Generalized case-control sampling under generalized linear models

Abstract

A generalized case-control (GCC) study, like the standard case-control study, leverages outcome-dependent sampling (ODS) to extend to nonbinary responses. We develop a novel, unifying approach for analyzing GCC study data using the recently developed semiparametric extension of the generalized linear model (GLM), which is substantially more robust to model misspecification than existing approaches based on parametric GLMs. For valid estimation and inference, we use a conditional likelihood to account for the biased sampling design. We describe analysis procedures for estimation and inference for the semiparametric GLM under a conditional likelihood, and we discuss problems with estimation and inference under a conditional likelihood when the response distribution is misspecified. We demonstrate the flexibility of our approach over existing ones through extensive simulation studies, and we apply the methodology to an analysis of the Asset and Health Dynamics Among the Oldest Old study, which motives our research. The proposed approach yields a simple yet versatile solution for handling ODS in a wide variety of possible response distributions and sampling schemes encountered in practice.

1 |. INTRODUCTION

With the growing availability of electronic health records and publicly available research databases, researchers who seek to answer novel medical and public health questions may have access to outcome (e.g., disease status or a quantitative phenotype) and covariate data, but may be missing the key exposure variable (e.g., a biomarker derived from an assay or a manual electronic health record review). Often, such exposure ascertainment is expensive and cannot be completed on all members of a cohort. It is well known that in many such “expensive-exposure” scenarios, random sampling can be inefficient for estimating exposure-outcome associations. It can therefore be beneficial to oversample portions of the population via outcome-dependent sampling (ODS). The most common ODS design is the case-control study, which became popular in the 1920s and led to successes in finding associations between lip cancer and pipe smoking (Broders, 1920), breast cancer and reproductive history (Lane-Claypon, 1926), and oral cancer and pipe smoking (Lombard and Doering, 1928). Methods of analysis for case-control studies are well established (Cornfield, 1951; Anderson, 1972; Prentice and Pyke, 1979; Breslow and Day, 1980; Breslow, 1996), and generally rely on the logistic regression model. Although extensions of case-control sampling designs to nonbinary distributions have been proposed (Lawless et al., 1999; Scott and Wild, 2011), doing so in a unified way has been challenging under the standard generalized linear model (GLM; McCullagh and Nelder, 1989) framework because it requires tailoring the implementation to each distinct outcome distribution; each unique response requires a new implementation that can be tedious and computationally burdensome.

To fix ideas, we frame the ODS problem in terms of a two-phase study (Breslow and Day, 1980; Breslow and Cain, 1988; Breslow, 1996; Breslow and Holubkov, 1997; Breslow and Chatterjee, 1999). A typical two-phase study is conducted as follows: in Phase 1, a representative sample of the response (and possibly inexpensive covariates) is collected for a, potentially large, number of observations. After Phase 1, a sampling plan is developed based on observed response information. In Phase 2, using the proposed sampling plan, a subset of Phase 1 observations is selected to collect expensive exposure information. As an example, consider a hypothetical electronic health record setting where the investigator is interested in the relationship between disease status and a (relatively expensive) genetic marker. Available health record information includes phenotype and other standard information, but does not readily contain the genetic marker data. In this example, the data available in the electronic health record are the Phase 1 sample. Based on the observed phenotype, which may have a more complicated structure than binary, we design a sampling scheme to identify the Phase 2 sample in which the genetic marker is ascertained. Following this sampling scheme, we would then collect genetic marker information for selected subjects.

Whereas in some ODS settings, both Phase 1 and Phase 2 data are available for analysis, we restrict our attention here to the commonly observed setting where Phase 2 data are available but Phase 1 data are not. That is, we only have access to the “complete-case” data set (Lawless et al., 1999), and not to Phase 1 responses. Many examples of this are given via the database for genotypes and phenotypes (dbGaP), where much of the available data are from studies only retaining observations selected into Phase 2. There are many reasons why only Phase 2 data would be retained. For example, one might need to obtain consent from patients to include them in publicly available data repositories and that may be more challenging for those not selected into Phase 2. Additionally, there are costs associated with data cleaning, and researchers may not be enthusiastic to clean and manage Phase 1 inexpensive covariates and data from the unsampled subjects in whom Phase 2 exposure data were not collected. An example of a study with only Phase 2 data is the CHARGE study (Lin et al., 2014), where Phase 1 and Phase 2 data were collected, but only Phase 2 data are freely available. Many available methods for ODS assume the inclusion of incomplete Phase 1 data, and are not readily adaptable to the case where only Phase 2 data are available (e.g., Chatterjee et al. (2003); Weaver and Zhou (2005); Tao et al. (2017)).

Unique to the analysis of case-control study data, logistic regression yields valid and efficient inferences on exposure-outcome associations (Cornfield, 1951; Anderson, 1972; Prentice and Pyke, 1979; Breslow and Day, 1980; Breslow, 1996) even if we ignore the case-control sampling plan. In general, however, analyzing the complete case data naive to the ODS sampling scheme is not valid. Lawless et al. (1999) describe the conditional likelihood calculations that explicitly acknowledge the design within the likelihood. Importantly, even though the approach allows for nonbinary responses, it still requires a known parametric distributional form for the response, which may lead to incorrect estimation and inference for parameters of interest if the parametric form is misspecified.

