Statistical Methods in Medical Research 4ed

310 Analysing non-normal data
mG ˆ antilog(E(log X)). If ln X is exactly normal (i.e. X has a log-normal
distribution), with mean and standard deviation on the log scale of mL and sL,
the geometric mean of X is exp…mL† whereas the arithmetic mean is exp
…mL† exp…1 2 s2L †, which exceeds the geometric mean, as is always the case. As
mL is the median of ln X, exp…mL† is also the median of X.
Differences between arithmetic means on the log scale, for example differences
between treatment groups, will estimate differences between logged geometric
means; so, for example, when comparing groups 1 and 2, the difference in
arithmetic means on the log scale will estimate
log mG…Group 1† log mG…Group 2† ˆlog mG…Group 1†
mG…Group 2† :
Thus, the antilog of the differences of the arithmetic means on the log scale
estimates a ratio of geometric means.
While the antilogs of arithmetic means computed on the log scale are readily
interpreted, there is no straightforward interpretation available for the antilog of
the standard deviation of the log Xi. If confidence intervals for geometric means
or ratios of geometric means are required, then the usual confidence intervals
should be computed on the log scale. Taking antilogs of the ends of these
intervals will yield confidence intervals on the original scale or for the ratio.
Example 10.10
The treatment of some skin diseases involves exposing patients to doses of ultraviolet
radiation. Too high a dose of radiation will burn the patient, so before treatment
commences it is important to expose small areas of skin to increasing doses to establish
at what dose a patient will burn. The minimum dose which just causes reddening of the
skin is known as the minimum phototoxic dose, MPD (in J=cm 2 ). Data on MPD were
collected from 51 patients with fair skins and from 44 patients with darker skin (categorized
using standard dermatological criteria): the data have kindly been made available by
Dr P.M. Farr. The mean and standard deviations are:
Arithmetic mean (J=cm 2 ) Standard deviation (J=cm 2 )
Fairer skin 2 60 2 52
Darker skin 3 34 3 56
While patients with fairer skin appear to burn at lower doses, it is also clear that the data
are far from normally distributed. Negative MPDs are impossible and the standard
deviations are approximately equal to the means. This hampers formal assessment of
the difference between the groups. Taking logarithms, to base 10, and computing means
and standard deviations of the logged values gives the second and third columns of the
following table. MPD has been extensively studied and there is good external evidence