IDENTIFICATION AND INFERENCE FOR MARGINAL AVERAGE TREATMENT EFFECT ON THE TREATED WITH AN INSTRUMENTAL VARIABLE

Abstract

In observational studies, treatments are typically not randomized and therefore estimated treatment effects may be subject to confounding bias. The instrumental variable (IV) design plays the role of a quasi-experimental handle since the IV is associated with the treatment and only affects the outcome through the treatment. In this paper, we present a novel framework for identification and inference using an IV for the marginal average treatment effect amongst the treated (ETT) in the presence of unmeasured confounding. For inference, we propose three different semiparametric approaches: (i) inverse probability weighting (IPW), (ii) outcome regression (OR), and (iii) doubly robust (DR) estimation, which is consistent if either (i) or (ii) is consistent, but not necessarily both. A closed-form locally semiparametric efficient estimator is obtained in the simple case of binary IV and outcome and the efficiency bound is derived for the more general case.

1. Introduction

Epidemiology studies and social sciences often aim to evaluate the effect of a treatment. For practical reasons, the average treatment effect among treated individuals (ETT) is sometimes of greater interest than the treatment effect in the population. In epidemiology studies concerning the toxic effects of a new drug or the treatment effect only on those who take the treatment, the ETT is the parameter of interest and it is known as “the effect of exposure on the exposed,” or “standardized morbidity” (Miettinen, 1974; Greenland and Robins, 1986). In econometrics, ETT is often used to evaluate the effects of a policy among those whom the policy is applied to. For example, Angrist (1995) evaluated the average effect of military service on the civilian earnings for veterans. Heckman et al. (1997, 1998) evaluated the average effect of job training for the program participants.

In observational or randomized studies with non-compliance, a primary challenge is the presence of unmeasured confounding, i.e. outcomes between treatment groups may differ not only due to the treatment effect, but also because of unmeasured factors that may affect the treatment selection.

Instrumental variables (IV) are useful in addressing unmeasured confounding. An IV is a variable that is associated with the treatment and it affects the outcome only through the treatment. The key idea of the IV method is to extract exogenous variation in the treatment that is unconfounded with the outcome and to take advantage of this bias-free component to make causal inference about the treatment effect (Robins, 1989; Angrist et al., 1996; Heckman, 1997).

The development of the IV approach can be traced back to Wright (1928) and Goldberger (1972) under linear structural equations in econometrics. Imbens and Angrist (1994), Angrist et al. (1996) and Heckman (1997) formalized the IV approach within the framework of potential outcomes or counterfactuals. Under additive and multiplicative structural nested models (SNMs), Robins (1989) and Robins (1994) evaluated the corresponding average treatment effect among treated individuals (ETT) conditional on the IV and observed covariates. Identification is achieved by assuming a certain degree of homogeneity with regard to the IV in an SNM of the conditional ETT (Hernán and Robins, 2006). Mainly, the assumption states that the magnitude of the conditional ETT does not vary with the IV. This is also referred to as the no-current treatment value interaction assumption. Under a similar identifying assumption, Vansteelandt and Goetghebeur (2003), Robins and Rotnitzky (2004), Tan (2010), Clarke et al. (2015) and Matsouaka and Tchetgen Tchetgen (2014) investigated estimation of this conditional causal effect using additive, multiplicative and logistic SNMs.

The literature mentioned above has some limitations. First of all, the literature focuses on the ETT conditional on the IV and observed covariates. The identification of such conditional ETT was achieved by specifying a functional form of the treatment causal effect. This is unattractive since it places constraints directly on the main parameter of interest and the misspecification of this functional form would lead to biased result. Second, the available inference methods require the treatment propensity score to be correctly specified even for an outcome regression-based estimator (Tan, 2010).

In this paper, we remedy these limitations in a novel framework for identification and estimation using an IV of the marginal ETT in the presence of unmeasured confounding. By targeting directly the marginal ETT, we allow the conditional causal effect to remain unrestricted. Our methods are particularly valuable when the primary goal is to obtain an accurate estimate of the treatment effect. Additionally, we propose a new identification strategy which is applicable to any type of outcome, and provides necessary and sufficient global identification conditions. Moreover, for inference, we propose three different semiparametric estimators allowing for flexible covariate adjustment, (i) inverse probability weighting (IPW), (ii) outcome regression (OR) and (iii) doubly robust (DR) estimation which is consistent if either (i) or (ii) is consistent but not necessarily both.

The outline for the paper is as follows. In Section 2, we introduce the notation and state the main assumptions. We study the nonparametric identification of ETT in Section 3. We introduce IPW, OR as well as DR estimators in Section 4. In Section 5, we assess the performance of various estimators in a simulation study. In Section 6, we further illustrate the methods with a study concerning the impact of participation in a 401(k) retirement programs on savings. We conclude with a brief discussion in Section 7.

2. Preliminary Results

Suppose that one observes independently and identically distributed data O = (A, Y, Z, C), where A is a binary treatment, Y is the outcome of interest, which may be dichotomous, polytomous, discrete or continuous, and the candidate IV Z and covariates C are both pre-exposure variables. Let a, y, z, c denote the possible values that A, Y, Z, C could take. Let Y_az denote the potential outcome if A and Z are set to a and z and let Y_a denote the potential outcome only A is set to a. We formalize the IV assumptions using potential outcomes:

(IV.1) Stochastic exclusion restriction:

$$Y_{az}=Y_a\mathrm{almost\; surely\; for\; all}a\mathrm{and}z;$$

(IV.2) Unconfounded IV-outcome relation:

$$f_{Y_0|Z,C}\left(y|z,c\right)=f_{Y_0|C}\left(y|c\right)\mathrm{for\; all}z\mathrm{and}c;$$

(IV.3) IV relevance:

$$\mathrm{Pr}\left(A=1|Z=z,C=c\right)\ne \mathrm{Pr}\left(A=1|Z=0,C=c\right)\mathrm{for\; all}z\ne 0\mathrm{and}c.$$

Assumption (IV.1) states that Z does not have a direct effect on the outcome Y thus we use Y_a to denote the potential outcome under treatment a for a = 0, 1. Assumption (IV.2) is ensured under physical randomization but will hold more generally if C includes all common causes of Z and Y. Assumptions (IV.1)–(IV.2) together imply that conditional on C, the IV is independent of the potential outcome for the unexposed, i.e., Y₀ ⫫ Z|C. Assumption (IV.3) states that A and Z have a non-null association conditional on C, even if the association is not causal. If assumptions (IV.1)–(IV.3) are satisfied, Z is said to be a valid IV.

