Testing the non-unity of rate ratio under inverse sampling

Inverse sampling is considered to be a more appropriate sampling scheme than the usual binomial sampling scheme when subjects arrive sequentially, when the underlying response of interest is acute, and when maximum likelihood estimators of some epidemiologic indices are undefined. In this article, we study various statistics for testing non-unity rate ratios in case-control studies under inverse sampling. These include the Wald, unconditional score, likelihood ratio and conditional score statistics. Three methods (the asymptotic, conditional exact, and Mid-P methods) are adopted for P-value calculation. We evaluate the performance of different combinations of test statistics and P-value calculation methods in terms of their empirical sizes and powers via Monte Carlo simulation. In general, asymptotic score and conditional score tests are preferable for their actual type I error rates are well controlled around the pre-chosen nominal level, and their powers are comparatively the largest. The exact version of Wald test is recommended if one wants to control the actual type I error rate at or below the pre-chosen nominal level. If larger power is expected and fluctuation of sizes around the pre-chosen nominal level are allowed, then the Mid-P version of Wald test is a desirable alternative. We illustrate the methodologies with a real example from a heart disease study.     

Extended Mantel-Haenszel estimating procedure for multivariate logistic regression models

A class of estimating functions is proposed for the estimation of multivariate relative risk in stratified case-control studies. It reduces to the well-known Mantel-Haenszel estimator when there is a single binary risk factor. Large-sample properties of the solutions to the proposed estimating equations are established for two distinct situations. Efficiency calculations suggest that the proposed estimators are nearly fully efficient relative to the conditional maximum likelihood estimator for the parameters considered. Application of the proposed method to family data and longitudinal data, where the conditional likelihood approach fails, is discussed. Two examples from case-control studies and one example from a study on familial aggregation are presented.     

Use of external rates in nested case-control studies with application to the international radiation study of cervical cancer patients.

A method is proposed for analysis of nested case-control studies that combines the matched comparison of covariate values between cases and controls and a comparison of the observed numbers of cases in the nesting cohort with expected numbers based on external rates and average relative risks estimated from the controls. The former comparison is based on the conditional likelihood for matched case-control studies and the latter on the unconditional likelihood for Poisson regression. It is shown that the two likelihoods are orthogonal and that their product is an estimator of the full survival likelihood that would have been obtained on the total cohort, had complete covariate data been available. Parameter estimation and significance tests follow in the usual way by maximizing this product likelihood. The method is illustrated using data on leukemia following irradiation for cervical cancer. In this study, the original cohort study showed a clear excess of leukemia in the first 15 years after exposure, but it was not feasible to obtain dose estimates on the entire cohort. However, the subsequent nested case-control study failed to demonstrate significant differences between alternative dose-response relations and effects of time-related modifiers. The combined analysis allows much clearer discrimination between alternative dose-time-response models.     

A comparative study of conditional maximum likelihood estimation of a common odds ratio

The finite-sample properties of various point estimators of a common odds ratio from multiple 2 X 2 tables have been considered in a number of simulation studies. However, the conditional maximum likelihood estimator has received only limited attention. That omission is partially rectified here for cases of relatively small numbers of tables and moderate to large within-table sample sizes. The conditional maximum likelihood estimator is found to be superior to the unconditional maximum likelihood estimator, and equal or superior to the Mantel-Haenszel estimator in both bias and precision.     

Capture-recapture when time and behavioral response affect capture probabilities

We consider a capture-recapture model in which capture probabilities vary with time and with behavioral response. Two inference procedures are developed under the assumption that recapture probabilities bear a constant relationship to initial capture probabilities. These two procedures are the maximum likelihood method (both unconditional and conditional types are discussed) and an approach based on optimal estimating functions. The population size estimators derived from the two procedures are shown to be asymptotically equivalent when population size is large enough. The performance and relative merits of various population size estimators for finite cases are discussed. The bootstrap method is suggested for constructing a variance estimator and confidence interval. An example of the deer mouse analyzed in Otis et al. (1978, Wildlife Monographs 62, 93) is given for illustration.     

