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

calculational sense method of estimation of those parameters, themethod of maximal likelihood. It can also be realized and it is believedthat, for the data given, both methods provide results very close toeach other. It is interesting, however, to see what practical conclusionswere made in the cited source. After estimating somehow the values ofthe parameters, we can apply formula (2.9) to find out the approximatevalue of the probability ˆp of developing the IHD for each examinedperson during the next 12 years.The highest probabilities were observed for men of 30 – 39 and 40 –49 years (0.986 and 0.742 that the IHD developed) and 50 – 62 years(0.770 did not develop). For women the probabilities of developing thedisease were 0.838 for ages 30 – 49 and 0.773 for ages 50 – 62 [that itdid not develop?]. To a certain extent these results refute the classicalsaying which served as the title of this § 2.2. True, it should be bornein mind that formally these figures concern forecasting alreadyhappened events. Such a forecast of future events is only possible ifthe coefficients of function (2.9) are roughly the same in another placeor time as those established by the authors. Such a supposition isprobable but not yet verified.Then, having arranged the set of values ˆp for each examinedperson, they can be subdivided into several equally numerous groups(of ten people, for example) such that those with the lowest values ofˆp are placed in the first of them, people having higher, still higher, ...values comprise the second, third, ... group. Had there been noconnection between the considered linear combination of risk factorswith the IHD, the number of cases of that disease in all groups wouldhave been approximately the same, but actually the emerged picture isdifferent in principle, see for example Table 3 borrowed from Truett etal (1967). The expected number of cases of the IHD was determinedby summing up the probabilities ˆp of all people in the appropriategroup.In that table, we are surprised first of all by the great difference(amounting to a few dozen times) between the sickness rate in groupsof high and low risk. Second, in spite of the obvious non-normality ofthe distributions, the results obtained by means of a normality modelagree well with the actual data. The isolation of groups of people witha higher danger of developing the IHD is thus possible by issuing fromthe most simple clinical examination (providing the listed above riskfactors). The same conclusion can be made when considering the datarepresented in separate age groups.However, it should not be thought that those results are reallysuitable for individual forecasts. Those can only be successful forcases of very high or very low individual risk ˆp but for all the totalitythe result would have been bad. This is connected with the IHDoccurring nevertheless rarely (11.8% in the mean for 12 years).Indeed, issuing from the values of ˆp we can only forecast the diseasein people with a sufficiently high ˆp and the opposite for all the others.When choosing the boundary of the group with the highest risk inTable 3 as the critical value of ˆp , a forecast of the disease would havebeen wrong in 100 – 37.5 = 62.5% of cases. And on the other hand110