We make the consistency assumption Y = AY₁ + (1 − A)Y₀. The marginal treatment effect on the treated is ETT = E(Y₁ − Y₀|A = 1). Because E(Y₁|A = 1) = E(Y|A = 1) can be consistently estimated from the average observed outcome of treated individuals, throughout, we focus on making inferences about ψ where

$$\psi =E\left(Y_0|A=1\right).$$

Suppose there exist unmeasured variables denoted by U such that controlling for (U, Z, C) suffices to account for confounding, i.e. Y₀ ⫫ A|(U, Z, C), however,

$$Y_0\cancel{⫫}A|\left(Z,C\right),$$ (1)

where ⫫ denotes statistical independence. As pointed out by Robins et al. (2000), potential outcomes can be viewed as the ultimate unmeasured confounders. This is because by the consistency assumption, the observed outcome Y is a deterministic function of the treatment and the potential outcomes. Thus, given (Y₀, Y₁), U does not contain any further information about Y. To make explicit use of (1), we define the extended propensity score

$$\pi \left(Y_0,Z,C\right)=\mathrm{Pr}\left(A=1|Y_0,Z,C\right),$$

as a function of Y₀.

3. Nonparametric Identification

While assumptions (IV.1)–(IV.3) suffice to obtain a valid test of the sharp null hypothesis of no treatment effect (Robins, 1994) and can also be used to test for the presence of confounding bias (Pearl, 1995), ETT is not uniquely determined by the observed data without any additional restriction. For simplicity, we first consider the situation where covariates are omitted and outcome and IV are both binary. From the observed data, one can identify the quantities Pr(Y₀, Z|A = 0), Pr(Z|A = 1) and Pr(A = 0). These quantities are functions of the unknown parameters: Pr(Z = 1), Pr(Y₀ = 1), and Pr(A = 0|Y₀, Z). Without imposing any additional assumption, there are six unknown parameters (one for Pr(Z = 1), one for Pr(Y₀ = 1) and four for Pr(A = 0|Y₀, Z)), however, only five degrees of freedom are available from the observed data (one for Pr(A = 0), one for Pr(Z| A = 1) and three for Pr(Y, Z|A = 0)). As a result, the joint distribution f (A, Y₀, Z) is not uniquely identified. Particularly, ψ is not identified.

For identification purposes, additional assumptions, such as Robins’ no-current treatment value interaction assumption (Hernán and Robins, 2006), must be imposed to reduce the set of candidate models for the joint distribution f(A, Y₀, Z, C). Below, we give a general necessary and sufficient condition for identification. Let ${\mathcal{P}}_{A|Y_0,Z,C}$ and ${\mathcal{P}}_{Y_0|C}$ denote the collections of candidates for Pr(A = 0|Y₀, Z, C) and f(Y₀|C), which are known to satisfy (IV.1) and (IV.2).

Condition 1. Any two distinct elements ${\mathrm{Pr}}_1(A=0|Y_0,Z,C),{\mathrm{Pr}}_2(A=0|Y_0,Z,C)\in {\mathcal{P}}_{A|Y_0,Z,C}$ and $f_1\left(Y_0\right|C),f_2\left(Y_0\right|C)\in {\mathcal{P}}_{Y_0|C}$, satisfy the inequality:

$$\frac{{\mathrm{Pr}}_1\left(A=0|Y_0,Z,C\right)}{{\mathrm{Pr}}_2\left(A=0|Y_0,Z,C\right)}\ne \frac{f_2\left(Y_0|C\right)}{f_1\left(Y_0|C\right)}.$$

The following proposition states that condition 1 is a necessary and sufficient condition for identifiability of the joint distribution of (A, Y₀, Z, C), where Y₀ and Z may be dichotomous, polytomous, discrete or continuous.

Proposition 1. The joint distribution of (A, Y₀, Z, C) is identified in the model defined by ${\mathcal{P}}_{A|Y_0,Z,C}$ and ${\mathcal{P}}_{Y_0|C}$ if and only if condition 1 holds.

It is convenient to check condition 1 for parametric models, but it may be harder for semiparametric and nonparametric models, since ${\mathcal{P}}_{A|Y_0,Z,C}$ and ${\mathcal{P}}_{Y_0|C}$ can be complicated. The following corollary gives a more convenient condition.

Corollary 1. Suppose that for any two candidates ${\mathrm{Pr}}_1(A=0|Y_0,Z,C),{\mathrm{Pr}}_2(A=0|Y_0,Z,C)\in {\mathcal{P}}_{A|Y_0,Z,C}$, the ratio Pr₁(A = 0|Y₀, Z, C)/Pr₂(A = 0|Y₀, Z, C) is either a constant or varies with Z. Then the joint distribution of (A, Y₀, Z, C) is identified.

Although the condition provided in Corollary 1 is a sufficient condition for identification, it allows identification of a large class of models. We further illustrate Proposition 1 and Corollary 1 with several examples. For simplicity, we again omit covariates, however, we show at the end of this section that similar results with covariates can be derived. For simplicity, we first consider the case of binary outcome with binary IV.

Example 1. Consider a model ${\mathcal{P}}_{A|Y_0,Z}=\{\mathrm{Pr}(A=0|Y_0,Z):\mathrm{logit}\mathrm{Pr}(A=0|Y_0,Z;{\theta }_1,{\theta }_2,{\eta }_1,{\eta }_2)={\theta }_1+{\theta }_{2}Z+{\eta }_1{Y}_0+{\eta }_2{Y}_{0}Z,{\theta }_1,{\theta }_2,{\eta }_1,{\eta }_2\in (-\infty ,\infty )\}$. The model is saturated since ${\mathcal{P}}_{A|Y_0,Z}$ contains all possible treatment mechanisms. It can be shown that neither the joint distribution nor ψ is identified even under the assumptions (IV.1)–(IV.3).

Example 1 shows that the joint density f (A, Y₀, Z) is not identified when the treatment selection mechanism is left unrestricted under (IV.1)–(IV.3). However, we show that the joint density f (A, Y₀, Z) is identified assuming separable treatment mechanism on the additive scale.

Example 2. Consider a model ${\mathcal{P}}_{A|Y_0,Z}=\{\mathrm{Pr}(A=0|Y_0,Z):\mathrm{logit}\mathrm{Pr}(A=0|Y_0,Z;{\theta }_1,{\theta }_2,{\eta }_1)={\theta }_1+{\theta }_{2}Z+{\eta }_1{Y}_0;{\theta }_1,{\theta }_2,{\eta }_1\in (-\infty ,\infty )\}$. The model is separable since ${\mathcal{P}}_{A|Y_0,Z}$ excludes an interaction between Y₀ and Z. It can be shown that both the joint distribution and ψ is identified under assumptions (IV.1)–(IV.3).

Example 2 agrees with the intuition that identification follows from having fewer parameters than the saturated model. Under the assumed model, we have five unknown parameters and five available degrees of freedom from the empirical distribution. We show in the next example that the joint distribution and ψ can be identified in a general separable model when the outcome and instrument are both continuous.

