Risk difference when group membership is misrecorded

When the group a person is recorded in (an officer's perception of race, say) differs at random from the true one, with the same error rates whatever the outcome, the measured groups are mixtures of the true ones. The measured risk difference is then the true one times the factor (se + sp - 1) p (1 - p) / (d1 d0), where d1 and d0 are the shares recorded in each group. For any sensitivity and specificity it is never larger in size than the true difference, so a disparity in recorded data understates the true one; with se + sp >= 1 it never has the opposite sign, while with se + sp < 1 a difference other than 0 has it. See proof("misrd").

Usage

misrd(r1, r0, prev, se, sp)

Arguments

r1

The true risk in group 1: one number from 0 to 1.

r0

The true risk in group 0: one number from 0 to 1.

prev

The true share in group 1, strictly between 0 and 1.

se

Sensitivity: the chance a member of group 1 is recorded in group 1, from 0 to 1.

sp

Specificity: the chance a member of group 0 is recorded in group 0, from 0 to 1.

Value

A named numeric vector: true (r1 - r0), measured (the difference between the recorded groups), factor (measured / true whenever true is not 0), d1 and d0.

References

Bross, I. (1954). Misclassification in 2 x 2 tables. Biometrics, 10(4), 478-486. doi:10.2307/3001619

Examples

misrd(r1 = 0.3, r0 = 0.1, prev = 0.4, se = 0.8, sp = 0.9)
#>      true  measured    factor        d1        d0 
#> 0.2000000 0.1426146 0.7130730 0.3800000 0.6200000