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Biometrika. Mar 2010; 97(1): 171–180.

Published online Dec 8, 2009. doi: 10.1093/biomet/asp062

PMCID: PMC3412601

Eric J. Tchetgen Tchetgen and James M. Robins

Department of Biostatistics, Harvard School of Public Health, 677 Huntington Avenue, Boston, Massachusetts 02115, U.S.A., Email: etchetge/at/hsph.harvard.edu, ; Email: robins/at/hsph.harvard.edu

Department of Economics, Universidad Torcuato Di Tella, Sáenz Valiente 1010, 1428 Buenos Aires, Argentina, ; Email: arotnitzky/at/utdt.edu

Received September 2008; Revised May 2009

Copyright © 2009 Biometrika Trust

This article has been cited by other articles in PMC.

We consider the doubly robust estimation of the parameters in a semiparametric conditional odds ratio model. Our estimators are consistent and asymptotically normal in a union model that assumes either of two variation independent baseline functions is correctly modelled but not necessarily both. Furthermore, when either outcome has finite support, our estimators are semiparametric efficient in the union model at the intersection submodel where both nuisance functions models are correct. For general outcomes, we obtain doubly robust estimators that are nearly efficient at the intersection submodel. Our methods are easy to implement as they do not require the use of the alternating conditional expectations algorithm of Chen (2007).

Given a random vector *O* = (*Y, A, L*) the conditional odds ratio function *γ* (*Y, A, y*_{0}, *a*_{0}, *L*) between *A* and *Y* given *L* at a given base point (*a*_{0}, *y*_{0}) is

where the vectors *Y* and *A* can take either discrete values, continuous values, or a mixture of both, *L* is a high-dimensional vector of measured auxiliary covariates, (*a*_{0}*, y*_{0}) is a user specified point in the sample space and *f* (*Y* | *A, L*)*, g*(*A* | *Y, L*) and *h*(*A, Y* | *L*) are, respectively, the conditional densities of *Y* given *A* and *L*, the conditional density of *A* given *Y* and *L* and the joint conditional density of *A* and *Y* given *L* with respect to a dominating measure *μ*. The odds ratio function is a particularly useful measure of association when *Y* and *A* take both discrete and continuous values. For instance, *A* and *Y* could each be a mixture of a discrete component encoding, say, the presence or absence of a given bacterium and a continuous component encoding the bacterial counts when it is present. In such a case, as argued by Chen (2007), a complete characterization of the association between bacterium *A* and bacterium *Y* given *L* would require separate comparisons of the probabilities of absence of one bacterium when the other bacterium is either absent or present at a particular concentration, and of the concentration distribution for one bacterium when the other bacterium is either absent or present at a particular concentration. Instead, the direct estimation of the odds ratio function relating bacterium *A* to bacterium *Y* given covariates *L* provides a unified solution to this problem and obviates the need for separate analyses.

Given *n* independent and identically distributed copies of *O*, Chen (2007) proposed a locally efficient iterative estimator of the parameter *ψ*_{0} in a semiparametric model that specifies (i) *γ* (*Y, A, L*) is equal to a known function *γ* (*Y, A, L; ψ*) evaluated at the unknown true *p*-dimensional parameter vector *ψ*_{0}, i.e.

(1)

where *γ* (*Y, A, L; ψ*) takes the value 1 if *A* = *a*_{0}*, Y* = *y*_{0}, or *ψ* = 0, so *ψ*_{0} = 0 encodes the null hypothesis that *Y* and *A* are conditionally independent given *L*, and (ii) either but not necessarily both, (a) a given parametric model *f* (*Y* | *a*_{0}, *L; θ*) for *f* (*Y* | *a*_{0}*, L*) or (b) a parametric model *g*(*A* | *y*_{0}*, L; α*) for *g*(*A* | *y*_{0}*, L*) is correct. Model is referred to as a union model because it is the union of the model that assumes that (i) and (iia) are true and the model that assumes that (i) and (iib) are true. An estimator of *ψ*_{0} that is consistent and asymptotically normal under this union model is referred to as doubly robust because, given equation (1), the estimator is consistent and asymptotically normal for *ψ*_{0} if one has succeeded in specifying either a correct model *f* (*Y* | *a*_{0}*, L*) or a correct model for *g*(*A* | *y*_{0}*, L*), thus giving the data analyst two chances rather than one chance to obtain valid inference for *ψ*_{0}.