Example 3. Consider the logistic separable treatment mechanism: ${\mathcal{P}}_{A|Y_0,Z}=\{\mathrm{Pr}(A=0|Y_0,Z):\mathrm{logit}\mathrm{Pr}(A=0|Y_0,Z)=q(Z)+h(Y_0)\}$, where q and h are unknown differentiable functions with h(0) = 0. It can be shown that ${\mathcal{P}}_{A|Y_0,Z}$ satisfies condition 1 and thus the joint distribution is identified under (IV.1)–(IV.3).

These results can be generalized to include covariates C. For instance, by allowing both q and h to depend on C in example 3:

$${\mathcal{P}}_{A|Y_0,Z,C}=\{\mathrm{Pr}(A=0|Y_0,Z,C):\mathrm{logit}\mathrm{Pr}(A=0|Y_0,Z,C)=q(Z,C)+h(Y_0,C)\},$$

where h(0, C) = 0, the joint distribution is identified whenever the interaction term of Y₀ and Z is absent.

In the Supplementary Materials, we present proofs for the above examples, and additional examples, such as the case of continuous outcome with binary IV, Probit link and a separable treatment mechanism.

4. Estimation

While nonparametric identification conditions are provided in Section 3, such conditions will seldom suffice for reliable statistical inference. Typically in observational studies, the set of covariates C is too large for nonparametric inference, due to the curse of dimensionality (Robins and Ritov, 1997). To make progress, we posit parametric models for various nuisance parameters, and provide three possible approaches for semiparametric inference that depend on different subsets of models. We describe an IPW, an OR and a DR estimator of the marginal ETT under assumptions (IV.1)–(IV.2) and condition 1. Throughout, we posit a parametric model f_Z|C (z|c) = Pr(Z = z|C = c; ρ) for the conditional density of Z given C. Let $\widehat{\rho }$ denote the maximum likelihood estimator (MLE) of p. Let ${\mathbb{P}}_n$ denote the empirical measure, that is ${\mathbb{P}}_{n}f(O)=n^{-1}{\sum }_{i=1}^{n}f(O_i)$. Let Ê denote the expectation taken under the empirical distribution of C and let $\widehat{\mathrm{Pr}}(A=1)={\sum }_{i=1}^n{A}_i/n$ denote the empirical probability of receiving treatment.

4.1. IPW estimator

For estimation, we first propose an IPW IV approach which extends standard IPW estimation of ETT to an IV setting. We make the positivity assumption that for all values of Y₀, Z and C, the probability of being unexposed to treatment is bounded away from 0. The IPW approach relies on the crucial assumption that the extended propensity score model π(Y₀, Z, C; γ) is correctly specified with unknown finite dimensional parameter γ and the following representation of ETT,

$$E\left(Y_0|A=1\right)=E\left\{\frac{\pi \left(Y_0,Z,C\right)Y\left(1-A\right)}{\mathrm{Pr}\left(A=1\right)\left\{1-\pi \left(Y_0,Z,C\right)\right\}}\right\}.$$ (2)

A derivation of the above equation is given in the Supplementary Materials. We solve the following equations to obtain an estimator $\widehat{\gamma }$ of γ:

$${\mathbb{P}}_n\{\frac{1-A}{1-\pi \left(Y_0,Z,C;\widehat{\gamma }\right)}-1\}=0,$$ (3)

$${\mathbb{P}}_n[\frac{1-A}{1-\pi \left(Y_0,Z,C;\widehat{\gamma }\right)}\left\{h_1\left(Z,C\right)-E(h_1\left(Z,C\right)|\left(C;\widehat{\rho }\right)\right\}]=0,$$ (4)

$${\mathbb{P}}_n[\frac{1-A}{1-\pi \left(Y_0,Z,C;\widehat{\gamma }\right)}\{h_2\left(C\right)-\widehat{E}\left(h_2\left(C\right)\right)\}]=0,$$ (5)

$${\mathbb{P}}_n[\frac{1-A}{1-\pi \left(Y_0,Z,C;\widehat{\gamma }\right)}t\left(Y,C\right)\{l\left(Z,C\right)-E(l\left(Z,C\right)|\left(C;\widehat{\rho }\right)\}]=0,$$ (6)

where ${(h_1^T,h_2^T,l^T)}^T$ satisfies the regularity condition (A.1) described in the Supplementary Materials. Equations (4) and (5) identify the association between (Z, C) and A in π(0, Z, C). If there is no selection bias, equations (3)–(5) are adequate to estimate the propensity score. By utilizing the IV property (IV.1)–(IV.2), equation (6) identifies the degree of selection bias encoded in the dependence of π on Y₀. Both equation (4) and (6) require the conditional density of IV Z, f_Z|C (z|c; ρ) to be correctly modeled. By equation (2), an extended propensity score estimator leads to an estimator of ψ. We have the following result:

Proposition 2. Under (IV.1)–(IV.2) and condition 1, suppose the extended propensity score model π(Y₀, Z, C; γ) and f_Z|C(z|c; ρ) are correctly specified, then the IPW estimator

$${\widehat{\psi }}^{ip\omega }={\mathbb{P}}_n\frac{\pi \left(Y_0,Z,C;\widehat{\gamma }\right)Y\left(1-A\right)}{\widehat{\mathrm{Pr}}\left(A=1\right)\left\{1-\pi \left(Y_0,Z,C;\widehat{\gamma }\right)\right\}},$$

is consistent for ψ.

We emphasize that the extended propensity score model can use any well-defined link function (e.g., logit, probit), and if condition 1 holds, Proposition 2 still holds. The functions h₁, h₂, t and l can be chosen based on the model for the extended propensity score. For example, assuming logit π(Y₀, Z, C; γ) = θ₀ + θ₁Z + θ₂C + ηY₀ where $\tilde{\eta }={({\theta }_1,{\theta }_2,\eta )}^T$ is a k-dimensional parameter vector. The k-dimensional function (h₁, h₂, t)^T can be chosen as ${(h_1,h_2,t)}^T=\partial \mathrm{logit}\pi (Y_0,Z,C;\gamma )/\partial \tilde{\eta }={(Z,C,Y_0)}^T$ and l can be chosen as any scalar function of (Z, C), e.g., l(Z, C) = Z. Thus we have exactly k + 1 estimating equations. The choice of h₁, h₂, t and l will generally impact efficiency but should not affect consistency as long as the identification conditions hold and the required models are correctly specified. The choice of h₁, h₂, t and l that leads to the most efficient IPW estimator can be derived using results in Newey and McFadden (1994). Due to space constraints, we only illustrate in details the choice of similar functions for efficient DR estimator in the section 4.3 and a similar derivation could be made here.

