Relative risk from an odds ratio: the interval that arrest-only data allow

Case-control and arrest-only samples identify the odds ratio of an outcome between two groups but not the two risks themselves. The relative risk nevertheless lies between 1 and the odds ratio (Research.P2.rr_between), since \(OR = RR (1-b)/(1-a)\) (or_eq_rr_mul), and the rare-outcome substitution \(RR \approx OR\) always overstates the relative risk in log magnitude (or_overstates). With a base rate supplied the two risks and the relative risk are point identified.

Usage

morie_relative_risk_from_or(odds_ratio, base_rate = NULL, exposed_share = NULL)

Arguments

odds_ratio: Identified odds ratio, positive.
base_rate: Optional overall outcome rate \(P(y=1)\); with exposed_share it point-identifies the two risks.
exposed_share: Share of the population with \(x = 1\).

Value

A list with rr_bounds (between 1 and the odds ratio), overstatement_factor (\(OR/RR\) when identified), risks (when identified) and theorems.

Examples

morie_relative_risk_from_or(3)
#> $rr_bounds
#> lower upper 
#>     1     3 
#> 
#> $theorems
#> [1] "Research.P2.or_eq_rr_mul"  "Research.P2.rr_between"   
#> [3] "Research.P2.or_overstates"
#> 
morie_relative_risk_from_or(3, base_rate = 0.2, exposed_share = 0.3)
#> $rr_bounds
#> lower upper 
#>     1     3 
#> 
#> $theorems
#> [1] "Research.P2.or_eq_rr_mul"  "Research.P2.rr_between"   
#> [3] "Research.P2.or_overstates"
#> 
#> $risks
#>       exposed     unexposed relative_risk 
#>     0.3333333     0.1428571     2.3333333 
#> 
#> $overstatement_factor
#> [1] 1.285714
#>