Mendelian Randomization Studies for a Continuous Exposure Under Case-Control Sampling

Abstract

In this article, we assess the impact of case-control sampling on mendelian randomization analyses with a dichotomous disease outcome and a continuous exposure. The 2-stage instrumental variables (2SIV) method uses the prediction of the exposure given genotypes in the logistic regression for the outcome and provides a valid test and an approximation of the causal effect. Under case-control sampling, however, the first stage of the 2SIV procedure becomes a secondary trait association, which requires proper adjustment for the biased sampling. Through theoretical development and simulations, we compare the naïve estimator, the inverse probability weighted estimator, and the maximum likelihood estimator for the first-stage association and, more importantly, the resulting 2SIV estimates of the causal effect. We also include in our comparison the causal odds ratio estimate derived from structural mean models by double-logistic regression. Our results suggest that the naïve estimator is substantially biased under the alternative, yet it remains unbiased under the null hypothesis of no causal effect; the maximum likelihood estimator yields smaller variance and mean squared error than other estimators; and the structural mean models estimator delivers the smallest bias, though generally incurring a larger variance and sometimes having issues in algorithm stability and convergence.

In observational studies, causal effect of an exposure on a disease outcome is often masked by unmeasured confounding. mendelian randomization studies, using inherited genetic variants as instrumental variables to facilitate causal inference (1, 2), are increasingly popular. The key concept is that genetic predisposition is split randomly at meiosis and therefore largely independent of nongenetic confounding variables. If the genetic variant is associated, preferably strongly, with the exposure and there is no alternative pathway from the genetic variant to the disease outcome, then the test of causality from the exposure to the disease can be inferred from the test of association between the genetic variant and the disease outcome (1, 3). For continuous outcomes and under additional model assumptions, the 2-stage least-squares method yields a consistent and asymptotically normal estimator of the causal effect (4, 5).

Compared with the classic instrumental variable regression in econometrics, mendelian randomization analyses in genetic epidemiology have the unique feature that disease outcomes are predominantly case-control status. Recently, substantial works have been developed to estimate the causal odds ratio for dichotomized outcomes, ranging from the 2-stage instrumental variables (2SIVs) approximation estimator that replaces the second-stage least-squares regression in 2-stage least squares by logistic regression (6, 7), to the related control function estimator (6), to the consistent estimator using double-logistic structural mean models (SMMs) under the potential outcomes framework (8, 9). Among these methods, the 2SIV estimator is often the de facto inference method, because it is straightforward to implement and approximates the causal effect that is defined in structural equation models (6, 7). The double-logistic estimator, though consistent in large samples, can be computationally difficult and sometimes suffers from poor or lack of identification in finite samples (10). The other drawback of the double-logistic SMMs is that the postulated association model may be uncongenial with the causal model, especially when there are adjusting covariates (9). Alternatively for SMMs, a selection-bias function can be parameterized to avoid the uncongenial issue (11), though computationally demanding in the presence of covariates. Refer, for example, to a review by Clarke and Windmeijer (12).

In epidemiologic studies, case-control sampling has been widely used for its cost effectiveness and statistical efficiency. A notable example of mendelian randomization using the case-control sampling design is the study of causal relationship between sex hormone–binding globulin (SHBG) and type 2 diabetes (13). Women with newly diagnosed type 2 diabetes and their 1:1 matched controls were sampled from the Women's Health Study for measuring genetic polymorphisms in the sex hormone-binding globulin gene, SHBG, and the plasma SHBG level. These analyses (13) used the naïve 2SIV method and ignored the case-control sampling scheme. Unlike the classic case-control analyses in which case-control sampling can be conveniently ignored while still preserving statistical efficiency (14), the validity of such naïve analyses in case-control mendelian randomization analyses needs to be carefully evaluated.

The issue of case-control sampling to the 2SIV procedure arises in the first stage, where the genetic risk score for the exposure is computed from a linear model. Because the genotypes and the plasma SHBG level were ascertained by case-control sampling and because the plasma SHBG level is strongly associated with the diabetes status, the first stage of the 2SIV procedure is therefore effectively a secondary-trait association, which has been recently investigated outside of the realm of mendelian randomization (15–18). Except in a few tangential situations, a naïve secondary-trait association without adjustment for case-control sampling can be severely biased, even under the null hypothesis of no secondary trait association. We note that, once the case-control sampling was accounted for in the first stage of 2SIV, the second stage presents no additional problem because the case-control sampling can be ignored for estimating slope parameters in a logistic regression (14).

Recently, there have been some discussions on this topic. For double-logistic SMMs, case-control sampling can be accounted for by the generalized method of moments (GMM) without much difficulty (19). For 2SIV estimators, it was noted that the naïve 2SIV method may not work under case-control sampling, and adjustment for case-control sampling bias can be performed by the inverse probability weighted (IPW) estimator, as suggested previously (20, 21).

However, the IPW estimator is poor in statistical efficiency in estimating secondary-trait association, because it does not leverage the information relating the disease outcome, the secondary trait, and the genotype. Use of the maximum likelihood (ML) estimator in the first-stage secondary-trait association, such as those developed previously (16, 17), would improve the efficiency of the resulting 2SIV estimator, although sensitive to model misspecification. For mendelian randomization studies, the efficiency-robustness consideration is quite different from that in secondary-trait association in a genome-wide association study. First, genetic instrumental variables are often chosen from known susceptibility loci that were identified and replicated a priori for the intermediate phenotype (13, 22). The false-positive genotype-exposure association is less a concern. Second, the instrumental variable regression requires some stringent assumptions to start with. For example, there is no direct effect from the gene to the outcome, nor is there interaction between gene and biomarker on the outcome. There are extensive discussions on limitations of these assumptions (7, 9, 23) that are not the intended topic of this work. If investigators made these assumptions and conducted mendelian randomization analyses, model misspecification may be less a concern to the ML estimator of secondary-trait association. In this case, the consideration may tip over to the potential efficiency gain, because mendelian randomization analyses are often plagued by weak genetic instruments and lack of power.

