New version of episensr available on CRAN! In the same vein than the other new function based on the work from Schneeweiss, this new version allows the computation of the E-value as proposed by VanderWeele and Ping, 2017.
The E-value is the minimum strength of association, on the risk ratio scale, that an unmeasured confounder would need to have with both the treatment and the outcome to fully explain away a specific exposure-outcome association, conditional on the measured covariates. Let’s go through the example provided in the paper by VanderWeele and Ping.
The example is based on the study by Victora et al., 1987, looking at the association between breastfeeding and infant death by respiratory infection. They found that, after adjusting for various covariates, infants fed with formula were 3.9 [1.8, 8.7] times more likely to die of respiratory infections than those who were breastfed. However, there was no control for smoking status, which could be associated with less breastfeeding and greater respiratory death.
The E-value expresses the magnitude of the confounder associations that could produce a confounding bias equal to the observed exposure-outcome association. With this example, we have:
## Registered S3 methods overwritten by 'ggplot2': ## method from ## [.quosures rlang ## c.quosures rlang ## print.quosures rlang
confounders.evalue(est = 3.9, type = "RR")
## ## --E-value-- ## Point estimate CI closest to H_0 ## RR: 3.900000 ## E-value: 7.263034
The risk ratio that was observed, 3.9, could be explained away by an unmeasured confounder associated with both the exposure and the outcome by a risk ratio of 7.26-fold each, above and beyond the measured confounders, but weaker confounding could not do so. I.e. you need a really strong confounder to explain away the association found between infant diet and mortality, of a RR = 3.9.
You can also report the limit of the confidence interval closest to the null.
confounders.evalue(est = 3.9, lower_ci = 1.8, upper_ci = 8.7, type = "RR")
## ## --E-value-- ## Point estimate CI closest to H_0 ## RR: 3.900000 1.800000 ## E-value: 7.263034 3.000000
So an unmeasured confounder associated with respiratory death and breastfeeding by a risk ratio of 3-fold each could explain away the lower confidence limit, but not a weaker confounder.
You can also compute E-values for non-null hypotheses. How large both unmeasured confounding associations has to be to shift the estimate from RR = 3.9 to a RR of 2?
confounders.evalue(est = 3.9, lower_ci = 1.8, upper_ci = 8.7, type = "RR", true_est = 2)
## ## --E-value-- ## Point estimate CI closest to H_0 ## RR: 3.900000 1.800000 ## E-value: 3.311066 1.462475
For an unmeasured confounder to shift the observed RR of 3.9 to a RR of 2, an unmeasured confounder associated with both the breastfeeding and respiratory death of RR = 3.3-fold each is required, but a weaker confounder could not.
Besides risk ratio (and odds ratio or hazard ratio for rare outcome, i.e. < 15%), E-values can also be computed for odds ratio and hazard ratio for common outcome, and differences in continuous outcomes.