In this report, we describe a unifying framework—in the sense that we have a single analysis strategy for many different forms of responses—for ODS from two-phase studies by leveraging an approach that will appeal to those familiar with GLMs and the classic case-control design. We consider a semiparametric extension to the GLM (SPGLM; Rathouz and Gao, 2009) that provides a full-likelihood alternative to the quasi-likelihood (QL; Wedderburn, 1974; McCullagh, 1983) modeling framework. The SPGLM nonparametrically models the response distribution given covariates, but similar to QL and other estimating function-based approaches, parametrically models the mean of the response given covariates. To highlight the connection to case-control studies, we refer to studies with beyond binary responses as generalized case-control (GCC) studies (Schildcrout et al., 2019). We are focusing on the setting where only Phase 2 data, as well as the sampling plan used to collect these data, are available. We apply the SPGLM framework to analyze data collected from GCC studies. The major strength to our approach is that it allows the experimenter to analyze data from GCC studies in a unified way covering a wide variety of response distributions, while also adding robustness by not having to assume a particular form for that distribution. This robustness is important under GCC sampling because, as we will show, fully parametric approaches can be sensitive to misspecification of the response distribution. As an added feature, the response distribution under ODS takes essentially the same functional form as it does under simple random sampling, thus simplifying the extension from simple random sampling to ODS.

The paper is organized as follows. In Section 2, we review the SPGLM and GCC designs. In Section 3, we develop the estimation and inference procedures under GCC studies using the SPGLM framework. In Section 4, we conduct simulation studies to examine the validity and robustness of the SPGLM approach to an analogous parametric approach under misspecification. Finally, in Section 5, we demonstrate the use of our methods using the AHEAD study wherein independent variables required an extensive interviews by an expert, and is therefore time-consuming and expensive to collect. Furthermore, the response for the AHEAD study does not follow one of the standard exponential family forms, meaning that standard parametric assumptions are not easy to implement. We conclude with a discussion and summary.

2 |. MODELING AND SAMPLING FRAMEWORK

2.1 |. Model

Rathouz and Gao (2009) proposed a semiparametric extension of traditional GLMs (McCullagh and Nelder, 1989). As with traditional GLMs, their model focuses on inference with respect to a parametric mean model. However, the SPGLM operates under nonparametric estimation of the response distribution. As such, it provides a full, albeit semiparametric, likelihood alternative to QL. Although the SPGLM has been shown to perform comparably to QL in terms of estimation and inference, the specification of a full likelihood has a number of advantages. These advantages include an objective function for model comparison, the ability to do prediction, and potential robustness under the traditional missing-at-random (MAR) assumption. Conveniently, through application of Bayes’ theorem, the model also adapts easily to ODS designs, the major theme of this paper, and one which we will study in detail.

Although, in contrast to standard GLMs, the SPGLM only requires specification of the mean model (i.e., it does not require specification of the response distribution); however, it still maintains the same interpretation of mean model parameters. The user specifies the linear predictor, and then selects the link function that corresponds to the desired model interpretation. Once the mean model is specified, the remaining characteristics of the conditional response distribution are estimated nonparametrically. Using minor distributional assumptions, the model potentially accounts for a large variety of analyses with nontraditional responses without concern for misspecifying the response distribution, yielding a unified approach across a range of distributional forms.

To fix ideas, let X_i represent a p × 1 vector of covariates for the ith subject, and let $Y={\left(Y_i\right)}_{i=1}^n$ represent a n × 1 vector of observations, where each observation Y_i = y_i is an independent realization sampled from

$$f\left(y_i\mid X_i;\beta ,f_0\right)=\text{exp}\left\{y_i{\theta }_i-b\left({\theta }_i\right)+\text{log}{f}_0\left(y_i\right)\right\},$$ (1)

for y_i in support $𝒴\subset ℛ$. The vector β contains the coefficients for the linear predictor, and f₀(·) is a nonparametric reference distribution on $𝒴$, to be estimated from the data. However, for fixed f₀, (1) is a natural exponential family (Morris, 1982) with canonical parameter θ_i and cumulant generating function

$$b\left(\theta ;f_0\right)≔b(\theta )=\text{log}{\int }_{u\in 𝒴}{f}_0(u)e^{\theta u} \text{d}u.$$

For given f₀, we therefore recover the standard results from GLMs, E(Y_i|X_i) = b′(θ_i) and Var(Y_i|X_i) = b″ (θ_i). The focus of inference is on the conditional mean of Y_i given X_i,

$$\begin{gathered} g^{-1}\left({\eta }_i\right)≔{\mu }_i≔E\left(Y_i\mid X_i;\beta \right)=b^{\prime }\left({\theta }_i\right) \\ =\frac{{\int }_{u\in 𝒴}u{f}_0(u)e^{{\theta }_{i}u} \text{d}u}{{\int }_{u\in 𝒴}{f}_0(u)e^{{\theta }_{i}u} \text{d}u,} \end{gathered}$$ (2)

where ${\eta }_i=X_i^T\beta$ is the linear predictor and g(·) is a known link function, strictly increasing on the interval (m, M), m = inf $𝒴$ and M = sup $𝒴$. An important note is that the free parameters here are β and f₀; β determines μ_i, which, together with f₀, determines θ_i through solution to (2). The resulting score and information for β is given by

$$S(\beta )=\sum _{i=1}^n\left(y_i-{\mu }_i\right)\frac{1}{b^{″}\left({\theta }_i\right)g^{\prime }\left({\mu }_i\right)}{X}_i.$$ (3)

and

$${ℐ}_{\beta \beta }=\sum _{i=1}^n{X}_i\frac{1}{b^{″}\left({\theta }_i\right){\left\{g^{\prime }\left({\mu }_i\right)\right\}}^2}{X}_i^T.$$ (4)

For fixed f₀, S(β) and ${ℐ}_{\beta \beta }$ arise from standard GLM theory.

When f₀ is not known, the fitting procedure for this model works via an extension of Fisher scoring by iteratively estimating β and f₀ while holding the other parameter fixed; see Wurm and Rathouz (2018). The response distribution for Y_i given covariates X_i is then estimated by plugging the estimated mean parameter, $\widehat{\mathit{\beta }}$, and reference distribution, ${\widehat{f}}_0$, into (1).