4.2. OR and DR estimators

Since Y₀ is never observed for the treated group, we use the following equation to decompose E[Y₀|A = 1, Z, C] into two parts: one can be estimated directly using restricted MLE and the other can be computed by solving an estimating equation. Specifically, we have

$$E\left\{g\left(Y_0,C\right)|A=1,Z,C\right\}=\frac{E\left[\exp \left\{\alpha \left(Y,Z,C\right)\right\}g\left(Y,C\right)|A=0,Z,C\right]}{E\left[\exp \left\{\alpha \left(Y,Z,C\right)\right\}|A=0,Z,C\right]},$$ (7)

where g is any function of Y₀ and C and α(Y₀, Z, C) is the generalized odds ratio function relating A and Y₀ conditional on Z and C as

$$\alpha \left(Y_0,Z,C\right)=\log \frac{f\left(Y_0|A=1,Z,C\right)f\left(Y_0=0|A=0,Z,C\right)}{f\left(Y_0|A=0,Z,C\right)f\left(Y_0=0|A=1,Z,C\right)}.$$

Since the association between Y₀ and A is attributed to unmeasured confounding, α(Y₀, Z, C) can be interpreted as the selection bias function. Thus, we express the conditional mean function E{g(Y₀, C)|A = 1, Z, C} in terms of f(Y|A = 0, Z, C) and α(Y₀, Z, C). We prove the equation (7) in the Supplementary Materials.

Let f(Y|A = 0, Z, C; ξ) denote a model for the density of the outcome among the unexposed conditional on Z and C, and let $\widehat{\xi }$ denote the restricted MLE of ξ obtained using only data among the unexposed. Let η denote the parameter indexing a parametric model for the selection bias function α as α(Y₀, Z, C; η). We obtain an estimator for η by solving:

$${\mathbb{P}}_n\left[\left\{\omega \left(Z,C\right)-E\left(\omega \left(Z,C\right)|C;\widehat{\rho }\right)\right\}\left\{AE[g\left(Y_0,C\right)|A=1,Z,C;\eta ,\widehat{\xi }]+\left(1-A\right)g\left(Y,C\right)\right\}\right]=0,$$ (8)

for any choice of functions ω and g such that the regularity condition (A.2) stated in the Supplementary Materials holds. Intuitively, the left hand side of equation (8) is an empirical estimator of the expected conditional covariance between ω(Z, C) and g(Y₀, C) given C, which should be zero by (IV.1)–(IV.2). Equation (8) requires the conditional density of IV Z, f_Z|C (z|c; ρ) to be correctly modeled. Based on equation (7), we can construct an estimator for ψ based on $\widehat{\eta },\widehat{\xi }$ and $\widehat{\rho }$.

Proposition 3. Under (IV.1)–(IV.2) and condition 1, suppose α(Y₀, Z, C; η), f_Z|C (z|c; ρ) and f (Y|A = 0, Z, C; ξ) are correctly specified, then the OR estimator

$${\widehat{\psi }}^{reg}={\mathbb{P}}_n\frac{A}{\widehat{\mathrm{Pr}}\left(A=1\right)}\frac{E[\exp \{\alpha (Y,Z,C;\widehat{\eta })\}Y|A=0,Z,C;\widehat{\xi }]}{E[\exp \{\alpha (Y,Z,C;\widehat{\eta })\}|A=0,Z,C;\widehat{\xi }]},$$

is consistent for ψ.

Functions g and ω in equation (8) can be chosen based on the model we posit for α(Y₀, Z, C). For example, assuming

$$\alpha \left(Y_0,Z,C;\eta \right)=\eta Y_0,$$ (9)

g can be chosen as g(Y₀, C) = ∂α(Y₀, Z, C; η)/∂η = Y₀ and ω can be chosen as any scalar function of (Z, C), e.g., ω(Z, C) = Z. The choice of g and ω may impact efficiency but does not affect consistency as long as the identification conditions hold and the required models are correctly specified. The choice of g and ω that leads to the most efficient OR estimator can be derived using Newey and McFadden (1994).

Tan (2010) proposed an OR estimator for the conditional ETT, which requires correctly specified models for both the treatment propensity score and the outcome regression function. In contrast, we circumvent the dependence of the regression estimator on the propensity score.

The proposed estimator for nuisance parameter η is closely related to the regression estimator proposed by Vansteelandt and Goetghebeur (2003) when Y is binary. Vansteelandt and Goetghebeur (2003) developed a two-stage logistic estimator which combines a logistic SMM at the first stage and a logistic regression association model at the second stage. Specifically, Vansteelandt and Goetghebeur (2003) focused on estimating ζ(Z, C) = logit Pr(Y₁ = 1|A = 1, Z, C) − logit Pr(Y₀ = 1|A = 1, Z, C), which encodes the conditional ETT given Z and C. Let ν denote the parameter indexing a model for ζ(Z, C) as ζ(Z, C; ν). They proposed to estimate ν in the estimating equation

$${\mathbb{P}}_n\left[\{\omega (Z,C)-E(\omega (Z,C)|C;\widehat{\rho })\}\left\{A\mathrm{expit}\{\vartheta (Z,C;\widehat{\varrho })-\zeta (Z,C;\nu )\}+(1-A)Y\right\}\right]=0.$$ (10)

where expit(x) = exp(x)/{1 + exp(x)} and ϑ(Z, C; ϱ) = logit Pr(Y = 1|A = 1, Z, C; ϱ).

Recall that we obtain an estimator of η indexing α(Y₀, Z, C; η) in the equation (8), which can be re-expressed as

$${\mathbb{P}}_n\left[\{\omega (Z,C)-E(\omega (Z,C)|C;\widehat{\rho })\}\left\{A\mathrm{expit}\{\delta (Z,C;\widehat{\xi })+\alpha (1,Z,C;\eta )\}+(1-A)Y\right\}\right]=0,$$ (11)

where δ(Z, C; $\widehat{\xi }$) = logit Pr(Y₀ = 1|A = 0, Z, C). Equations (10) and (11) mainly differ in the way Pr(Y₀ = 1|A = 1, Z, C) is estimated. More specifically, (10) obtains Pr(Y₀ = 1|A = 1, Z, C) using Pr(Y₁ = 1|A = 1, Z, C) as a baseline risk for the model while (11) uses Pr(Y₀ = 1|A = 0, Z, C) as baseline risk. This difference is important since Vansteelandt and Goetghebeur (2003) failed to obtain a DR estimator of ζ(Z, C) while as we show next, our choice of parameterization yields a DR estimator of the marginal ETT.

