Update with differetn covariable values

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

@@ -6,30 +6,34 @@
 ```
 Unmeasured or uncontrolled confounding is a common problem in
 observational studies. This is a challenge to observational research
-even in the analysis of total effects
+even in the analysis of total effects.
 
 When we are interested in pathways and direct and indirect effects, the
 assumptions about confounding that are needed to identify these effects
 are even stronger than for total effects.
 
 We might be worried that these assumptions are violated and that our
-estimates are biased
+estimates are biased.
 
 **Sensitivity analysis techniques can help assess HOW ROBUST results are
-to violations in the assumptions being made** These techniques assess
-the extent to which an unmeasured variable (or variables) would have to
-affect both the exposure and the outcome in order for the observed
-associations between the two to be attributable solely to confounding
-rather than a causal effect of the exposure on the outcome It can also
-be useful in assessing a plausible range of values for the causal effect
-of the exposure on the outcome corresponding to a plausible range of
-assumptions concerning the relationship between the unmeasured
-confounder and the exposure and outcome
-
-##Sensitivity analysis for unmeasured confounding for total effects
-Consider the following figure in which U represents an unmeasured
+to violations in the assumptions being made.**
+
+These techniques assess the extent to which an unmeasured variable (or
+variables) would have to affect both the exposure and the outcome in
+order for the observed associations between the two to be attributable
+solely to confounding rather than a causal effect of the exposure on the
+outcome.
+
+It can also be useful in assessing a plausible range of values for the
+causal effect of the exposure on the outcome corresponding to a
+plausible range of assumptions concerning the relationship between the
+unmeasured confounder and the exposure and outcome.
+
+## Sensitivity analysis for unmeasured confounding for total effects
+
+Consider the following figure in which *U* represents an unmeasured
 confounder, *C* measured covariables, *A* the exposure and *Y* the
-outcome
+outcome.
 
 ```{r, echo = FALSE}
 # Creating The causal diagram for a mediation model
@@ -73,66 +77,108 @@ measured covariables *C* versus the true causal effect.
 ### Continuous outcomes
 
 Suppose then we have obtained an estimate of the effect of the exposure
-A on the outcome Y conditional on measured covariables C using
+*A* on the outcome *Y* conditional on measured covariables *C* using
 regression analysis.
 
-We will define the bias factor B_add(*c*) on the additive scale as the
-difference between the expected differences in outcomes comparing A = a
-and A = a\* conditional on covariables *C* = *c* and what we would have
-obtained had we been able to adjust for *U* as well.
+We will define the bias factor **B_add(*c*)** on the additive scale as
+the difference between the expected differences in outcomes comparing
+*A* = *a* and *A* = *a*^\*^ conditional on covariables *C* = *c* and
+what we would have obtained had we been able to adjust for *U* as well.
 
-If the exposure is binary, then we simply have *a* = 1 and *a* \* = 0.
+If the exposure is binary, then we simply have *a* = 1 and *a*^\*^ = 0.
 
 A simple approach to sensitivity analysis is possible if we assume that
-(A3.1) *U* is binary and (A3.2) that the effect of *U* (on the additive
-scale) is the same for those with exposure level *A* = *a* and exposure
-level *A* = *a* \* (no *U* × *A* interaction).
+**(A8.1.1)** *U* is binary and **(A8.1.2)** that the effect of *U* (on
+the additive scale) is the same for those with exposure level *A* = *a*
+and exposure level *A* = *a*^\*^ (no *U* × *A* interaction).
 
 If these assumptions hold, let γ be the effect of *U* on *Y* conditional
 on *A* and *C*, that is:
 
-$γ = E(Y|a, c,U = 1)$ − $E(Y|a, c,U = 0)$
+$γ = E(Y|a,c,U = 1)$ − $E(Y|a,c,U = 0)$
+
+Note that by assumption **(A8.1.2)**,
 
-Note that by assumption (A3.2), $γ = E(Y|a, c,U = 1)$ −
-$E(Y|a, c,U = 0)$ is the same for both levels of the exposure of
-interest. Note also that *γ* is the effect of *U* on *Y* already having
-adjusted for *C*; that is, in some sense the effect of *U* on *Y* not
-through *C*
+$γ = E(Y|a,c,U = 1)$ − $E(Y|a,c,U = 0)$
+
+is the same for both levels of the exposure of interest.
+
+Note also that *γ* is the effect of *U* on *Y* already having adjusted
+for *C*; that is, in some sense the effect of *U* on *Y* not through *C*
 
