"Recent new data on old age mortality point to a particular model for the way in which the probability of dying increases with age. The model is found to fit not only modern data but also some widely spaced historical data for the 19th and 17th centuries, and even some estimates for the early mediaeval period. The results show a pattern which calls for explanation. The model can also be used to predict a probability distribution for the highest age which will be attained in given circumstances. The results are relevant to the current debate about whether there is a fixed upper limit to the length of human life." A discussion of the paper by several researchers and a reply by the author are included.
Kannisto (2001) has shown that as the frequency distribution of ages at death has shifted to the right, the age distribution of deaths above the modal age has become more compressed. In order to further investigate this old-age mortality compression, we adopt the simple logistic model with two parameters, which is known to fit data on old-age mortality well (Thatcher 1999). Based on the model, we show that three key measures of old-age mortality (the modal age of adult deaths, the life expectancy at the modal age, and the standard deviation of ages at death above the mode) can be estimated fairly accurately from death rates at only two suitably chosen high ages (70 and 90 in this study). The distribution of deaths above the modal age becomes compressed when the logits of death rates fall more at the lower age than at the higher age. Our analysis of mortality time series in six countries, using the logistic model, endorsed Kannisto’s conclusion. Some possible reasons for the compression are discussed.
Given the number of successes in a random sample, prediction limits can be determined for the number which will be observed in a second sample, in a way which does not depend on any assumption or inference about the unknown proportion in the population. Such "confidence limits" for the prediction are found to correspond to Bayesian solutions based on two particular prior distributions, and are related to Laplace's rule of succession. The results suggest a possible type of "prediction strategy".
The paper describes an application of Bayes’ Theorem to the problem of estimating from past data the probabilities that patients have certain diseases, given their symptoms. The data consist of hospital records of patients who suffered acute abdominal pain. For each patient the records showed a large number of symptoms and the final diagnosis, to one of nine diseases or diagnostic groups. Most current methods of computer diagnosis use the “Simple Bayes” model in which the symptoms are assumed to be independent, but the present paper does not make this assumption. Those symptoms (or lack of symptoms) which are most relevant to the diagnosis of each disease are identified by a sequence of chi-squared tests. The computer diagnoses obtained as a result of the implementation of this approach are compared with those given by the “Simple Bayes” method, by the method of classification trees (CART), and also with the preliminary and final diagnoses made by physicians.
Frequency distributions of the wages and salaries of individual employees can be obtained from wagecensuses, tax records and the Family Expenditure Survey. These various sources are described and compared. Some old and new data are presented which illustrate current variations in earnings and also confirm previous findings. Some of the main distributions are of the lognormal and Champernowne forms, and their relative dispersions show a very marked stability over time. In the case of manual workers, the weekly and hourly earnings of both men and women appear to have almost identical distributions, apart from scale factors. Existing theories do not seem to provide a complete explanation of the empirical results. 1968]THATCHER -Distribution of Earnings of Employees 135 3. SOURCES OF DATA The Board of Trade and Ministry ofLabour SurveysThe earliest surveys of the distribution of weekly earnings were known as "Wage Censuses". These were confined to manual workers, and were made by obtaining details from large numbers of employers about the earnings of each of the manual workers on their payrolls. This method was first used by the Board of Trade in 1886. Owingto the expense involved, full-scale surveys of this type have been very infrequent and only three others have been held, in 1906, 1938 and 1960. Their coverage was extensivebut not complete: in particular they did not include agriculture, coal mining, distributive trades or miscellaneous services, though in 1960 distributions for coal miners and agricultural workers were available separately.
The 'law of mortality' proposed by Heligman and Pollard is compared with the law of Gompertz and with English Life Table No. 14. Some new mathematical results are derived, including specific equations for the curve of deaths. Some numerical illustrations are given.
The pattern of mortality at the highest ages has been considered by many authors, including Redington (1969), Humphrey (1970) and Benjamin (1964, 1982). The questions raised have included the following:(a) Is there a definite upper limit to the span of human life, so that qx reaches unity at a finite age? Or does qx tend gradually to unity as age tends to infinity, as happens under the Gompertz and Makeham laws? Or does qx tend to a constant less than unity, as under the Perks formula or the formula which was used to graduate the English Life Tables No. 11 and 12?(b) Has the fall in mortality rates at lower ages been accompanied by a similar fall at the very highest ages? Has the upper tail of the curve of death (μxlx) shifted?(c) Does the lower mortality of females compared with males extend to the very highest ages, or do the rates eventually tend to converge?
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