1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

a linear function is not restricted. We will therefore take a necessarystep to further complication supposing thatp(x 1 , ..., x k ) = f(a 0 + a 1 x 1 + ... + a k x k )where f is a function of one variable changing from 0 to 1.There still remains the problem of choosing f; many considerationsof simplicity show that most convenient is the so-called logisticfunction1f ( y) = .y1 + e −Finally, changing notation to bring it in correspondence with the citedwork, we have the main hypothesis in the form1p( x1,..., xk) =.k1−exp[ α β x ]− −∑i=1ii(2.9)This function is called the multidimensional logistic function. Wehave certainly not proved that the probability sought, p, must beexpounded by (2.9), but arrived at that function without making anylogical absurdities.After formulating the main hypothesis (2.9) the parameters of thatfunction ought to be estimated and it is here that the authorsdeliberately admit a logical contradiction. They suggest the model of amultidimensional normal distribution for the results of the examination(2.8). This is obviously impossible because two of the seven factors,NNo. 6 and 7, are measured in discrete units so that normality isformally impossible. Then, it is rather strange to suppose that age isnormally distributed. In general, unlike the small illogicalities ofchoosing a statistical test when data are already available (§ 2.1), herewe see a serious corruption of logic which can only be exonerated bythe result obtained (cf. the proverb: Victors are not judged).More precisely, the main hypothesis consisted in that there are twomany-dimensional normal totalities, one consisting of the observationsof the risk factors for those who were not taken ill during the next 12years, the second concerned those who were. A problem is formulatedabout the methods of distinguishing these totalities.The classical supposition of the discriminant analysis states that thecovariance matrices of both totalities are the same which leads (arather remarkable fact!) to the expression (2.9) which we arrived at byconsiderations of simplicity. Nowadays this probability is understoodas the posterior probability of being taken ill given that theobservations provided values (2.8) of risk factors. This time, however,the authors also obtained a method of estimating the parameters of(2.9). It is illogical to the same extent as the supposition of normality.Our own reasoning which first led us to the expression (2.9) wouldhave led us to a quite another and more complicated in the109