# Sensitivity Analysis for Confidence Intervals and Point Estimates from Weighted Rank Statistics in Block Designs Uses a weighted rank statistic to perform a sensitivity analysis for an I x J observational block design in which each of I blocks contains one treated individual and J-1 controls. Inverts the test to obtain an estimate and a confidence interval for an additive treatment effect, tau; see Rosenbaum (1993, 2002, 2007). ```r wgtRankCI(y, phi = "u868", phifunc = NULL, gamma = 1, alternative="greater", alpha=0.05, eps = 0.00001) ``` ## Arguments - `y`: A matrix or data frame with I rows and J columns. Column 1 contains the response of the treated individuals and columns 2 throught J contain the responses of controls in the same block. An error will result if y contains NAs. - `phi`: The weight function to be applied to the ranks of the within block ranges. The options are: (i) "wilc" for the stratified Wilcoxon test, which gives every block the same weight, (ii) "quade" which ranks the within block ranges from 1 to I, and is closely related to Quade's (1979) statistic; see also Tardif (1987), (iii) "u868" based on Rosenbaum (2011), (iv) u878 based on Rosenbaum (2011). Note that phi is ignored if phifunc is not NULL. - `phifunc`: If not NULL, a user specified weight function for the ranks of the within block ranges. The function should map [0,1] into [0,1]. The function is applied to the ranks divided by the sample size. - `gamma`: A single number greater than or equal to 1. gamma is the sensitivity parameter. Two individuals with the same observed covariates may differ in their odds of treatment by at most a factor of gamma; see Rosenbaum (2002, Chapter 4; 2017, Chapter 9). - `alternative`: Must equal "greater", "less", or "twosided". This determines whether the confidence interval is one-sided or two-sided. - `alpha`: Coverage of the confidence interval is 1-alpha. - `eps`: A small number. In searching for a root, a difference of eps will be regarded as negligible. A larger eps will produce an answer more quickly, while a smaller eps will produce a more accurate answer. ## Details This test is developed and evaluated in Rosenbaum (2024), and it is inverted for point estimates and confidence intervals in the usual way. Understand the test first. The two-sided interval is the intersection of two one-sided intervals, each with coverage 1-alpha/2; see Cox (1977, Section 4.2). ## Returns - **estimate**: The interval of point estimates, reducing to a single number when gamma=1. - **confidence**: The one-sided or two-sided confidence interval. ## References Brown, B. M. (1981). Symmetric quantile averages and related estimators. Biometrika, 68(1), 235-242. Cox, D. R. (1977). The role of significance tests [with discussion and reply]. Scandinavian Journal of Statistics, 4, 49-70. Gastwirth, J. L., Krieger, A. M., and Rosenbaum, P. R. (2000). Asymptotic separability in sensitivity analysis. Journal of the Royal Statistical Society B 2000, 62, 545-556. Hodges, J. L. and Lehmann, E. L. (1963). Estimates of Location Based on Rank Tests. The Annals of Mathematical Statistics, 34(2), 598-611. Lehmann, E. L. (1975). Nonparametrics: Statistical Methods Based on Ranks. San Francisco: Holden-Day. Quade, D. (1979). Using weighted rankings in the analysis of complete blocks with additive block effects. Journal of the American Statistical Association, 74, 680-683. Rosenbaum, P. R. (1993). Hodges-Lehmann point estimates of treatment effect in observational studies. Journal of the American Statistical Association, 88(424), 1250-1253. Rosenbaum, P. R. (2002). Observational Studies (2nd Edition). New York: Springer. Rosenbaum, P. R. (2007). Sensitivity analysis for M‐estimates, tests, and confidence intervals in matched observational studies. Biometrics, 63(2), 456-464. Rosenbaum, P. R. (2011). A new U‐Statistic with superior design sensitivity in matched observational studies. Biometrics, 67(3), 1017-1027. Rosenbaum, P. R. (2014) Weighted M-statistics with superior design sensitivity in matched