diff --git a/sessions/causal-mediation-analysis-sensitivity-analysis.qmd b/sessions/causal-mediation-analysis-sensitivity-analysis.qmd index 93c8235..53b5971 100644 --- a/sessions/causal-mediation-analysis-sensitivity-analysis.qmd +++ b/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 @@ -429,17 +478,19 @@ 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* @@ -447,6 +498,15 @@ 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}