Dahabreh 2020 - Outcome Model, Inverse Odds Weighting and Doubly Robust Estimators

Summary

Three families of estimators of $\mu (a)=E\{E[Y\mid X,S=1,A=a]\mid S=0\}$: outcome-model (OM), inverse-odds weighting (IOW1 unnormalized, IOW2 normalized), and three doubly robust estimators (DR1-DR3). Each is a solution to a missing-data problem; DR is consistent if either the outcome model or the participation model is right.

Overview

Under consistency the unknown $Y^a$ for target individuals is a missing outcome, with missingness indicator $I(S=1,A=a)$. Parametric working models throughout: outcome $g_a(X;\theta )\approx E[Y\mid X,S=1,A=a]$; participation $p(X;\beta )\approx \mathrm{Pr}[S=1\mid X]$; trial treatment $e_a(X;\gamma )\approx \mathrm{Pr}[A=a\mid X,S=1]$. Treatment probabilities are known by design (true values can be used; estimating them can reduce standard errors).

Main Content

Outcome model-based (OM) estimator (eq. 1)

$${\hat{\mu }}_{OM}(a)=(\sum _{i=1}^n(1-S_i){)}^{-1}\sum _{i=1}^n(1-S_i)g_a(X_i;\hat{\theta }).$$

Fit the outcome regression in trial participants (separately by arm, i.e., all treatment-by-covariate interactions) and average predictions over target covariates. Consistent when the outcome model is correct.

Inverse-odds weights

$${\hat{w}}_a(X_i,S_i,A_i)=\frac{1-p(X_i;\hat{\beta })}{p(X_i;\hat{\beta })e_a(X_i;\hat{\gamma })}I(S_i=1,A_i=a).$$

The factor $(1-p)/p$ is the inverse of the odds of participation. Interpretation: treat the trial as a survey sample from the target with covariate-dependent, estimated sampling probabilities.

IOW estimators (eqs. 2-3)

Unnormalized: ${\hat{\mu }}_{IOW1}(a)=({\sum }_i(1-S_i){)}^{-1}{\sum }_i{\hat{w}}_a(X_i,S_i,A_i)Y_i$. Normalized (ratio/Hajek form): ${\hat{\mu }}_{IOW2}(a)=({\sum }_i{\hat{w}}_a{)}^{-1}{\sum }_i{\hat{w}}_a{Y}_i$. IOW2 equals the intercept (and intercept + treatment coefficient) of a saturated weighted least squares regression of $Y$ on $A$. It always lies in the support of $Y$ (e.g., within $[0,1]$ for binary $Y$), giving much better finite-sample behaviour with variable weights. Both are consistent if the participation model is right.

Why the IO weighting estimator targets $\mu (a)$

Since $E[I(S=1,A=a)Y\mid X]=p(X)e_a(X)g_a(X)$,

$$E[\frac{1-p}{p{e}_a}I(S=1,A=a)Y]=E[(1-p(X))g_a(X)]=E[(1-S)g_a(X)],$$

so dividing by $\mathrm{Pr}[S=0]$ gives $E[g_a(X)\mid S=0]=\mu (a)$. (Derived here from the definitions; it is the expectation behind eq. 2.)

Doubly robust estimators (eqs. 4-6)

$${\hat{\mu }}_{DR1}(a)=(\sum _i(1-S_i){)}^{-1}\sum _i\{{\hat{w}}_a(Y_i-g_a(X_i;\hat{\theta }))+(1-S_i)g_a(X_i;\hat{\theta })\}$$ $${\hat{\mu }}_{DR2}(a)=\left(\sum _i{\hat{w}}_a{)}^{-1}\sum _i{\hat{w}}_a\right(Y_i-g_a(X_i;\hat{\theta }))+(\sum _i(1-S_i){)}^{-1}\sum _i(1-S_i)g_a(X_i;\hat{\theta })$$ $${\hat{\mu }}_{DR3}(a)=(\sum _i(1-S_i){)}^{-1}\sum _i(1-S_i)g_a(X_i;{\hat{\theta }}_w)$$

where ${\hat{\theta }}_w$ comes from the outcome regression among trial participants weighted by ${\hat{w}}_a$. DR3 is doubly robust when the outcome is modelled with a linear exponential family quasi-likelihood and canonical link.

Double robustness

DR estimators are consistent for $\mu (a)$ and asymptotically normal when either the participation model or the outcome model is correctly specified (formal result in the Appendix, absent from the PDF; refs [36],[40]). In practice DR estimators are usually more precise than IOW and nearly as precise as OM. Caveat: with a misspecified outcome model and highly variable weights, DR can do worse than non-DR estimators using the same misspecified outcome model (refs [41],[34]).

Special cases (paper's argument, Sec. 6.2)

IOW is DR with a grossly misspecified outcome model $g_a\equiv 0$; OM is DR with grossly misspecified participation $p\equiv 1$ (weights $\equiv 0$). So correct IOW or OM performance in simulation already demonstrates the double robustness property.

Inference

All estimators are partial M-estimators, so sandwich variances are available (R package geex is cited); the nonparametric bootstrap is easier and “will often be preferred in practice”.

Examples

Runnable sketch - OM, IOW2 and DR2 on simulated trial + cohort data

import numpy as np
rng = np.random.default_rng(0)
n = 5000
X = rng.standard_normal((n, 3))
# trial selection depends on X1 (an effect modifier), X2, X3
logit = -2.5 + X @ np.array([1., 1., 1.])
p_true = 1 / (1 + np.exp(-logit)); S = rng.random(n) < p_true
A = rng.random(n) < 0.5
tau = 1 + 2 * X[:, 0]                 # effect modification by X1
Y = X @ np.array([1., 1., 1.]) + A * tau + rng.standard_normal(n)
 
def fit_logit(X, s):
    Z = np.c_[np.ones(len(X)), X]; b = np.zeros(Z.shape[1])
    for _ in range(50):
        p = 1 / (1 + np.exp(-Z @ b)); b += np.linalg.solve(Z.T @ (Z * (p * (1 - p))[:, None]), Z.T @ (s - p))
    return 1 / (1 + np.exp(-Z @ b))
 
p = fit_logit(X, S.astype(float))
def mu(a):
    sel = S & (A == a)
    Z = np.c_[np.ones(n), X]
    beta = np.linalg.lstsq(Z[sel], Y[sel], rcond=None)[0]
    g = Z @ beta
    w = np.where(sel, (1 - p) / (p * 0.5), 0.0)      # known e_a = 0.5
    om = g[~S].mean()
    iow2 = (w * Y).sum() / w.sum()
    dr2 = (w * (Y - g)).sum() / w.sum() + g[~S].mean()
    return om, iow2, dr2
est = np.array(mu(1)) - np.array(mu(0))
print("target ATE: OM, IOW2, DR2 =", est.round(3), "| truth", tau[~S].mean().round(3),
      "| trial-only", (Y[S & A].mean() - Y[S & ~A].mean()).round(3))

The trial-only contrast is biased upward (selection over-represents high $X_1$); the three adjusted estimates recover the target ATE (Monte Carlo summaries in Dahabreh 2020 - Simulation Study of Estimators).

Connections

Compare to Bayesian treatments of weighting in Bayesian Inverse Probability Weighting; the AIPW/DR structure is the same as in Doubly-Robust Estimands for ATT(g,t) (with $S$ playing the role of treatment).

See Also

• Dahabreh 2020 - Identifiability Conditions and Identification
• Dahabreh 2020 - Simulation Study of Estimators
• Dahabreh 2020 - Practical Issues, Diagnostics and Discussion