2.2 |. Generalized case-control studies

As previously mentioned, case-control sampling is used in studies with retrospective sampling of a binary response. If one level of the response, for example, disease status, is rare, case-control sampling is much more efficient in terms of sample size than an equivalent equal probability sampling (EPS) design, which samples at random with respect to the response given predictors. In principle, there is nothing restricting the case-control concept to binary outcomes. For example, if the response of interest is a count, we may design a study to oversample extreme levels of the count to increase observed covariability between the exposure of interest and response. We refer to such designs as GCC studies, and propose analysis for such studies when the underlying model is a member of the GLM family.

In order that we may precisely define a GCC study, we consider a setting where the researcher provides us with a collection of observations ${\left(Y_i,X_i\right)}_{i=1}^n$, where X_i includes both exposure data and adjustment variables, from Phase 2 of a two-phase study. The challenge is that the exposure data were collected via ODS, encoded by a function ξ(·) on the support of Y_i, and defined as

$$\text{P}\left(S_i=1\mid Y_i,X_i\right)=P\left(S_i=1\mid Y_i\right)=\xi \left(Y_i\right)\text{ since }{S}_i\coprod X_i\mid Y_i.$$ (5)

Here, S_i is the sampling indicator for the ith subject. The setup is very similar to the well-known case-control study; however for most scenarios beyond binary response with logistic regression, the investigator must also supply ξ(y) to the analyst.

There are many different ways to define the sampling scheme via the function ξ(·). For example, each level of Y_i could receive a unique sampling probability. Alternatively, the investigator may choose to make ξ(·) a smooth function of Y_i, or the experimenter may choose to group particular levels of the response, and sample according to

$$\xi (y)=\{\begin{array}{ll}{\xi }_1,\hfill & y<a\hfill \\ {\xi }_2,\hfill & a\le y\le b\hfill \\ {\xi }_3,\hfill & b<y,\hfill \end{array}$$

where b > a are cut-points. This flexibility allows experimenters to pick simple designs while still retaining the complexity of the response variable for modeling purposes.

3 |. ESTIMATION AND INFERENCE

3.1 |. Estimation

In this section, we develop estimation and inference for coefficients of the mean model in the SPGLM, accounting for GCC sampling. Beginning with Model (1), sampling based on the observed values of outcome Y induces a new response distribution in the sampled data. To study this induced distribution, we leverage the sampling scheme introduced in (5). Using Bayes’ rule, we obtain the following density for outcome Y conditional on being sampled under ODS,

$$\begin{array}{l}{f}^{\ast }(y\mid S=1,X)=\frac{f(y\mid X)\xi (y)}{{\int }_{𝒴}f(u\mid X)\xi (u)\text{d}u}\\ =\frac{\text{exp}\left\{y\theta +\text{log}\left(f_0^{\ast }\right)\right\}}{{\int }_{𝒴}\text{exp}\left\{u\theta +\text{log}\left(f_0^{\ast }\right)\right\}\text{d}u}\\ =\text{exp}\left\{y\theta -b^{\ast }(\theta )+\text{log}\left(f_0^{\ast }\right)\right\},\end{array}$$ (6)

where we define $f_0^{\ast }(y)=f_0(y)\xi (y)$ and $b^{\ast }(\theta )=\text{log}\left({\int }_{𝒴}\text{exp}\left\{u\theta +\text{log}\left(f_0^{\ast }\right)\right\}\text{d}u\right)$. The induced mean under ODS is now μ∗= b∗′(θ), which is defined as

$$b{\ast }^{\prime }\left({\theta }_i\right)=\frac{{\int }_{u\in 𝒴}u{f}_0^{\ast }(u)e^{{\theta }_{i}u} \text{d}u}{{\int }_{u\in 𝒴}{f}_0^{\ast }(u)e^{{\theta }_{i}u} \text{d}u},$$

and the induced variance is b∗″ (θ). It important to note that μ_i and ${\mu }_i^{\ast }$ (and the respective variances) are with respect to different distributions: μ_i is respect to the response distribution under EPS and ${\mu }_i^{\ast }$ is with respect to the induced distribution from performing GCC sampling.

Focusing for the moment on estimation of β with fixed f₀(y), and using (6), the conditional log-likelihood function under ODS is

$$\begin{array}{l}{l}^{\ast }(\beta )=\text{log}\left[\prod _{i=1}^n\text{exp}\left\{y_i{\theta }_i-b^{\ast }\left({\theta }_i\right)+\text{log}\left(f_0^{\ast }\right)\right\}\right]\\ =\sum _{i=1}^n\left\{y_i{\theta }_i-b^{\ast }\left({\theta }_i\right)+\text{log}\left(f_0^{\ast }\right)\right\}.\end{array}$$

To derive the score for β, we recall that ${\eta }_i=X_i^T\beta$ and g(μ_i) = η_i and apply the chain rule to obtain

$$S(\mathit{\beta })=\frac{\partial l^{\ast }}{\partial \mathit{\beta }}=\sum _{i=1}^n\frac{\partial l_i^{\ast }}{\partial {\theta }_i}\frac{\partial {\theta }_i}{\partial {\mu }_i}\frac{\partial {\mu }_i}{\partial {\eta }_i}\frac{\partial {\eta }_i}{\partial \mathit{\beta }},$$

Where $\frac{\partial l_i^{\ast }}{\partial {\theta }_i}=\left\{y_i-b^{{\ast }^{\prime }}\left({\theta }_i\right)\right\}=\left(y_i-{\mu }_i^{\ast }\right)$, $\frac{\partial {\theta }_i}{\partial {\mu }_i}=\mathrm{Var}{\left(Y_i\mid X_i\right)}^{-1}=\frac{1}{b^{″}\left({\theta }_i\right)}$, $\frac{\partial {\mu }_i}{\partial {\eta }_i}={\left\{g^{\prime }\left({\mu }_i\right)\right\}}^{-1}$, and $\frac{\partial {\eta }_i}{\partial \beta }=X_i$. Note that the derivatives involving the mean with respect to EPS (μ_i) are not with respect to the induced mean under GCC sampling (${\mu }_i^{\ast }$). Finally, we obtain the score for β under ODS,

$$S(\beta )=\sum _{i=1}^n\left(y_i-{\mu }_i^{\ast }\right)\frac{1}{b^{″}\left({\theta }_i\right)g^{\prime }\left({\mu }_i\right)}{X}_i.$$ (7)

