including demography, epidemiology, and actuarial science. The main objective of this article is to study the effect of retrospective ascertainment on inference regarding recurrent events of time to pregnancy (TTP) data. For the particular TTP dataset that we consider, couples are included retrospectively based on their first pregnancy and then followed prospectively to a second pregnancy or to end of study. We consider a conditional model for the recurrent events data where the second TTP is included only if it is observed and a full model where the nonobserved second TTPs are included as suitably right censored. We furthermore consider two different approaches to modeling the dependencies of the recurrent events. A traditional frailty model, where the frailty enters the model as an unobserved covariate, and a marginal frailty model are applied. We find that efficiency is gained from including the second TTPs, with the full model being the most efficient. Further, the marginal frailty model is preferred over the traditional frailty model because estimates of covariate effects are easier to interpret and are more robust to changes in the frailty distribution.

Dichotomization of continuous measurements using generalized additive modelling--application in predicting intrapartum caesarean delivery.

In prediction model development, continuous variables are often dichotomized at empirically chosen thresholds to simplify calculations and facilitate decision making. However, these choices are often made in the absence of estimated covariate effects and may be inaccurate, thus weakening the models. To improve this approach, generalized additive modelling that allows non-parametric estimation of the true covariate effects is used for threshold selection. In this study, this approach is illustrated by development of prediction models for intrapartum Caesarean deliveries. The prediction performance of the models thus developed is significantly better than that developed using empirically chosen thresholds for dichotomization.

Hierarchical Generalized Linear Models in the Analysis of Variations in Health Care Utilization

In recent years many studies have reported large differences in the use of medical treatments and procedures across geographic regions, hospitals, and individual health care providers. Beyond reporting on the extent of observed variations, these studies examine the role of contributing factors including patient, regional, and provider characteristics. In addition, they may assess the relation between health care processes and outcomes, such as patient mortality, morbidity, and functioning. Studies of variations in health care utilization and outcomes involve the analysis of multilevel clustered data; for example, data on patients clustered by hospital and/or geographic region. The goals of the analysis include the estimation of cluster-specific adjusted responses, covariate effects, and components of variance. The analytic strategy needs to account for correlations induced by clustering and to handle the presence of large variations in cluster size. In this article we formulate a broad class of hierarchical generalized linear models (HGLMs) and discuss their applications to the analysis of health care utilization data. The models can incorporate covariates at each level of the hierarchical data structure, can account for greater variation than what is allowed by the variance in a one-parameter exponential family, and permit the use of heavy-tailed distributions for the random effects. We develop a Bayesian approach to fitting HGLMs using Markov chain Monte Carlo methods and discuss several methods for model checking. The HGLM analysis is presented in the context of two examples of applications to the study of variations in the utilization of medical procedures for elderly Medicare beneficiaries who sustained a heart attack. The first example involves the analysis of clustered longitudinal data with binomial responses and examines geographic and temporal trends in the utilization of coronary angiography across the United States during the 4-year period 1987–1990. The second example involves the analysis of multilevel, clustered data with Poisson responses and examines hospital variations in the utilization of coronary artery bypass graft surgery in 1990. The HGLM analysis incorporates state-level and hospital-level covariates and makes it possible to estimate covariate effects and cluster-specific rates of utilization for both hospitals and states.