Cinelli Hazlett 2020 - Multiple and Nonlinear Confounders

Summary

A set of omitted confounders $Z$ (possibly non-linear, interacted, or standing in for functional-form misspecification of $X$) induces the same bias as the single composite $Z^{\ast }=Z\hat{\gamma }$. Since $Z^{\ast }$ has the same outcome partial $R^2$ as $Z$ but a no-larger treatment partial $R^2$, the single-confounder formulas are conservative upper bounds for the multivariate case.

Overview

Allow $Z$ to be an $n\times k$ matrix with coefficient vector $\hat{\gamma }$ in the full model

$$Y=\hat{\tau }D+X\hat{\beta }+Z\hat{\gamma }+{\hat{\epsilon }}_{full}.(23)$$

Main Content

Multiple confounders reduce to one composite (Eqs. 23-24)

Let $Z^{\ast }:=Z\hat{\gamma }$. Regressing on $(D,X,Z^{\ast })$ gives the same $\hat{\tau }$ as regressing on $(D,X,Z)$, and $R_{Y\sim Z^{\ast }|D,X}^2=R_{Y\sim Z|D,X}^2$. But $R_{D\sim Z^{\ast }|X}^2\le R_{D\sim Z|X}^2$. Therefore the bias from omitting $Z$ is at most the value of the single-confounder formula evaluated at $(R_{Y\sim Z|D,X}^2,R_{D\sim Z|X}^2)$.

Proof sketch. $\hat{\gamma }$ is chosen to maximise the fit with the outcome, not with the treatment. The composite $Z^{\ast }$ is a linear combination of the columns of $Z$, so the treatment can be explained at most as well by $Z^{\ast }$ as by $Z$ (a regression on more regressors can only increase $R^2$). Hence $R_{D\sim Z^{\ast }|X}^2\le R_{D\sim Z|X}^2$, and since bias is increasing in $R_{D\sim Z|X}^2$ (Eq. 13), the bound follows. (An alternative proof is in Hosman et al. 2010, Sec. 4.1.)

Degrees of freedom. For multivariate $Z$ use $df-k$ in the SE formula (Eq. 12) instead of $df-1$; but the exercise is hypothetical (“one can always imagine to have measured $Z^{\ast }$”), so the correction is a formality.

Non-linear confounders and functional-form misspecification

The columns of $Z$ are arbitrary, so the result covers non-linearity. Let the correct model be $Y=\hat{\tau }D+\hat{\beta }X+{\hat{\gamma }}_{1}Z+{\hat{\gamma }}_2{Z}^2+{\hat{\gamma }}_3(Z\times X)+{\hat{\gamma }}_4{X}^2+\hat{\epsilon }$. If the analyst did not measure $Z$ nor think $X$ enters quadratically, set $Z=(Z_1=Z,Z_2=Z^2,Z_3=Z\times X,Z_4=X^2)$ and all previous arguments follow. Thus the same plot can be read as a sensitivity bound to all omitted variables acting together, including misspecification of the observed covariates’ functional form.

What “conservative” means

Sensitivity statements are exact for a single linear confounder and conservative for multiple or non-linear ones: any confounder collection with the same pair of partial $R^2$ values (outcome side, treatment side of the collection) produces at most that much bias.

Examples

Wealth and Political Attitudes together

In Darfur, a skeptic worries Wealth and Political Attitudes (possibly interacted) confound the harm-peace relation. Treat $Z=(\text{Wealth},\text{PolAtt},\text{Wealth}\times \text{PolAtt})$. The analyst never needs to specify their joint distribution: she asks whether the group could explain, say, $12\%$ of residual outcome variance and $1\%$ of residual treatment variance (the Female bound), and reads the answer off the contour plot.

# Composite-confounder check: Z* = Z @ gamma has the same outcome partial R2 but no larger treatment partial R2
import numpy as np
rng = np.random.default_rng(0); n = 20000
X = rng.normal(size=(n, 2)); Z = rng.normal(size=(n, 3))
D = X @ [.5, .2] + Z @ [.4, -.3, .2] + rng.normal(size=n)
Y = 0.2*D + X @ [.3, .1] + Z @ [.5, .5, -.4] + rng.normal(size=n)
def resid(v, M): A = np.column_stack([np.ones(n), M]); return v - A @ np.linalg.lstsq(A, v, rcond=None)[0]
gamma = np.linalg.lstsq(np.column_stack([np.ones(n), D, X, Z]), Y, rcond=None)[0][-3:]
zs = Z @ gamma
r2 = lambda a, b: np.corrcoef(a, b)[0,1]**2
Dx = resid(D, X)
print("R2_D~Z*|X  :", r2(Dx, resid(zs, X)))
Zres = np.column_stack([resid(Z[:,i], X) for i in range(3)])
print("R2_D~Z|X   :", 1 - resid(Dx, Zres).var()/Dx.var())   # >= the line above

Connections

• Marketing: an MMM or lift regression omits many drivers at once (seasonality, promotions, competitor actions). The multivariate result lets one summarise them as a single worst-case pair of partial $R^2$ values; see Cinelli Hazlett 2020 - Planted Confounders and Bayesian Sketch.
• Related: Missing Data Models (assumptions about unobserved parts cannot be tested).

See Also

Cinelli Hazlett 2020 - Bounding Confounder Strength with Observed Covariates, Cinelli Hazlett 2020 - Making Sense of Sensitivity - Overview