Note that the score in (7), obtained under ODS closely resembles the score for β under EPS shown in (3), the only difference being that μ_i is replaced with ${\mu }_i^{\ast }$. Next, we consider the information for β under ODS. Using standard information calculations,

$${ℐ}_{\beta \beta }=\sum _{i=1}^n\frac{b^{{\ast }^{″}}\left({\theta }_i\right)}{b^{″}\left({\theta }_i\right)}{X}_i\frac{1}{b^{″}\left({\theta }_i\right){\left\{g^{\prime }\left({\mu }_i\right)\right\}}^2}{X}_i^T.$$ (8)

Notice that in (8), the information involves a ratio of Var(Y_i|X_i) under GCC sampling and the corresponding value under EPS, namely, b∗″(θ_i)/b″(θ_i). This gives us reason to believe that if we use GCC sampling to sample in such a way that the empirical variance of Y (given covariates) is increased relative to under EPS, we will realize increased efficiency for estimating β. This is only approximately true, as we will see, owing to the need to also estimate the reference distribution f₀. However, this ratio is the driving factor to determine efficiency gains from GCC sampling compared to equivalent studies sampling at random with respect to the response.

Under EPS, the SPGLM has an interesting feature where the parameters β in the mean model are orthogonal to the parameters for estimating the reference distribution f₀ (Huang and Rathouz, 2017). Examining (7), we can see that $\left(y_i-{\mu }_i^{\ast }\right)$ is a function of f₀(y) and as such the orthogonality between β and f₀(y) under EPS no longer exists under ODS. As is true with all nuisance parameter problems, this implies that estimation of f₀(y) under ODS will now affect the asymptotic variance of $\widehat{\mathit{\beta }}$. However, this also creates another issue in the context of GCC sampling; note that the mean model is being estimated with respect to β, and consequently, μ_i, but the score function for β using the conditional likelihood is a function of ${\mu }_i^{\ast }$. This means when solving (7) equal to zero, we must use the sampling plan and the estimated response distribution to translate μ_i to ${\mu }_i^{\ast }$. As the only difference in (7) for parametric GLMs is that f₀ is assumed to have a particular parametric form, this means that misspecifying f₀ leads to potential bias (as well as incorrect standard errors) for β under fully-parametric GLMs. As shown in Section 4, the SPGLM is flexible enough to eliminate this bias in many cases. In addition to flexibility, the SPGLM is easier to implement than likelihoods that assume an unbounded support for the response because the integral in the denominator of (6) becomes a sum over the observed support. In the next section, we show how to obtain correct inferences using the SPGLM with a conditional likelihood.

3.2 |. Inference

We will focus on inference with respect to the coefficients in the mean model. Although we have already calculated an information matrix for β, we do not yet have a method to conduct correct inferences for β because these calculations do not reflect simultaneous estimation of the parameter f₀ under ODS. To address this issue, we note that the effective information for β in the presence of estimating f₀ is

$${ℐ}_{\beta \beta }^{\text{eff}}={ℐ}_{\beta \beta }-{ℐ}_{\beta f_0}{ℐ}_{f_0{f}_0}^{-1}{ℐ}_{f_0\beta }$$ (9)

(see, for example, Shao, 2003). Above, ${ℐ}_{\beta \beta }$ represents the β corner of the joint information matrix of β and f₀. A similar interpretation holds for ${ℐ}_{f_0{f}_0}$, and ${ℐ}_{\beta f_0}$ represents the cross-information between the two parameters. As previously stated, in the case of ODS, the cross-information between β and f₀ is nonzero, and therefore, the orthogonality between f₀ and β available under EPS is not obtained under ODS.

We restrict the inference to the case where Y has finite support, denote $𝒴={\left(s_1,\dots ,s_K\right)}^T$. Similarly, let $f_0^{\ast }={\left(f_1^{\ast },\dots ,f_K^{\ast }\right)}^T$, where the mth entry of $f_0^{\ast }$ is the induced reference distribution evaluated at S_m. Now, we write the empirical (conditional) likelihood (see Owen, 1991) for f₀,

$$l^{\ast }\left(f_0\right)=\sum _{i=1}^n\left\{y_i{\theta }_i-\text{log}\left(\sum _{k=1}^K{f}_k^{\ast }{e}^{{\theta }_i{s}_k}\right)+\sum _{k=1}^{K}I\left(y_i=s_k\right)\text{log}\left(f_k^{\ast }\right)\right\}.$$

We denote the mth component of the score for f₀(y) by S(f_m) and use the chain rule,

$$S\left(f_m\right)=\frac{\partial l^{\ast }}{\partial f_m^{\ast }}\frac{\partial f_m^{\ast }}{\partial f_m}+\frac{\partial l^{\ast }}{\partial \theta }\frac{\partial \theta }{\partial f_m},$$

yielding

$$\begin{array}{l}S\left(f_m\right)=\sum _{i=1}^n\{\frac{I\left(y_i=s_m\right)}{f_m}-\frac{e^{{\theta }_i{s}_m}}{e^{b\ast \left({\theta }_i\right)}}\xi \left(s_m\right)\\ -\left(y_i-{\mu }_i^{\ast }\right)\frac{e^{{\theta }_i{s}_m}\left(s_m-{\mu }_i\right)}{e^{b\left({\theta }_i\right)}{b}^{″}\left({\theta }_i\right)}\}.\end{array}$$ (10)

