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: $E\left[Y^{a=1,c=0}\right]$ - $E\left[Y^{a=0,c=0}\right]$

• 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

• $E\left[Y^{a=1,c=0}|L\right]$ - $E\left[Y^{a=0,c=0}|L\right]$

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: $Pr[A=1|Y^{a=0},L]$ = $Pr[A=1|L]$
• $logitPr[A=1|Y^{a=0},L]={\alpha }_0+{\alpha }_1{Y}^{a=0}+{\alpha }_{2}L$ and so ${\alpha }_1=0$

14.3 Structural nested mean models

• Assuming no censoring: $E\left[Y^{a=1}|L\right]$ - $E\left[Y^{a=0}|L\right]$
• Structural model for the conditional causal effect: $E[Y^{a=1}-Y^{a=0}|L]$ = ${\beta }_1\ast a$
• Structural model for the conditional causal effect with EMM by L: $E[Y^{a=1}-Y^{a=0}|L]$ = ${\beta }_1\ast a+{\beta }_2\ast a\ast L$
  • under conditional exchangeability: $E[Y^{a=1}-Y^{a=0}|L,A=a]$ = ${\beta }_1\ast a+{\beta }_2\ast a\ast L$ (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: $E\left[Y^{a=1,c=0}|L\right]$ - $E\left[Y^{a=0,c=0}|L\right]$
• 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:
  • $E[Y^{a=1,c=0}|L,A]$ - $E[Y^{a=0,c=0}|L,A]$
  • = $E[Y^{a=1,c=0}-Y^{a=0,c=0}|L,A]$
  • = ${\beta }_1\ast a+{\beta }_2\ast a\ast L$

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: $Y_i^{a=1}-Y_i^{a=1}={\psi }_{1}a+{\psi }_{2}a\ast a\ast L_i$ with ${\psi }_1+{\psi }_2\ast l$ 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: $E[Y^{a=1}-Y^{a=0}|L,A=a]$ = ${\beta }_1\ast a$
• No ${\beta }_2\ast a\ast L$ 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 $Y_i^{a=1}-Y_i^{a=0}={\psi }_{1}a$ or $Y^{a=0}=Y_i^{a=1}-{\psi }_{1}a$

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: $Y^{a=0}=Y_i^{a=1}-{\psi }_{1}a$ is $Y^{a=0}=Y-{\psi }_{1}A$
• ${\psi }_1$ 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: $H\left({\psi }^{\ast }\right)=Y-{\psi }^{\ast }A$
• If conditional exchangeability holds: ${\alpha }_1=0$ in the logistic regression model for treatment:
  • $logitPr[A=1|H({\psi }^{\ast }),L]={\alpha }_0+{\alpha }_{1}H{\psi }^{\ast }+{\alpha }_2\ast L$
  • The guess that gets us closer to ${\alpha }_1=0$ is the most correct guess
• 95% CI for ${\psi }^{\ast }$ 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 $\beta \ast a\ast V$: $E[Y^{a=1}-Y^{a=0}|L,A=a]$ = ${\beta }_1\ast a+\beta \ast a\ast V$ corresponding to the rank preserving model $Y_i^{a=1}-Y_i^{a=0}={\psi }_{1}a+{\psi }_2\ast a\ast V_i$
• The logistic model for treatment them will be: $logitPr[A=1|H({\psi }^{\ast },L)={\alpha }_0+{\alpha }_{1}H({\psi }^{\ast })+{\alpha }_{2}H({\psi }^{\ast })V+{\alpha }_{3}L]$, where $H\left({\psi }^{\ast }\right)=({\psi }_1^{\ast },{\psi }_2^{\ast })$
• The values of ${\psi }_1^{\ast }$ and ${\psi }_2^{\ast }$ produce $H0\left({\psi }^{\ast }\right)\perp \perp A|L$, 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