wgtRank {weightedRank}

Sensitivity Analysis for Weighted Rank Statistics in Block Designs

Description

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.

Usage

wgtRank(y, phi = "u868", phifunc = NULL, gamma = 1)

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) "u858", "u868", "u878", or "u888" 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. 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).

Details

This method is developed and evaluated in Rosenbaum (2024).

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).

Value

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.

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.

Author(s)

Paul R. Rosenbaum

References

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.

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.

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

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)