Then, the maximum empirical likelihood estimate is the solution to S(f_m) = 0, ∀m, and the resulting information is

$$\begin{array}{l}{ℐ}_{f_k{f}_m}=\sum _{i=1}^n\frac{I(k=m)e^{{\theta }_i{s}_k}\xi \left(s_k\right)}{e^{b\ast \left({\theta }_i\right)}{f}_k}-\frac{e^{{\theta }_i\left(s_k+s_m\right)}\xi \left(s_k\right)\xi \left(s_m\right)}{e^{2b\ast \left({\theta }_i\right)}}\\ -\frac{\left(s_k-{\mu }_i^{\ast }\right)\left(s_m-{\mu }_i\right)e^{{\theta }_i\left(s_k+s_m\right)}\xi \left(s_k\right)}{e^{b\ast \left({\theta }_i\right)}{e}^{b\left({\theta }_i\right)}{b}^{″}\left({\theta }_i\right)}\\ -\frac{\left(s_k-{\mu }_i\right)\left(s_m-{\mu }_i^{\ast }\right)e^{{\theta }_i\left(s_k+s_m\right)}\xi \left(s_m\right)}{e^{b\ast \left({\theta }_i\right)}{e}^{b\left({\theta }_i\right)}{b}^{″}\left({\theta }_i\right)}\\ +\frac{\left(s_k-{\mu }_i\right)\left(s_m-{\mu }_i\right)e^{{\theta }_i\left(s_k+s_m\right)}b{\ast }^{″}(\theta )}{e^{2b\left({\theta }_i\right)}{\left\{b^{″}\left({\theta }_i\right)\right\}}^2}.\end{array}$$

See Online Appendix A for score and information calculations.

Finally, The resulting cross-information is given by

$$\begin{array}{l}{ℐ}_{\beta f_m}=\sum _{i=1}^n{X}_i\frac{e^{{\theta }_i{s}_m}}{g^{\prime }\left({\mu }_i\right)b^{″}\left({\theta }_i\right)}\\ \times \left\{\frac{\left(s_m-{\mu }_i^{\ast }\right)\xi \left(s_m\right)}{e^{b\ast \left({\theta }_i\right)}}-\frac{\left(s_m-{\mu }_i\right)b^{{\ast }^{″}}\left({\theta }_i\right)}{e^{b\left({\theta }_i\right)}{b}^{″}\left({\theta }_i\right)}\right\}.\end{array}$$ (11)

Note that a term without the star notation is the corresponding term as if the data were sampled at random with respect to the response. The first thing we notice about (11) is that if we reduce this equation to the EPS setting, the term in brackets equals zero, which matches the expected result.

Due to a technical issue regarding identifiability, the model in (1) has three constraints imposed on the reference distribution, which requires us to calculate a “constrained” information for f₀. In essence, this requires us to perform a projection of the information into a subspace that accounts for these constraints. However, this does not have a meaningful impact on the interpretation of our results, and is therefore left to Online Appendix B.

Because we restrict our setting to the case where the response has finite support, we may use the inverse effective information (9) for β and standard maximum likelihood results to construct asymptotically correct standard errors for β under ODS. Together with the results for estimation in Section 3.1, we can conduct asymptotically correct analysis for ODS from an arbitrary response distribution with a unified approach.

4 |. SIMULATIONS

4.1 |. Zero-inflated truncated poisson

A main strength of our approach is that we simultaneously estimate the response distribution given covariates in the population, while also accounting for GCC sampling. This gives us a valid and robust method for handling data from GCC studies under many response distributions. Our goal with the present simulation study is to assess how the SPGLM and traditional GLM approaches under GCC work in cases where standard model distributional assumptions (e.g., Poisson, Binomial) are violated. We also want to assess if there is a penalty in efficiency for estimating the response distribution using the SPGLM in cases where a standard GLM is correctly specified. To study this, we begin with a mean model of the form

$$\text{log}{\mu }_i={\beta }_0+{\beta }_1{x}_i,$$ (12)

where each y_i is an independent sample from model (1). As a reference distribution, we consider two cases; the first is a modified Poisson with rate parameter equal to 1, truncated at y = 5, and three times the original mass on y = 0. We refer to this as a zero-inflated, truncated Poisson. This distribution is intended to represent data one might encounter in practice, in the sense that it is an overdispersed count of a finite number of items, but is not well captured with standard GLMs. Furthermore, this data-generating model is similar to the “instrumental activities of daily living” from the AHEAD study. For the second case, we simply generate data from a Poisson distribution with mean μ_i. For each setting, we let $X_i~\text{Unif}\left(-\frac{\sqrt{12}}{4},\frac{\sqrt{12}}{4}\right)$; these bounds set sd(X_i) = 1/2. Finally, we set β₀ = 0.2, let β₁ = 0.2 or 0.7, and allow sample size to vary as shown in Table 1. We consider a two-phase design that samples 100,000 observations at random during Phase 1, and then uses GCC sampling to sample, on average, equal numbers from two response strata: y < 4 (controls) and y ≥ 4 (cases), where within each stratrum, we sample, on average, equal numbers from each level of the response. For example, if we sample 1000 observations in total, approximately 500 observed values will be such that y < 4 and y ≥ 4, respectively. Within the “controls” stratum, approximately 125 will come from each level of y, whereas within the “cases” stratum, approximately 250 will come from each level of y. The prevalence of cases in the population of the zero-inflated, truncated Poisson is approximately 6% in the small effect setting (when β₁ = 0.2) and 8% in the large effect setting (when β₁ = 0.7), and slightly larger in the standard Poisson setting.

TABLE 1.

