The correct application of Mage's Johnson SB procedure for fitting exposure data

Abstract

Flynn (2006) made an excellent review of different methods of applying the Johnson S B distribution, also known as the four-parameter lognormal distribution, to exposure data. He applied the Mage (1980a) percentile solution procedure for estimating S B distribution parameters (Johnson, 1949) and compared it with four other methods for fitting the S B model. However, he concluded that Mage's percentile procedure did not always return values for the four S B parameters, and that ''the quantile and method-of-moments fitting procedures may provide performance superior to that of the percentile method.'' For y ¼ (x-X min )/(X max -x), where x is an exposure variable bounded between a minimum value (X min ) and a maximum value (X max ), the S B probability density function p(x) is written as Eq. (1), where m and s are the mean and SD of log(y), respectively: pðxÞ ¼ ffi p ð2p s 2 Þ À1 ½ðX max À xÞ À1 þ ðx À X min Þ À1 expð À 0:5½ðlogðyÞ À mÞ=s 2 Þ;