Addedd binary outcomes and NDE and NIE

sessions/causal-mediation-analysis-sensitivity-analysis.qmd

@@ -202,10 +202,10 @@ covariables *C* obtained from a logistic regression model.
 The bias factor is defined as **B~*mult*~(*c*)** on the multiplicative
 scale as the ratio of:
 
-1- the risk ratio (or odds ratio, with a **rare outcome**) comparing *A*
+1- The risk ratio (or odds ratio, with a **rare outcome**) comparing *A*
 = *a* and *A* = *a*^\*^ conditional on covariables *C*= *c* and
 
-2- what we would have obtained as the risk ratio (or odds ratio) had we
+2- What we would have obtained as the risk ratio (or odds ratio) had we
 been able to condition on both C and U.
 
 We now make the simplifying assumptions that **(A8.1.3)** *U* is binary
@@ -216,11 +216,11 @@ the same for those with exposure level *A* = *a* and exposure level *A*
 If these assumptions hold, we will let *γ* be the effect of *U* on *Y*
 conditional on *A* and *C* on the risk ratio scale, that is:
 
-$\frac{γ = P(Y = 1\|a, c,U = 1)}{P(Y = 1\|a, c,U = 0)}$
+$γ = \frac{ P(Y = 1\|a, c,U = 1)}{P(Y = 1\|a, c,U = 0)}$
 
 By assumption **(A8.1.2b)**
 
-$\frac{γ = P(Y=1\|a,c,U=1)}{P(Y=1\|a,c,U=0)}$
+$γ = \frac{P(Y=1\|a,c,U=1)}{P(Y=1\|a,c,U=0)}$
 
 is the same for both levels of the exposure.
 
@@ -237,19 +237,19 @@ We can use the bias formula by specifying the effect of *U* on *Y* on
 the risk ratio scale and the prevalence of *U* among those with exposure
 levels *A* = *a* and *A* = *a*^\*^.
 
-Once we have calculated the bias term B\~*mult\~*(*c*), we can estimate
-our risk ratio controlling only for *C* (if the outcome is rare, fit a
-logistic regression) and we divide our estimate by B~mult~(*c*) to get
-the corrected estimate for risk ratio—that is, what we would have
-obtained if we had adjusted for *U* as well.
+Once we have calculated the bias term **B~*mult*~(*c*)**, we can
+estimate our risk ratio controlling only for *C* (if the outcome is
+rare, fit a logistic regression) and we divide our estimate by
+**B~*mult*~(*c*)** to get the corrected estimate for risk ratio—that is,
+what we would have obtained if we had adjusted for *U* as well.
 
 Under the simplifying assumptions of (A8.1.1) and (A8.1.2b), we can also
 obtain corrected confidence intervals by dividing both limits of the
-confidence interval by B\~*mult\~*(*c*).
+confidence interval by **B~*mult*~(*c*)**.
 
 Note that to use the bias factor in (8.2), we must specify the
-prevalence of the unmeasured confounder in both exposure groups *P*(*U*
-= 1\|*a*, *c*) and *P*(*U* = 1\|*a*^\*^, *c*), not just the difference
+prevalence of the unmeasured confounder in both exposure groups *P*(*U*=
+1\|*a*, *c*) and *P*(*U* = 1\|*a*^\*^, *c*), not just the difference
 between these two prevalences as in (8.1) for outcomes on the additive
 scale.
 
@@ -261,7 +261,9 @@ explain away an effect and also
 2- the sensitivity analysis parameters that would be required to shift
 the confidence interval to just include the null.
 
-## Sensitivity analysis for controled direct effects for a continuous outcome
+## Sensitivity analysis for controled direct effects
+
+### Continuous outcomes
 
 Assume that controlling for (*C,U*) would suffice to control for
 exposure--outcome and mediator--outcome confounding but that no data are
@@ -299,7 +301,7 @@ If we have not adjusted for *U*, then our estimates controlling only for
 We will consider estimating the controlled direct effect, *CDE*(*m*),
 with the mediator fixed to *m* conditional on the covariables *C* = *c*.
 
-Let \*B\^{CDE}\_{add}*(*m*\|*c\*) denote the difference between:
+Let $B^{CDE}_{add}(m|c)$ denote the difference between:
 
 1- the estimate of the *CDE* conditional on *C*
 
@@ -332,11 +334,11 @@ Under assumptions (A8.1.1) and (A8.2.2b), the bias factor is simply
 given by the product of these two sensitivity-analysis parameters
 (VanderWeele, 2010a):
 
-\*B\^{CDE}\_{add}*(*m*\|*c*) =* δmγm\*
+$B^{CDE}_{add}(m|c) = δmγm$ **8.3**
 
 This formula states that under assumptions (A8.1.1) and (A8.2.2b) the
-bias factor B\^{CDE}\_{add}(*m*\|*c*) for the *CDE*(*m*) is simply given
-by the product *δmγm*.
+bias factor $B^{CDE}_{add}(m|c)$ for the *CDE*(*m*) is simply given by
+the product *δmγm*.
 
 Under these simplifying assumptions, this gives rise to a particularly
 simple sensitivity analysis technique for assessing the sensitivity of
@@ -359,10 +361,9 @@ on *Y*) and by varying *δm*, interpreted as the prevalence difference of
 *U*, comparing exposure levels *a* and *a*^\*^ conditional on *M* = *m*
 and *C* = *c*.
 