In this article, we give a rigorous treatment on the impact of case-control sampling on the testing and estimation of causal effect of a continuous exposure, typically a biomarker, on a rare disease outcome. This scenario is common in mendelian randomization studies (13, 22, 24), encompassing a majority of cardiovascular diseases and cancers. We first define the causal effect in structural equation models. We show that the naïve 2SIV estimator provides a valid test for the causal effect under random sampling and, surprisingly, under case-control sampling. We then develop asymptotic distributions for various 2SIV estimators in a unified fashion for the first-stage naïve, IPW, and ML estimators. For completeness, we also include in the comparison the double-logistic SMM because of its easy implementation by the GMM function in R (R Foundation for Statistical Computing, Vienna, Austria). Such numerical comparison between 2SIV and SMM is rarely seen in the literature. The assessment of bias and efficiency for various estimators under case-control sampling is conducted by extensive simulations. We close by a discussion on the merits and limitations of the aforementioned estimators.

METHODS

Consider a mendelian randomization study of the potential causal effect of a continuous intermediate phenotype X on a dichotomous disease outcome Y. Denote by G a set of genotypes used as instrumental variables for inferring the causality. Denote by U the collection of confounding variables that mask the causal effect of X on Y. Suppose U is indeed measured; then, under a random sampling scheme, a cohort of N subjects with independent and identically distributed random variables (Y_i, X_i, G_i, U_i) is generated from the following models sequentially:

$$X_i={\alpha }_0+{\alpha }_1{G}_i+{\alpha }_2{U}_i+{ϵ}_i,$$ (1)

$$\text{logit}\{\text{Pr}(Y_i=1|X_i,U_i)\}={\theta }_0+{\theta }_1{X}_i+{\theta }_2{U}_i,$$ (2)

where ${ϵ}_i$ is identically distributed random error that is independent of G_i and U_i. By construction, both U_i and ${ϵ}_i$ have mean 0. Models 1 and 2 implicitly assumed that Y ⊥ G|(X, U). The models also imply that there is no interaction between G and U in generating X; there is no interaction between X and U in generating Y; and the error $ϵ$ is homoscedastic. Often in genetic studies, there are adjusting covariates in models 1 and 2 conditional upon which the validity of instrumental variables holds. The results presented in this paper can be extended to more realistic settings with covariates. For ease of exposition, we omit adjusting covariates hereafter.

The causal effect of X on Y is defined as the log odds ratio θ₁ in model 2, conditional on all confounding variables U. The challenge of directly estimating θ₁ is that U is not measured. Further complicating the estimation of the causal effect, a case-control sampling mechanism takes place so that only a subset of subjects have (G, X) measured. Let R_i denote the indicator of whether a subject is included in the case-control subset. The observed data, therefore, contain (Y_i, R_i, X_iR_i, G_iR_i), i = 1, … N. We assume that Pr(R_i = 1|Y_i = k, X_i, G_i) = Pr(R_i = 1|Y_i = k) = π_i, so that G and X are missing at random in the sense described elsewhere (25). Following the Bernoulli sampling (26), the participants are first inspected for Y_i sequentially, and the ith subject is selected for observation (G_i, X_i) with prespecified positive probabilities π_i. Let n denote the number of subjects who are in the case-control samples.

Instrumental variables and the 2-stage estimator

If the sampling scheme is simply random and the outcome Y is continuous, the difficulty of estimating causal effect in the presence of unmeasured confounding can be overcome by exploiting instrumental variables. Because G ⊥ U, α₁ ≠ 0, and Y ⊥ G|(X, U), then G is qualified as an instrumental variable. The classic 2-stage least-squares estimator proceeds by first regressing X on G and then regressing Y on the predicted value $\hat{X}$ from the first stage (4, 5). When there is only 1 instrumental variable, the 2-stage least-squares estimate of θ₁ is merely the ratio of the instrumental variable effect on Y and the instrumental variable effect on X.

For a dichotomous Y, the 2SIV estimator can be similarly obtained by fitting the 2-stage models sequentially,

$$\mathbb{E}(X_i|G_i)={\alpha }_0+{\alpha }_1{G}_i,$$ (3)

$$\text{logit}[\mathbb{E}\{Y_i|\hat{\mathbb{E}}(X_i|G_i)\}]={\theta }_0^{\ast }+{\theta }_1^{\ast }\hat{\mathbb{E}}(X_i|G_i).$$ (4)

The 2SIV estimator based on models 3 and 4 converges to the solution of the expected estimation equation derived from the model

$$\text{logit}[\mathbb{E}\{Y_i|\mathbb{E}(X_i|G_i)\}]={\theta }_0^{\ast }+{\theta }_1^{\ast }({\alpha }_0+{\alpha }_1{G}_i),$$

which is clearly inconsistent for θ₁. Nonetheless, we show in the following corollary that stronger statements can be put forth concerning the properties of ${\hat{\theta }}_1^{\ast }$.

Corollary 1

Suppose that models 1 and 2 are data-generating and that the two working models, 3 and 4, were fitted. If G is qualified as an instrumental variable then ${\theta }_1^{\ast }=0$ if and only if θ₁ = 0 and ${\theta }_1^{\ast }$ has the same sign as θ₁.

The proof can be found in the Appendix. This result suggests that, although the naïve 2SIV estimator for a causal odds ratio is generally not consistent, the corresponding testing procedure is valid and consistent for testing whether the causal effect θ₁ = 0. This result holds regardless of whether the disease is rare or not and thus establishes the general utility of 2SIV in mendelian randomization analyses.

