Mathematical aspects of the paternity index I = X/Y, especially in relation to the chance of non-exclusion of non-fathers

In a previous paper the author mentioned some aspects of the paternity index I (= X/Y): Among false triplets the frequency of those with I equal to or higher than an (observed) I value of Ix is considerably lower than 1/Ix; among false triplets the mean value of I is equal to 1, and among non-excluded non-fathers it is equal to the inverse of the chance of non-exclusion; among true triplets the mean value of 1/I (= i) is equal to the chance of non-exclusion of non-fathers. In a statistical material rather strong deviations from some of these expectations were observed. In the present paper further characteristics of the distribution of I values were taken into consideration, and especially those that should hold if lnI would fit in with a normal distribution. It was supposed that with the aid of such a distribution the deviations mentioned above could be recognized as chance variability. It appears, however, that neither the logarithms of the paternity index, nor those of the zygosity index of twins (chosen as an analogous model that is more easily analysable than the paternity index) are really normally distributed. This, in turn, makes that estimates of probability of paternity, based on such a supposition, are of doubtful reliability. Besides it is concluded that also for other reasons other estimates than Essen-Möller's W (or I or i), as probability of first type errors, lead in practice to conclusions that are equally subdue to a priori suppositions as are W values and may be, in fact, much more erroneous than those. Special attention is paid to the statistical analysis of paternity studies with more than one alleged father, and it is concluded that in such cases the general formula that may be considered to be equivalent with Essen-Möller's formula for one-man paternity cases, i.e., W = X/(X + Y) or I/(I + 1), must be W1 = I1/(sigma I + n); W2 = I2/(sigma I + n) etc. and certainly not W1 = I1/(sigma I + 1); W2 = I2/(sigma + 1) etc.