An example of a simple parametric model for the odds ratio function is the bilinear log-odds ratio model (Chen, 2003, 2004). It assumes that *γ* (*Y, A, L; ψ*_{0}) = exp{*ψ*_{0}(*Y* − *y*_{0}) (*A* − *a*_{0})}, where is the direct product. This model includes all of the generalized linear regression models with canonical link functions as special cases. In the case of stratified 2 × 2 tables, it implies homogeneous odds ratios, but is easily extended to the case of nonhomogeneous odds ratios. Other interesting examples of odds ratio models are given by Chen (2007).

Unfortunately, Chen’s aforementioned locally efficient doubly robust estimator of *ψ*_{0} under model is computationally very demanding, especially when *A* and *Y* have multiple continuous components. The main contribution of our paper is to provide novel and highly efficient doubly robust estimators of *ψ*_{0} that are substantially easier to compute than those of Chen.

Before describing our new approach, we briefly summarize Chen’s results. He considered the following parametric and semiparametric approaches to the estimation of *ψ*_{0}: a prospective likelihood approach under the model that assumes that one has correctly modelled the nuisance baseline function *f* (*Y* | *a*_{0}*, L*); a retrospective likelihood approach under the model that assumes that one has correctly specified a model for the nuisance baseline function *g*(*A* | *y*_{0}*, L*); a joint likelihood approach under the intersection model that assumes that both models and are correct; and a doubly robust locally semiparametric efficient approach under the union model of § 1.

In his doubly robust approach, Chen establishes that in the semiparametric model characterized by the sole restriction (1), the density *h*(*A, Y* | *L*) can be written as *h*(*A, Y* | *L; ψ*_{0}), where

(2)

*f* (*y* | *L, A* = *a*_{0}) and *g*(*a* | *Y* = *y*_{0}, *L*) are the unknown conditional densities that generated the data and are solely restricted by ∫ *γ* (*y, a, L*) *f* (*y* | *L, A* = *a*_{0})*g*(*a* | *Y* = *y*_{0}*, L*)*dμ*(*a, y*) < ∞ almost everywhere. Then, he specifies parametric models *f* (*Y* | *a*_{0}*, L; θ*) and *g*(*A* | *y*_{0}*, L; α*) for the unknown nuisance baseline functions *f* (*y* | *a*_{0}*, L*) and *g*(*a* | *y*_{0}*, L*), obtains profile estimates (*ψ*) and (*ψ*) of the nuisance parameters *θ* and *α* and calculates the efficient score *Ŝ*_{eff} (*ψ*) * S*_{eff} {(*ψ*), (*ψ*)*, ψ*} for *ψ* in the semiparametric model evaluated at the law [*γ* (*y, a, l; ψ*)*, f* {*y* | *a*_{0}*, l; *(*ψ*)}, *g*{*a* |*y*_{0}*,l; *(*ψ*)}] indexed by {(*ψ*)*, * (*ψ*)*, ψ*}. Next, he estimates *ψ*_{0} with the solution _{eff} to *P _{n}*{

By definition, the efficient score *S*_{eff} = Π (*S _{ψ}* |
) for a parameter

(3)

contains all functions that have zero-mean conditional on both (*A, L*) and (*Y, L*). When both *A* and *Y* contain continuous components and *ψ*_{0} ≠ 0, Chen (2007) finds that this projection and therefore *S*_{eff} do not exist in closed form and must be computed using the iterative alternating conditional expectations algorithm. Each iteration requires the evaluation, by numerical integration, of conditional expectations, which seriously limits the practicality of Chen’s approach, particularly when *A* and/or *Y* have two or more continuous components.