Mendelian randomization under case-control sampling

The issue of case-control sampling to the 2SIV estimation lies in the first stage, which is essentially a secondary-trait association previously studied (15–17). Let α = (α₀, α₁) represent the parameters in model 3. If the intermediate phenotype X and the case-control status Y are correlated, the distribution of X in the ascertained case-control sample is likely to be different from that in the general population because of oversampling of cases. So the naïve estimation of α ignoring the sampling scheme may be biased. In what follows, we first establish the asymptotic distribution of the 2SIV estimator based on an approximately consistent estimator of $\hat{\alpha }$ in a unified fashion for rare disease outcomes; second, we discuss the properties of $\hat{\alpha }$ in the naïve 2SIV estimator and the IPW- or ML-based 2SIV estimators that account for selection bias; then, we include in our discussion the double logistic SMM estimators.

Suppose Y is indeed rare, and so the logistic regression in model 4 can be approximated by log-linear regression. For the naïve, IPW, or ML approach, the 2SIV method proceeds by first computing an estimated value ${\hat{X}}_i$ and then plugging it into the log-linear model

$$\log \{\mathbb{E}(Y_i|{\hat{X}}_i)\}={\theta }_0^{\ast }+{\theta }_1{\hat{X}}_i$$ (5)

in the case-control sample. Denote ${\hat{\theta }}_{2\mathrm{SIV}}$ as the 2SIV estimator $({\hat{\theta }}_0^{\ast },{\hat{\theta }}_1)$.

Define the log-linear regression model of Y and the instrumental variables G as

$$\log \{\mathbb{E}(Y_i|G_i)\}={\gamma }_0+{\gamma }_1{G}_i.$$ (6)

It follows that γ₁ = θ₁α₁ and ${\gamma }_0={\theta }_1{\alpha }_0+{\theta }_0^{\ast }$, because

$$\begin{array}{rr}\mathbb{E}(Y_i|G_i)& ={\mathbb{E}}_U{\mathbb{E}}_{X|G,U}\mathbb{E}(Y_i|X_i,G_i,U_i)\\ & ={\mathbb{E}}_U{\mathbb{E}}_{ϵ}\text{exp}\{{\theta }_0+{\theta }_1({\alpha }_0+{\alpha }_1{G}_i+{\alpha }_2{U}_i+{ϵ}_i)+{\theta }_2{U}_i\}\\ & =\text{exp}({\theta }_1{\alpha }_0+{\theta }_0^{\ast }+{\theta }_1{\alpha }_1{G}_i),\end{array}$$

where ${\theta }_0^{\ast }={\theta }_0+\text{log}\mathbb{E}\{\text{exp}({\theta }_{2}U+{\theta }_1{\alpha }_{2}U+{\theta }_1ϵ)\}$. This result does not require $ϵ$ to be normally distributed as long as the error is homoscedastic (7). In matrix notation, denote $\theta =({\theta }_0^{\ast },{\theta }_1)$, γ = (γ₀, γ₁), and $\mathcal{A}=\left(\begin{array}{cc}1& {\alpha }_0\\ 0& {\alpha }_1\end{array}\right)$; then $\gamma =\mathcal{A}\theta$. The slope component of ${\hat{\theta }}_{2\mathrm{SIV}}$ is approximately consistent for θ₁, and the joint distribution is asymptotically normal with the limiting distribution expressed in the following corollary.

Corollary 2

The asymptotic expansion of ${\hat{\theta }}_{2\mathrm{SIV}}$ is

$$\sqrt{n}({\hat{\theta }}_{2\mathrm{SIV}}-\theta )=({\mathcal{A}}^T{A}_2\mathcal{A}{)}^{-1}{\mathcal{A}}^T{A}_2\{\sqrt{n}(\hat{\gamma }-\hat{\mathcal{A}}\theta )\}+o_{\mathrm{p}}(1),$$

where A₂ is the information matrix for $\hat{\gamma }$ in model 6.

The proof can be found in the Appendix. Corollary 2 provides a unified way to compute the variance of the ${\hat{\theta }}_{2\mathrm{SIV}}$ based on the joint asymptotic distribution of $(\hat{\alpha },\hat{\gamma })$, whether $\hat{\alpha }$ is derived from the naïve, IPW, or ML estimator. Note that ${\hat{\theta }}_{2\mathrm{SIV}}$ is a member of so-called minimal distance estimators that has been studied in the econometrics literature on limited dependent variable models (27). In the Appendix, we provide the derivation for the joint limiting distribution of $(\hat{\alpha },\hat{\gamma })$.

Naïve estimator

The naïve estimator for α is obtained by solving

$$\sum _i{R}_i{\mathcal{G}}_i^T\{X_i-\mathbb{E}(X_i|{\mathcal{G}}_i)\}=0,$$

where G = (1, G) is the design matrix of model 3, and R_i is the indicator of observing G. The naïve estimator is generally biased for estimating α (15–17). However, one interesting observation from our simulation study is that, when there is no causal effect (θ₁ = 0), the naïve estimator of α₁ is actually consistent. This is surprising in that, even if θ₁ = 0, Y and X can still be correlated because of unmeasured confounding U. Observe that

$$\begin{array}{rr}& \mathbb{E}\{R_i{\mathcal{G}}_i^T(X_i-{\alpha }_0-{\alpha }_1{G}_i)\}\\ & =\sum _{k=0,1}{q}_k{p}_k\mathbb{E}\{{\mathcal{G}}_i^T({\alpha }_2{U}_i+{ϵ}_i)|Y_i=k\},\end{array}$$ (7)

where p_k is the probability of Y_i = k, and q_k is the sampling probability in the stratum Y_i = k, for k = 0, 1. Because under the null hypothesis θ₁ = 0, Y ⊥ G|U, and because as an instrumental variable G ⊥ U, then Y ⊥ G and by the Bayes rule U ⊥ G|Y. Moreover, under the causal null hypothesis, $ϵ$ and Y are independent. Therefore model 7 becomes

$$\mathbb{E}\left[{\mathcal{G}}_i^T\left\{\sum _k{q}_k{p}_k{\alpha }_2\mathbb{E}(U_i|Y_i=k)\right\}\right].$$

This suggests that the solution of the naïve estimating function is $({\alpha }_0^{\ast },{\alpha }_1)$, where ${\alpha }_0^{\ast }={\alpha }_0+q_k{p}_k{\alpha }_2\mathbb{E}(U_i|Y_i=k)$. The naïve estimator for α₁ is consistent under the null hypothesis for the causal effect, even if the case-control sampling is ignored. Note that this result is derived under several assumptions in the data-generating model 1, that there is no interaction between G and U, and that $ϵ$ is random homoscedastic error. These assumptions are not needed for the test of G-Y association acting as a valid test of causal effect, as originally proposed by Katan (1) and Didelez and Sheehan (3). Because the G-Y association is assessed in logistic regression, such a test is not affected by case-control sampling either.