observational studies with multiple controls. Journal of the American Statistical Association, 109(507), 1145-1158. Rosenbaum, P. R. (2015). Bahadur efficiency of sensitivity analyses in observational studies. Journal of the American Statistical Association, 110(509), 205-217. Rosenbaum, P. R. (2017). Observation and Experiment: An Introduction to Causal Inference. Cambridge, MA: Harvard University Press. Rosenbaum, P. R. (2018). Sensitivity analysis for stratified comparisons in an observational study of the effect of smoking on homocysteine levels. The Annals of Applied Statistics, 12(4), 2312-2334. Rosenbaum, P. R. (2024) Bahadur efficiency of observational block designs. Journal of the American Statistical Association. 119, 1871-1881. Rosenbaum, P. R. (2025) An Introduction to the Theory of Observational Studies. Switzerland: Springer. Tardif, S. (1987). Efficiency and optimality results for tests based on weighted rankings. Journal of the American Statistical Association, 82(398), 637-644. ## Author(s) Paul R. Rosenbaum ## Note The computations use the separable approximation discussed in Gastwirth et al. (2000) and Rosenbaum (2018). Compare with the method in Rosenbaum (2014) and the R package sensitivitymw. ## See Also An alternative approach avoids rank tests and uses weighted M-statistics instead, as in the sensitivitymw package and Rosenbaum (2014). However, Bahadur efficiency calculations are available for weighted rank statistics; see Rosenbaum (2024). ## Examples ```r data(aHDL) y<-t(matrix(aHDL$hdl,4,406)) # Without sensitivity analysis wgtRankCI(y,phi="wilc",gamma=1) # Stratified Wilcoxon rank sum test # With sensitivity analysis # The CI excludes 0 if the test rejects 0. wgtRank(y,phi="wilc",gamma=3.7) # Stratified Wilcoxon rank sum test wgtRankCI(y,phi="wilc",gamma=3.7) wgtRank(y,phi="u878",gamma=3.7) # U-statistic with weights (8,7,8) wgtRankCI(y,phi="u878",gamma=3.7) # A user defined weight function, brown, analogous to Brown (1981). brown<-function(v){((v>=.333)+(v>=.667))/2} wgtRankCI(y,phifunc=brown,gamma=3.7) # # COMPARISONS WITH STANDARD METHODS # y2<-c(-17, -3, -1, 2, 8, 9, 10, 11, 12, 16, 20) y2a<-cbind(y2,-y2)/2 # # Compare with the usual Hodges-Lehmann estimate, # namely the median of the Walsh averages (yi+yj)/2 # hl<-function(y){ w<-NULL I<-length(y) for (i in 1:I) for (j in i:I) w<-c(w,(y[i]+y[j])/2) median(w) } hl(y2) wgtRankCI(y2a,phi="quade")$estimate # # Compare with wilcox.test confidence interval in the stats package # stats::wilcox.test(y2,conf.int=TRUE,correct=FALSE,exact=FALSE, alternative = "greater")$conf.int wgtRankCI(y2a,phi="quade",alternative="greater")$confidence stats::wilcox.test(y2,conf.int=TRUE,correct=FALSE,exact=FALSE, alternative = "less")$conf.int wgtRankCI(y2a,phi="quade",alternative="less")$confidence stats::wilcox.test(y2,conf.int=TRUE,correct=FALSE,exact=FALSE, alternative = "two.sided")$conf.int wgtRankCI(y2a,phi="quade",alternative="twosided")$confidence # # Compare with senWilcox in the DOS2 package # # Note: senWilcox reports only one-sided estimates with 1-sided tests # DOS2::senWilcox(y2,conf.int=TRUE,alternative="greater",gamma=1.25) wgtRankCI(y2a,phi="quade",gamma=1.25,alternative = "greater") DOS2::senWilcox(y2,conf.int=TRUE,alternative="less",gamma=1.25) wgtRankCI(y2a,phi="quade",gamma=1.25,alternative = "less") DOS2::senWilcox(y2,conf.int=TRUE,alternative="twosided",gamma=1.25) wgtRankCI(y2a,phi="quade",gamma=1.25,alternative = "twosided") # # Compare with senU in the DOS2 package # # Note: senU reports only one-sided estimates with 1-sided tests # DOS2::senU(y2,gamma=1.25,m=8,m1=6,m2=8,alternative = "greater", exact=FALSE,conf.int = TRUE) wgtRankCI(y2a,phi="u868",gamma=1.25,alternative = "greater") DOS2::senU(y2,gamma=1.25,m=8,m1=6,m2=8,alternative = "less", exact=FALSE,conf.int = TRUE) wgtRankCI(y2a,phi="u868",gamma=1.25,alternative = "less") DOS2::senU(y2,gamma=1.25,m=8,m1=6,m2=8,alternative = "twosided", exact=FALSE,conf.int = TRUE) wgtRankCI(y2a,phi="u868",gamma=1.25,alternative = "twosided") ```

wgtRankCI function