-We can subtract the bias factor \*B\^{CDE}\_{add}*(*m*\|*c*) =* δmγm\*
-from the observed estimate to obtain a corrected estimate of the effect
-(what we would have obtained had it been possible to adjust for *U* as
-well).
+We can subtract the bias factor $B^{CDE}_{add}(m|c)$ from the observed
+estimate to obtain a corrected estimate of the effect (what we would
+have obtained had it been possible to adjust for *U* as well).
 
 Under the simplifying assumptions (A8.1.1) and (A8.2.2b), we could also
 subtract this bias factor from both limits of a confidence interval to
@@ -376,6 +377,54 @@ If there is no interaction between the effects of *A* and *M* on *Y*,
 then this simple sensitivity analysis technique based on using formula
 above will also be applicable to natural direct effects as well.
 
+### Binary outcomes
+
+We will consider estimating the controlled direct effect odds ratio from
+Chapter 2, $OR^{CDE}(m)$, with the mediator fixed at level *m*,
+conditional on the covariates *C* = *c*.
+
+This approach will assume a rare outcome but can also be used for risk
+ratios with a common outcome. Let $B^{CDE}_{mult}(m|c)$ denote the ratio
+of
+
+1- The estimate of the controlled direct effect conditional on *C* (
+
+2- What would have been obtained had adjustment been made for *U* as
+well.
+
+Suppose that (A8.1.1) *U* is binary and that (A8.1.2d) the effect of *U*
+on *Y* on the ratio scale, conditional on exposure, mediator, and
+covariables (*A,M,C*), is the same for both exposure levels *A = a* and
+*a*^\*^.
+
+Let *γm* be the effect of *U* on *Y* conditional on *A, C*, and *M = m*,
+that is:
+
+$γm = \frac{P(Y = 1|a, c,m,U = 1)}{P(Y = 1|a, c,m,U = 0)}$
+
+Note that by (A8.1.2), *γm* is the same for both levels of the exposure
+of interest.
+
+Under assumptions (A8.1.1) and (A8.1.2d), the bias factor on the
+multiplicative scale is given by:
+
+$B^{CDE}_{mult}(m|c) = \frac{1+(γm −1)P(U= 1|a,m, c)}{1+(γm−1)P(U = 1\|a∗,m, c)}$
+**(8.4)**
+
+Once we have calculated the bias term $B^{CDE}_{mult}(m|c)$, we can
+estimate the *CDE* risk ratio controlling only for *C* (if the outcome
+is rare), we fit a logistic regression) and we divide our estimate and
+confidence intervals by the bias factor $B^{CDE}_{mult}(m|c)$ to get the
+corrected estimate for CDE risk ratio and its confidence interval—that
+is, what we would have obtained if we had adjusted for *U* a well.
+
+We have to specify the two prevalences of U, namely $P(U = 1|a,m, c)$
+and $P(U = 1|a∗,m, c)$, in the different exposure groups conditional on
+*M* and *C*.
+
+As with *CDE* on an additive scale, the issue of conditioning on *M* in
+the interpretation of these prevalences is important
+
 ## Sensitivity analysis for natural direct and indirect effects
 
 ### Sensitivity analysis for natural direct and indirect effects in the abscence of exposure-mediator interaction
@@ -429,24 +478,35 @@ Because a mediator--outcome confounder does not confound the
 exposure-outcome relationship, we can still obtain valid estimates of
 the total effect.
 
-And, it turns out that the combination of the DE and IE do constitute a
-consistent estimator of the total effect, even though the *DE* and *IE*
-estimators will themselves be biased for the true *NDE* and *NIE*.
+And, it turns out that the combination of the *DE* and *IE* do
+constitute a consistent estimator of the total effect, even though the
+*DE* and *IE* estimators will themselves be biased for the true *NDE*
+and *NIE*.
 
-Knowing that the DE and IE estimates combine to a valid estimate of the
-total effect then allows us to employ the sensitivity analysis
+Knowing that the *DE* and *IE* estimates combine to a valid estimate of
+the total effect then allows us to employ the sensitivity analysis
 techniques for *CDE* for *NIE* as well.
 
-To do so, we use the negation (on the additive scale) of the bias
-formulas that we used for *CDE* (and *NDE*). Thus on the additive scale,
-for a continuous outcome, our bias factor for the *NDE* would simply be:
+To do so, we use the negation (on the additive scale) or the inverse (on
+the multiplicative ratio scale) of the bias formulas used for *CDE* (and
+*NDE*). Thus on the additive scale, for a continuous outcome, our bias
+factor for the *NDE* would simply be:
 
 −*δmγm*
 
 and we could subtract this from the estimate and both limits of the
 confidence interval to obtain a corrected estimate and confidence
 interval for the *NIE*.
 
+For a binary outcome, on the odds ratio scale with rare outcome or risk
+ratio scale with common outcome, our bias factor for the *NIE* would be
+the inverse of that in **(8.4)**:
+
+$\frac{1+(γm−1)P(U=1|a∗,m,c)}{1+(γm−1)P(U=1|a,m,c)}$
+
+and we could divide our *NIE* estimates and its confidence interval by
+this bias factor to obtain a corrected estimate and confidence interval.
+
 We first load the nhanes data:
 
 ```{r}