Results for estimation and inference from simulation study described in Section 4.1 (only β₁). We show results from 2000 replicates with varying sample size and effect size of predictor, as well as two different data-generating models/analysis strategies

Sample size	Data-generating model	Analysis strategy	True value	AE	ESE	AESE	CP
n = 100	Poisson GLM	Poisson GLM	0.2	0.206	0.120	0.117	0.950
		SPGLM		0.208	0.125	0.121	0.953
	Zero-inflated, truncated Poisson	Poisson GLM	0.2	0.166	0.132	0.122	0.922
		SPGLM		0.204	0.164	0.163	0.956
n = 500	Poisson GLM	Poisson GLM	0.2	0.199	0.0518	0.0516	0.952
		SPGLM		0.200	0.0531	0.0528	0.955
	Zero-inflated, truncated Poisson	Poisson GLM	0.2	0.161	0.0589	0.0537	0.854
		SPGLM		0.198	0.0730	0.0720	0.946
n = 1000	Poisson GLM	Poisson GLM	0.2	0.200	0.0368	0.0364	0.947
		SPGLM		0.200	0.0378	0.0372	0.944
	Zero-inflated, truncated Poisson	Poisson GLM	0.2	0.161	0.0420	0.0379	0.794
		SPGLM		0.199	0.0520	0.0508	0.941
n = 100	Poisson GLM	Poisson GLM	0.7	0.712	0.139	0.141	0.959
		SPGLM		0.713	0.156	0.156	0.951
	Zero-inflated, truncated Poisson	Poisson GLM	0.7	0.592	0.148	0.135	0.847
		SPGLM		0.711	0.185	0.183	0.954
n = 500	Poisson GLM	Poisson GLM	0.7	0.701	0.0629	0.0618	0.952
		SPGLM		0.701	0.0702	0.0688	0.942
	Zero-inflated, truncated Poisson	Poisson GLM	0.7	0.581	0.0669	0.0597	0.491
		SPGLM		0.700	0.0821	0.0808	0.947
n = 1000	Poisson GLM	Poisson GLM	0.7	0.700	0.0448	0.0436	0.954
		SPGLM		0.700	0.0500	0.0485	0.940
	Zero-inflated, truncated Poisson	Poisson GLM	0.7	0.581	0.0465	0.0421	0.223
		SPGLM		0.701	0.0576	0.0571	0.949

Abbreviations: AE, average estimate; ESE, empirical standard error; AESE, average estimated standard error; CP, coverage probability, 95% confidence intervals.

To investigate the robustness property of the semiparametric GLM, we consider two different analysis strategies. First, we analyze the resulting Phase 2 data from the previously described data-generating methods using the SPGLM accounting for GCC sampling. We also analyze the same data with a Poisson GLM (assuming ξ(y) = 0 for y > 5), also accounting for GCC sampling. This is done to (1) investigate the impact of misspecifying the response distribution on the conditional likelihood approach, and (2) characterize the loss of efficiency for estimating the response distribution compared to correctly specifying it. To perform the analysis using the SPGLM, we use the methodology developed in Section 3. To perform the corresponding Poisson analysis, as discussed in Rathouz and Gao (2009), any GLM can be expressed using (1), for example, for a Poisson GLM, f₀(y) = 1/(ey!). Therefore, instead of estimating f₀, we perform the Poisson analysis by supplying the correct f₀.

In Table 1, we report the empirical mean and empirical standard deviations of the coefficient estimates for 2000 replications, the average estimated standard error, and coverage probability associated with 95% confidence intervals for the slope coefficient. Results regarding the intercept are shown in Table C.1, which is available in Online Appendix C. Results are shown for both data-generating models, as well as both analysis strategies. For the SPGLM, we observe minimal bias in parameter and uncertainty estimates, even for n = 100, and the coverage probabilities approximately obtain their 95% nominal values in each setting. On the other hand, for the Poisson GLM analysis, when there is model misspecification, there is both bias in parameters of interest, as well as incorrect coverage. When the Poisson model is correctly specified, the Poisson GLM model behaves as expected, achieving minimal bias and correct coverage probabilities. Additionally, the SPGLM achieves minimal bias, and only has around a 5% loss of efficiency in the cases with a small effect size, which is when performing ODS has the greatest opportunity to increase efficiency. The largest loss of efficiency is approximately 20% (in the large effect size), but still achieves correct coverage probabilities.

4.2 |. Overdispersed binomial

Next, we aim to investigate the effects of model mispecification in another setting, using an overdispersed binomial distribution. This set of simulations is important because it allows us to vary the amount of model mispecification by varying the amount of overdispersion as a hyperparameter to the data-generating model. This allows us to characterize how sensitive GLM-based approaches are to distributional mispecification. To be specific, data were generated as follows:

1. Generate $X_i~\text{Unif}\left(-\frac{\sqrt{12}}{4},\frac{\sqrt{12}}{4}\right).$

2. Calculate ${\mu }_i=\frac{1}{1+e^{-{\eta }_i}}$, where η_i = β₀ + β₁X_i (β₀ = −0.6 or 0.2 and β₁ = 0.2).

3. Generate P_i ~ Beta(a, b), where a and b are selected such that E(P_i) = μ_i and Var(P_i) is selected to give the desired amount of overdispersion (relative to a binomial distribution with m = 9), for example, Var(P_i) = μ_i(1 − μ_i)/{α ∗ (m − 1)} yields {(1/α) × 100}% overdispersion.

4. Generate Y_i ~ Bin(m = 9, p = p_i), where m and p denote the number of trials and probability of success, respectively.

This results in binomial data with overdispersion, where the amount of overdispersion is governed by the parameter α. In Figure 1, we show the resulting beta-binomial distribution for 25%, 50%, and 100% overdispersion compared to binomial responses without overdispersion from Bin(m = 9, p = μ_i). The setting with 25% overdispersion is considered a setting with a “very minor” amount of overdispersion. In our simulation, we vary the intercept term as either equal to −0.6 or 0.2 to give a “skewed” or “symmetric” response distribution.

FIGURE 1.

