Difficulties with the lognormal model in mean estimation and testing

A frequent assumption in environmental risk assessment is that the underlying distribution of an analyte concentration is lognormal. However, the distribution of a random variable whose log has a t-distribution has infinite mean. Because of the proximity of the standard normal and t-distribution, this suggests that a distribution such as the gamma or truncated normal, with smaller right tail probabilities, might make a better statistical model for mean estimation than the lognormal. In order to assess the effect of departures from lognormality on lognormal-based statistics, we simulated complete lognormal, truncated normal, and gamma data for various sample sizes and coefficients of variation. In these cases, departures from lognormality were not easily detected with the Shapiro-Wilk test. Various lognormal-based estimates and tests were compared with alternate methods based on the ordinary sample mean and standard error. The examples were also considered in the presence of random left censoring with the mean and standard error of the product limit estimate replacing the ordinary sample mean and standard error. The results suggest that in the estimation of or tests about a mean, if the assumption of lognormality is at all suspect, then lognormal-based approaches may not be as good as the alternative methods.