The main contribution of our paper is to show that, even though the projection Π(*R* |
) of a given random variable *R* = *r*(*Y, A, L*) into the orthocomplement
does not exist in closed form when both *A* and *Y* contain continuous components, the set
does have a closed-form representation, which appears to be new. We use our representation to obtain doubly robust estimators, i.e. consistent and asymptotically normal estimators of *ψ*_{0} in the union model , that are nearly as efficient as _{eff} under the intersection submodel, yet do not require the alternating conditional expectations algorithm. Moreover, our closed-form representation of
is of independent interest, with applications beyond the present paper. For example, Vansteelandt et al. (2008) use our representation to construct multiple robust estimators of the parameter encoding the interaction on an additive and multiplicative scale between two exposures *A*_{1} and *A*_{2} in their effects on an outcome *Y*.

In the special situation where either *Y* or *A* has finite support, Bickel et al. (1993) provide a closed-form expression for Π(*R* |
), which Chen, however, did not use to give a closed-form expression for *S*_{eff}. We remedy this oversight and obtain doubly robust locally-efficient closed-form estimating functions when *Y* and/or *A* has finite support; some emphasis is given to the important case of dichotomous Y which, incidentally, coincides with the semiparametric logistic regression model.

In the following, for a vector *υ* we write *υ*^{2} = *υυ*^{T}. To simplify notation, we suppose *y*_{0} =0 and *a*_{0} = 0 throughout, so that *γ* (*Y,* 0*, L; ψ*) = *γ* (0*, A, L; ψ*) = *γ* (*Y, A, L*; 0) = 1. We shall also use the following definition.

Definition 1. *Given conditional densities f*^{†}(*Y* | *L*) *and g ^{†}*(

As noted previously, under model characterized by restriction (1), Chen showed that
is given by the set (3). We now provide a new closed-form representation of this set. To do so, for a fixed choice of admissible independence density *h ^{†}*(

with *h* (*Y, A* | *L; ψ*) defined in (2) and *d ^{†}*(

Theorem 1. *Given an admissible independence density h ^{†}, an alternative representation of the set*

*Proof.* One can verify by explicit calculation that {*U*(*ψ*_{0}; *d, h ^{†}*) :

*Remark.* We give an alternative, more abstract, proof of the fact that *U*(*ψ*_{0}; *d, h ^{†}*) {

By standard semiparametric theory (Bickel et al., 1993), Theorem 1 implies that if is a regular and asymptotically linear estimator of *ψ*_{0} in model , then given any admissible independence density *h ^{†}*, there exists a

To do so, we adopt the notational convention introduced in § 1 that given a function such as *U*(*ψ*_{0}; *d, h ^{†}*) which depends on the unknown law

*d*(*Y, A, L*) is a user-supplied function,

is the profile maximum likelihood estimator of *α* at a fixed *ψ, g*(*A* | *Y, L; ψ, α*) = *γ*(*Y, A, L; ψ*)*g*(*A* | *Y* = 0*, L; α*)*/∫g*(*a* | *Y* = 0*, L; α*)*γ* (*Y, a, L; ψ*) *dμ*(*a*),

is the profile maximum likelihood estimator of *θ* at a fixed *ψ, f* (*Y* | *A, L; ψ, θ*) = *γ*(*Y, A, L; ψ*) *f* (*Y* | *A* = 0*, L; θ*)*/∫γ* (*y, A, L; ψ*) *f* (*y* | *A* = 0*, L; θ*)*dμ*(*y*), and *ĥ ^{†}*(

Theorem 2. *Suppose ĥ ^{†}*(

(4)

*and thus converges in distribution to N*(0, ∑)*, where*

*with θ**(*ψ*) *and α**(*ψ*) *denoting the probability limits of *(*ψ*) *and *(*ψ*)*, respectively, and*

(5)

*where*
*and*
*are the scores for θ and α, respectively.*

A consistent estimator of Σ is

where is defined as *M* but with expectations replaced by their empirical version. Thus, can easily be used to obtain Wald-type confidence intervals for components of *ψ*_{0}.

