Designing efficient randomized trials: power and sample size calculation when using semiparametric efficient estimators

A.1 Details of the cross-fit AIPW estimator in randomized trials

Here we discuss the AIPW estimator and its semiparametric efficiency in the context of randomized trials. While the conclusions of this paper apply to any semiparametric efficient estimator, it may help the reader to understand semiparametric efficiency in the context of a single estimator. The details provided here are all available elsewhere in the literature or follow immediately from known results [14, 20, 27, 32]. We reframe them here to provide a quick reference and starting point for further reading.

Preliminaries. A scalar parameter of a distribution is a functional that ingests the distribution and returns a number. For example, consider the distribution of a scalar random variable Y defined by the PDF f _y . The mean of Y (which is a parameter) is the functional ∫yf _y (y)dy. An estimator of a (scalar) parameter is a function that takes data sampled from the distribution in question and returns a number which is meant to approximate the parameter. For example, an estimate of the mean of Y is 1 n ∑ i y i when y _i are assumed to be draws from the distribution of Y. An estimator is consistent if it recovers the true value of the parameter as the sample size grows. It is usually possible to construct many different consistent estimators of the same parameter, so we are interested in finding the one that has the smallest possible sampling variance. We call this the efficient estimator.

It becomes much easier to find the efficient estimator if we restrict our attention to the class of regular and asymptotically linear (RAL) estimators, and we actually don’t lose anything by doing so. It is not important to understand the precise mathematical definition of regularity, but heuristically, a regular estimator does not have anomalously bad performance for certain special values of the parameter. Thus restricting ourselves to regular estimators is a good and sensible thing to do. Effectively all estimators which can be used in practice are regular. Moreover, among regular estimators of some parameter, a result called the Hájek–Le Cam convolution theorem guarantees that the estimator with the smallest asymptotic variance is asymptotically linear, so we lose nothing by further restricting ourselves to asymptotically linear estimators once we’ve excluded irregular estimators [14].

The definition of asymptotic linearity for an estimator ψ ̂ of a parameter ψ is that there exists an influence function ϕ such that n ( ψ ̂ − ψ ) = E ̂ ϕ + o p ( 1 ) . In other words, the estimator ψ ̂ behaves like an IID average of some random variable ϕ in large samples. Asymptotic linearity immediately implies n ( ψ ̂ − ψ ) ⇝ N ( 0 , V ϕ ) by the central limit theorem. Therefore the asymptotic variance of any asymptotically linear estimator is given by the variance of its influence function.

Finding the most efficient RAL estimator thus boils down to finding the influence function with the smallest variance, which we call the efficient influence function (EIF). The EIF defines a lower bound on the achievable sampling variance when estimating a parameter. It so happens that it is often possible to characterize the space of all influence functions of RAL estimators of a given parameter in a generic generative model, which makes it possible to derive the EIF. Obtaining an efficient estimator is therefore a matter of deriving the EIF for the class of RAL estimators of a parameter and then constructing an estimator that has that influence function.

Application. In the semiparametric statistical model P(Y, W, X) = P(Y|X)P(W|X)P(X) (with P(Y|X), P(W|X), and P(X) free to be any distributions that satisfy mild regularity conditions), it turns out that the efficient influence function of the class of RAL estimators for the parameter μ w = E Y | W = w is ϕ w = W w π w ( X ) ( Y − μ w ( X ) ) + ( μ w ( X ) − μ w ) . Proving this is not simple (see Tsiatis [14]), but after the fact has been established all our subsequent theory requires only algebra and elementary tools from large-sample theory.

As might be expected, the EIF for the two-dimensional parameter μ = [ μ 0 , μ 1 ] ⊤ is ϕ μ = [ ϕ 0 , ϕ 1 ] ⊤ . Influence functions obey a “chain rule” such that if the EIF of ψ is ϕ, the EIF of g(ψ) is ∇g ^⊤ ψ. For us, that means that the EIF of τ = r(μ₀, μ₁) is ϕ = r₀′(μ₀, μ₁)ϕ₀ + r₁′(μ₀, μ₁)ϕ₁.

In a two arm randomized trial with treatment fractions π _w , one estimator of μ _w that has the efficient influence function is μ ̂ w * = E ̂ W w π w ( Y − μ w ( X ) ) + μ w ( X ) . The fact is easily verified by showing n ( μ ̂ w − μ w ) = E ̂ ϕ w + o p ( 1 ) with an application of the central limit theorem. An application of the delta method shows that τ ̂ * = r ( μ ̂ 0 * , μ ̂ 1 * ) attains the asymptotic variance ν 2 = V ϕ and is therefore efficient. We’ll call this the “oracle estimator”.

