# Sensitivity Analysis Focusing on Subgroups with Demonstrated Insensitivity to Unmeasured Bias Sensitivity analysis using a d-statistic employing conditional inference to focus on those subgroups with demonstrated insensitivity to unmeasured biases. ```r dstat(y, qs = c(1/3, 2/3), gamma = 1, f = NULL, fscore = NULL, fr = 1, alpha = 0.05) ``` ## Arguments - `y`: A numeric vector of treated-minus-control matched pair differences in outcomes. - `qs`: Quantiles of |y| that partly define the d-statistic. Each coordinate of qs must be a number strictly between 0 and 1; otherwise, an error will result. See Details. - `gamma`: The sensitivity parameter, a number gamma>=1. - `f`: If f is not NULL, then it must be a factor that further subdivides y beyond the subdivisions implied by qs. If f is not NULL, then the length of f must equal the length of y; otherwise, an error will result. - `fscore`: If fscore is not NULL, then fscore contains integer scores to be attached to the levels of f. If f is not NULL but fscore is NULL, then the levels of f are viewed as nominal with equal emphasis. An error will result if fscore is not NULL but: (i) the scores are not integers, (ii) f is NULL, or (iii) the number of scores does not equal the number levels of f. - `fr`: A nonnegative number. If fr=0, then the test is simply a group-rank test, using every category, without conditional inference. The recommended default of fr=1 uses a category only if the proportion of positive y's in this category is at least equal to gamma/(1+gamma), and the conditional inference corrects for selection of categories based on y. In general, a category is used if the proportion of positive y is at least fr*gamma/(1+gamma), reducing to all categories if fr=0. - `alpha`: Of limited importance, a text message interprets numerical results in terms of rejection or not of the null hypothesis of no treatment effect in a one-sided, level-alpha test in the presence of a bias in treatment assignment of at most gamma>=1. ## Details The method is from Rosenbaum (2019). The example reproduces aspects of this manuscript. The default values of qs, 1/3 and 2/3, are from Brown (1981)'s test. See Markowski and Hettmansperger (1982) for discussion of other choices. See Rosenbaum (2015) for comparisons of performance of different fixed choices of qs; here, a fixed choice is obtained by setting fr=0. If a pair difference in y is zero, it falls in the lowest quantile of pairs and therefore receives weight zero along with other pair differences with small |y|. ## Returns - **T**: The test statistic - **comp2**: The sharp upper bound on the one-sided, exact P-value testing the null hypothesis of no treatment effect in the presence of a bias in treatment assignment of at most gamma. - **scores**: A vector reminding you of the scores, fscore, that you may have attached to the levels of f. - **table**: A table showing how individual categories contribute to the overall test. The notation in this table is from Rosenbaum (2019). - **summary**: A text summary of the conclusion. ## References Brown, B. M. (1981). Symmetric quantile averages and related estimators. Biometrika, 68(1), 235-242. Lalive, R., Van Ours, J., & Zweimüller, J. (2006). How changes in financial incentives affect the duration of unemployment. The Review of Economic Studies, 73, 1009-1038. Markowski, E. P., Hettmansperger, T. P. (1982). 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). 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. (1999). Using quantile averages in matched observational studies. Journal of the Royal Statistical Society: Series C (Applied Statistics), 48(1), 63-78. Rosenbaum, P. R. and Silber, J. H. (2009) Amplification of sensitivity analysis in observational studies. Journal of the American Statistical Association, 104, 1398-1405. Rosenbaum, P. R. (2010) The power of a sensitivity analysis and its limit. Chapter 14 of Design of Observational Studies. NY: Springer. 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. (2019). A highly adaptive test for matched observational studies. Manuscript. ## Author(s) Paul R. Rosenbaum ## Examples ```r # First example is from Rosenbaum (2019) data("lalive") attach(lalive) y<-log2((1+dur[after==1])/52)-log2((1+dur[after==0])/52) dstat(y,qs=c(1/3,2/3),fr=0,gamma=1.15) # Brown's (1981) test dstat(y,qs=c(2/3),fr=0,gamma=1.25) # Noether's (1973) Test #Amplification: see Rosenbaum and Silber (2009), Rosenbaum (2017, Table 9.1) amplify(1.25,2) bothseasonal<-(2==(seasonal[after==1]+seasonal[after==0]))*1 bothseasonal<-factor(bothseasonal,levels=1:0, labels=c("S","O"),ordered=TRUE) straddle<-1*((pmin(dur[after==1],dur[after==0])<=(bdur[after==0])) &(pmax(dur[after==1],dur[after==0])>(bdur[after==0]))) straddle<-factor(straddle,levels=c(0,1), labels=c("N","Y"),ordered=TRUE) dose<-1*((bdur[after==1]-bdur[after==0])>9.5) dose<-factor(dose,levels=c(0,1), labels=c("L","H"),ordered=TRUE) f<-bothseasonal:dose:straddle dstat(y,qs=c(1/3,2/3),f=f,gamma=1.25) # Reproduces Table 2 in Rosenbaum (2019) dstat(y,qs=c(1/3,2/3),f=f,gamma=1.45) amplify(1.45,c(2,2.5,3,4)) # Doubling the weight for high-dose matched pairs levels(f) fs<-c(1,1,2,2,1,1,2,2) dstat(y,qs=c(1/3,2/3),f=f,fscore=fs,gamma=1.45) rm(y,f,dose,straddle,bothseasonal) detach(lalive) rm(lalive) # Second example uses dental data data(dental) attach(dental) f<-age:dose levels(f) # Doubles the weight at high dose using fscore # For qs = (.4,.8), see Markowski and Hettmansperger (1982) dstat(y,qs=c(.4,.8),gamma=4.25,f=f,fscore=c(1,2,1,2)) rm(f) detach(dental) rm(dental) ```

dstat function