*Remark.* When * _{f}* and/or

We first consider the case in which both *A* and *Y* contain continuous components. Chen’s estimator _{eff} solving *P _{n}*{

When *ψ*_{0} is not known to be nearly zero, we adopt a general approach proposed by Newey (1993). We take a basis system *ϕ _{j}* (

where
is a generalized inverse of Γ* _{K}* =

Neither of these two strategies is needed if *Y* or *A* have finite support as an explicit form for the efficient score in this case was given by Bickel et al. (1993). Without loss of generality, assume *Y* has finite support say {*y*_{0}*, y*_{1}*, . . ., y _{M}*

Consider the vector {*I* (*Y* = *y*_{1})*, . . ., I* (*Y* = *y _{M}*

Furthermore, Bickel et al. (1993) show that *Ũ* {*ψ*_{0}; *k*_{eff} (*ψ*_{0})} is the efficient score function of *ψ* in model , where *k*_{eff} (*ψ*_{0}) equals *k*_{eff} (*ψ*_{0}) = [ log{*ρ*^{T}(*A, L; ψ*)}*/ψ*] |_{ψ= ψ0}, with *ρ* (*A, L; ψ*) defined to be the (*M* − 1) × 1 vector with the *j*th component equal to *γ* (*y _{j}, A, L, ψ*)

We next derive a doubly robust locally-efficient estimating function *U*(*ψ, θ, α; d*_{eff}*, h ^{†}*) in our class that equals

with *y*_{1} = 1 and *η*(*L*) = log[Pr(*Y* = 1 | *A* = 0*, L*)/{1 − Pr(*Y* = 1 | *A* = 0*, L*)}] is an unrestricted function of *L*. Since *Y* is binary, any function *d*(*Y, A, L*) may be written as *Ym*(*A, L*) + *n*(*A, L*) with *m*(*A, L*) = *d*(1*, A, L*) − *d*(0*, A, L*) and *n*(*A, L*) = *d*(0*, A, L*). Given an admissible independence density *h ^{†}*(

Furthermore, by

we have that

Thus, since *S*_{eff} = *Ũ* (*k*_{eff}) is the efficient score, we conclude that *V* {*ψ*_{0}; *r*_{eff} (*h ^{†}*;

Therefore, the solution to either of the following estimating equations is doubly robust locally semiparametric efficient
, or
where *d*_{eff} (*Y, A, L; ψ, θ, α, ĥ ^{†}*) =

Although the common variation independent parameterization of *h*(*A, Y* | *L*) *f _{L}* (

Theorem 3. *Under the common parameterization by* (*ψ, f, g, f _{L}*)

In the Appendix, we prove this result for discrete *A*, thereby avoiding technicalities that arise in the continuous case.

Andrea Rotnitzky and James Robins were funded by grants from the U.S. National Institutes of Health. The authors wish to thank the reviewers for helpful comments. Andrea Rotnitzky is also affiliated with the Harvard School of Public Health.

*Proof of Theorem* 2. We assume that the regularity conditions of Theorem 1A of Robins et al. (1992) hold for *U*(*ψ*_{0}; *θ, α, h ^{†}*)

since

Then, under the assumed regularity conditions the formulae (4) and (5) follow from standard Taylor series arguments, whenever *E*[*M*{*ψ, θ**(*ψ*)*, α**(*ψ*); *d, h ^{†}*}

*Proof of Theorem* 3. The proof is by contradiction: if *S*(*ψ, f***, g**) were doubly robust, then, for every *f**, *S*(*ψ, f***, g**) would be an unbiased estimating function for *ψ* with power against local alternatives in the submodel _{g*} of model in which *g* = *g** is known a priori. Hence, it suffices to prove that model _{g*} does not admit such unbiased estimating functions. Noting that model _{g*} can be parameterized by (*ψ, f, f _{L}*), we need to prove there is no function

where *Ũ* (*ψ, f ; k*) = (*ψ, f*)[*k*(*A, L*) − *Ẽ*{*k*(*A, L*) | *L; ψ, f*}] is *Ũ* (*ψ; k*) defined in § 4 with the dependence on *f* now made explicit. Suppose *Q*(*ψ*) existed. Then *Q*(*ψ*) = *T* (*k _{f}, υ_{f}*;

Thus, for a function *r*(*L*), *k _{f}*

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