On the Problem of Comparing the Means and Medians of Two Independent Lognormal Distributions

Jos´E A Montoya,Gudelia Figueroa,Gerardo Navarro

doi:10.6339/jds.201710_15(4).00004

Abstract

Inferences about the ratio of two lognormal means δ can depend on plausible values of ρ, the ratio of the normal standard deviations associated to these distributions. This aspect is not usually considered in most of the analyses carried out in some applied sciences. In this paper we propose a profile likelihood approach that allows the comparison of two independent lognormal data sets in a more exhaustive way. Inferences about δ, ρ and (δ, ρ) are jointly analyzed through a simple closed-form expression obtained for the profile likelihood function of the parameter vector (δ, ρ). A similar analysis is done for ψ and ρ, where ψ is the ratio of two lognormal medians, obtaining also a simple closed-form expression for the profile likelihood function of these parameters. These expressions allow us to construct likelihood contour plots that capture most of the information provided by the samples and become valuable to identify if a trade-off between the parameters under study occurs; in case of that, individual inferences should be analyzed carefully. A detailed series of Monte Carlo simulations are included; they illustrate the performance of profile likelihood and parametric bootstrap approaches, for different sample sizes and parameter values.

Highlights

Lognormal distributions occurs frequently in various applications coming from ecology, biology, hydrology, medicine, human behavior and many other scientific fields, where measurements under analysis are positive, with a small mean and a large variance
The simple closed-form expression developed in Section 2 and all the features included in the likelihood contour plot, afford the comparison of two independent lognormal data sets in a more exhaustive way
As we have seen, elongated or triangular contour shapes are indicative of a trade-off between parameters δ and ρ, so caution must be taken when summarizing individual inferences by using a single p-value or its corresponding confidence interval

Summary

Introduction

Lognormal distributions occurs frequently in various applications coming from ecology, biology, hydrology, medicine, human behavior and many other scientific fields, where measurements under analysis are positive, with a small mean and a large variance. That explains the large list of procedures that have been proposed to prove the equality of these population means. These approaches usually consider the difference or the ratio of these means, as the parameter of interest, and a decision is generally taken by constructing a confidence interval or by doing a hypothesis test for one of these parameters. A procedure focused on a difference of these means is the one proposed by Chen (1994), where the author explored the advantages of a new test over a t-test based on log-data. Comparing the means of two lognormal distributions is still a problem of interest and different test statistics are proposed and compared to measure, in a certain way, their performance (Jiang et al, 2014)

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