Inferences on the means of lognormal distributions using generalized p-values and generalized confidence intervals

The lognormal distribution is widely used to describe the distribution of positive random variables; in particular, it is used to model data relevant to occupational hygiene and to model biological data. A problem of interest in this context is statistical inference concerning the mean of the lognormal distribution. For obtaining confidence intervals and tests for a single lognormal mean, the available small sample procedures are based on a certain conditional distribution, and are computationally very involved. Occupational hygienists have in fact pointed out the difficulties in applying these procedures. In this article, we have first developed exact confidence intervals and tests for a single lognormal mean using the ideas of generalized p-values and generalized confidence intervals. The resulting procedures are easy to compute and are applicable to small samples. We have also developed similar procedures for obtaining confidence intervals and tests for the ratio (or the difference) of two lognormal means. Our work appears to be the first attempt to obtain small sample inference for the latter problem. We have also compared our test to a large sample test. The conclusion is that the large sample test is too conservative or too liberal, even for large samples, whereas the test based on the generalized p-value controls type I error quite satisfactorily. The large sample test can also be biased, i.e., its power can fall below type I error probability. Examples are given in order to illustrate our results. In particular, using an example, it is pointed out that simply comparing the means of the logged data in two samples can produce a different conclusion, as opposed to comparing the means of the original data. (C) 2002 Elsevier Science B.V. All rights reserved.