Heretofore, we have constructed estimators in two different approaches. Both approaches assume correct models for α(Y₀, Z, C; η) and f_Z|C (z|c; ρ). The IPW approach further relies on a consistent estimator of the baseline extended propensity score β(Z, C) = logit Pr(A = 1|Y₀ = 0, Z, C), which under the logit link and together with α(Y₀, Z, C; η), provides a consistent estimator of the extended propensity score π(Y₀, Z, C; γ) = expit {α(Y₀, Z, C; η) + β(Z, C; θ)}. The OR approach further relies on a consistent estimator of f(Y|A = 0, Z, C), which together with α(Y₀, Z, C; η), provides a consistent estimator of Pr(Y₀ = 1|A = 1, Z, C) by (7). Define ${\mathcal{M}}_a$ as the collection of laws with parametric models f_Z|C(z|c; ρ), α(Y₀, Z, C; η) and β(Z, C; θ) while f(Y|A = 0, Z, C) is unrestricted. Likewise, define ${\mathcal{M}}_y$ as the collection of laws with parametric models f_Z|C(z|c; ρ), α(Y₀, Z, C; η) and f(Y|A = 0, Z, C; ξ) while β(Z, C) is unrestricted. The main appeal of a doubly robust estimator is that it remains consistent if either β(Z, C; θ) or f(Y|A = 0, Z, C; ξ) is correctly specified. To derive a DR estimator for ψ in the union space ${\mathcal{M}}_a\cup {\mathcal{M}}_y$, we first propose a DR estimator for the parameter η of the selection bias model α(Y₀, Z, C; η). For notational convenience, let

$$\begin{array}{l}{Q}_g\le \left(Y,A,Z,C;\gamma ,\xi \right)\\ =\frac{\left(1-A\right)\pi \left(Y,Z,C;\gamma \right)}{1-\pi \left(Y,Z,C;\gamma \right)}\left[g\left(Y,C\right)-\frac{E\left[\exp \left\{\alpha \left(Y,Z,C;\eta \right)\right\}g\left(Y,C\right)|A=0,Z,C;\xi \right]}{E\left[\exp \left\{\alpha \left(Y,Z,C;\eta \right)\right\}|A=0,Z,C;\xi \right]}\right]\\ +A\frac{E\left[\exp \left\{\alpha \left(Y,Z,C;\eta \right)\right\}g\left(Y,C\right)|A=0,Z,C;\xi \right]}{E\left[\exp \left\{\alpha \left(Y,Z,C;\eta \right)\right\}|A=0,Z,C;\xi \right]}.\end{array}$$ (12)

Consider the estimating equation for the selection bias parameter $\tilde{\eta }$

$${\mathbb{P}}_n\left[\left[\omega \left(Z,C\right)-E\left\{\omega \left(Z,C\right)|C;\widehat{\rho }\right\}\right]{\tilde{Q}}_g\left(Y,A,Z,C;\tilde{\gamma },\widehat{\xi }\right)\right]=0,$$ (13)

where

$$\begin{array}{l}{\tilde{Q}}_g\left(Y,A,Z,C;\tilde{\gamma },\widehat{\xi }\right)\\ =Q_g\left(Y,A,Z,C;\tilde{\gamma },\widehat{\xi }\right)+\left(1-A\right)g\left(Y,C\right)\\ =\frac{1-A}{1-\pi \left(Y,Z,C;\gamma \right)}g\left(Y,C\right)\\ +\frac{A-\pi \left(Y,Z,C;\gamma \right)}{1-\pi \left(Y,Z,C;\gamma \right)}\frac{E\left[\exp \left\{\alpha \left(Y,Z,C;\eta \right)\right\}g\left(Y,C\right)|A=0,Z,C;\xi \right]}{E\left[\exp \left\{\alpha \left(Y,Z,C;\eta \right)\right\}|A=0,Z,C;\xi \right]}.\end{array}$$

Equation (13) is key to obtaining a DR estimation of the selection bias function and thus of ETT. Intuitively, the left hand side of equation (13) is also an empirical estimator of the expected conditional covariance between ω(Z, C) and g(Y₀, C) given C, which should be zero by (IV.1)–(IV.2). In addition to the model f_Z|C(z|c; ρ) for IV, equation (8) only involves outcome regression model, while equation (13) involves both outcome regression and propensity score models. Hence, the parameter $\widehat{\eta }$ obtained from (8) depends on the correct specification of outcome regression, while as we show in the following proposition, the parameter estimate for η obtained from (13) is doubly robust. We solve equation (13) jointly with equations (3)–(5) with $\widehat{\gamma }$ replaced by $\tilde{\gamma }=({\widehat{\eta }}^{\mathit{DR}},\tilde{\theta })$. The choice of h₁, h₂, g and ω can be decided as in Sections 4.1 and 4.2.

Proposition 4. Under (IV.1)−(IV.2) and condition 1, ${\widehat{\eta }}^{DR}$ and ${\widehat{\psi }}^{DR}$ are consistent in the union model ${\mathcal{M}}_a\cup {\mathcal{M}}_y$, where ${\widehat{\psi }}^{DR}={\widehat{\mathbb{P}}}_n{Q}_{\tilde{g}}(Y,A,Z,C;\tilde{\gamma },\widehat{\xi })/\widehat{\mathrm{Pr}}(A=1)$ and $\tilde{g}(Y,C)=Y$

Proposition 4 implies that ${\widehat{\eta }}^{DR}$ and ${\widehat{\psi }}^{DR}$ are both DR estimators since their consistency only requires either the extended propensity score or the outcome regression model to be correctly specified but not necessarily both.

4.3. Local efficiency

The large sample variance of doubly robust estimators ${\widehat{\eta }}^{DR}$ and ${\widehat{\psi }}^{DR}$ at the intersection submodel ${\mathcal{M}}_a\cap {\mathcal{M}}_y$ where all models are correctly specified, is determined by the choice of g(Y, C) and ω(Z, C) in equation (13). In the Supplementary Materials, we derive the semiparametric efficient score of (η, ψ) in a model ${\mathcal{M}}_{\mathit{np}}$ that only assumes that Z is a valid IV and the selection bias function α(Y₀, Z, C; η) is correctly specified. As discussed in the Supplementary Materials, the efficient score is generally not available in closed-form, except in special cases, such as when Z and Y are both polytomous. Here, we illustrate the result by constructing a locally efficient estimator of (η, ψ) when Z and Y are both binary. In this vein, similar to the definition of ${\tilde{Q}}_{\upsilon }\left(Y,A,Z,C;\gamma ,\xi \right)$, define

$$\begin{array}{l}{\tilde{Q}}_{\upsilon }\left(Y,A,Z,C;\gamma ,\xi \right)=\frac{\left(1-A\right)\upsilon \left(Y,Z,C\right)}{1-\pi \left(Y,Z,C;\gamma \right)}\\ +\frac{A-\pi \left(Y,Z,C;\gamma \right)}{1-\pi \left(Y,Z,C;\gamma \right)}\frac{E\left[\exp \left\{\alpha \left(Y,Z,C;\eta \right)\right\}\upsilon \left(Y,Z,C\right)|A=0,Z,C;\xi \right]}{E\left[\exp \left\{\alpha \left(Y,Z,C;\eta \right)\right\}|A=0,Z,C;\xi \right]},\end{array}$$

where υ is any function of (Y₀, Z, C).

