# [R] Calculating Goodman-Kurskal's gamma using delta method

Frank E Harrell Jr f.harrell at vanderbilt.edu
Fri Sep 2 15:03:02 CEST 2005

```Wuming Gong wrote:
> Dear list,
>
> I have a problem on calculating the standard error of
> Goodman-Kurskal's gamma using delta method. I exactly follow the
> method and forumla described in Problem 3.27 of Alan Agresti's
> Categorical Data Analysis (2nd edition). The data I used is also from
> the job satisfaction vs. income example from that book.
>
> job <- matrix(c(1, 3, 10, 6, 2, 3, 10, 7, 1, 6, 14, 12, 0, 1, 9, 11),
> nrow = 4, ncol = 4, byrow = TRUE, dimnames = list(c("< 15,000",
> "15,000 - 25,000", "25,000 - 40,000", "> 40,000"), c("VD", "LD", "MS",
> "VS")))
>
> The following code is for calculating gamma value, which is consistent
> with the result presented in section 2.4.5 of that book.
>
> C <- 0
> D <- 0
> for (i in 1:nrow(job)){
> 	for (j in 1:ncol(job)){
> 		pi_c <- 0
> 		pi_d <- 0
> 		for (h in 1:nrow(job)){
> 			for (k in 1:ncol(job)){
> 				if ((h > i & k > j) | (h < i & k < j)){
> 					pi_c <- pi_c + job[h, k]/sum(job)
> 				}
>
> 				if ((h > i & k < j) | (h < i & k > j)){
> 					pi_d <- pi_d + job[h, k]/sum(job)
> 				}
> 			}
> 		}
>
> 		C <- C + job[i, j] * pi_c
> 		D <- D + job[i, j] * pi_d
> 	}
> }
> gamma <- (C - D) / (C + D) # gamma = 0.221 for this example.
>
> The following code is for calculating stardard error of gamma.
> sigma.squared <- 0
> for (i in 1:nrow(job)){
> 	for (j in 1:ncol(job)){
> 		pi_c <- 0
> 		pi_d <- 0
> 		for (h in 1:nrow(job)){
> 			for (k in 1:ncol(job)){
> 				if ((h > i & k > j) | (h < i & k < j)){
> 					pi_c <- pi_c + job[h, k]/sum(job)
> 				}
>
> 				if ((h > i & k < j) | (h < i & k > j)){
> 					pi_d <- pi_d + job[h, k]/sum(job)
> 				}
> 			}
> 		}
> 		phi <- 4 * (pi_c * D - pi_d * C) / (C + D)^2
>
> 		sigma.squared <- sigma.squared + phi^2
> 	}
> }
>
> se <- (sigma.squared/sum(job))^.5 # 0.00748, which is different from
> the SE 0.117 given in section 3.4.3 of that book.
>
> I am not able to figure out what is the problem with my code... Could
> anyone point out what the problem is?
>
> Thanks.
>
> Wuming

Save your time (and much execution time) by using the Hmisc package's
rcorr.cens function with the argument outx=TRUE.  rcorr.cens using a
standard U-statistic variance estimator.

--
Frank E Harrell Jr   Professor and Chair           School of Medicine
Department of Biostatistics   Vanderbilt University

```