IPW estimator

The simple IPW estimator solves

$$\sum _i\frac{R_i}{{\pi }_i}{\mathcal{G}}_i^T\{X_i-\mathbb{E}(X_i|{\mathcal{G}}_i)\}=0.$$ (8)

In the context of secondary trait association, it was noted that the IPW estimator is robust against model misspecification (15, 17). For example, even if the fitted model is wrong, the IPW estimator converges to the solution of, $\mathbb{E}[{\mathcal{G}}_i^T\{X_i-\mathbb{E}(X_i|{\mathcal{G}}_i)\}]=0$, which is a well-defined population quantity, as long as the sampling probability π is correctly specified. Compared with the ML estimator we describe below, the consistency of IPW does not depend on correctly specifying the distribution of X or the primary trait regression model. However, it is known to be inefficient for secondary trait association (15, 17, 18), because it does not exploit the information in X and G that is related to primary outcome Y. When there is a parametric model regressing Y on X and G, the ML estimator for secondary trait association is more efficient. If the regression of Y on X and G is left nonparametric, ML has no efficiency gain over IPW (17). Furthermore, when there are additional data in the cohort beyond case-control status, which is often true, Tapsoba et al. (17) discussed ways to improve the efficiency of the IPW estimator, by using for example the estimated weight ${\hat{\pi }}_i$ and adding an augmented term to estimating equation 8.

ML estimator

The likelihood approach to secondary trait association is efficient (15, 16), provided the model specification is correct. The likelihood for regression parameters in the 2-phase sampling plan, given the observed data (Y, RX, RG), is proportional to

$$\begin{array}{rr}& \{\text{P}{\text{r}}_{\eta }(Y|X,G)\text{P}{\text{r}}_{\alpha }(X|G)\text{P}{\text{r}}_f(G){\}}^{R=1}\\ & \times {\left\{{\int }_{X,G}\text{P}{\text{r}}_{\eta }(Y|X,G)\text{P}{\text{r}}_{\alpha }(X|G)\text{P}{\text{r}}_f(G)\text{d}g\text{d}x\right\}}^{R=0},\end{array}$$ (9)

where α indexes the secondary-trait model 3, η = (η₀, η₁, η₂) indexes the nuisance primary trait working model

$$\text{logit}\{\mathbb{E}(Y|G,X)\}={\eta }_0+{\eta }_{1}X+{\eta }_{2}G,$$ (10)

and f describes the distribution of G. For example, refer to the likelihood development for 2-phase sampling reported by Lawless et al. (26). The sampling process does not depend on the regression parameter and therefore is omitted in the likelihood. Working model 10 contains the association of G due to unmeasured confounding U, which can be expressed as a function of X, G, and $ϵ$ based on the data-generating model 1. As emphasized elsewhere (17), the consistency of the ML estimator depends on the correct specification of both Pr_α(X|G) and Pr_η(Y|X, G). In the context of mendelian randomization analyses, because of the assumptions and data-generating models one has to make a priori to infer causality, model misspecification may be less an issue to the ML approach. For example, omitting an interaction term between G and X in the primary case-control regression model when there is indeed an interaction can be devastating to the error control of secondary-trait association (17). If model 2 is the true data-generating model, we can express U as a function of $(X,G,ϵ)$. Under the rare disease assumption and by integrating out $ϵ$, model 10 is the correct model and the interaction term is not needed. On the other hand, the distribution Pr_α(X|G) needs to be integrated in the likelihood. For X being a gaussian distribution, the integration can be well approximated by the Gauss-Hermite quadrature. For nongaussian distributions, we found in simulations that the quadrature integration method is robust and yields little bias.

Double-logistic SMM estimator

Aside from structural equation models, the causal effect can be alternatively defined by the potential outcomes approach (28). Let Y(x) denote the potential outcome of Y when X is experimentally altered to an arbitrary value x within the set of all attainable values. Two assumptions are commonly made: the “consistency assumption” that Y(x) = Y with probability 1 when X = x, and the “stable unit treatment value assumption,” known as SUTVA, that potential outcomes of any subject are not related to other subjects' potential outcomes. The causal odds ratio can be defined (8, 11) as follows:

$$\frac{\text{Odds}(Y=1|X,G)}{\text{Odds}(Y(0)=1|X,G)}=\text{exp}({\theta }_{1}X),$$

where Y(0) is the potential outcome for a subject when his or her X value is controlled at 0. Vansteelandt and Goetghebeur (8) proposed a double-logistic structural mean model, essentially the generalized method of moments, for estimating the causal odds ratio in randomized trials. For estimation, a logistic model for the observed data similar to model 10 is needed in addition to the moment conditions generated by properties of instrumental variables. For mendelian randomization studies with case-control sampling, the same estimation principle applies, and little change is needed in estimation (19). In particular, when the instrumental variable is dichotomous and there are no covariates, as in our simulation model later, no congeniality problem arises. Define a set of estimating functions as follows:

$$\{\begin{array}{c}0=\sum _{i=1}^n{Y}_i-\text{expit}({\beta }_0+{\beta }_1{X}_i+{\beta }_2{G}_i)\\ 0=\sum _{i=1}^n\{Y_i-\text{expit}({\beta }_0+{\beta }_1{X}_i+{\beta }_2{G}_i)\}{X}_i\\ 0=\sum _{i=1}^n\{Y_i-\text{expit}({\beta }_0+{\beta }_1{X}_i+{\beta }_2{G}_i)\}{G}_i\\ 0=\sum _{i=1}^n\frac{1}{{\pi }_i}\text{expit}\{{\beta }_0+{\beta }_1{X}_i+{\beta }_2{G}_i\\ -\log (q_1/q_0)-{\theta }_1{X}_i\}\{G_i-\mathbb{E}(G_i)\}.\end{array}$$

