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OTE/SPH
OTE/SPH
August 31, 2006
JWBK119-12
Introduction to the Analysis of Categorical Data
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Table 12.6 Relative risks and 95% confidence intervals
Conditional probability Relative 95% confidence
risk interval

P Employed in Private Sector as RSE
no - PhD Qualification

3.78 (3.55, 4.03)
P Employed in Private Sector as RSE
PhD Qualification

P Employed in Private Sector as RSE
PhD Qualification

0.26 (0.25, 0.28)

P Employed in Private Sector as RSE
no - PhD Qualification

P Employed in Public Sector as RSE
no - PhD Qualification

0.34 (0.33, 0.35)
P Employed in Public Sector as RSE
PhD Qualification

P Employed in Public Sector as RSE
PhD Qualification

2.96 (2.87, 3.05)

P Employed in Public Sector as RSE
no - PhD Qualification

possible relative risk measures are shown in Table 12.6, together with their confidence
intervals. Since a relative risk measure of 1 indicates that there are no differences in the
probabilities conditional upon the qualification levels, it can be observed that the prob-
ability of finding employment in the private sector is 2.78 times (=3.78 − 1) higher
without PhD qualification. Assuming the frequency counts follow independent bi-
nomial distributions for both categories of the explanatory variable, the confidence
levels indicate that the probability of finding employment in the private sector is at
least 2.55 times (=3.55 − 1) higher if the RSE does not have a PhD.
Another commonly used measure of association for contingency tables is the odds
ratio. In contrast to relative risk which is a ratio of two probabilities, odds ratio is a
ratio of two odds. Given that an explanatory variable for a two-way contingency table
has I levels with i representing the ith level of the explanatory variable, the odds of
a success for the ith category is defined as
π i1
odds i = . (12.10)
1 − π i1
It can be observed that the odds are essentially the ratio of successes to failures for
each category of the explanatory variable. The odds ratio is then defined as the ratio
of these odds:
odds 1 π 11 /π 12 π 11 π 22
θ = = = . (12.11)
odds 2 π 21 /π 22 π 12 π 21
The sample estimate of the odds ratio can be expressed using the frequency counts in
each cell:
odds 1 n 11 n 12
ˆ θ = = . (12.12)
odds 2 n 12 n 21
For the example on the sectoral employment of RSEs by qualification levels with
data shown in Table 12.3, using the same definition of ‘success’ for each level of the
explanatory variable (with and without PhD qualifications), the sample odds of a