Comparison of two medians using a two‐sample lognormal model in environmental contexts

A test procedure is derived for comparing medians in a lognormal two‐sample context where some data may be left‐censored owing to non‐detects. This procedure is an extension of the method introduced by Harris (1991). It is assumed that two independent samples of sizesn1andn2are available from two lognormal populations LN(μ1,σ1) and LN(μ2,σ2). The EM algorithm is used to obtain maximum likelihood estimates of the parameters in four cases:H1(μ1= μ2,σ1= σ2),H2(μ1≠ μ2,σ1= σ2),H3(μ1= μ2,σ1≠ σ2), andH4(μ1≠ μ2,σ1≠ σ2). A protocol is described for testing the equality of the two medians (H0: μ1= μ2versusHr: μ1≠ μ2). Two procedures are recommended for this test, depending on whether the population skewness parameters are equal: (H0: σ1= σ2) or not (Hr: σ1≠ σ2). Asymptotic chi‐square tests are used in the recommended protocol. Similarities and differences between these lognormal methods and their two‐sample normal symmetric counterparts are discussed with particular reference to the role of the assumption σ1= σ2in both contexts. A detailed case study example of the method is provided using data contained in Millard and Deverel (1988). Comparisons are made between the lognormal methodology introduced here and the non‐parametric methods used by Milard and Deverel (1988) in the analysis of these data.