A one-step locally efficient estimator of η in ${\mathcal{M}}_{\mathit{np}}$ is given by

$${\widehat{\eta }}^{eff}={\widehat{\eta }}^{DR}-{\left\{E\left({\nabla }_{\eta }{\widehat{S}}_{\eta }^{eff}|\widehat{\gamma },\widehat{\xi }\right)\right\}}^{-1}E\left({\widehat{S}}_{\eta }^{eff}|\widehat{\gamma },\widehat{\xi }\right),$$

where $\overline{\upsilon }(Y,Z,C)=\{Y-E(Y\left|C\right)\left\}\right\{Z-E\left(Z\right|C\left)\right\},\Delta \left(\eta \right)={\tilde{Q}}_{\overline{\upsilon }}\left(Y,A,Z,C;\gamma ,\xi \right)$ and

$${\widehat{S}}_{\eta }^{eff}=E{\left\{\Delta \left(\eta \right)\Delta {\left(\eta \right)}^T|C;\widehat{\gamma },\widehat{\xi }\right\}}^{-1}E\left\{\partial \Delta \left(\eta \right)/\partial {\eta }^T|C;\widehat{\gamma },\widehat{\xi }\right\}\Delta \left({\widehat{\eta }}^{eff}\right)$$

is the efficient score of η evaluated at the estimated intersection submodel ${\mathcal{M}}_a\cap {\mathcal{M}}_y$. Further, let ${\widehat{\psi }}^{DR}({\widehat{\eta }}^{eff})$ denote a DR estimator for ψ evaluated at the estimated intersection submodel ${\mathcal{M}}_a\cap {\mathcal{M}}_y$ with ${\widehat{\eta }}^{eff}$ substituted in for ${\widehat{\eta }}^{DR}$. Then the efficient estimator of ψ is given by

$${\widehat{\psi }}^{eff}={\widehat{\psi }}^{DR}\left({\widehat{\eta }}^{eff}\right)-E{\left\{{\Delta }^2\left({\widehat{\eta }}^{eff}\right)|C;\widehat{\gamma },\widehat{\xi }\right\}}^{-1}E\left\{{\widehat{\psi }}^{DR}\left({\widehat{\eta }}^{eff}\right)\Delta \left({\widehat{\eta }}^{eff}\right)|C;\widehat{\gamma },\widehat{\xi }\right\}\Delta \left({\widehat{\eta }}^{eff}\right).$$

5. Simulations

Simulations for both binary and continuous outcomes were conducted to evaluate the finite sample performance of the causal effect estimators derived in Sections 4.1 and 4.2. Let ${\mathcal{M}}_a^c$ denote the complement space of ${\mathcal{M}}_a$ and likewise define ${\mathcal{M}}_y^c$. Simulations were conducted under three scenarios: (i) ${\mathcal{M}}_a\cap {\mathcal{M}}_y$, that is both outcome regression and extended propensity score are correctly specified, (ii) ${\mathcal{M}}_a\cap {\mathcal{M}}_y^c$ that is only the extended propensity score is correctly specified and (iii) ${\mathcal{M}}_a^c\cap {\mathcal{M}}_y$ that is only the outcome regression model is correctly specified.

Simulations were first carried out for a binary outcome. For scenario (i), the simulation study was conducted in the following steps:

Step 1: A hypothetical study population of size n = 1000 (or n = 5000) was generated and each individual had baseline covariates C₁ and C₂ generated independently from Bernoulli distributions with probability 0.4 and 0.6 respectively. Then the IV Z was generated from the model: logit Pr(Z = 1|C) = 0.2 + 0.4C₁ − 0.5C₂ and potential outcomes Y₀, Y₁ from models logit Pr(Y₀ = 1|Z, C) = 0.6 + 0.8C₁ − 2C₂ and logit Pr(Y₁ = 1|Z, C) = 0.7 − 0.3C₁. The treatment variable A was generated from logit Pr(A = 1|Y₀, Z, C) = 0.4 + 2Z + 0.8C₁ − 0.6Y₀ − 1.6C₁Z, and the observed outcome was Y = Y₀(1 − A) + Y₁A.

Step 2: The following extended propensity score model was estimated and the parameters γ = (θ₁, θ₂, θ₃, θ₄, η) in the model

$$\mathrm{logit}\mathrm{Pr}\left(A=1|Y_0,Z,C;\gamma \right)={\theta }_1+{\theta }_{2}Z+{\theta }_3{C}_1+{\theta }_4{C}_{1}Z+\eta Y_0$$ (14)

were estimated using estimating equations (3)–(6) with h₁(Z, C) = (Z, C₁Z)^T, h₂(C) = C₁, t(Y, C) = Y and l(Z, C) = Z and ${\widehat{\psi }}^{\mathit{ipw}}$ was evaluated.

Step 3: The selection bias function was correctly specified as in (9), ξ in the regression outcome model

$$\mathrm{logit}E\left(Y|A=0,Z,C;\xi \right)={\xi }_1+{\xi }_2{C}_1+{\xi }_3{C}_2+{\xi }_{4}Z+{\xi }_5{C}_{1}Z$$ (15)

was estimated by restricted MLE, and α was estimated by solving equation (8) with ω(Z, C) = Z and g(Y, C) = Y and ${\widehat{\psi }}^{reg}$ was evaluated.

Step 4: The selection bias function was correctly specified as in (9), ξ in equation (15) was estimated by restricted MLE, parameters γ in (14) was estimated using (3)–(5) and (13) where h, t, l, ω, g are chosen as in Step 2 and Step 3 and ${\widehat{\psi }}^{DR}$ was evaluated.

Step 5: Steps 1–4 were repeated 1000 times.

The data generating mechanism described in Step 1 satisfies the assumptions (IV.1)–(IV.2) for both a = 0, 1. As shown in example 1, ψ is identified from the observed data since the treatment mechanism is a separable logit model. Also in the Supplementary Materials, we verify that model (15) for E(Y|A = 0, C, Z) contains the true data generating mechanism. Simulations for scenario (ii) were similar to scenario (i) except that (14) was replaced with

$$\mathrm{logit}\mathrm{Pr}\left(A=1|Y_0,Z,C;\gamma \right)={\theta }_1+{\theta }_{2}Z+{\theta }_3{C}_1+\eta Y_0,$$ (16)

which is misspecified if θ₄ ≠ 0 in equation (14). For scenario (iii), the potential outcome model (15) was replaced with

$$\mathrm{logit}E\left(Y|A=0,Z,C;\widehat{\xi }\right)={\xi }_1+{\xi }_2{C}_1+{\xi }_{4}Z,$$ (17)

which is misspecified if ξ₃ ≠ 0 and ξ₅ = 0 in equation (15). We use the R package BB (Varadhan and Gilbert, 2009) to solve the nonlinear estimating equations. Simulation results for 1000 Monte Carlo samples are reported in Figure 1 and empirical coverage rates are presented in Table 1. Under correct model specification, all estimators have negligible bias which diminishes with increasing sample size. In agreement with our theoretical results, the IPW and regression estimators are biased with poor empirical coverages when the extended propensity score or the outcome model is mis-specified, respectively. The DR estimator performs well in terms of bias and coverage when either model is mis-specified but the other is correct. When all models are correctly specified, the relative efficiency of the locally semiparametric efficient estimator compared to the DR estimator of η and ψ are 0.840 and 0.810 respectively, based on Monte Carlo standard errors at sample size n = 5000. This shows that substantial efficiency gain may be possible at the intersection submodel when using the locally efficient score.

