Sensitivity Analysis for 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.
wgtRank(y, phi = "u868", phifunc = NULL, gamma = 1)
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. A 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 rates. The function should map [0,1] into [0,1]. The function is applied to the ranks divided by the sample size. See the example.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 (1987; 2017, Chapter 9).This method is developed and evaluated in Rosenbaum (2022).
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).
pval: 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.
detail: Details of the computation of pval: the standardized deviate, the test statistic, its null expectation, its null variance and the value of gamma.
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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. (2022). Bahadur efficiency of observational block designs. Manuscript.
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.
Paul R. Rosenbaum
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.
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 (2022).
data(aHDL) y<-t(matrix(aHDL$hdl,4,406)) wgtRank(y,phi="wilc",gamma=3.5) # Stratified Wilcoxon rank sum test wgtRank(y,phi="quade",gamma=3.5) # Quade's test wgtRank(y,phi="quade",gamma=4.5) # Quade's test, larger gamma wgtRank(y,phi="quade",gamma=4.6) # Quade's test, larger gamma wgtRank(y,phi="u868",gamma=5.4) # New U-statistic weights (8,6,8) wgtRank(y,phi="u878",gamma=6) # New U-statistic weights (8,7,8) # As an aid to interpreting gamma, see the amplify function. amplify(3.5,8) amplify(4.6,8) amplify(5.4,8) amplify(6,8) # A user defined weight function, brown, analogous to Brown (1981). brown<-function(v){((v>=.333)+(v>=.667))/2} wgtRank(y,phifunc=brown,gamma=4.7)
Useful links