E-Value Sensitivity Analysis

Computes the E-value for a point estimate and optionally for a confidence interval bound, following VanderWeele & Ding (2017). The E-value is the minimum strength of association (on the risk-ratio scale) that an unmeasured confounder would need with both the treatment and the outcome, conditional on measured covariates, to fully explain away the observed association.

Usage

sa_evalue(
  estimate,
  ci_lo = NULL,
  ci_hi = NULL,
  type = c("RR", "OR", "HR", "IRR", "RD"),
  p0 = NULL,
  rare = NULL
)

Arguments

estimate: Numeric. The point estimate (RR, OR, HR, or RD on the original scale). Must be > 0 for RR/OR/HR; any finite value for RD.
ci_lo: Optional numeric. Lower confidence interval bound.
ci_hi: Optional numeric. Upper confidence interval bound. Both ci_lo and ci_hi must be supplied together to compute the CI E-value, and must satisfy ci_lo <= estimate <= ci_hi.
type: One of "RR" (default), "OR", "HR", "IRR", or "RD". "IRR" (incidence rate ratio) uses the same formula as "RR"; it is provided as a distinct value for clarity in count-outcome studies.
p0: Baseline outcome probability in the unexposed group. Required when type = "RD", and used only then. Must be in (0, 1).
rare: TRUE or FALSE; required when type = "OR" or "HR". Whether the outcome is rare in the study population (prevalence under about 15% at the end of follow-up). TRUE uses the OR or HR as the RR; FALSE applies the common-outcome conversion. The default NULL aborts for OR and HR. Ignored for other types.

Value

An object of class c("sa_evalue", "ps_data"). There is no input data, so $data is an empty data frame and $models an empty list.

$meta$evalue_estimate: E-value for the point estimate.
$meta$evalue_ci: E-value for the confidence interval bound closest to the null. NA if ci_lo/ci_hi are not supplied. 1 if the CI already crosses the null.
$meta$rr_estimate: The point estimate converted to the risk-ratio scale (same as estimate when type = "RR").
$meta$type: The measure type supplied.
$meta$rare: For type = "OR" or "HR", the rare-outcome choice used for the conversion; NA for other types.
$meta$estimate, $meta$ci_lo, $meta$ci_hi, $meta$p0: The inputs, NA where not supplied.
$tables$evalue: Data frame with one row for the estimate and one for the CI bound closest to the null: quantity, value on the original scale, rr on the risk-ratio scale (NA when the CI crosses the null) and evalue.

Details

Odds ratios and hazard ratios are converted to the risk-ratio scale before the E-value formula is applied, and the conversion depends on whether the outcome is rare (VanderWeele & Ding 2017; VanderWeele 2017):

Rare outcome (prevalence at most about 15%): the OR or HR approximates the RR and is used directly.
Common outcome: RR = sqrt(OR) for an odds ratio, and RR = (1 - 0.5^sqrt(HR)) / (1 - 0.5^sqrt(1 / HR)) for a hazard ratio.

There is no default for this choice: for type = "OR" or "HR" the caller must set rare. Rarity refers to the outcome's prevalence in the study population (under about 15% at the end of follow-up), and it cannot be inferred from the baseline risk p0 alone: with OR = 4 and p0 = 0.10 the exposed risk is about 0.31, and reversing the exposure coding would flip any rule based on p0. The same conversion is applied to the confidence interval bounds.

For risk differences a baseline outcome probability p0 must be supplied, and the conversion RR = (RD + p0) / p0 treats p0 as fixed: its own sampling uncertainty is not carried into the CI E-value.

References

VanderWeele, T. J., & Ding, P. (2017). Sensitivity analysis in observational research: Introducing the E-value. Annals of Internal Medicine, 167(4), 268-274. doi:10.7326/M16-2607

VanderWeele, T. J. (2017). On a square-root transformation of the odds ratio for a common outcome. Epidemiology, 28(6), e58-e60. doi:10.1097/EDE.0000000000000733

Examples

# Risk ratio of 2.5 -- how robust is the result?
sa_evalue(2.5, ci_lo = 1.8, ci_hi = 3.5)
#> <sa_evalue>
#>   Estimate    : RR = 2.5 (CI 1.8 to 3.5)
#>   RR scale    : 2.5
#>   E-value     : 4.43649 (estimate), 3 (CI)

# Odds ratio of 0.6 (protective effect) for a common outcome
sa_evalue(0.6, ci_lo = 0.4, ci_hi = 0.9, type = "OR", rare = FALSE)
#> <sa_evalue>
#>   Estimate    : OR = 0.6 (CI 0.4 to 0.9)
#>   Rare outcome: FALSE
#>   RR scale    : 0.774597
#>   E-value     : 1.90392 (estimate), 1.292878 (CI)

# Hazard ratio of 1.8 for an outcome under 15% at end of follow-up
sa_evalue(1.8, ci_lo = 1.2, ci_hi = 2.7, type = "HR", rare = TRUE)
#> <sa_evalue>
#>   Estimate    : HR = 1.8 (CI 1.2 to 2.7)
#>   Rare outcome: TRUE
#>   RR scale    : 1.8
#>   E-value     : 3 (estimate), 1.689898 (CI)

# Risk difference of 0.10 with baseline probability 0.20
ev <- sa_evalue(0.10, ci_lo = 0.02, ci_hi = 0.18, type = "RD", p0 = 0.20)
ev$meta$evalue_estimate
#> [1] 2.366025
ev$tables$evalue
#>   quantity value  rr   evalue
#> 1 estimate  0.10 1.5 2.366025
#> 2       ci  0.02 1.1 1.431662