noether function

Sensitivity Analysis Using Noether's Test for Matched Pairs

Computes a sensitivity analysis for treated-minus-control matched pair differences in observational studies.

noether(y, f = 2/3, gamma = 1, alternative = "greater")

Arguments

• y: A vector of treated-minus-control matched pair differences.
• f: A nonnegative number strictly less than 1. Suppose that there are I matched pair differences, length(y)=I. Rank the absolute pair differences from 1 to I with average ranks for ties. Noether's statistic looks at the roughly (1-f)I pair differences with absolute ranks that are at least fI, and computes the sign test from these fI pair differences. With f=0, Noether's statistic is the usual sign test statistic. With f=2/3, Noether's statistic focuses on the 1/3 of pairs with the largest absolute pair differences. In his article, Noether suggested f=1/3 for randomized matched pair differences from a Normal distribution, but f=2/3 is better for sensitivity analyses in observational studies. Pair differences that are zero are not counted, but this is uncommon for f=2/3.
• gamma: A number greater than or equal to 1. gamma is the sensitivity parameter, where gamma=1 for a randomization test, and gamma>1 for a sensitivity analysis.
• alternative: The possible alternatives are "greater", "less" or "two.sided"; however, "two.sided" is available only for gamma=1.

Details

Noether's (1973) strengthens the sign test. In a randomized experiment, it increase power. In an observational study, it increases design sensitivity and the Bahadur efficiency of a sensitivity analysis.

Because the test has a binomial null distribution in both a randomized experiment and in an observational study, Noether's test is used in a number of problems in Introduction to the Theory of Observational Studies.

Noether's test is related to methods of Gastwirth (1966), Brown (1981), and Markowski and Hettmansperger (1982). Its properties in an observational study are discussed Rosenbaum (2012, 2015).

Returns

• number.pairs: Number of pairs used by Noether's statistic, roughly fI.

• positive.pairs: Number of positive pair differences among used pairs.

• pval: P-value testing the null hypothesis of no treatment effect. Obtained from the binomial distribution.

References

Brown, B. M. (1981) doi:10.1093/biomet/68.1.235 Symmetric quantile averages and related estimators. Biometrika, 68(1), 235-242.

Gastwirth, J. L. (1966) doi:10.1080/01621459.1966.10482185 On robust procedures. Journal of the American Statistical Association, 61(316), 929-948.

Markowski, E. P. and Hettmansperger, T. P. (1982) doi:10.1080/01621459.1982.10477905 Inference based on simple rank step score statistics for the location model. Journal of the American Statistical Association, 77(380), 901-907.

Noether, G. E. (1973) doi:10.1080/01621459.1973.10481411 Some simple distribution-free confidence intervals for the center of a symmetric distribution. Journal of the American Statistical Association, 68(343), 716-719.

Rosenbaum, P. R. (2012) <10.1214/11-AOAS508> An exact adaptive test with superior design sensitivity in an observational study of treatments for ovarian cancer. Annals of Applied Statistics, 6, 83-105.

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.

Author(s)

Paul R. Rosenbaum

Note

As noted in the Preface to Introduction to the Theory of Observational Studies, Noether's statistic is used in a sequence of Problems that appear in various chapters.

Examples

set.seed(1)
y<-rnorm(1000)+.5
noether(y,f=0,gamma=3)
noether(y,f=2/3,gamma=3)