Figure 1:

Performance of the IPW, OR and DR estimators of ψ with binary outcomes.

Note: In each boxplot, the true value ψ₀ is marked by the horizontal lines, white boxes are for n = 1000 and grey boxes are for n = 5000.

Table 1:

Empirical coverage rates based on 95% Wald confidence intervals for both binary and continuous outcomes

	Binary Y		Cont. Y
sample size (n)	1000	5000	1000	5000
(i) both π and μ are correct
${\widehat{\psi }}^{ipw}$	0.86	0.90	0.96	0.95
${\widehat{\psi }}^{reg}$	0.84	0.92	0.97	0.95
${\widehat{\psi }}^{DR}$	0.85	0.91	0.97	0.96
(ii) only π is correct
${\widehat{\psi }}^{ipw}$	0.86	0.90	0.96	0.95
${\widehat{\psi }}^{reg}$	0.79	0.60	0.39	0.00
${\widehat{\psi }}^{DR}$	0.86	0.91	0.97	0.95
(iii) only μ is correct
${\widehat{\psi }}^{ipw}$	0.78	0.53	0.39	0.00
${\widehat{\psi }}^{reg}$	0.84	0.92	0.97	0.95
${\widehat{\psi }}^{DR}$	0.85	0.92	0.96	0.96

The coverage was evaluated under three scenarios: (i) both outcome regression and the extended propensity score are correctly specified, in (ii) only the extended propensity score is correct and in (iii) only the outcome regression model is correct.

Simulations for a continuous outcome were conducted similarly as for the binary outcome in the following steps.

Step 1*: Covariates C₁ and C₂ were generated as in Step 1, Z was generated from model logit Pr(Z = 1|C) = 0.7 + 0.8C₁ − C₂, and Y₀, Y₁ from models Y₀|Z, C ~ N(0.5 + C₁ + 3C₂, 1) and Y₁|Z, C ~ N(1.1 − 1.3C₁, 1), A was generated from logit Pr(A = 1|Y₀, Z, C) = −0.2 − 3Z −3C₁ + 0.3Y₀ + 4C₁Z, and Y = Y₀(1 − A) + Y₁A.

Step 2*: Same as Step 2.

Step 3*: Same as Step 3 except the following regression outcome models were fit to the data.

$$E\left\{Y\exp \left(\eta Y\right)|A=0,Z,C;\xi \right\}={\xi }_1+{\xi }_2{C}_1+{\xi }_3{C}_2+{\xi }_{4}Z+{\xi }_5{C}_{1}Z+{\xi }_6{C}_{2}Z+{\xi }_7{C}_1{C}_2+{\xi }_8{C}_1{C}_{2}Z.$$ (18)

$$E\left\{Y\exp \left(\eta Y\right)|A=0,Z,C;\xi \right\}={\xi }_9+{\xi }_{10}{C}_1+{\xi }_{11}{C}_2+{\xi }_{12}Z+{\xi }_{13}{C}_{1}Z+{\xi }_{14}{C}_{2}Z+{\xi }_{15}{C}_1{C}_2+{\xi }_{16}{C}_1{C}_{2}Z.$$ (19)

Step 4*: Same as Step 4 except that (15) was replaced by (18) and (19).

Step 5*: Same as Step 5

Simulation for a continuous outcome under scenario (ii) was carried out similarly as that for scenario (i) except that (14) was replaced by (16). For scenario (iii), the potential outcome models (18) and (19) were replaced with the linear models

$$E\left\{Y\exp \left(\eta Y\right)|A=0,Z,C;\xi \right\}={\xi }_1+{\xi }_2{C}_1+{\xi }_{4}Z.$$ (20)

$$E\left\{\exp \left(\eta Y\right)|A=0,Z,C;\xi \right\}={\xi }_9+{\xi }_{10}{C}_1+{\xi }_{12}Z.$$ (21)

We use the R package nleqslv (Hasselman, 2014) to solve the nonlinear estimating equations.

We verify in the Example A.1 of the Supplementary Materials that ψ is identified from the observed data. The simulation results for 1000 Monte Carlo samples are reported in Figure 2 and empirical coverage rates are presented in Table 1. Results are similar to the those for the binary outcome. Under correct model specification, all estimators have negligible bias which diminishes with increasing sample size. The IPW and OR estimators are biased with poor empirical coverages when the corresponding model is mis-specified. The DR estimator performs well in terms of bias and coverage when either the extended propensity score or the outcome regression model is correctly specified.

Figure 2:

Performance of the IPW, OR and DR estimators of ψ with continuous outcomes

Note: In each boxplot, the true value ψ₀ is marked by the horizontal lines, white boxes are for n = 1000 and grey boxes are for n = 5000.

6. Application

Since the 1980s, tax-deferred programs such as individual Retirement Accounts (IRAs) and the 401(k) plan have played an important role as a channel for personal savings in the United States. Aiming to encourage investment for future retirement, the 401(k) plan offers tax deductions on deposits into retirement accounts and tax-free accrual of interest. The 401(k) plan shares similarities with IRAs in that both are deferred compensation plans for wage earners but the 401(k) plan is only provided by employers. The study includes 9275 people and once offered the 401(k) plan, individuals decide whether to participate in the program. However, participants usually have a stronger preference for savings which suggests the presence of selection bias. This was addressed as individual heterogeneity by Abadie (2003) and it has been pointed out that a simple comparison of personal savings between participants and non-participants may yield results that were biased upward. It was also postulated that given income, the 401(k) eligibility is unrelated to the individual preferences for savings thus can be used as an instrument for participation in 401(k) program (Poterba and Venti, 1994; Poterba et al., 1995). The complier causal effect for the 401(k) plan was studied by Abadie (2003). Here, we reanalyze these data to illustrate the proposed estimators of the marginal ETT.

