14.1 The causal question revisited

• generalized methods for treatment contrasts that vary over time
• models whose parameters are estimated via g-estimation are structural nested models
• each g-method has a different set of modeling assumptions

14.1 The causal question revisited

• Total effect and no censoring: \(E[Y^{a=1, c=0}]\) - \(E[Y^{a=0, c=0}]\)

• Effect in the stratum: \(E[Y^{a=1, c=0}|sex=1]\) - \(E[Y^{a=0, c=0}|sex=1]\)

  • IPT-weighted MSM with interaction term
  • Standardisation in the stratum of interest
  • g-estimation

14.1 The causal question revisited

• g-estimation to estimate the average causal effect of smoking cessation A on weight gain Y in each strata defined by the covariates L

• \(E[Y^{a=1, c=0}|L]\) - \(E[Y^{a=0, c=0}|L]\)

14.2 Exchangeability revisited

• Conditional exchangeability: the outcome distribution in the treated and the untreated would be the same if both groups had received the same treatment level: \(Y^a \perp\perp A|L\) for \(a=0\) and \(a=1\)
• Knowing the value of \(Y^{a=0}\) does not provide information (help distinguish) between exposure levels: \(Pr[A=1|Y^{a=0}, L]\) = \(Pr[A=1|L]\)
• \(logitPr[A=1|Y^{a=0}, L] = \alpha_0 + \alpha_1Y^{a=0} + \alpha_2L\) and so \(\alpha_1=0\)

14.3 Structural nested mean models

• Assuming no censoring: \(E[Y^{a=1}|L]\) - \(E[Y^{a=0}|L]\)
• Structural model for the conditional causal effect: \(E[Y^{a=1} - Y^{a=0}|L]\) = \(\beta_1*a\)
• Structural model for the conditional causal effect with EMM by L: \(E[Y^{a=1} - Y^{a=0}|L]\) = \(\beta_1*a + \beta_2*a*L\)
  • under conditional exchangeability: \(E[Y^{a=1} - Y^{a=0}|L, A=a]\) = \(\beta_1*a + \beta_2*a*L\) (structural nested mean model)
• \(\beta_1\) and \(\beta_2\) are estimated using g-estimation
• Nested model means it is "nested" when the treatment is time-varying

14.3 Structural nested mean models

• Structural nested models are semiparametric
• Agnostic about the intercept and the effect of L t(no parameter \(\beta_0\) and no parameter \(\beta_3\))
• Fewer assumptions and potentially more robust to model misspecification than g-computation of the parametric g-formula

14.3 Structural nested mean models

• Assuming censoring: \(E[Y^{a=1, c=0}|L]\) - \(E[Y^{a=0, c=0}|L]\)
• G-estimation can be used to adjust for confounding but not selection bias
• Need IP weighting for selection bias, first
• Nonstabilized IP weights for construction of pseudo-population:
  • $W^C = 1/pr[C=0|L, A
• In pseudo-population without censoring:
  • \(E[Y^{a=1, c=0}|L, A]\) - \(E[Y^{a=0, c=0}|L, A]\)
  • = \(E[Y^{a=1, c=0} - Y^{a=0, c=0}|L, A]\)
  • = \(\beta_1*a + \beta_2*a*L\)

14.4 Rank preservation

• Rank each subject according to their observed Y
• Rank according to subjects' counterfactual outcomes \(Y^a\)
• Had both lists \(Y^{a=0}\) and \(Y^{a=1}\) were in the same, there was rank preservation
• Treatment has no effect on no one's outcome (sharp null hypothesis), the rank preservation holds
• The conditional rank preservation holds when effect of A on Y is the same
• Rank preserving structural model: \(Y^{a=1}_i - Y^{a=1}_i = \psi_1a + \psi_2a * a*L_i\) with \(\psi_1 + \psi_2*l\) is the causal effect for all individuals \(i\) with \(L=l\)

• However, the outcomes nearly never are expected to be constant on an individual level and so (additive) rank preservation is implausible.
• Average causal effects are agnostic about the the individual causal effects.
• Never use rank preservation model in practice, however, it is used in the section 14.5

14.5 G-estimation

• Estimate parameters of the structural nested mean model: \(E[Y^{a=1} - Y^{a=0}|L, A=a]\) = \(\beta_1*a\)
• No \(\beta_2*a*L\) term assumes constant treatment effect across strata of \(L\) (no additive effect measure modification (EMM) by \(L\))
• Additive rank preservation model assumes that effect of \(A\) is constant and the same for all individuals and average causal effect \(\beta_1\) is the same as \(\psi_1\) individual causal effect (although implausible in real life)
• Rank preserving model \(Y^{a=1}_i - Y^{a=0}_i = \psi_1a\) or \(Y^{a=0} = Y^{a=1}_i - \psi_1a\)

14.5 G-estimation

• 1st step: linking the model and the observed data
• By consistency: \(Y^{a=1} = Y\) among \(A=1\) and \(Y^{a=0} = Y\) among \(A=0\)
• Under rank preservation and correctly specified model: \(Y^{a=0} = Y^{a=1}_i - \psi_1a\) is \(Y^{a=0} = Y - \psi_1A\)
• \(\psi_1\) is unknown and the aim of the analysis is to estimate \(\psi_1\)
• To compute \(\psi_1\) make guesses: 1) -20 2) 0 3) 10
  • Then compute: \(H(\psi^*) = Y - \psi^*A\)
• If conditional exchangeability holds: \(\alpha_1 = 0\) in the logistic regression model for treatment:
  • \(logitPr[A=1|H(\psi^*), L] = \alpha_0 +\alpha_1H\psi^* + \alpha_2*L\)
  • The guess that gets us closer to \(\alpha_1 = 0\) is the most correct guess
• 95% CI for \(\psi^*\) is computed based on a set of guesses that are tested against the H0 and results with P-values > 0.05 are the limits that form 95% CI

14.6 Structural nested models with two or more parameters

• In the presence of the EMM, the model in the previous slides was misspecified and the answer given was incorrect
• The EMM by \(V\) requires estimation of the term \(\beta *a*V\): \(E[Y^{a=1} - Y^{a=0}|L, A=a]\) = \(\beta_1*a + \beta *a*V\) corresponding to the rank preserving model \(Y^{a=1}_i - Y^{a=0}_i = \psi_1a + \psi_2*a*V_i\)
• The logistic model for treatment them will be: \(logitPr[A=1|H(\psi^*, L) = \alpha_0 + \alpha_1 H(\psi^*) + \alpha_2H(\psi^*)V + \alpha_3 L]\), where \(H(\psi^*)=(\psi^*_1, \psi^*_2)\)
• The values of \(\psi^*_1\) and \(\psi^*_2\) produce \(H0(\psi^*) \perp\perp A |L\), or we need to find guesses that result in \(\alpha_1\) and \(\alpha_2\) both being zero.

References

1. Hernán MA, Robins JM (2020). Causal Inference: What If. Boca Raton: Chapman & Hall/CRC (v. 30mar21)
2. https://remlapmot.github.io/cibookex-r/g-estimation-of-structural-nested-models.html