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

From: Frank E Harrell Jr <f.harrell_at_vanderbilt.edu>
Date: Fri 02 Sep 2005 - 23:03:02 EST

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

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