Estimation of Risk Ratios in Cohort Studies with a Common Outcome: A Simple and Efficient Two-stage Approach

8.1 Proof of Result 1

Let denote the (probability) limiting value of To show that the result holds, it suffices to show that is an unbiased estimating function; that is, we need to show that Now,

To establish the large sample behavior of we perform a standard Taylor expansion

By the law of large numbers and an application of Slutzky’s theorem, we conclude that has large sample distribution equal to the distribution of

since We may further conclude that the large sample variance of is given by

because

where Furthermore, because covariance matrices are positive-definite, we may conclude that is conservative for the variance-covariance matrice in the positive-definite sense, that for any non-zero constant vector t

and therefore is a conservative estimator of Whereas is consistent for where

8.2 Proof of Result 2

Consider the semiparametric model given solely by restriction (3); then it is well known that all regular and asymptotically linear estimators of are fully characterized by the set of influence functions:

See Bickel et al. [12] and Robins and Rotnitzky [16]. It is straightforward to verify that this set is equivalently written:

Now, the score for in this model is given by

therefore, the efficient score of i.e., the orthogonal projection of onto is ,with in other words,

since , and for all

The proof is completed by noting that

where

Then, a theorem due to Bickel et al. [12] states that for any initial consistent estimator of an efficient estimator can be constructed by a one-step update of in the direction of the estimated efficient score by using the following formula

where is an empirical version of obtained by replacing all expectations by empirical expectations, with estimated by and estimated by the simple plug-in estimator is a similarly constructed estimator of the expected derivative of the efficient score, with respect to evaluated at It is straightforward to verify that reduces to the formula provided in the main text. Furthermore, the theorem of Bickel et al. [12] further states that under standard regularity conditions, is asymptotically normal with mean zero and variance

which is also the semiparametric efficiency bound of Finally, is an empirical version of which converges to the latter in probability.

In order to prove Result 3, we first establish a more general result, for which we allow to be continuous, and for the model to incorporate a possible interaction between the exposure and the covariates, say a component of Specifically, we suppose that

[7]

and let denote a working model for the mean of the exposure given covariates; where is the identity link for continuous and g is the logit link for binary Define the estimating function

Then, we have the following lemma.

Lemma 1: Under model (7),

[8]

if either but not necessarily both of the following conditions hold,

1. or

2. and model (5) holds.

8.3 Proof of Lemma 1

which is certainly zero if (1) holds since then .If (2) holds, we have

since the last quantity is part of the first order condition used to estimate either by ordinary least-squares when is continuous or by logistic regression in the binary case.

8.4 Proof of Result 3

The result immediately follows from Lemma 1 since when is binary, it is straightforward toverify that equation (8) is equivalent to

Therefore, if either (1) holds, and thus converges to or (2) holds and thus converges to we have that converges to