The logarithmic transformation and the geometric mean in reporting experimental IgE results: what are they and when and why to use them?

BACKGROUND

Immunologic data, such as IgE and interleukin 4, tend to have positively skewed distributions with a long tail of larger values. This renders analyses based on normal distribution theory questionable (eg, t tests and analysis of variance) and distorts the sample mean as a measure of central tendency. These problems can be addressed through analysis of log-transformed data. Data analyzed in this fashion are summarized with the geometric mean.

OBJECTIVE

To elucidate the use of the logarithmic transform and the geometric mean in the analysis of immunologic data.

METHODS

The analysis may be conducted by transforming the data to a logarithmic scale to achieve a bell-shaped (approximately normal) distribution. The bell-shaped distribution needed to validate statistical inferences is only achieved in the transformed scale. In summarizing the research findings, the statistical analyst usually will transform means and confidence intervals from the logarithmic scale back to the original scale of measurement. Statistical inferences in the log scale remain valid for the data. The result of back transforming the mean of logarithmic values to the original scale is the geometric mean. This statistic is less subject to distortion by the unusually large values in the tail of the positively skewed distribution of the data.

RESULTS

A brief example is used to illustrate this type of analysis.

CONCLUSIONS

Logarithmic transformation permits valid statistical inference for positively skewed immunologic data. A result of this analysis is the geometric mean, which is a better measure of central tendency of this data type than the usual sample mean.