Franks 2019 - Tukey Factorization and Marginal Contrast Estimands

Summary

Tukey’s factorization writes the joint law of $(T,Y(t))$ as an identified observed part times an unidentified selection function ratio. For both potential outcomes it adds a conditional copula for $(Y(0),Y(1))$ given $T$, but Theorem 1 shows ATE, ATT, ATC and QTE (marginal-contrast estimands) depend only on the two marginal selection functions, so the study can be treated as two independent missing-data problems.

Overview

Two technical conditions are needed. They hold automatically for the logistic-mEF family of Franks 2019 - Logistic Selection with Exponential Family Mixtures (Condition 1) or must be argued (Condition 2).

Main Content

One potential outcome (Sec. 3.1)

Tukey factorization for one arm (Eq. 3)

$$f_{\psi }(Y(t),T|X)=f^{obs}(Y(t)|T=t,X)f(T=t|X)\cdot \frac{f_{\psi }(T|Y(t),X)}{f_{\psi }(T=t|Y(t),X)}.$$

The first two factors constitute the identified observed-data density; the last (the selection function ratio) is unidentified but interpretable and determines the relation between observed and missing outcome distributions.

Condition 1 (integral constraint, Eq. 4)

$${\int }_{\mathcal{Y}}{f}^{obs}(Y(t)|T=t,X)\frac{f_{\psi }(T=1-t|Y(t),X)}{f_{\psi }(T=t|Y(t),X)}dY(t)=\frac{f(T=1-t|X)}{f(T=t|X)}.$$

Arises by integrating (3) over $Y(t)$ (the odds of being in the other arm must match the marginal odds). It restricts the selection functions usable; it has caused technical difficulty in earlier work (Robins et al. 2000).

Condition 2 (outcome overlap)

$P(Y(t)\in A|T=1-t,X)>0\Rightarrow P(Y(t)\in A|T=t,X)>0$ for all measurable $A$. The support of the missing outcomes lies inside the support of the observed ones, so the missing distribution is an extrapolation (re-weighting) of the observed one. Unlike Condition 1 it is an assumption about the true data-generating process and cannot be enforced by model specification.

Both potential outcomes (Sec. 3.2)

Under Conditions 1-2 the joint density is determined by the observed data plus three unidentified models: treatment given $Y(1)$ alone, treatment given $Y(0)$ alone, and a copula.

General factorization (Eq. 5)

$$f(T,[Y(0),Y(1)]|X)=[f(Y(0)|T=0,X)f(T=0|X)\frac{f(T|Y(0),X)}{f(T=0|Y(0),X)}][f(Y(1)|T=1,X)f(T=1|X)\frac{f(T|Y(1),X)}{f(T=1|Y(1),X)}]\frac{c(F(Y(0)|T,X),F(Y(1)|T,X)|T,X)}{f(T|X)},$$

with marginal selection factors $f(T|Y(1),X)$, $f(T|Y(0),X)$ and conditional copula density $c=\frac{f(Y(0),Y(1)|T,X)}{f(Y(0)|T,X)f(Y(1)|T,X)}$. Hence the missing outcome given the observed one (Eq. 6): $f^{mis}(Y(t)|T=1-t,Y(1-t),X)\propto f^{obs}(Y(t)|T=t,X)\frac{f(T=1-t|Y(t),X)}{f(T=t|Y(t),X)}c(\cdot )$.

Marginal contrast estimand (Definition 1)

$\tau$ is a marginal contrast iff it is a function only of the marginals $f(Y(0)|X)$ and $f(Y(1)|X)$. Includes ATE and QTE (and ATT, ATC, odds ratios); excludes the proportion with a positive effect, the variance of the treatment effect, and unit-level effects.

Theorem 1

If $f(T,[Y(0),Y(1)]|X)$ admits Tukey’s factorization (5) and $\tau$ is a marginal contrast, then $\tau$ is uniquely determined by the marginal selection factors $f(T|Y(1),X)$ and $f(T|Y(0),X)$; it is invariant to the conditional copula.

Proof (Appendix A). It suffices to show $f(Y(t)|T)$ is free of copula parameters. $f(Y(t)|T=t)$ is modelled directly. For $T=1-t$:

$$f(Y(t)|T=1-t)=\int f(Y(t),Y(1-t)|T=1-t)dY(1-t)\propto f(Y(t)|T=t)\frac{f(T=1-t|Y(t))}{f(T=t|Y(t))}\int c(\cdot )f(Y(1-t)|T=1-t)dY(1-t),$$

and by the definition of the copula density the integral equals $\int f(Y(1-t)|Y(t),T)dY(1-t)=1$. So the copula integrates out.

Corollary 1.1 (Bayesian invariance)

If, in addition, the copula parameters are a priori independent (“distinct”) of the parameters of all other factors in (5), the posterior of any marginal-contrast estimand is invariant to the copula specification in the likelihood.

Consequence: only the two selection factors must be specified, which matters because dependence between potential outcomes is unobservable “even in experiments”. Non-marginal estimands (individual effects, optimal regimes; Klausch et al. 2018) would require the copula and are left to future work.

Examples

Perfect-doctor, Rubin (2003)

A doctor who perfectly predicts $Y(0),Y(1)$ and assigns the better arm: the dependence between $Y(0),Y(1)$ matters for individual effects but, by Theorem 1, not for the ATE once the two selection functions (${\gamma }_0<0$, ${\gamma }_1>0$ in the logistic parameterisation of the next notes) are fixed.

# Tilting an observed sample to the missing distribution via self-normalised importance weights (Eq. 1)
import numpy as np
def tilt_weights(y, gamma, sign=+1):
    w = np.exp(sign * gamma * y)           # odds f(T=1-t|y)/f(T=t|y) for a logistic selection in y
    return w / w.sum()
y_obs = np.random.default_rng(0).normal(10, 2, 1000)
w = tilt_weights(y_obs, 0.2)
print("observed mean", y_obs.mean(), " implied missing mean", (w*y_obs).sum())

Connections

• The selection function is the Bayesian analogue of the “confounding function” in Robins and Blackwell sensitivity analyses; in marketing, $f(T|Y(0))$ asks “are high-baseline-demand markets more likely to receive the media push?” (hidden targeting).
• Used with logistic selection in Franks 2019 - Logistic Selection with Exponential Family Mixtures.

See Also

Franks 2019 - Flexible Sensitivity Analysis Without Observable Implications - Overview, Copula Estimation, Potential Outcomes Framework