Conditional likelihood methods for haplotype-based association analysis using matched case-control data

Genetic epidemiologists routinely assess disease susceptibility in relation to haplotypes, that is, combinations of alleles on a single chromosome. We study statistical methods for inferring haplotype-related disease risk using single nucleotide polymorphism (SNP) genotype data from matched case-control studies, where controls are individually matched to cases on some selected factors. Assuming a logistic regression model for haplotype-disease association, we propose two conditional likelihood approaches that address the issue that haplotypes cannot be inferred with certainty from SNP genotype data (phase ambiguity). One approach is based on the likelihood of disease status conditioned on the total number of cases, genotypes, and other covariates within each matching stratum, and the other is based on the joint likelihood of disease status and genotypes conditioned only on the total number of cases and other covariates. The joint-likelihood approach is generally more efficient, particularly for assessing haplotype-environment interactions. Simulation studies demonstrated that the first approach was more robust to model assumptions on the diplotype distribution conditioned on environmental risk variables and matching factors in the control population. We applied the two methods to analyze a matched case-control study of prostate cancer.     

Maximum likelihood estimation for Cox's regression model under nested case-control sampling

Nested case-control sampling is designed to reduce the costs of large cohort studies. It is important to estimate the parameters of interest as efficiently as possible. We present a new maximum likelihood estimator (MLE) for nested case-control sampling in the context of Cox's proportional hazards model. The MLE is computed by the EM-algorithm, which is easy to implement in the proportional hazards setting. Standard errors are estimated by a numerical profile likelihood approach based on EM aided differentiation. The work was motivated by a nested case-control study that hypothesized that insulin-like growth factor I was associated with ischemic heart disease. The study was based on a population of 3784 Danes and 231 cases of ischemic heart disease where controls were matched on age and gender. We illustrate the use of the MLE for these data and show how the maximum likelihood framework can be used to obtain information additional to the relative risk estimates of covariates.     

Logistic regression methods for retrospective case-control studies using complex sampling procedures

There are a number of possible designs for case-control studies. The simplest uses two separate simple random samples, but an actual study may use more complex sampling procedures. Typically, stratification is used to control for the effects of one or more risk factors in which we are interested. It has been shown (Anderson, 1972, Biometrika 59, 19-35; Prentice and Pyke, 1979, Biometrika 66, 403-411) that the unconditional logistic regression estimators apply under stratified sampling, so long as the logistic model includes a term for each stratum. We consider the case-control problem with stratified samples and assume a logistic model that does not include terms for strata, i.e., for fixed covariates the (prospective) probability of disease does not depend on stratum. We assume knowledge of the proportion sampled in each stratum as well as the total number in the stratum. We use this knowledge to obtain the maximum likelihood estimators for all parameters in the logistic model including those for variables completely associated with strata. The approach may also be applied to obtain estimators under probability sampling.     

Conditional logistic analysis of case-control studies with complex sampling

Methods for the analysis of unmatched case-control data based on a finite population sampling model are developed. Under this model, and the prospective logistic model for disease probabilities, a likelihood for case-control data that accommodates very general sampling of controls is derived. This likelihood has the form of a weighted conditional logistic likelihood. The flexibility of the methods is illustrated by providing a number of control sampling designs and a general scheme for their analyses. These include frequency matching, counter-matching, case-base, randomized recruitment, and quota sampling. A study of risk factors for childhood asthma illustrates an application of the counter-matching design. Some asymptotic efficiency results are presented and computational methods discussed. Further, it is shown that a 'marginal' likelihood provides a link to unconditional logistic methods. The methods are examined in a simulation study that compares frequency and counter-matching using conditional and unconditional logistic analyses and indicate that the conditional logistic likelihood has superior efficiency. Extensions that accommodate sampling of cases and multistage designs are presented. Finally, we compare the analysis methods presented here to other approaches, compare counter-matching and two-stage designs, and suggest areas for further research.To whom correspondence should be addressed.     

Bias and efficiency in family-based gene-characterization studies: conditional, prospective, retrospective, and joint likelihoods

We revisit the usual conditional likelihood for stratum-matched case-control studies and consider three alternatives that may be more appropriate for family-based gene-characterization studies: First, the prospective likelihood, that is, Pr(D/G,A second, the retrospective likelihood, Pr(G/D); and third, the ascertainment-corrected joint likelihood, Pr(D,G/A). These likelihoods provide unbiased estimators of genetic relative risk parameters, as well as population allele frequencies and baseline risks. The parameter estimates based on the retrospective likelihood remain unbiased even when the ascertainment scheme cannot be modeled, as long as ascertainment only depends on families' phenotypes. Despite the need to estimate additional parameters, the prospective, retrospective, and joint likelihoods can lead to considerable gains in efficiency, relative to the conditional likelihood, when estimating genetic relative risk. This is true if baseline risks and allele frequencies can be assumed to be homogeneous. In the presence of heterogeneity, however, the parameter estimates assuming homogeneity can be seriously biased. We discuss the extent of this problem and present a mixed models approach for providing consistent parameter estimates when baseline risks and allele frequencies are heterogeneous. The efficiency gains of the mixed-model prospective, retrospective, and joint likelihoods relative to the efficiency of conditional likelihood are small in the situations presented here.     