Example of resulting response distribution for varying amount of overdispersion (25%, 50%, and 100%) compared to generating binomial responses with probability equal to μ_i. This figure appears in color in the electronic version of this article, and any mention of color refers to that version

For these simulations, we generate Phase 1 data at random with respect to the response. In this setting, the Phase 1 data have 1 million observations; this is so that there are enough samples in each level of the response to conduct our GCC sampling plan. We perform GCC sampling such that in each unique level of the response, we expect 50 samples, resulting in an expected sample size of 500. For each setting, we perform two analyses: one assuming a binomial response distribution (as explained in Section 4.1, except now f₀(y) = (1/2)^m), and another with our methodology developed for the SPGLM. The analysis strategy is such that we analyze the data according to the mean model

$$\text{log}\left(\frac{{\mu }_i}{m-{\mu }_i}\right)={\beta }_0+{\beta }_1{x}_i,$$

with the corresponding distributional assumption, accounting for GCC sampling. The simulation study is replicated 2000 times. We show the results for the slope parameter in Table 2. Results regarding the intercept are shown in Table C.2, which is available in Online Appendix C. In summary, the SPGLM achieves approximately valid estimation and inference in all settings. However, the binomial analysis yields inaccurate results, both in terms of estimation and inference, under GCC sampling.

TABLE 2.

Results for simulation study investigating robustness (only β₁), as described in Section 4.2

Symmetry	Data-generating model	Analysis strategy	True calue	AE	ESE	AESE	CP
Symmetric	Binomial	Binomial GLM	0.2	0.201	0.0326	0.0327	0.954
		SPGLM		0.201	0.0338	0.0341	0.953
	25% Overdispersion	Binomial GLM	0.2	0.230	0.0438	0.0383	0.846
		SPGLM		0.204	0.0398	0.0411	0.962
	50% Overdispersion	Binomial GLM	0.2	0.246	0.0579	0.0428	0.733
		SPGLM		0.205	0.0489	0.0485	0.946
	100% Overdispersion	Binomial GLM	0.2	0.253	0.0766	0.0495	0.690
		SPGLM		0.207	0.0628	0.0634	0.953
Skewed	Binomial	Binomial GLM	0.2	0.201	0.0326	0.0328	0.954
		SPGLM		0.201	0.0341	0.0345	0.953
	25% Overdispersion	Binomial GLM	0.2	0.226	0.0466	0.0387	0.852
		SPGLM		0.196	0.0418	0.0413	0.940
	50% Overdispersion	Binomial GLM	0.2	0.241	0.0595	0.0434	0.757
		SPGLM		0.197	0.0495	0.0488	0.946
	100% Overdispersion	Binomial GLM	0.2	0.243	0.0806	0.0499	0.722
		SPGLM		0.195	0.0649	0.0636	0.945

Abbreviations: AE, average estimate; ESE, empirical standard error; AESE, average estimated standard error; CP, coverage probability, 95% confidence intervals.

It is important to note that even though the SPGLM achieves correct estimation and inference, the beta-binomial data generated in these simulations are not a member of the assumed model for the SPGLM, as covered in Section 2.1. Even so, the flexibility of the SPGLM captures the distribution sufficiently to yield reliable statistical results. On the other hand, very mild deviations from the binomial distribution yield inaccurate analyses using standard GLM approaches.

5 |. THE AHEAD STUDY

To test our method, we implement GCC sampling using data from the Asset and HEAlth Dynamics among the oldest old (AHEAD) study (Soldo et al., 1997). The AHEAD study, part of the HRS (Health and Retirement Study), is sponsored by the National Institute on Aging (grant number NIA U01AG009740) and is conducted by the University of Michigan. The AHEAD study was a national longitudinal study of individuals aged 70 and older. The objectives of AHEAD were to monitor transitions in physical, functional, and cognitive health, and to study the relationship of late-life changes in health to patterns of dissaving and income. The AHEAD study is of particular interest because an exposure of interest, household net worth, was ascertained by in-depth interview of the subject by a specialist, which was expensive and time-consuming.

For our purposes, we use the complete-case baseline data from 1993, N = 6, 441. The variables of interest are: number of instrumental activities of daily living tasks for which the subject reports some difficulty (ranges from zero to five), age, sex, immediate word recall (number of words out of 10 that subjects can list immediately after hearing them read), and categorical values of net worth. We perform two separate analyses, one regressing the number of instrumental activities of daily living on age, sex, immediate word recall, and categorical values of net worth and another analysis regressing immediate word recall on age, sex, and categorical values of net worth. For the analysis with number of instrumental activities of daily living as the outcome, we use a log link function, and for the analysis with immediate word recall as the outcome, we use a logistic link with 11 categories (i.e., g(μ_i) = log{μ_i/(10 − μ_i)}). These analyses are chosen to mirror those from Rathouz and Gao (2009) under EPS. The distribution of the response values is given in Table 3, along with the proposed number of samples and corresponding sampling probabilities for each level. Note that for these candidate designs, we are oversampling levels with relatively low counts compared to the full data.

TABLE 3.

Distribution of response data and proposed GCC sampling design for AHEAD study

Response value	0	1	2	3	4	5	6	7	8	9	10
Number of instrumental activities of daily living
Count	4,806	1,043	343	157	64	28	—	—	—	—	—
Sampling probability	10%	30%	50%	70%	90%	100%	—	—	—	—	—
Expected sample size	481	313	172	110	58	28	—	—	—	—	—
Immediate word recall
Count	154	195	526	1,001	1,450	1,355	954	445	196	105	60
Sampling probability	100%	90%	70%	50%	30%	10%	30%	50%	70%	90%	100%
Expected sample size	154	176	368	501	435	136	286	223	137	95	60

We first analyze the complete data using the semiparametric GLM under EPS as a gold standard. Then, we select a subsample from the complete data according to the sampling plans in Table 3. We then analyze the sampled data using either the SPGLM or a standard GLM-based analysis, both accounting for the appropriate sampling scheme. In the log-linear model (instrumental activities of daily living as outcome), a Poisson distribution is assumed for the response for the GLM-based analysis. Similarly, a binomial distribution is assumed for the corresponding analysis with immediate word recall as the outcome.

