Job satisfaction

Description

From For Ordinal Variables

A 4×4 cross-classification of income level (ordinal) against job-satisfaction level (ordinal) from a US survey. Used to illustrate ordinal-association measures (Kendall’s ${\tau }_b$, Goodman-Kruskal $\gamma$) — the analogues of Pearson’s correlation for ordered categorical data.

	Very Dissat.	Little Dissat.	Mod. Sat.	Very Sat.
<15,000	1	3	10	6
15,000–25,000	2	3	10	7
25,000–40,000	1	6	14	12
>40,000	0	1	9	11

Ordinal association

From For Ordinal Variables and Definition 4.1

A pair of subjects is:

• concordant — subject ranked higher on $X$ is also ranked higher on $Y$
• discordant — subject ranked higher on $X$ is ranked lower on $Y$
• tied — same rank on $X$ and/or $Y$

Let $C$ = # concordant pairs, $D$ = # discordant pairs.

Goodman-Kruskal $\gamma$: $(C-D)/(C+D)$ — easy to interpret; ignores ties.
Kendall’s ${\tau }_b$: similar numerator, with a normalising denominator that handles ties; less sensitive to cut-points defining the categories.

Both measures lie in $[-1,1]$:

• $\approx 0$ → weak trend
• near $\pm 1$ → strong monotone (positive / negative) association

See Gamma Tau for a longer reference on these measures.

R code

x <- matrix(c(1, 3, 10, 6,
              2, 3, 10, 7,
              1, 6, 14, 12,
              0, 1, 9, 11), ncol=4, byrow=TRUE)
dimnames(x) <- list(c("<15,000", "15,000-25,000",
                      "25,000-40,000", ">40,000"),
                    c("Very Dissat.", "Little Dissat.",
                      "Mod. Sat.", "Very Sat."))
us_svy_tab <- as.table(x)
 
library(DescTools)
output <- Desc(x, plotit = FALSE, verbose = 3)
output[[1]]$assocs

Python code

import numpy as np
from scipy import stats
 
us_svy_tab = np.array([[1, 3, 10, 6],
                       [2, 3, 10, 7],
                       [1, 6, 14, 12],
                       [0, 1, 9, 11]])
dim1 = us_svy_tab.shape
x, y = [], []
for i in range(dim1[0]):
    for j in range(dim1[1]):
        for _ in range(us_svy_tab[i, j]):
            x.append(i); y.append(j)
 
kt_output = stats.kendalltau(x, y)
print(f"Estimate of tau-b: {kt_output.statistic:.4f}.")
# The estimate of tau-b is 0.1524.

Results: Goodman-Kruskal $\gamma \approx 0.22$, Kendall’s ${\tau }_b\approx 0.15$. Both point to a weak positive association: higher-income respondents tend to report higher job satisfaction, though the effect is modest and the lower confidence limit is close to zero — borderline significant.

Note that in Python, scipy.stats.kendalltau expects two rank vectors, so we reconstruct them by “unrolling” the contingency table. R’s DescTools::Desc works directly on the table.

---

See also: L4 Exploring Categorical Data · Gamma Tau