The first 3 estimating equations are derived from the observational model, and the last is based on the property that G shall be orthogonal to the odds of Y(0), with proper adjustment for case-control sampling, including 1/π_i and log(q₁/q₀) for differential sampling probabilities in cases and controls. The estimation can be immediately accomplished by the R function GMM. The optimization method chosen in the GMM function is “BFGS,” a gradient-based searching algorithm with the maximum iteration number 3,000. In simulations, we found that though the GMM function is easy to use, the estimation sometimes does not converge and can be unstable, for example, producing a huge variance estimate.

NUMERICAL STUDY

We conducted simulation studies to compare the finite-sample behavior of the naïve, IPW, ML, and GMM estimators. A random cohort with sample size of either 5,000 or 20,000 was first generated with independent and identically distributed data (G, U, X, Y) following the data-generating models: G ∼ Bernoulli(0.4), X = −ln(1.5)G + U, and ℰ(Y) = expit(−θ₀ +θ₁X + U). We varied the distribution of U to be either N(0, 1) or a standardized χ²(5) in order to assess the impact of X deviating from a normal distribution. The causal effect θ₁ is chosen to be 0, ln 1.5 or ln 2. A case-control sample that contained all cases and the same number of controls was then selected from the cohort. The value of θ₀ was picked such that the proportion of cases in the cohort was approximately 4.5%, so approximately 225 (or 900) cases and 225 (or 900) controls were sampled from the cohort of size 5,000 (or 20,000). Specifically, θ₀ was chosen as (−3.5, −3.8, −4.1) and (−3.6, −4, −4.4) for normal and χ² confounding, respectively. Under each parameter setting, 1,000 independent data sets were simulated. Operating characteristics for ${\hat{\alpha }}_1$ were assessed by using all data sets; those for ${\hat{\theta }}_1$ were based on the data sets when the estimation algorithm converged and the variance estimate appears regular (i.e., <50). The proportion left out in the summary was computed as the convergence rate.

Our simulation settings are similar to those reported by Bowden and Vansteelandt (19): We omitted the random error term $ϵ$ in the data-generating model 1 when simulating data sets. The purpose is that, through reparameterization, the causal effect θ₁ in the structural equation models is also the causal odds ratio in SMM, setting the stage for a side-by-side comparison of 2SIV and double-logistic SMM estimators. Observe that U = X + ln(1.5)G, odds(Y = 1) = exp{−θ₀ +(θ₁ + 1)X + ln(1.5)G}. By SMM notation, odds(Y(0) = 1) =exp{−θ₀ + X + ln(1.5)G} = −θ₀ + U, and hence G ⊥ Y(0). Therefore by construction, θ₁ is also the causal effect under SMM.

Tables 1 and 2 show the performance of various estimators when X is indeed a gaussian distribution. From Table 1, it seems that, under the null hypothesis of no causal effect, ${\hat{\alpha }}_1$ values from all 3 methods (naïve, IPW, and ML) are virtually unbiased, the means of the estimated variance are close to the sample variances, and the coverage probability of 95% confidence interval is correct around 0.95. The result is that the naïve estimator is unbiased under the null of no causal effect as predicted by theoretical development but actually counterintuitive, because under the null G and X in this example both are correlated with Y when unmeasured confounding U is not conditioned upon. As the effect size increases from 0 to ln(1.5) or to ln(2), the naïve $\hat{\alpha }$ becomes biased, and the coverage probabilities of 95% confidence intervals are smaller than 95%. Both IPW and ML remain unbiased estimators under the null, or under the alternative, the bias of ML is generally smaller than that of IPW. Furthermore, the ML estimator is more efficient than the IPW estimator, with a nearly 50% reduction of variance when N = 5,000.

Table 1.

Performance of the Naïve, IPW, ML, and GMM Estimators of α₁ in 1,000 Simulated Data Sets^a

	N = 5,000/n ≈ 450					N = 20,000/n ≈ 1,800
	Bias	$\mathbf{V}\mathbf{a}\mathbf{r}({\hat{\alpha }}_1)$	$\widehat{\mathbf{V}\mathbf{a}\mathbf{r}}({\hat{\alpha }}_1)$	95% CP	MSE	Bias	$\mathbf{V}\mathbf{a}\mathbf{r}({\hat{\alpha }}_1)$	$\widehat{\mathbf{V}\mathbf{a}\mathbf{r}}({\hat{\alpha }}_1)$	95% CP	MSE
θ1 = 0
Naïve	−0.003	0.011	0.011	0.949	0.011	0	0.003	0.003	0.946	0.003
IPW	−0.001	0.016	0.017	0.957	0.016	−0.002	0.004	0.004	0.947	0.004
ML	−0.003	0.009	0.009	0.953	0.009	0	0.002	0.002	0.947	0.002
θ1 = ln(1.5)
Naïve	−0.038	0.013	0.012	0.924	0.014	−0.029	0.003	0.003	0.907	0.004
IPW	−0.002	0.016	0.016	0.943	0.016	−0.002	0.004	0.004	0.938	0.004
ML	−0.005	0.009	0.008	0.940	0.009	0	0.002	0.002	0.954	0.002
θ1 = ln(2)
Naïve	−0.063	0.013	0.013	0.911	0.017	−0.056	0.003	0.003	0.834	0.006
IPW	0	0.016	0.016	0.949	0.016	−0.001	0.004	0.004	0.949	0.004
ML	−0.003	0.008	0.008	0.953	0.008	0	0.002	0.002	0.952	0.002

Abbreviations: CP, coverage probability; GMM, generalized method of moments; IPW, inverse probability weighted; ML, maximum likelihood; MSE, mean squared error.

