# One-step and Two-step Signed Ranks. Calculates either one-step or two-step signed ranks as proposed by Noether (1973), Brown (1981) and Markowski and Hettmansperger (1982). ```r bmhranks(y, q1 = 1/3, q2 = 2/3) ``` ## Arguments - `y`: A vector of matched pair differences. - `q1`: Quantile at which step-ranks rise from 0 to 1. - `q2`: Quantile at which step-ranks rise from 1 to 2. If q1=q2, then there is only one step, from 0 to 1. ## Returns A vector with the same length as y containing the step-ranks. ## References Brown, B. M. (1981) Symmetric quantile averages and related estimators. Biometrika, 68(1), 235-242. Markowski, E. P. and 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. (2012) An exact adaptive test with superior design sensitivity in an observational study of treatments for ovarian cancer. The Annals of Applied Statistics, 6(1), 83-105. Rosenbaum, P. R. (2015) Bahadur efficiency of sensitivity analyses in observational studies. Journal of the American Statistical Association, 110(509), 205-217. ## Author(s) Paul R. Rosenbaum ## Note The use of step-ranks in observational studies is discussed in Rosenbaum (1999, 2012, 2015). They can have larger design sensitivity than Wilcoxon ranks and higher Bahadur efficiency in a sensitivity analysis. ## See Also The function multrnk() computes an alternative family of signed ranks. ## Examples ```r data(smokerlead) attach(smokerlead) w<-rank(abs(llead)) # Wilcoxon ranks o<-order(w) brn<-bmhranks(llead) # Brown (1981) ranks plot(w[o],brn[o],type="n",xlab="Wilcoxon ranks",ylab="Step Ranks", main="Comparison of 3 Step Ranks") lines(w[o],brn[o],col="black") # The following two-step ranks were best for Normal data in # Table 2 of Markowski-Hettmansperger (1982). mhn<-bmhranks(llead,q1=.4,q2=.8) lines(w[o],mhn[o],col="blue") # Noether (1973) ranks take a single step. The case of a step # at q1=q2=2/3 was evaluated in Rosenbaum (2015, Table 2). noe<-bmhranks(llead,q1=2/3,q2=2/3) lines(w[o],noe[o],col="red") legend(20,1.75,c("Brown","MH","Noether"), lty=c(1,1,1),col=c("black","blue","red")) # Adaptive choice of Brown or Noether ranks was considered in # Rosenbaum (2012). In this case, an exact distribution # is available, but function tt() uses a limiting Normal # distribution instead. H<-cbind(brn,noe) tt(llead,H,gamma=1) tt(llead,H,gamma=3.7) rm(w,brn,mhn,noe,H) detach(smokerlead) ```

bmhranks function