Unfortunately, the oracle estimator τ ̂ * is infeasible in practice because the conditional means μ _w (X) are not known. However, it turns out that estimates can be substituted without sacrificing the optimality properties. Let μ ̂ w ( k ) = E ̂ W w π w ( Y − μ ̂ w ( − k ) ( X ) ) + μ ̂ w ( − k ) ( X ) k ( i ) = k . This is the marginal mean estimated from the kth fold of data using the conditional mean function estimated from the rest of the data. Let μ ̂ w * ( k ) be the oracle equivalent that uses the true conditional mean μ _w (X). Clearly, μ ̂ w = ∑ n ( k ) n μ ̂ w ( k ) where n^(k) is the number of observations in fold k (and similarly for μ ̂ w ). If we can show that n ( μ ̂ w ( k ) − μ ̂ w * ( k ) ) → p 0 , then Slutsky’s theorem and the delta method imply that τ ̂ has the same asymptotic properties as τ ̂ * , i.e. n ( τ ̂ − τ ) ⇝ N ( 0 , ν 2 ) . In other words, since the oracle estimator is efficient with a known asymptotic variance, the feasible estimator is also efficient and has the same asymptotic variance. It turns out that if we assume that E μ ̂ w ( − k ) ( X ) − μ w ( X ) 2 → 0 (a very weak condition), then it is possible to show n ( μ ̂ w ( k ) − μ ̂ w * ( k ) ) → p 0 as desired.

We begin by deriving an expression for the difference between the oracle and feasible estimators:

where in the last line all we’ve done is expand the E ̂ ⋅ notation and introduce I k = { i : k ( i ) = k } . Notice that if we condition on the dataset I ( − k ) , the estimated function μ ̂ w ( − k ) ( ⋅ ) becomes fixed because it does not depend on the data in fold k. Therefore E μ ̂ w ( k ) − μ ̂ w * ( k ) I ( − k ) = 0 because of the conditional independence between 1 − W w , i π w and μ ̂ w ( − k ) ( X i ) − μ w ( X i ) in fold k and the fact that E 1 − W w , i π w = 0 . This, in turn, implies that E μ ̂ w ( k ) − μ ̂ w * ( k ) 2 I ( − k ) = V μ ̂ w ( k ) − μ ̂ w * ( k ) I ( − k ) , which we will use shortly. Moreover, the terms within the sum of Eq. (10) are all IID conditional on I ( − k ) so we can pass the variance through the sum (and gain a 1/n^(k) in the process):

Our plan is to show n ( μ ̂ w ( k ) − μ ̂ w * ( k ) ) → L 2 0 , which implies n ( μ ̂ w ( k ) − μ ̂ w * ( k ) ) → p 0 because L² convergence implies convergence in probability. By the definition of L² convergence, that means we must show E n μ ̂ w ( k ) − μ ̂ w * ( k ) 2 → 0 , which we can do by using what we’ve derived above:

This must converge to 0 because n and n^(k) grow in proportion to each other and the expectation E μ ̂ w ( − k ) ( X i ) − μ w ( X i ) 2 converges to 0 by our mean-square consistency assumption (Eq. (3)).

By the arguments above, this is enough to establish the asymptotic normality of our estimator with the efficient asymptotic variance: n ( τ ̂ − τ ) ⇝ N ( 0 , ν 2 ) .

Similar arguments are required to show that the plug-in variance estimator is consistent (when cross-fitting is used). Some care is required to address the terms r w ′ ( μ ̂ 0 , μ ̂ 1 ) , etc., but the elementary tools of asymptotic statistics suffice.

A.2 Proof of Theorem 1

Beginning with Eq. (4), we have that

from expansion of the variance. We will use several tricks to analyze these terms. The first is that W _w Y = W _w Y _w , by the fact that Y = Y₀W₀ + Y₁W₁ (note W w 2 = W w and W₀W₁ = 0, which we will also use). Secondly, W w ⊥ Y w , μ ̂ w ( X ) by unconfoundedness, allowing for factorization of expectations.

These tricks and a few lines of algebra show that the variances in the first and second terms above (Eq. (13)) are

where σ w 2 ( X ) ≡ V Y w | X is the conditional variance function in treatment arm w. In the last line we define κ w 2 . ≡ E σ w 2 ( X ) (the average conditional variance) and σ w 2 ≡ V Y w (the marginal variance) to simplify notation.

Similar manipulation reduces the covariance term in Eq. (13) to

by exploitation of the law of total variance V Y w ︸ σ w 2 = E σ w 2 ( X ) ︸ κ w 2 + V μ w ( X ) and definition of γ ≡ Corr[μ₀(X), μ₁(X)]. Assembling the above and making explicit the known signs of r w ′ gives