Effect heterogeneity and variable selection for standardizing causal effects to a target population

Abstract

The participants in randomized trials and other studies used for causal inference are often not representative of the populations seen by clinical decision-makers. To account for differences between populations, researchers may consider standardizing results to a target population. We discuss several different types of homogeneity conditions that are relevant for standardization: Homogeneity of effect measures, homogeneity of counterfactual outcome state transition parameters, and homogeneity of counterfactual distributions. Each of these conditions can be used to show that a particular standardization procedure will result in an unbiased estimate of the effect in the target population, given assumptions about the relevant scientific context. We compare and contrast the homogeneity conditions, in particular their implications for selection of covariates for standardization and their implications for how to compute the standardized causal effect in the target population. While some of the recently developed counterfactual approaches to generalizability rely upon homogeneity conditions that avoid many of the problems associated with traditional approaches, they often require adjustment for a large (and possibly unfeasible) set of covariates.

Appendices

Appendix 1

Proofs of identifying expressions from Table 2. We note that these proofs are not new to this paper, and are included here only for completeness:

Approach 1

$$\begin{aligned} \begin{aligned}&\sum _v{\left[ {\text {RR}}_{s,v}\times {\text {Pr}}(V=v \vert Y^{a=0}=1, P=t)\right] }\\&\quad =\sum _v{\left[ {\text {RR}}_{t,v}\times {\text {Pr}}(V=v \vert Y^{a=0}=1, P=t)\right] } (\because {{\text {RR}}_{s,v}={\text {RR}}_{t,v})}\\&\quad =\sum _v{\left[ \frac{{\text {Pr}}(Y^{a=1}=1 \vert V=v, P=t) \times {\text {Pr}}(V=v \vert Y^{a=0}=1, P=t)}{{\text {Pr}}(Y^{a=0}=1 \vert V=v, P=t)} \right] } \\&\quad =\sum _v{\left[ \frac{{\text {Pr}}(Y^{a=1}=1 \vert V=v, P=t)\times {\text {Pr}}(Y^{a=0}=1 \vert V=v, P=t)\times {\text {Pr}}(V=v \vert P=t)}{{\text {Pr}}(Y^{a=0}=1 \vert V=v, P=t)\times {\text {Pr}}(Y^{a=0}=1 \vert P=t)}\right] }&\\&\quad =\sum _v{\left[ \frac{{\text {Pr}}(Y^{a=1}=1\vert V=v, P=t)\times {\text {Pr}}(V=v \vert P=t)}{{\text {Pr}}(Y^{a=0}=1 \vert P=t)}\right] }\\&\quad =\frac{\sum _v{\left[ {\text {Pr}}(Y^{a=1}=1\vert V=v, P=t)\times {\text {Pr}}(V=v \vert P=t)\right] }}{{\text {Pr}}(Y^{a=0}=1 \vert P=t)}\\&\quad =\frac{{\text {Pr}}(Y^{a=1}=1 \vert P=t)}{{\text {Pr}}(Y^{a=0}=1 \vert P=t)}\\&\quad ={\text {RR}}_t \end{aligned} \end{aligned}$$

(1)

Approach 2

$$\begin{aligned} \begin{aligned}&\sum _v{ \left[ {\text {Pr}}(Y^{a=0}=1 \vert V=v, P=t) \times {\text {RR}}_{s,v} \times {\text {Pr}}(V=v \vert P=t) \right] }\\&\quad =\sum _v{ \left[ {\text {Pr}}(Y^{a=0}=1 \vert V=v, P=t) \times \text {RR}_{t,v} \times {\text {Pr}}(V=v \vert P=t) \right] (\because {\text {RR}_{s,v}=\text {RR}_{t,v})}}\\&\quad =\sum _v{ \left[ {\text {Pr}}(Y^{a=0}=1 \vert V=v, P=t) \times \frac{{\text {Pr}}(Y^{a=1}=1 \vert V=v, P=t)}{{\text {Pr}}(Y^{a=0}=1 \vert V=v, P=t)} \times {\text {Pr}}(V=v \vert P=t) \right] }\\&\quad =\sum _v{ \left[ {\text {Pr}}(Y^{a=1}=1 \vert V=v, P=t) \times {\text {Pr}}(V=v \vert P=t) \right] }\\&\quad ={\text {Pr}}(Y^{a=1}=1 \vert P=t) \end{aligned} \end{aligned}$$

(2)

Approach 3

(3)

Approach 4

We are here assuming that Y is a binary variable, the proof generalizes readily to settings with continuous or time-to-event outcomes. In order to simplify the logic, we will further assume that the same set of baseline covariates V is sufficient to control both for confounding for A, and for differences between populations. In other words, we will assume conditional exchangeability in the study population ( ) and conditional effect homogeneity in distribution ( ). Before we begin, it is useful to note that $\frac{{\text {Pr}}(A=a, V=v, P=s)}{{\text {Pr}}(A=a \vert P=s, V=v) \times {\text {Pr}}(P=s \vert V=v)} = {\text {Pr}}(V=v)$. This follows from sequential application of the definition of conditional probability.

(4)

Approach 5

The proof of approach 5 is closely related to that for approach 4. Westreich et al. [28] provide a full proof in the appendix.

Appendix 2

Here, we prove that if there is effect homogeneity in distribution between the groups $W=1$ and $W=0$, then the parameter $\beta _2$ must be equal to zero in the regression model

$$\begin{aligned} {\text {logit\,Pr}} (Y = 1 \vert A,W, P=s) = \beta _0 + \beta _1 A + \beta _2 W \end{aligned}$$

(5)

Note here that we are discussing a regression model fit within the study population, and where the homogeneity assumption is between groups of baseline covariate W. In contrast to the rest of the paper, we are therefore using the homogeneity assumption rather than .

Additionally, we will make the following assumptions:

By consistency and exchangeability, the model can be rewritten as a structural model:

$$\begin{aligned} {\text {logit\,Pr}}(Y^a = 1 \vert W, P=s) = \beta _0 + \beta _1 a + \beta _2 W \end{aligned}$$

(6)

If $W = 0$, we have:

$$\begin{aligned} {\text {logit\, Pr}}(Y^a = 1 \vert W=0, P=s) = \beta _0 + \beta _1 a \end{aligned}$$

(7)

If $W = 1$, we have:

$$\begin{aligned} {\text {logit\, Pr}}(Y^a = 1 \vert W=1, P=s) = \beta _0 + \beta _1 a + \beta _2 \end{aligned}$$

(8)

By the assumption of effect homogeneity in distribution, we can set these equal:

$$\begin{aligned} \beta _0 + \beta _1 a = \beta _0 + \beta _1 a + \beta _2 \end{aligned}$$

Solving this for $\beta _2$ we get $\beta _2 = 0$.