 Now let *δ* denote the difference in the prevalence of the unmeasured
-confounder *U* for those with *A*=*a* versus those with *A* = *a* \*,
+confounder *U* for those with *A*=*a* versus those with *A* = *a*^\*^,
 that is:
 
-$δ = P(U = 1|a, c)$ − $P(U = 1|a*, c)$
+$δ = P(U = 1|a,c)$ − $P(U = 1|a^*,c)$
 
-Under assumptions (A3.1) and (A3.2), the bias factor is simply given by
-the product of these two sensitivity analysis parameters:
+Under assumptions **(A8.1.1)** and **(A8.2.2)**, the bias factor is
+simply given by the product of these two sensitivity analysis
+parameters:
 
-$B_add(c) = γ δ$
+$B_add(c) = γδ$
 
 Thus to calculate the bias factor we only need to specify the effect of
-U on Y and the prevalence difference of U between the two exposure
+*U* on *Y* and the prevalence difference of *U* between the two exposure
 groups and then take the product of these two parameters.
 
-Once we have calculated the bias term B_add(c), we can simply estimate
-our causal effect conditional on C and then subtract the bias factor to
-get the "corrected estimate"--- that is, what we would have obtained if
-we had controlled for C and U.
+Once we have calculated the bias term **B_add(c)**, we can simply
+estimate our causal effect conditional on *C* and then subtract the bias
+factor to get the "corrected estimate"--- that is, what we would have
+obtained if we had controlled for *C* and *U*.
 
-Under these simplifying assumptions (A3.1) and (A3.2), we can also get
-adjusted confidence intervals by simply subtractingγ δ from both limits
-of the estimated confidence intervals
+Under these simplifying assumptions **(A8.1.1)** and **(A8.1.2)**, we
+can also get adjusted confidence intervals by simply subtracting *γδ*
+from both limits of the estimated confidence intervals.
 
-We may not believe any particular specification of the parameters γ and
-δ, but we could vary these parameters (based on expert knowledge or
-previous studies reporting estimates of the associations of the C and Y)
+We may not believe any particular specification of the parameters *γ*
+and *δ*, but we could vary these parameters (based on expert knowledge
+or previous reported estimates of the associations of the *C* and *Y*)
 over a range of plausible values to obtain what were thought to be a
 plausible range of corrected estimates.
 
 Using this technique, we could also examine how substantial the
 confounding would have to be to explain away an effect (we could do this
 for the estimate and confidence interval).
 
+### Continuous Outcome with Different Sensitivity Analysis Parameters for Different Covariate Values
+
+Suppose now that instead of focusing on effects conditional on a
+particular covariate value *C* = *c* or specifying the sensitivity
+analysis parameters *γ* and *δ* to be the same for each covariable *C*,
+we were interested in the overall marginal effect averaged over the
+covariables and we wanted to specify different sensitivity analysis
+parameters for different covariable levels.
+
+Suppose then for each level of the covariates of interest *C* = *c* we
+specified a value for the effect of *U* on *Y*
+
+$γ(c) = E(Y|a, c,U = 1) − E(Y|a, c,U = 0)$
+
+and also a value for the prevalence difference of *U* between those with
+exposure status *A* = *a* and *A* = *a*^\*^ and covariables *C* = *c*
+
+$δ(c) = P(U = 1|a, c)−P(U = 1|a^*, c)$
+
+We could then obtain an overall bias factor, **Badd**, by taking the
+product of the bias factors in each strata of *C* and then averaging
+these over *C*, weighting each strata of *C* according to what
+proportion of the sample was in that strata. The overall bias factor is
+then
+
+$Badd=\sum~c~\{γ(c)δ(c)\}P(C=c)$
+
+We could then subtract this overall bias factor from our estimate
+adjusted only for *C* to obtain a corrected estimate.
+
+In this case, however, we can now longer simply subtract the bias factor
+from both limits of the confidence intervals because this does not take
+into account the variability in our estimates of the proportion of the
+sample in each strata of the covariates *P(C = c*).
+
+Corrected confidence intervals could instead be obtained by
+bootstrapping.
+
 ```{=html}
 <!-- - binary outcome  
 
@@ -148,4 +194,3 @@ Coding part - Omar
 #check the mediation package and the CMAverse package for sensitivity analysis
 -->
 ```
-