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

Wuming Gong wuming.gong at gmail.com
Fri Sep 2 08:49:47 CEST 2005

```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

```