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StatXact Example 14
Do older employees get poorer job ratings?
In 1994, a major US company used a certain management program to determine which employees would be subject to RIF’s (“reductions in force,” or layoffs). Small groups of employees were considered together, and those with superior job ratings were retained. Terminations in one such group led to litigation, and a central statistical question arose: “Do older employees get lower performance ratings than one might expect if the association between age and rating were merely random?”
In the following table, rows are employees, and columns are their ages and ratings. To maintain confidentiality, these numbers have been altered somewhat, but the structure remains as it was in the legal case.
|
|
Performance Rating |
|
|
Age |
Excellent |
Good |
Fair |
Poor |
|
20 |
4 |
0 |
0 |
0 |
|
36 |
0 |
1 |
0 |
0 |
|
39 |
1 |
0 |
0 |
0 |
|
40 |
1 |
0 |
0 |
0 |
|
43 |
0 |
0 |
0 |
1 |
|
47 |
0 |
1 |
0 |
0 |
overall |
totals |
6 |
2 |
0 |
1 |
9 |
To test whether there is an association between age and performance, we use a linear-by-linear association, found in the StatXact menu under “Doubly ordered RxC Table tests.”
Here are the results:
LINEAR-BY-LINEAR ASSOCIATION TEST
Asymptotic One-sided p-value: 0.0512
Exact One-sided p-value: 0.0357
Clearly, the assumptions underlying the asymptotic procedure (large enough sample to reasonably be expected to approach the asymptotic distribution, few zero counts in the cells) are not met. The asymptotic p-value is 0.0512, notsignificant at the 0.05 level. But with data this sparse can you really trust the asymptotic p-value? Fortunately, you don't need to! The exact p-value is computed by StatXact to be 0.0357, significant at the 0.05 level.
A litigator relying on the asymptotic procedure (inappropriately) would reach the unjustified conclusion that there is no statistically significant relationship between age and performance ratings. The exact procedure reveals that there is, in fact, a statistically significant relationship between age and performance ratings!
If your data sets are small, unbalanced, or oddly distributed you should be using exact methods. Don’t let your data analysis suffer from any of these deficiencies. Exact methods are always appropriate, regardless of sample size, balance or distribution.
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