^a Data-generating models are X = −log(1.5)G + U, Pr(Y = 1) = expit(−θ₀ + θ₁X + U), where U ∼ X(0, 1).

Table 2.

Performance of the Naïve, IPW, ML, and GMM Estimators of θ₁ in 1,000 Simulated Data Sets^a

	Convergence Rate	Bias	$\mathbf{V}\mathbf{a}\mathbf{r}({\hat{\theta }}_1)$	$\widehat{\mathbf{V}\mathbf{a}\mathbf{r}}({\hat{\theta }}_1)$	95% CP	Type 1 Error/Power	MSE
N = 5,000/n ≈ 450
θ1 = 0
Naïve	0.998	−0.061	0.297	0.388	0.964	0.036	0.301
IPW	0.998	−0.020	0.341	0.533	0.985	0.015	0.342
ML	1	−0.012	0.279	0.323	0.977	0.023	0.279
GMM	0.937	−0.049	0.313	0.459	0.982	0.018	0.315
θ1 = ln(1.5)
Naïve	1	−0.106	0.215	0.266	0.978	0.178	0.226
IPW	0.995	−0.003	0.322	0.483	0.982	0.065	0.322
ML	1	−0.021	0.276	0.304	0.978	0.091	0.276
GMM	0.923	−0.039	0.285	0.745	0.991	0.087	0.286
θ1 = ln(2)
Naïve	1	−0.204	0.162	0.188	0.976	0.356	0.203
IPW	1	−0.008	0.434	0.665	0.957	0.133	0.434
ML	1	−0.070	0.278	0.305	0.960	0.214	0.283
GMM	0.908	−0.022	0.320	1.273	0.981	0.193	0.321
N = 20,000/n ≈ 1,800
θ1 = 0
Naïve	1	−0.016	0.063	0.063	0.958	0.042	0.063
IPW	1	−0.003	0.063	0.063	0.962	0.038	0.063
ML	1	−0.003	0.061	0.061	0.952	0.048	0.061
GMM	0.999	−0.014	0.077	0.081	0.969	0.031	0.077
θ1 = ln(1.5)
Naïve	1	−0.105	0.048	0.046	0.956	0.349	0.059
IPW	1	−0.068	0.065	0.065	0.947	0.279	0.069
ML	1	−0.070	0.062	0.060	0.939	0.286	0.066
GMM	0.995	−0.037	0.088	0.092	0.966	0.258	0.090
θ1 = ln(2)
Naïve	1	−0.205	0.038	0.036	0.839	0.736	0.080
IPW	1	−0.115	0.075	0.071	0.904	0.626	0.088
ML	1	−0.122	0.065	0.061	0.900	0.649	0.080
GMM	0.995	−0.017	0.103	0.143	0.972	0.606	0.103

Abbreviations: CP, coverage probability; GMM, generalized method of moments; IPW, inverse probability weighted; ML, maximum likelihood; MSE, mean squared error.

^a Data-generating models are X = −log(1.5)G + U, Pr(Y = 1) = expit(−θ₀ + θ₁X + U), where U ∼ X(0, 1).

Table 2 shows the performance of 4 estimators of the causal effect θ₁, including naïve, IPW, ML, and GMM. When θ₁ = 0, all 4 estimators are unbiased when N = 20,000, yet the naïve estimator and the GMM estimator seem to have small-sample bias when N = 5,000. The variance of the ML estimator is the smallest among the 4 estimators, as is the mean squared error for the ML estimator. Under the alternative, the naïve estimator shows some bias in a much bigger magnitude than the IPW and ML estimators. The ML estimator generally has lower variance than IPW, but the improvement is diminished when the sample size gets to 20,000. This is because the variance of estimated α₁ is much reduced in magnitude when N = 20,000 (Table 1), so small that a half reduction in the scale of the third decimal place does not meaningfully influence the variance of the estimated θ₁.

The GMM estimator appears unbiased under all settings, but its variance is much greater than the IPW and ML estimators, resulting in a bigger mean squared error. The consistency of GMM hinges on the moment condition that exploits the orthogonality of instrumental variables and unmeasured confounding. Its estimation is more complex, involving parameters in the working model for observed outcome in addition to the causal effect θ₁, thereby incurring a bigger variance. It is also notable that the convergence of GMM and the stability of variance estimation can be problematic when the sample size is small and the causal effect increases. As much as 10% of the simulated data sets could yield GMM estimators that did not converge. In a comparison of all 4 methods, the ML estimator has consistently the smallest mean squared error among all scenarios.

In Tables 3 and 4, we assess the performance of various estimators for α₁ and θ₁, respectively, when X is from a standardized χ²(5) distribution having mean 0 and variance 1. The nonnormality of X will affect the performance of the ML estimator because the gaussian quadrature integration used in ML estimation is not ideal for nongaussian distributions. Surprisingly in Table 3, the ML estimator for α₁ did not show much bias even though the numerical integration technique is not optimal. The variance of the ML estimator, however, is no longer superior to the IPW estimator. As a result, the ML estimator for θ₁ does not outperform the IPW estimator as shown in Table 4. The GMM estimator remains unbiased in all settings in Table 4, but its variance is substantially bigger than those of the ML and the IPW estimators. Under the smaller sample size (N = 5,000), the estimated variance for GMM can be quite unstable, with as much as 15% of the simulations that did not converge.

Table 3.

