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11.1 INTRODUCTION 537
TABLE 11.1.1 Data from a Random Sample of 25 Research
Subjects Tested for ≤-9-THC,
Example 11.1.1
Case No. Concentration ( Mg/ml) Log 10 Concentration ( Mg/ml)
1 .30 .52
2 2.75 .44
3 2.27 .36
4 2.37 .37
5 1.12 .05
6 .60 .22
7 .61 .21
8 .89 .05
9 .33 .48
10 .85 .07
11 2.18 .34
12 3.59 .56
13 .28 .55
14 1.90 .28
15 1.71 .23
16 .85 .07
17 1.53 .18
18 2.25 .35
19 .88 .05
20 .49 .31
21 4.35 .64
22 .67 .17
23 2.74 .44
24 .79 .10
25 6.94 .84
Box-and-whisker plots from SPSS software for these data are shown in Figure 11.1.1.
The raw data are clearly skewed, and an outlier is identified (observation 25). A log 10
transformation, which is often useful for such skewed data, removes the magnitude of
the outlier and results in a distribution that is much more nearly symmetric about the
median. Therefore, the transformed data could be used in lieu of the raw data for constructing
the regression model. Though symmetric data do not, necessarily, imply that
the data are normal, they do result in a more appropriate model. Formal tests of normality,
as previously mentioned, should always be carried out prior to analysis.
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Unequal Error Variances When the variances of the error terms are not equal,
we may obtain a satisfactory equation for the model, but, because the assumption that
the error variances are equal is violated, we will not be able to perform appropriate
hypothesis tests on the model coefficients. Just as was the case in overcoming the
non-normality problem, transformations of the regression variables may reduce the
impact of unequal error variances.