We illustrate the methods in the context of a dichotomous outcome defined as the indicator that a person falls in the first quartile of net savings of the observed sample (equal to −$500). The treatment variable is a binary indicator of participation in a 401(k) plan and the IV is a binary indicator of 401(k) eligibility. The covariates are standardized log family income (log₁₀ (income) −4.5), standardized age (age −41) and its square, marital status and family size. Age ranged from 25 to 64 years, marital status is binary indicator variable and family size ranges from 1 to 13 people. These covariates are thought to be associated with unobserved preferences for savings. Let ψ = E(Y₀|A = 1) denote for a family that actually participated in the 401(k) program, the probability that they would have had net financial assets above the first quartile, had possibly contrary to fact, they been forced not to participate in the program. The ETT = E(Y₁ − Y₀| A = 1) is the effect of 401(k) plan on the difference scale for the probability of family net financial assets above the first quartile among participants. Equivalently, ETT can also be interpreted as an effect of the intervention in reducing a person’s risk for poor savings performance as measured by falling below the first quartile of the empirical distribution of savings for the sample. Before implementing our IV estimators, we first obtained a standard IPW estimator of the ETT under an assumption of no unmeasured confounding, i.e. ${\widehat{\psi }}_0^{ipw}$ defined as ${\widehat{\psi }}^{ipw}$ with α = 0. Thus, the propensity score was modeled as:

$$\mathrm{logit}\mathrm{Pr}\left(A=1|Z,C\right)=1+Z+\log \left(\mathrm{income}\right)+\mathrm{married}+\mathrm{age}+\mathrm{fsize}+{\mathrm{age}}^2,$$

and estimated by standard maximum likelihood. The IPW estimate of ψ was ${\widehat{\psi }}_0^{ipw}=0.688$ with standard error (se) 0.014, where se was evaluated using the sandwich estimator accounting for all sources of variabilities. In comparison, the estimator based on the empirical estimate of E(Y|A = 1) was 0.883 (se = 0.006). Thus an estimate of ETT was $\widehat{\mathrm{ETT}}=0.194$ (se = 0.016), which suggests the 401(k) plan may have a significant effect on increasing the family net financial assets among participants.

However, this result may be spurious due to the suspicion that even after controlling for observed covariates, there may still exist unmeasured factors that confound the relationship between 401(k) plan and the family net financial assets. Assuming assumptions (IV.1)–(IV.2) and condition 1, we applied the methods proposed in Section 4 to estimate the ETT in the presence of unmeasured confounders. The following parametric models were considered:

$$\mathrm{logit}\mathrm{Pr}\left(Z=1|C\right)=1+\log \left(\mathrm{income}\right)+\mathrm{married}+\mathrm{age}+\mathrm{fsize}+{\mathrm{age}}^2,$$

$$\mathrm{logit}\mathrm{Pr}\left(Y=1|A=0,Z,C\right)=1+Z+\log \left(\mathrm{income}\right)+\mathrm{married}+\mathrm{age}+\mathrm{fsize}+{\mathrm{age}}^2,$$

We specified the selection bias function as in (9), thus the selection bias function was assumed to depend on Y₀ linearly. Possible deviations from this simple model was explored by allowing for potential interactions of Y₀ with observed covariates in the extended propensity score. Thus, we posited the following parametric model for the extended propensity score which satisfies identifying condition 1 as a submodel of the separable model:

$$\mathrm{logit}\mathrm{Pr}\left(A=1|Y_0,Z,C\right)=1+Z+Y_0+\log \left(\mathrm{income}\right)+\mathrm{married}+\mathrm{age}+\mathrm{fsize}+{\mathrm{age}}^2,$$

Table 2 reports point estimates and estimated standard errors for the IV, extended propensity score and the outcome regression models. Although the DR estimator also involves an outcome regression model among the unexposed, it is the same model as required for the regression estimator, thus these estimates are only repeated once. The instrument is strongly associated with family income (log OR = 2.823, se = 0.106), age (log OR = 0.007, se = 0.002) and age square (log OR = −0.002, se = 2e⁻⁴). The selection bias parameter was estimated to be 0.320 (se = 0.115) by IPW, 0.385 (se = 0.135) by OR and 0.280 (se = 0.101) by DR estimation. This provides strong evidence that unmeasured confounding may be present and the stronger saving preference one has, the more likely one would participate in the 401(k) plan. All three estimators of the marginal ETT also agree with each other: they are significant but with a smaller Z-score value than when the selection bias is ignored (for example, the IPW estimator suggests $\widehat{\mathrm{ETT}}=0.134$, se = 0.013). The efficient estimator for the selection bias parameter is 0.273 and for the ETT is 0.137, both in agreement with the other three estimators. Thus we may conclude that even after adjustment for unobserved preferences for savings, the 401(k) plan still can increase net financial assets among participants.

Table 2:

Point estimates and estimated se [in bracket] of IPW, OR and DR estimators for ETT of 401(k) plan as well as the parameters for IV, extended propensity score and outcome regression outcome models required by those estimators

	IV model	IPW propensity	regression	DR propensity
Intercept	−0.180 [0.058]	−8.685 [1.832]	1.307 [0.073]	−8.629 [1.796]
linc	2.695 [0.107]	1.626 [0.210]	0.618 [0.128]	1.633 [0.209]
age	0.007 [0.002]	−0.009 [0.005]	0.035 [0.003]	−0.009 [0.005]
fsize	−0.037 [0.019]	−0.004 [0.033]	−0.127 [0.022]	−0.005 [0.033]
marr	−0.145 [0.063]	−0.032 [0.108]	−0.133 [0.075]	−0.031 [0.108]
age2	0.002 [2e-04]	0.001 [4e-04]	6e-04 [3e-04]	0.001 [4e-04]
Z		9.150 [1.820]	−0.210 [0.074]	9.126 [1.781]
α		0.320 [0.115]	0.385 [0.135]	0.280 [0.101]
ψ = E(Y0|A = 1)		0.749 [0.012]	0.746 [0.012]	0.750 [0.012]
ETT		0.134 [0.013]	0.137 [0.014]	0.132 [0.014]

These findings roughly agree with results obtained by Abadie in the sense that the IV estimate corrects the observational estimate towards the null. However, it may be difficult to directly compare our findings to those of Abadie who reported the compliers average treatment effect under a monotonicity assumption of the IV-exposure relationship, and assuming no unmeasured confounding of this first stage relation. Our approaches rely on neither assumption, but instead rely on condition 1 encoded in the functional form of the extended propensity score model for identification. In order to assess the robustness of the selection bias model, additional functional forms were explored. We considered adding to α an interaction between Y₀ and each of the covariates: log income, marriage status, family size. There was no evidence in favor of any such interaction.

7. Discussion

In this paper, we establish that access to an IV allows for identification of an association between exposure to the treatment and the potential outcome when unexposed, which directly encodes the magnitude of selection bias into treatment due to confounding. We propose IPW, OR as well as DR estimators for the treatment effect amongst treated individuals. Vansteelandt and Goetghebeur (2003) and Robins (1994) proposed identification and inference approaches under no-current treatment value interaction assumption, thus their estimators remain consistent under the null hypothesis of no ETT. In contrast, the identification and inference approaches we proposed may be particularly valuable when an ITT analysis indicates a non-null treatment effect and thus Robins’ identification assumption of no-current treatment value interaction may be violated.

The proposed methods assume the treatment is binary. They can be generalized without much effort to categorical treatment. However, when the treatment is continuous (for example, A is treatment dose), then a parametric model for the treatment effect as well as a model for the density of A may be unavoidable for estimation. We leave this as a topic for future research.