Statistical Methods in Medical Research 4ed

that the logarithm of the MPD does closely follow a normal distribution (MacKenzie,
1983).
Arithmetic mean,
logged values
Standard deviation,
logged values
10.8 Transformations 311
Geometric mean,
original scale (J=cm 2 )
Fairer skin 0 225 0 414 1 68
Darker skin 0 268 0 478 1 85
The antilogs of the arithmetic means of the logged values, i.e. the geometric means, are
shown above and can be seen to be noticeably smaller than the arithmetic means in the
previous table. The standard deviations on the log scale are similar and the null hypothesis
that the population geometric means are equal can be tested using the usual two-sample t
test (see §4.3). Applying the t test to the logged values gives t ˆ 0 47, P ˆ 0 64. The
difference in the arithmetic means of the logged values is 0 043, and a 95% confidence
interval for the difference is … 0 224, 0 139†. Taking antilogs of the difference, and of
the limits of the confidence interval, gives 0 91 for the ratio of geometric means (confirmed
by direct calculation because 1 68=1 85 ˆ 0 91) and the 95% confidence interval for the
ratio is (0 60, 1 38).
Of course, a logarithmic transformation can only be applied to a positive
number, and associated quantities, such as geometric means are only defined for
positive variables. Even if the sample values are generally positive the application
of a log transformation can be ruled out because of occasional zero values. Even
if a value of zero is physiologically impossible recorded zeros can arise because
actual levels are below limits of detection. There are no simple solutions to this
difficulty. Ad hoc solutions, such as replacing zero values by the limit of detection
(or perhaps half the limit of detection) if the limit is known, may prove satisfactory.
Adding a constant k to each xi before taking logs is an alternative, although
it can be difficult to decide on the value for k and the results of the analysis can
be disconcertingly sensitive to the choice of k. Treating k as a parameter to be
estimated by the data is beset by theoretical problems.