Rosenbaum Sensitivity Bounds for Matched Analyses

Assesses how sensitive a matched-pair analysis is to unmeasured confounding using the Rosenbaum bounds framework. For each value of the sensitivity parameter Gamma – the maximum odds ratio for differential treatment assignment within a matched pair due to hidden bias – the function reports the upper and lower bounds on the p-value of the Wilcoxon signed-rank test.

Usage

sa_rosenbaum(
  x,
  outcome_col,
  gamma_max = 3,
  gamma_inc = 0.25,
  alpha = 0.05,
  treatment_col = NULL,
  pair_id_col = "pair_id",
  treated_level = NULL,
  alternative = c("auto", "greater", "less")
)

Arguments

x: A ps_match object (returned by ps_match()) or a plain data frame whose rows are matched patients.
outcome_col: Name of the numeric outcome column. Required.
gamma_max: Maximum Gamma to test. Default 3.
gamma_inc: Increment between Gamma values. Default 0.25.
alpha: Significance threshold. Default 0.05.
treatment_col: Name of the binary treatment column (0/1). Taken from x$meta$treatment_col when x is a ps_match object. Required when x is a plain data frame.
pair_id_col: Name of the pair-ID column created by ps_match(). Taken from x$meta$pair_id_col when recorded. Default "pair_id".
treated_level: The value of treatment_col that is treated. Read from the metadata of a ps_data object when it records one. NULL (default) requires a 0/1 or logical column and takes 1 or TRUE.
alternative: Direction of the test: "auto" (default; chosen from the sign of the median difference, see Details), "greater" (treated outcome above control) or "less".

Value

An object of class c("sa_rosenbaum", "ps_data"):

$data: The input data frame (x$data for a ps_match object).
$meta: Column names, treated_level, the settings (gamma, the grid tested, gamma_max, gamma_inc, alpha, alternative, the direction tested, and direction_from_data), and the results: sensitivity_value, the largest Gamma for which the upper p-value is below alpha (NA if the analysis is not significant at Gamma = 1), n_pairs, the complete matched pairs used, n_dropped, n_nonzero and the statistic t_plus.
$tables$bounds: Data frame with columns gamma, p_upper, p_lower, reject_upper (logical – is the upper bound < alpha?).
$tables$pairs: One row per complete pair: pair_id, treated and control outcomes, and difference.
$models: An empty list.

Details

A Gamma of 1 corresponds to no hidden bias (equivalent to perfect randomisation within pairs). The sensitivity value is the largest Gamma at which the upper p-value bound remains below alpha; a larger sensitivity value indicates a more robust result.

The statistic is the Wilcoxon signed-rank sum $T^+$ of the ranks of $|d_i|$ over pairs with $d_i > 0$, where $d_i$ is the treated minus the control outcome and zero differences are dropped. Tied $|d_i|$ take their average rank. Under hidden bias Gamma each pair is positive with probability at most $p^+ = \Gamma / (1 + \Gamma)$, so the worst-case null distribution has mean $p^+ \sum r_i$ and variance $p^+ (1 - p^+) \sum r_i^2$, and the best case uses $p^- = 1 / (1 + \Gamma)$. Using $\sum r_i^2$ rather than $n(n+1)(2n+1)/6$ corrects the variance for ties; without ties the two agree. The p-values use the normal approximation with a continuity correction of 0.5, so at Gamma = 1 they equal the one-sided stats::wilcox.test() p-value with exact = FALSE and correct = TRUE.

Rosenbaum's method assumes the direction of the test is fixed before the outcome is seen. Set alternative to "greater" (treated above control) or "less" to state it. The default "auto" keeps the earlier behaviour and picks the direction from the data: it tests the lower tail, with a warning, when the median nonzero difference is negative. A direction chosen from the data makes the bounds slightly optimistic, so prefer an explicit alternative in a pre-specified analysis.

Column names and the treated level are read from a ps_match() object's metadata. A supplied treatment_col, pair_id_col or treated_level must agree with that metadata; a conflict aborts.

References

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

Examples

dta <- sample_ps_data(n = 300, seed = 1)
m   <- ps_match(dta)
res <- sa_rosenbaum(m, outcome_col = "ef", gamma_max = 2, alternative = "greater")
res
#> <sa_rosenbaum>
#>   Outcome     : ef (greater, 300 pairs)
#>   Sensitivity : Gamma = 1.75 at alpha = 0.05
#>  gamma      p_upper      p_lower reject_upper
#>   1.00 5.990619e-11 5.990619e-11         TRUE
#>   1.25 7.884045e-07 2.220446e-16         TRUE
#>   1.50 2.257868e-04 0.000000e+00         TRUE
#>   1.75 7.391183e-03 0.000000e+00         TRUE
#>   2.00 6.394389e-02 0.000000e+00        FALSE
res$tables$bounds
#>   gamma      p_upper      p_lower reject_upper
#> 1  1.00 5.990619e-11 5.990619e-11         TRUE
#> 2  1.25 7.884045e-07 2.220446e-16         TRUE
#> 3  1.50 2.257868e-04 0.000000e+00         TRUE
#> 4  1.75 7.391183e-03 0.000000e+00         TRUE
#> 5  2.00 6.394389e-02 0.000000e+00        FALSE
res$meta$sensitivity_value
#> [1] 1.75