Page 194 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

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Case Study 179
and odds ratio. The use of these measures is illustrated using the sectoral employment
example.
Consider the contingency table shown in Table 12.3. Using the generic terms ‘suc-
cess’to represent employment in the private sector and ‘failure’to refer as employment
in the public sector (these labels can of course be interchanged), π 11 is defined as the
proportion of successes for RSEs with PhDs and π 21 is defined as the proportion of
successes for RSE without PhDs. The difference between these proportions essentially
compares the success probabilities (or the probability of being employed in the pri-
vate sector as an RSE for this example). A more formal way to define the difference
between these probabilities using conditional probabilities is:

P(Employed in Private Sector as RSE|PhD Qualification) −
P(Employed in Private Sector as RSE|No PhD Qualification).

Assuming that the counts in both rows follow independent binomial distributions and
usingthesampleproportions, p 11 and p 21 ,asestimatesforthesuccessprobabilities,the
sample difference (p 11 − p 21 ) estimates the population difference with the estimated
standard error given by

p 11 (1 − p 11 ) p 21 (1 − p 21 )
se(p 11 − p 21 ) = + .
n 1+ n 2+
Thus, a 95% confidence interval for the sample difference is

(p 11 − p 21 ) ± z α/2 se(p 11 − p 21 ).
The sample difference for the contingency table shown in Table 12.3 is −0.53 and
its corresponding confidence interval is (−0.55, −0.52). Since the interval contains
only negative values, it can be concluded that the PhD qualification has a negative
influence on the probability of successfully finding employment in the private sector.
Apart from the difference in the conditional probability of being employed in the
privatesectors,conditionalprobabilitiesofemploymentinthepublicsectorcanalsobe
elicited from the contingency table. The difference between conditional probabilities
for employment in the public sector as an RSE is 0.53 with a corresponding confidence
interval of (0.52, 0.55). This difference is formally expressed as

P(Employed in Public Sector as RSE|PhD Qualification) −
P(Employed in Public Sector as RSE|no - PhD Qualification).

Relative risk is a more elaborate measure than the difference between proportions of
successes. Rather than considering the absolute differences of proportions (π 11 − π 21 ),
itcapturestherelativedifferencesarethrougharatioofproportions(π 11 /π 21 ).Utilizing
this measure, bias due to scaling effect inherent in absolute differences is mitigated.
The sample estimate of the relative risk is p 11 /p 21 . As it is a ratio of two random
variables, its sampling distribution can be highly skewed and its confidence interval
relatively more complex. Both the relative risk and its corresponding confidence inter-
vals can be evaluated with software such as MINITAB. The sample estimates of four