James Robins (1999)
Marginal structural models target causal effects of treatment regimes when confounders both affect later treatment and are affected by earlier treatment. A simple model is
\[ E[Y^{\bar a}] = \beta_0+\beta_1 a_0+\beta_2 a_1. \]
Inverse probability weights create a pseudo-population in which treatment history is independent of measured confounder history:
\[ W_i=\prod_t \frac{1}{\Pr(A_{it}=a_{it}\mid \bar A_{i,t-1},\bar L_{it})}. \]
Then a weighted regression estimates the marginal causal model.
Simulate time-varying confounding: \(L_1\) is affected by \(A_0\) and also predicts \(A_1\) and \(Y\).
n = 1200
L0 = rng.normal(size=n)
p0 = special.expit(-0.1 + 0.9 * L0)
A0 = rng.binomial(1, p0)
L1 = 0.8 * L0 + 0.9 * A0 + rng.normal(size=n)
p1 = special.expit(-0.2 + 0.8 * L1 + 0.5 * A0)
A1 = rng.binomial(1, p1)
Y = 1.0 * A0 + 1.3 * A1 + 0.9 * L0 + 1.0 * L1 + rng.normal(size=n)
A0.mean(), A1.mean()(np.float64(0.4825), np.float64(0.5941666666666666))Estimate the treatment probabilities in the denominator of \(W_i\).
def fit_logit(X, d):
def obj(b):
xb = X @ b
return -np.sum(d * xb - np.logaddexp(0, xb))
return optimize.minimize(obj, np.zeros(X.shape[1])).x
ph0 = special.expit(np.c_[np.ones(n), L0] @ fit_logit(np.c_[np.ones(n), L0], A0))
ph1 = special.expit(np.c_[np.ones(n), L1, A0] @ fit_logit(np.c_[np.ones(n), L1, A0], A1))
w = (A0/ph0 + (1-A0)/(1-ph0)) * (A1/ph1 + (1-A1)/(1-ph1))
w.mean(), np.quantile(w, [0.05, 0.95])(np.float64(4.076379119622979), array([ 1.40580129, 10.03347031]))Fit the marginal structural model \(E[Y^{\bar a}]=\beta_0+\beta_1 a_0+\beta_2 a_1\) by weighted least squares.
ordinary = linalg.lstsq(np.c_[np.ones(n), A0, A1, L0, L1], Y)[0][1:3]
X_msm = np.c_[np.ones(n), A0, A1]
sw = np.sqrt(w / w.mean())
msm = linalg.lstsq(X_msm * sw[:, None], Y * sw)[0][1:3]
ordinary, msm(array([0.95949876, 1.21576968]), array([1.37548673, 1.03425169]))Compare ordinary adjusted coefficients with the IPW marginal structural estimates.
fig, ax = plt.subplots(figsize=(6, 3.4))
loc = np.arange(2)
ax.bar(loc - 0.18, ordinary, width=0.35, label="ordinary adjusted")
ax.bar(loc + 0.18, msm, width=0.35, label="IPW MSM")
ax.scatter(loc, [1.0, 1.3], color="#66ff99", marker="x", s=70, label="truth")
ax.set(xticks=loc, xticklabels=["A0", "A1"], ylabel="coefficient")
ax.legend()
plt.show()