A general approach for two-stage analysis of multilevel clustered non-Gaussian data

In this article, we propose a two-stage approach to modeling multilevel clustered non-Gaussian data with sufficiently large numbers of continuous measures per cluster. Such data are common in biological and medical studies utilizing monitoring or image-processing equipment. We consider a general class of hierarchical models that generalizes the model in the global two-stage (GTS) method for nonlinear mixed effects models by using any square-root-n-consistent and asymptotically normal estimators from stage 1 as pseudodata in the stage 2 model, and by extending the stage 2 model to accommodate random effects from multiple levels of clustering. The second-stage model is a standard linear mixed effects model with normal random effects, but the cluster-specific distributions, conditional on random effects, can be non-Gaussian. This methodology provides a flexible framework for modeling not only a location parameter but also other characteristics of conditional distributions that may be of specific interest. For estimation of the population parameters, we propose a conditional restricted maximum likelihood (CREML) approach and establish the asymptotic properties of the CREML estimators. The proposed general approach is illustrated using quartiles as cluster-specific parameters estimated in the first stage, and applied to the data example from a collagen fibril development study. We demonstrate using simulations that in samples with small numbers of independent clusters, the CREML estimators may perform better than conditional maximum likelihood estimators, which are a direct extension of the estimators from the GTS method.     

Sampling for conditional inference on case-control data

The problem of exact conditional inference for discrete multivariate case-control data has two forms. The first is grouped case-control data, where Monte Carlo computations can be done using the importance sampling method of Booth and Butler (1999, Biometrika86, 321-332), or a proposed alternative sequential importance sampling method. The second form is matched case-control data. For this analysis we propose a new exact sampling method based on the conditional-Poisson distribution for conditional testing with one binary and one integral ordered covariate. This method makes computations on data sets with large numbers of matched sets fast and accurate. We provide detailed derivation of the constraints and conditional distributions for conditional inference on grouped and matched data. The methods are illustrated on several new and old data sets.     

A semiparametric mixture cure survival model for left‐truncated and right‐censored data

In follow‐up studies, the disease event time can be subject to left truncation and right censoring. Furthermore, medical advancements have made it possible for patients to be cured of certain types of diseases. In this article, we consider a semiparametric mixture cure model for the regression analysis of left‐truncated and right‐censored data. The model combines a logistic regression for the probability of event occurrence with the class of transformation models for the time of occurrence. We investigate two techniques for estimating model parameters. The first approach is based on martingale estimating equations (EEs). The second approach is based on the conditional likelihood function given truncation variables. The asymptotic properties of both proposed estimators are established. Simulation studies indicate that the conditional maximum‐likelihood estimator (cMLE) performs well while the estimator based on EEs is very unstable even though it is shown to be consistent. This is a special and intriguing phenomenon for the EE approach under cure model. We provide insights into this issue and find that the EE approach can be improved significantly by assigning appropriate weights to the censored observations in the EEs. This finding is useful in overcoming the instability of the EE approach in some more complicated situations, where the likelihood approach is not feasible. We illustrate the proposed estimation procedures by analyzing the age at onset of the occiput‐wall distance event for patients with ankylosing spondylitis.     

Tests for Trends in Incidence Rate Ratios

Breslow (1984) described an efficient score test for trend in incidence density rate ratios for cohort studies under a conditional Poisson or binomial model employing maximum likelihood estimation of the rate parameters. In this communication, an alternative derivation of this statistic that is based on an unconditional approach is provided, along with an examination of associated goodness-of-fit tests and methods of confidence interval estimation. The procedures are illustrated by a cohort study of ischemic heart disease mortality following industrial exposure to carbon disulfide.     

Matched Case—Control Data Analysis with Selection Bias

Case-control studies offer a rapid and efficient way to evaluate hypotheses. On the other hand, proper selection of the controls is challenging, and the potential for selection bias is a major weakness. Valid inferences about parameters of interest cannot be drawn if selection bias exists. Furthermore, the selection bias is difficult to evaluate. Even in situations where selection bias can be estimated, few methods are available. In the matched case-control Northern Manhattan Stroke Study (NOMASS), stroke-free controls are sampled in two stages. First, a telephone survey ascertains demographic and exposure status from a large random sample. Then, in an in-person interview, detailed information is collected for the selected controls to be used in a matched case-control study. The telephone survey data provides information about the selection probability and the potential selection bias. In this article, we propose bias-corrected estimators in a case-control study using a joint estimating equation approach. The proposed bias-corrected estimate and its standard error can be easily obtained by standard statistical software.     

