Rosenbaum sensitivity bounds, Wilcoxon signed-rank

Formula: vary Gamma; compute the upper p-value

Usage

CnsRos(pairs, Gamma = 1)

Arguments

pairs: Within-pair differences; exact zeros are dropped.
Gamma: Sensitivity parameter, at least 1.

Value

List with estimate, p_upper, p_lower, W, mu_plus, sigma_plus, z_upper, n_pairs, Gamma, method.

Details

Under a bias of at most Gamma in the odds of treatment within a matched pair, the null distribution of the signed-rank statistic is bounded by the one where each pair contributes its rank with probability p+ = Gamma/(1+Gamma). The upper p-value uses the normal approximation with mean p+ sum(q) and variance p+(1-p+) sum(q^2). At Gamma = 1 that reduces to the ordinary Wilcoxon signed-rank test.

References

Rosenbaum (2002), Observational Studies, 2nd ed., Springer, section 4.3.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
CnsRos(V)
#> $estimate
#> [1] 0.005859343
#> 
#> $p_upper
#> [1] 0.005859343
#> 
#> $p_lower
#> [1] 0.005859343
#> 
#> $W
#> [1] 36
#> 
#> $mu_plus
#> [1] 18
#> 
#> $sigma_plus
#> [1] 7.141428
#> 
#> $z_upper
#> [1] 2.520504
#> 
#> $n_pairs
#> [1] 8
#> 
#> $Gamma
#> [1] 1
#> 
#> $method
#> [1] "Rosenbaum sensitivity bounds, Wilcoxon signed-rank"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"