Uses a weighted rank statistic to perform a sensitivity analysis for an I x J observational block design in which each of I blocks contain J individuals, some of whom are treated individuals and others are controls.
gwgtRank(y, z, phi = "u868", phifunc = NULL, gamma = 1, detail=FALSE)
Upper bound on the one-sided P-value when testing the null hypothesis of no treatment effect against the alternative hypothesis that treated responses are higher than control responses.
Details of the computation of pval: the standardized deviate, the test statistic, its null expectation, its null variance and the value of gamma.
A matrix or data frame of responses with I rows and J columns. An error will result if y contains NAs.
A matrix or data frame of treatment indicators with I rows and J columns. In z, the ith row and jth column is a 1 if this individual is treated, or a 0 if this individual is a control. If a row of z contains J ones or J zeros, that block will be removed from further computations as there is no one to compare, and a warning will appear. An error will result if z contains NAs.
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.
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. See the example.
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 (1987; 2017, Chapter 9).
If detail=FALSE, then brief output is given, similar to that provided by wgtRank(). In both cases, the asymptotically separable approximation of Gastwirth et al. (2000) is used. If detail=TRUE, then the separable approximation is compared with a bound in Rosenbaum (2018), and the output from the senstrat package is given. For moderate J and large I, the bound is likely to agree exactly with the separable approximation.
Paul R. Rosenbaum
The function gwgtRank() differs from wgtRank() in that wgtRank requires one treated individual and J-1 controls, whereas gwgtRank permits 1 to J-1 treated indivuals with the rest as controls.
This function uses the senstrat() function in the senstrat package to perform the calculations.
To test in the lower tail -- to test against the alternative that treated responses are lower than control responses, apply the function to -y. For a two-sided test, do both one-sided tests and apply the Bonferroni inequality, doubling the smaller of the two one-sided P-value bounds; see Cox (1977, Section 4.2).
Brown, B. M. (1981). <doi:10.1093/biomet/68.1.235> 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). <doi:10.1111/1467-9868.00249> Asymptotic separability in sensitivity analysis. Journal of the Royal Statistical Society B 2000, 62, 545-556.
Lehmann, E. L. (1975). Nonparametrics: Statistical Methods Based on Ranks. San Francisco: Holden-Day.
Quade, D. (1979). <doi:10.2307/2286991> 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. (1987). <doi:10.2307/2336017> Sensitivity analysis for certain permutation inferences in matched observational studies. Biometrika, 74(1), 13-26.
Rosenbaum, P. R. (2011). <doi:10.1111/j.1541-0420.2010.01535.x> A new U‐Statistic with superior design sensitivity in matched observational studies. Biometrics, 67(3), 1017-1027.
Rosenbaum, P. R. (2013). <doi:10.1111/j.1541-0420.2012.01821.x> Impact of multiple matched controls on design sensitivity in observational studies. Biometrics, 2013, 69, 118-127.
Rosenbaum, P. R. (2014) <doi:10.1080/01621459.2013.879261> 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). <doi:10.1080/01621459.2014.960968> Bahadur efficiency of sensitivity analyses in observational studies. Journal of the American Statistical Association, 110(509), 205-217.
Rosenbaum, P. (2017). <doi:10.4159/9780674982697> Observation and Experiment: An Introduction to Causal Inference. Cambridge, MA: Harvard University Press.
Rosenbaum, P. R. (2018). <doi:10.1214/18-AOAS1153> 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) <doi:10.1080/01621459.2023.2221402> Bahadur efficiency of observational block designs. Journal of the American Statistical Association, 119, 1871-1881.
Rosenbaum, P. R. (2025) <doi:10.1007/978-3-031-90494-3> An Introduction to the Theory of Observational Studies. Switzerland: Springer.
Tardif, S. (1987). <doi:10.2307/2289476> Efficiency and optimality results for tests based on weighted rankings. Journal of the American Statistical Association, 82(398), 637-644.
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).
data(aHDL)
y<-t(matrix(aHDL$hdl,4,406))
z<-matrix(0,dim(y)[1],dim(y)[2])
z[,1]<-1
gwgtRank(y,z,phi="quade",gamma=3.5) # Quade's test
gwgtRank(y,z,phi="quade",gamma=3.5,detail=TRUE) # Alternative output
# A user defined weight function, brown, analogous to Brown (1981).
brown<-function(v){((v>=.333)+(v>=.667))/2}
gwgtRank(y,z,phifunc=brown,gamma=4.7)
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