A Semiparametric Joint Model for Longitudinal and Survival Data with Application to Hemodialysis Study

Summary .  In many longitudinal clinical studies, the level and progression rate of repeatedly measured biomarkers on each subject quantify the severity of the disease and that subject's susceptibility to progression of the disease. It is of scientific and clinical interest to relate such quantities to a later time-to-event clinical endpoint such as patient survival. This is usually done with a shared parameter model. In such models, the longitudinal biomarker data and the survival outcome of each subject are assumed to be conditionally independent given subject-level severity or susceptibility (also called frailty in statistical terms). In this article, we study the case where the conditional distribution of longitudinal data is modeled by a linear mixed-effect model, and the conditional distribution of the survival data is given by a Cox proportional hazard model. We allow unknown regression coefficients and time-dependent covariates in both models. The proposed estimators are maximizers of an exact correction to the joint log likelihood with the frailties eliminated as nuisance parameters, an idea that originated from correction of covariate measurement error in measurement error models. The corrected joint log likelihood is shown to be asymptotically concave and leads to consistent and asymptotically normal estimators. Unlike most published methods for joint modeling, the proposed estimation procedure does not rely on distributional assumptions of the frailties. The proposed method was studied in simulations and applied to a data set from the Hemodialysis Study.     

Point-source modelling using matched case-control data

We describe an extension to matched case-control studies of the parametric modelling framework developed by Diggle (1990) and Diggle and Rowlingson (1994) to investigate raised risk around putative sources of environmental pollution. We use a conditional likelihood approach for the family of risk functions considered in Diggle and Rowlingson (1994). We show that the likelihood surface that results from these models may be highly irregular, and provide a Bayesian analysis in which we investigate the posterior distribution using Markov chain Monte Carlo. An analysis of one-one matched data that were collected to investigate the relationship between respiratory disease and distance to roads in East London is presented.     

Incorporating covariates into standard line transect analyses

An implicit assumption of standard line transect methodology is that detection probabilities depend solely on the perpendicular distance of detected objects to the transect line. Heterogeneity in detection probabilities is commonly minimized using stratification, but this may be precluded by small sample sizes. We develop a general methodology which allows the effects of multiple covariates to be directly incorporated into the estimation procedure using a conditional likelihood approach. Small sample size properties of estimators are examined via simulations. As an example the method is applied to eastern tropical Pacific dolphin sightings data.     

Homogeneity score test for the intraclass version of the kappa statistics and sample-size determination in multiple or stratified studies

When the intraclass correlation coefficient or the equivalent version of the kappa agreement coefficient have been estimated from several independent studies or from a stratified study, we have the problem of comparing the kappa statistics and combining the information regarding the kappa statistics in a common kappa when the assumption of homogeneity of kappa coefficients holds. In this article, using the likelihood score theory extended to nuisance parameters (Tarone, 1988, Communications in Statistics-Theory and Methods 17(5), 1549-1556) we present an efficient homogeneity test for comparing several independent kappa statistics and, also, give a modified homogeneity score method using a noniterative and consistent estimator as an alternative. We provide the sample size using the modified homogeneity score method and compare it with that using the goodness-of-fit method (GOF) (Donner, Eliasziw, and Klar, 1996, Biometrics 52, 176-183). A simulation study for small and moderate sample sizes showed that the actual level of the homogeneity score test using the maximum likelihood estimators (MLEs) of parameters is satisfactorily close to the nominal and it is smaller than those of the modified homogeneity score and the goodness-of-fit tests. We investigated statistical properties of several noniterative estimators of a common kappa. The estimator (Donner et al., 1996) is essentially efficient and can be used as an alternative to the iterative MLE. An efficient interval estimation of a common kappa using the likelihood score method is presented.     

On the use of concordant pairs in matched case-control studies

A new estimator of the common odds ratio in one-to-one matched case-control studies is proposed. The connection between this estimator and the James-Stein estimating procedure is highlighted through the argument of estimating functions. Comparisons are made between this estimator, the conditional maximum likelihood estimator, and the estimator ignoring the matching in terms of finite sample bias, mean squared error, coverage probability, and length of confidence interval. In many situations, the new estimator is found to be more efficient than the conditional maximum likelihood estimator without being as biased as the estimator that ignores matching. The extension to multiple risk factors is also outlined.