Working with transformed data.
Kjaergard and Als-Nielsen [ ref 1] have carried out an interesting
study. They have chosen an important and original topic, used an
appropriate study design and chosen the correct method of analysis -
analysis of variance (ANOVA). ANOVA assumes that within each group the
data come from a normal distribution, with equal variance in each group.
In order to achieve this, the investigators say that they used a
logarithmic transformation of their dependent variable, a six point scale
of authors’ conclusions.
At this stage, it seems to me that something has gone badly wrong.
The results presented in this paper could not have been obtained from an
analysis of such a logarithmically transformed scale. It is not possible
to calculate standard errors, p-values or other inferential statistics for
the un-transformed original scale from the transformed scale. (If it were
possible, there would be no point in making the transformation!).
Statements such as: ‘the mean difference between groups was 0.48 (SE
0.13), p = 0.014’ are impossible to interpret in this context. The authors
have either made some kind of error in moving between the logarithmically
transformed scale and the un-transformed scale, or they did not use a
transformed scale after all.
The correct procedure when doing ANOVA on transformed data is to do
the analysis and calculate any inferential statistics using the
transformed data. The ANOVA can be presented by using one or more standard
analysis of variance tables, including F statistics, with appropriate
degrees of freedom, and p - values. Any contrasts can also be investigated
using the transformed scale. In this case, a multiple testing procedure is
used and this could be tabulated to show which contrasts were examined,
what their p-values were and how this compares with the Bonferroni value.
When the analysis is completed, it is possible to make more sense of
what it means by transforming certain points from the log scale back to
the original scale (by taking the exponential or anti-logarithm). For
example, means can be translated back to the original scale along with the
end points of the confidence intervals for the means. Typically, this will
give asymmetric confidence interval, with the mean value closer to one end
than the other. This is how I know that the figure showing means and
confidence intervals cannot be correct - the confidence intervals are much
At present, the results presented in this paper cannot be
interpreted. The authors need to consult a statistician and submit
corrections along the lines that I have suggested above.
1 Kjaergard L L and Als-Nielsen B. Association between competing
interests and authors’ conclusions: epidemiological study of randomised
clinical trials reported in the BMJ. BMJ 2002; 325: 249 (3 August)
Competing interests: No competing interests