Ordinal variables

For a random sample of black males the General Social Survey in 1996 asked two questions:

• Q1: What is your yearly income (income)?

• Q2: How satisfied are you with your job (satisfaction)?

Both measurements are on an ordinal scale.

We might do a chi-square test to see whether Q1 and Q2 are associated, but the test does not exploit the ordinality.

We shall consider a test that incorporates ordinality.

Gamma coefficient

Consider a pair of respondents, where respondent1 is below respondent2 in relation to Q1.

• If respondent1 is also below respondent2 in relation to Q2 then the pair is concordant.

• If respondent1 is above respondent2 in relation to Q2 then the pair is disconcordant.

Let

• \(C=\) the number of concordant pairs in our sample.

• \(D=\) the number of disconcordant pairs in our sample.

We define the estimated gamma coefficient \[\hat{\gamma} = \frac{C-D}{C+D} = \underbrace{\frac{C}{C+D}}_{concordant~prop.} - \underbrace{\frac{D}{C+D}}_{discordant~prop.}\]

Gamma coefficient

Properties:

• Gamma lies between -1 og 1

• The sign tells whether the association is positive or negative

• Large absolute values correspond to strong association

The standard error \(se({\hat{\gamma}})\) on \({\hat{\gamma}}\) is complicated to calculate, so we leave that to software.

We can now determine a 95% confidence interval: \[{\hat{\gamma}}\pm 1.96se({\hat{\gamma}})\] and if zero is contained in the interval, then there is no significant association, when we perform a test with a 5% significance level.

Example

First, we need to install the package vcdExtra, which has the function GKgamma for calculating gamma. It also has the dataset on job satisfaction and income built-in:

library(vcdExtra)
JobSat

##         satisfaction
## income   VeryD LittleD ModerateS VeryS
##   < 15k      1       3        10     6
##   15-25k     2       3        10     7
##   25-40k     1       6        14    12
##   > 40k      0       1         9    11

GKgamma(JobSat, level = 0.90)

## gamma        : 0.221 
## std. error   : 0.117 
## CI           : 0.028 0.414

A positive association. Marginally significant at the 10% level, but not so at the 5% level.