The results of this are shown in Tables 4 and 5. For nearly every coefficient, estimates for the SPGLM analysis are closer to the complete data analysis than the equivalent GLM-based analysis; in general, the SPGLM-based methods yield valid estimates for coefficients under GCC sampling, whereas there is a small amount of bias for coefficients of interest under the GLM-based analysis. Additionally, and importantly, there are significant reductions in standard error due to GCC sampling. For example, in the first analysis, the standard error of the coefficient for the first level of net worth from the EPS analysis is reduced by a factor of 1.37 (0.0780/0.134 times the square root of the sampling fraction) by performing GCC sampling. Therefore, the GCC design is almost twice (1.37² = 1.88) as efficient as EPS; similar results hold for other regression parameters. Inspecting the same coefficient in the analysis considering immediate word recall as the response, we see that the standard error is reduced by a factor of 1.20. This efficiency gain is smaller due to a different sampling plan and model; however, there is still an efficiency gain of 1.45 (1.20²). Differences in estimates from the complete data to the corrected approach could be due to issues with mean model specification or violations in the model in (1). It appears that the only parameter that is markedly different is the coefficient of sex in the analysis involving instrumental activities of daily living, and it is possible that the form of the mean model is adequate for age, immediate word recall, and net worth, but the form does not hold as well for comparison of females to males, leading to less consistent results for the sex coefficient.

TABLE 4.

Results from study implementing GCC sampling on AHEAD data with number of instrumental activities of daily living as response. Below we show estimated coefficients and estimated standard error from the full data. We also show estimated coefficients with estimated standard error under SPGLM and GLM-based analyses implementing the GCC sampling plan from Table 3

Coefficient	Full data (n = 6,441) Estimate (SE)	SPGLM GCC Estimate (SE)	GLM GCC Estimate (SE)
(Intercept)	−3.606 (0.337)	−3.588 (0.546)	−5.176 (0.307)
age	0.0496 (0.00389)	0.0492 (0.00634)	0.0620 (0.00359)
sex:female	0.158 (0.0518)	0.0434 (0.0856)	0.195 (0.0470)
immed. word recall	−0.207 (0.0141)	−0.182 (0.0229)	−0.252 (0.0130)
netwc:1–24k	−0.256 (0.0780)	−0.263 (0.134)	−0.355 (0.0718)
netwc:25–74k	−0.450 (0.0800)	−0.450 (0.135)	−0.602 (0.0732)
netwc:75–199k	−0.692 (0.0814)	−0.745 (0.135)	−0.871 (0.0741)
netwc:200k+	−0.763 (0.0899)	−0.859 (0.152)	−0.936 (0.0816)

TABLE 5.

Results from study implementing GCC sampling on AHEAD data with immediate word recall as response. Below we show estimated coefficients and estimated standard error from the full data. We also show estimated coefficients with estimated standard error under SPGLM- and GLM-based analyses implementing the GCC sampling plan from Table 3

Coefficient	Full data (w = 6441) Estimate (SE)	SPGLM GCC Estimate (SE)	GLM GCC Estimate (SE)
(Intercept)	2.220 (0.134)	2.047 (0.172)	1.403 (0.0972)
age	−0.0393 (0.00167)	−0.0372 (0.00217)	−0.0249 (0.00121)
sex:female	0.215 (0.0188)	0.206 (0.0236)	0.133 (0.0133)
netwc:1–24k	0.279 (0.0413)	0.314 (0.0547)	0.188 (0.0308)
netwc:25–74k	0.388 (0.0401)	0.433 (0.0535)	0.259 (0.0298)
netwc:75–199k	0.549 (0.0387)	0.550 (0.0523)	0.357 (0.0288)
netwc:200k+	0.687 (0.0397)	0.673 (0.0533)	0.440 (0.0293)

6 |. DISCUSSION

The results for the AHEAD study imply that there are potentially huge benefits to planning and performing GCC studies, but it is also of importance to correctly specify the response distribution. We have proposed an asymptotically correct method for analyzing data arising from GCC studies under a wide variety of settings. We developed a consistent estimator in GCC studies, and showed how to obtain correct standard errors and inferences for each of these design cases. Furthermore, we have shown that standard methods are highly sensitive to assumptions regarding response distributions, but our methodology is flexible enough to hold for a wide variety of possible response distributions, whereas standard methods fail. This flexibility allows for a single analysis method covering a large variety of possible response structures.

Now that we have established a unified approach for the analysis of GCC data, future work can address how to select designs within a given setting for this class of studies. For example, a future aim is to give researchers guidelines and tools for selecting the best sampling probabilities across possible response values. Additionally, we may aim to develop methods that use our method, but leverage all the data in Phase 1 via a full-likelihood approach (as opposed to our conditional likelihood approach proposed here) to increase efficiency in parameter estimates of interest when covariate information besides the exposure of interest is available (see, for example, Robins et al., 1995; Chatterjee et al., 2003; Weaver and Zhou, 2005; Tao et al., 2017). Leveraging the remainder of data in Phase 1 requires modeling the distribution of the covariate of interest, which is usually done nonparametrically. The additional Phase 1 data require different methods that are not directly comparable to the conditional likelihood methods described here. It is of interest to extend the SPGLM to this setting because current methods assume that the distribution of response given covariates is fully parametric. Another line of future research can be how to use our work to develop tools to check modeling assumptions from GCC studies. Finally, an additional area of future research may be to extend the SPGLM to account for other complex designs such as multistage and partial questionnaire designs (Wacholder et al., 1994; Whittemore and Halpern, 1997).

Our framework is a natural extension of the case-control study, one of the most commonly used designs in both medicine and public health. Using this novel methodology, practitioners—in particular epidemiologists and clinical investigators—will have more flexibility in experimental design, while also having a tool that yields a similar interpretation to those currently used in practice.

DATA AVAILABILITY STATEMENT

The data that support the findings in this paper are available in the Health and Retirement Study Repository at (http://hrsonline.isr.umich.edu/index.php?p=avail). These data were derived from the following resources available in the public domain: AHEAD 1993 Core, (http://hrsonline.isr.umich.edu/index.php?p=shoavail&iyear=BC).