Performance of the Naïve, IPW, and ML Estimators of α₁ in 1,000 Simulated Data Sets^a

	N = 5,000/n ≈ 450					N = 20,000/n ≈ 1,800
	Bias	$\mathbf{V}\mathbf{a}\mathbf{r}({\hat{\alpha }}_1)$	$\widehat{\mathbf{V}\mathbf{a}\mathbf{r}}({\hat{\alpha }}_1)$	95% CP	MSE	Bias	$\mathbf{V}\mathbf{a}\mathbf{r}({\hat{\alpha }}_1)$	$\widehat{\mathbf{V}\mathbf{a}\mathbf{r}}({\hat{\alpha }}_1)$	95% CP	MSE
θ1 = 0
Naïve	0.007	0.020	0.019	0.943	0.020	0.003	0.005	0.005	0.958	0.005
IPW	0.009	0.014	0.015	0.955	0.014	−0.002	0.004	0.004	0.953	0.004
ML	0.007	0.017	0.016	0.947	0.017	0.002	0.004	0.004	0.951	0.004
θ1 = ln(1.5)
Naïve	−0.033	0.022	0.022	0.943	0.023	−0.021	0.006	0.005	0.932	0.006
IPW	−0.006	0.014	0.013	0.949	0.014	0.002	0.003	0.003	0.950	0.003
ML	−0.004	0.016	0.016	0.943	0.016	0.006	0.004	0.004	0.952	0.004
θ1 = ln(2)
Naïve	−0.044	0.023	0.024	0.944	0.025	−0.039	0.006	0.006	0.935	0.007
IPW	0.003	0.012	0.013	0.961	0.012	0.001	0.004	0.003	0.947	0.004
ML	0.007	0.016	0.016	0.957	0.016	0.009	0.004	0.004	0.949	0.004

Abbreviations: CP, coverage probability; IPW, inverse probability weighted; ML, maximum likelihood; MSE, mean squared error.

^a Data-generating models are X = −ln(1.5)G + U, Pr(Y = 1) = expit(−θ₀ + θ₁X + U), where U ∼ standardized χ²(5).

Table 4.

Performance of the Naïve, IPW, ML, and GMM Estimators of θ₁ in 1,000 Simulated Data Sets^a

	Convergence Rate	Bias	$\mathbf{V}\mathbf{a}\mathbf{r}({\hat{\theta }}_1)$	$\widehat{\mathbf{V}\mathbf{a}\mathbf{r}}({\hat{\theta }}_1)$	95% CP	Type 1 Error/Power	MSE
N = 5,000/n ≈ 450
θ1 = 0
Naïve	0.986	−0.067	0.389	0.753	0.956	0.044	0.394
IPW	0.996	−0.011	0.333	0.513	0.982	0.018	0.333
ML	0.995	−0.042	0.381	0.564	0.976	0.024	0.382
GMM	0.867	−0.020	0.412	0.713	0.977	0.023	0.412
θ1 = ln(1.5)
Naïve	0.995	−0.179	0.254	0.552	0.985	0.177	0.286
IPW	0.999	−0.076	0.292	0.387	0.972	0.068	0.297
ML	0.996	−0.108	0.269	0.429	0.990	0.118	0.281
GMM	0.845	−0.072	0.370	0.605	0.980	0.137	0.375
θ1 = ln(2)
Naïve	0.997	−0.338	0.221	0.468	0.992	0.310	0.335
IPW	1	−0.185	0.331	0.484	0.941	0.120	0.365
ML	1	−0.236	0.312	0.482	0.972	0.212	0.367
GMM	0.845	−0.126	0.490	1.781	0.966	0.188	0.506
N = 20,000/n ≈ 1,800
θ1 = 0
Naïve	1	−0.032	0.060	0.070	0.973	0.027	0.061
IPW	1	−0.016	0.055	0.062	0.971	0.029	0.055
ML	1	−0.020	0.057	0.065	0.972	0.028	0.058
GMM	0.983	−0.042	0.095	0.124	0.970	0.030	0.097
θ1 = ln(1.5)
Naïve	1	−0.143	0.047	0.049	0.970	0.321	0.068
IPW	1	−0.107	0.061	0.062	0.930	0.226	0.072
ML	1	−0.113	0.059	0.060	0.960	0.263	0.072
GMM	0.978	−0.029	0.111	0.130	0.960	0.268	0.112
θ1 = ln(2)
Naïve	1	−0.306	0.036	0.038	0.670	0.594	0.130
IPW	1	−0.244	0.065	0.064	0.794	0.465	0.125
ML	1	−0.246	0.056	0.057	0.829	0.533	0.116
GMM	0.988	−0.054	0.139	0.273	0.944	0.465	0.142

Abbreviations: CP, coverage probability; GMM, generalized method of moments; IPW, inverse probability weighted; ML, maximum likelihood; MSE, mean square error.

^a Data-generating models are X = −In(1.5)G + U, Pr(Y = 1) = expit(−θ₀ + θ₁X + U), where U ∼ standardized χ²(5).

DISCUSSION

Several new observations arise in this paper. First, the commonly used naïve 2SIV estimator, ignoring biased sampling such as reported previously (13), yields a biased estimate of the causal effect as one would predict. Yet surprisingly, the test for whether there is a causal effect remains valid. In other words, biased sampling does not affect the validity of the 2SIV testing, nor does it affect the validity of testing the G-Y association as a test of causal effect. This result is interesting as mendelian randomization is plagued by concerns of violating assumptions (23). Focusing on testing causal effect, rather than estimation, may alleviate these concerns (23). The other observation is that, for unbiased estimation, the case-control sampling can be accounted for by using IPW or ML in the first stage of 2SIV, the latter of which is more efficient in a small-to-moderate sample size. Finally, the weighted double-logistic SMM estimator generally yields a smaller bias but a larger variance, therefore resulting in a larger mean squared error than does the 2SIV estimator.

