A Rosenbaum Gamma bound is not a test of ignorability

For a matched-pairs sign test with n discordant pairs and T treated-case pairs, the worst-case expectation of \(T\) at sensitivity \(\Gamma\) is \(n\Gamma/(1+\Gamma)\), non-decreasing in \(\Gamma\) (Research.P5.Gamma.worst_mean_mono); \(\Gamma^* = T/(n-T)\) is the value where it reaches \(T\) (gamma_star_kills) and depends on the table alone (gamma_star_table_only).

Usage

morie_gamma_bound(T, n, Gamma = NULL)

Arguments

T

Number of discordant pairs in which the treated unit is the case.

n

Number of discordant pairs.

Gamma

Optional sensitivity values at which to evaluate the worst case.

Value

A list: gamma_star and, with Gamma, a data frame worst with Gamma, worst_mean and survives (T above the worst-case mean).

References

Rosenbaum PR (2002). Observational Studies, 2nd ed. Springer.

Examples

morie_gamma_bound(T = 70, n = 100, Gamma = c(1, 1.5, 2, 2.5))
#> $gamma_star
#> [1] 2.333333
#> 
#> $worst
#>   Gamma worst_mean survives
#> 1   1.0   50.00000     TRUE
#> 2   1.5   60.00000     TRUE
#> 3   2.0   66.66667     TRUE
#> 4   2.5   71.42857    FALSE
#> 
#> attr(,"theorem")
#> [1] "Research.P5.Gamma.gamma_star_kills"