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

Treating observations whose results depend on many factors isfraught with an absolutely general difficulty and overcoming it waspossibly the main finding of the work done. The point is that the resultof observation (in this case, the emergence of the IHD) is generallyconnected with the values of the risk factors in a barely understoodway. When there are a few such factors, one or two, say, the data areusually divided into intervals according to their value; in the mostsimple case, into two, but this is very crude and it is better to havemore.If each factor is subdivided into several levels, all theircombinations should be applied to form the appropriate groupsproviding the frequencies of the IHD being estimates of theprobabilities. These will indeed adequately describe the data(somewhat roughly because the values of the risk factors areconsidered approximately).For example, the contents of cholesterol can be considered on fourlevels [...], the values of the systolic blood pressure also on four levels[...]. We then arrange a two-dimensional classification [...] and obtain16 groups with the frequency of the emergence of the IHD calculatedin each of them not for all 4856 observations, but for their number inthe group which is 16 times smaller in the mean. Joining men andwomen together will likely be thought inadmissible so that the numberof observations becomes about twice smaller.In general, a modest number of observations of the order of ahundred (when having a great many total number of observations) willbe left for each frequency. But what happens if we add three moregroups of different ages? And four more according to the intensity ofsmoking? [..] As a result, we will obtain a classification with eachgroup containing at best one observation and cases of no observationsat all are not excluded. Consequently, we will be unable to determineany probabilities. [...]The same difficulty occurs in many technical problems concerningthe reliability of machinery established by several types of checks.Suppose that the results of the checks arex 1 , ..., x k (2.8)and we would like to derive the probability of failure-free work as afunction p(x 1 , ..., x k ). The attempt to achieve this by multivariateanalysis will be senseless.Let us see how this problem was solved in the Framinghaminvestigation. As far as it is possible to judge, its solution had anindisputable part, but the other part was absolutely illogical. This doesnot mean that it is in essence wrong, but that it possibly needs somespecification. The first part can be thus expounded.When having to do with several variables, their only well studiedfunction is the linear function; there exists an entire pertinent science,linear algebra, which also partly studies the function of the seconddegree. It would therefore be expedient to represent the unknownprobability p(x 1 , ..., x k ) by a linear function. This, however, isobviously impossible because probability changes from 0 to 1 whereas108