Beyond IPW, there are a number of other methods that account for sampling bias in secondary association analyses. We show the ML method as a mere example of improving the efficiency of 2SIV through improving its first stage regression. When there are additional adjusting covariates in models 1 and 2, ML can be much harder to develop, because one has to model the distribution of such covariates. The profile likelihood, similar to that reported by Lin and Zeng (16), can be developed to treat the covariate distribution nonparametrically. Furthermore, the distributional assumption of secondary trait X can be further relaxed by using recently proposed estimation equation approaches (18, 29).

We have considered a simple unmatched case-control sampling scheme nested within a cohort, where both G and X are missing at random. For matched case-control studies, the methods being discussed can be used by adding matching variables into regression models. The IPW method and the weighted double-logistic SMM method require sampling probabilities to be known, which is feasible for nested case-control samples in a cohort. On the other hand, the ML method can be formulated for general case-control studies without knowing sampling fractions (16). Sometimes X is observed for everyone in the cohort, whereas G is almost always ascertained by case-control sampling. In such a setting, the ML estimator of α₁ is much more advantageous relative to the IPW estimator, because it will leverage the complete information for X. The estimation is quite simple because numerical integration is no longer needed. Refer to the report by Tapsoba et al. (17) for more discussion.

Alternative instrumental variable estimators to the 2SIV procedure include the control functions approach, which adds the first stage residual as an additional regressor in model 5. In doing so, the bias of the instrumental variance estimator can be attenuated (6). The idea of using ML in the first stage of 2SIV to improve the efficiency of causal effect estimates is also applicable to the control functions approach. In future work, we plan to explore the asymptotic distribution of the resulting causal estimates using the control functions approach, as well as its comparison with other approaches.

APPENDIX

Proof of Corollary 1

Suppose θ₁ = 0, then from model 2 we have Y ⊥ G|U. Because G ⊥ U, this leads to Y ⊥ G. Therefore, fitting model 4 leads to ${\theta }_1^{\ast }=0$. Conversely, if θ₁ ≠ 0, and because α₁ ≠ 0, we have $\text{Pr}(Y|G,U,ϵ)\ne \text{Pr}(Y|U,ϵ)$. Without loss of generality, suppose G = 0 is the baseline genotype group, and suppose θ₁α₁ > 0, so $\text{Pr}(Y|G=g,U,ϵ)>\text{Pr}(Y|G=0,U,ϵ)$ for every U and $ϵ$. Because G ⊥ U and $ϵ$ is independent error, Pr(Y|G = g) > Pr(Y|G = 0), so ${\theta }_1^{\ast }{\alpha }_1>0$. A similar argument holds for the setting where θ₁α₁ < 0. Therefore, θ₁ = 0 if and only if θ* = 0; θ₁ and ${\theta }_1^{\ast }$ have the same sign.

Derivation for the joint limiting distribution of $(\hat{\alpha },\hat{\gamma })$

The estimators for both can be represented by asymptotically linear estimators (30, 31). For many approaches to estimating α, including IPW and ML we detail below, $\hat{\alpha }$ can be written as $\sqrt{n}(\hat{\alpha }-\alpha )=n^{-1/2}{\sum }_{i=1}^n{B}_{1i}+o_{\mathrm{p}}(1)$, where B₁ is often referred to as the influence function of α, ℰ(B_1i) = 0, and $\mathbb{E}(B_{1i}^T{B}_{1i})<\mathrm{\infty }$. To estimate γ in model 6, $\hat{\gamma }$ can be represented in terms of influence function B₂ without much difficulty. Then, by the central limit theorem, Sluskey's theorem, and the Cramer-Wold device, the joint limiting distribution of $\hat{\alpha }$ and $\hat{\gamma }$ is

$$\sqrt{n}\left(\begin{array}{c}\hat{\alpha }-\alpha \\ \hat{\gamma }-\gamma \end{array}\right){\to }_{d}N\left(0,\left[\begin{array}{cc}\mathbb{E}(B_{1i}^T{B}_{1i})& \mathbb{E}(B_{2i}^T{B}_{1i})\\ \mathbb{E}(B_{1i}^T{B}_{2i})& \mathbb{E}(B_{2i}^T{B}_{2i})\end{array}\right]\right).$$

In what follows, we discuss in detail each of the 3 estimators of α we consider, how to compute the covariate matrix, and the implication of $\hat{\alpha }$ to estimating θ₁.

Proof of Corollary 2

To obtain ${\hat{\theta }}_{2\mathrm{SIV}}$, one first computes the predicted value $\hat{X}$ in model 3 and then solves the estimating equation $S_n(\hat{X};\theta )$ in model 4. Because γ = Aθ, where A is a matrix composed of elements in α, the 2-stage least-squares estimating function can be equivalently presented as ${\sum }_{i=1}^n\hat{\mathcal{A}}{S}_{2i}(\hat{\mathcal{A}}\theta )/n^{1/2}$, where S_2i is the estimating function for model 6. Taylor series expansion with respect to both α and θ yields

$${\mathcal{A}}^T{\text{A}}_2\mathcal{A}{n}^{1/2}(\hat{\theta }-\theta )+{\mathcal{A}}^T{A}_2{n}^{1/2}(\hat{\mathcal{A}}-\mathcal{A})\theta .$$

On the other hand, the estimating function for ${\hat{\theta }}_{2\mathrm{SIV}}$ also equals ${\mathcal{A}}^T{A}_2{n}^{1/2}(\hat{\gamma }-\gamma )+o_{\mathrm{p}}(1)$, because $\hat{\mathcal{A}}\hat{\theta }$ is consistent for γ. It follows that

$$n^{1/2}({\hat{\theta }}_{2\mathrm{SIV}}-\mathit{\theta })=({\hat{\mathcal{A}}}^T{A}_2\hat{\mathcal{A}}{)}^{-1}{\hat{\mathcal{A}}}^T{A}_2\{n^{1/2}(\hat{\gamma }-\hat{\mathcal{A}}\theta )\}+o_{\mathrm{p}}(1).$$