Olivier Jake, Johnson William D, Marshall Gailen D
Division of Clinical Immunology and Allergy, Department of Preventive Medicine, University of Mississippi Medical Center, Jackson, MS, USA.
Ann Allergy Asthma Immunol. 2008 Apr;100(4):333-7. doi: 10.1016/S1081-1206(10)60595-9.
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.
To elucidate the use of the logarithmic transform and the geometric mean in the analysis of immunologic data.
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.
A brief example is used to illustrate this type of analysis.
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.
免疫数据，如免疫球蛋白E和白细胞介素4，往往具有正偏态分布，有一个较大值的长尾。这使得基于正态分布理论的分析（如t检验和方差分析）存在问题，并扭曲了作为集中趋势度量的样本均值。这些问题可以通过对对数转换后的数据进行分析来解决。以这种方式分析的数据用几何平均数进行汇总。
阐明对数转换和几何平均数在免疫数据中的应用。
分析可通过将数据转换为对数尺度来实现钟形（近似正态）分布。验证统计推断所需的钟形分布仅在转换后的尺度上实现。在总结研究结果时，统计分析人员通常会将均值和置信区间从对数尺度转换回原始测量尺度。对数尺度上的统计推断对数据仍然有效。将对数值的均值转换回原始尺度的结果是几何平均数。该统计量较少受到数据正偏态分布尾部异常大值的扭曲。
用一个简短的例子来说明这种类型的分析。
对数转换允许对正偏态免疫数据进行有效的统计推断。该分析的结果是几何平均数，它比通常的样本均值更能作为这种数据类型集中趋势的度量。
