Reducing Parameters by Developing a New Mortality Model

Rajesh K. Chauhan, Planning Department, Uttar Pradesh, India

Efforts to model human mortality started over 277 years ago when De Moivre discussed a one-parameter model in 1725. Mathematicians have long given attention to modelling this phenomenon; demographers, of course, made a larger contribution to mortality modelling, and actuarial scientists also contributed to developing suitable calculation procedures. A range of models have been developed by various researchers; these models fit well in certain situations, such as for specific age segments or for developed-country data. The models are used in the graduation or smoothing of different functions of life tables or aggregated sets of mortality values. There are three major classes of mortality graduation: model life tables; relational models; and laws of mortality (mathematical equations). Model life tables are useful in providing some idea of mortality in the absence of adequate data. Though they are carefully constructed by analysis and evaluation of available data, they suffer from the errors of approximation in computing indices. In modern days, through censuses and demographic and health surveys, better data are more commonly available. Also developing countries today have had different demographic experiences from the developed countries because of the exchange of medical and contraceptive technology in the era of globalisation. Their mortality patterns therefore may differ from those of the now industrially advanced countries when they were developing medical technology. In fact, developing countries have more detailed data available than are required for choosing a model life table so these models are not worth pursing. But this model has been shown to be of limited value for modelling Indian mortality since the fits were not good at both ends of the age scale As the Brass logit life table system relates any mortality experience to the standard one. But this model has been shown to be of limited value for modelling Indian mortality since the fits were not good at both ends of the age scale. Although the Zaba and Ewbank et al. models are improvements on Brass, these models are not suitable for forecasting and the data limitations do not allow them to be applied to developing countries mortality. The Lee-Carter model, for which the first vector usually explains more than 95 per cent of the overall variation, allows forecasting from a time series of mortality. Data for many developing countries may not be available according to time series, this method can not be tried with them. Notably the method has not been tested for any developing-country data so far. Though this model represents the time index of mortality, it assumes that the age component remains constant over time. Among the laws of mortality, the Heligman-Pollard (H-P) model was successfully tested on the various data sets. Its logical capacity to model the three different components of mortality gives the model a theoretical base for the modelling. Unlike many other attempts it covers the entire age range. Among the others, which cover the entire age ranges, the H-P model has been fitted extensively to various mortality experiences across countries. The Heligman-Pollard model fits age patterns of mortality well, but problems arise concerning parameter variability; for example the parameters D takes on negative values. The values of the parameters E and F are also highly unstable. This instability seems to be resolved when the parameters A, B and C assume values of the right order (as suggested by Heligman and Pollard for Australian data). There seems to be a need to investigate this instability of estimation and possibly improve the model to increase the stability. When the H-P model is fitted to various data, it has been observed that the model provides a good fit in many cases but may also provide some negative parameters owing to the non-linear-weighted regression estimations. While weighted regression estimates depend on the values from the three different parts of the curve, it may be possible that stability in estimation may be achieved by a more parsimonious option. It was suggested by various researchers that by fixing the values of one or two parameters to a feasible constant, stability was achieved in estimation. Many researchers noticed that overparameterisation is a concern with the H-P model. Thus, reducing the number of parameters will help to achieve a more parsimonious model and hence resolve the problem of negative parameter estimation. It is the first step to achieving the parsimony by having the minimum possible number of parameters without reducing the accuracy in fitting. As discussed above, the Heligman-Pollard (H-P) model requires improvement to reduce the variation in parameter estimates when fitting to actual data. It is clear that the H-P model is correct in defining three phases of age-related mortality change. For the last phase, older-age mortality, as applied by the H-P model, two parameters are required: one showing the base mortality and the other the geometric rise in mortality. Again as specified by the H-P model, the middle phase, where the accident hump occurs, needs three parameters to show the location, severity and spread of the hump. This paper shows that the early, declining-mortality phase can be modelled with fewer parameters than three. Empirical evidence shows that the minimum of the mortality curve is within the age range 9-15 years in almost all populations. Mortality declines typically just after birth until it reaches its minimum level. During infancy the decline is much faster and this decline has been captured by mathematical formulae. In this paper a new statistical model for the first phase of mortality is proposed and tested. This model has the capacity to capture the declining phase of mortality until its minimum. The added advantages are that the new model effectively requires estimation of only one parameter when fitted as the first part in isolation and two parameters when fitting for the entire age range. The model itself is a probability density function. Data for Australian, Swedish and Japanese mortality have been used for testing. The new model of early-age mortality used in this study is similar to the H-P model; however, the new model has certain advantages over the H-P because of its robust fitting procedure. The new model remains simple, as only a single parameter estimation is required, contrary to past models which require the estimation of up to three parameters to achieve the same goal. When used with the entire age range, the new model also needs a scalar parameter included in the model to place the model in the right place on the vertical axis. The new model, a finite range model, was initially introduced for reliability analysis by Mukherjee & Islam (1983); it was used successfully to model the distribution of deaths in infancy by Chauhan (1997) using data from Sweden, the USA and India. Later Krishnamoorthy & Rajna (1999) affirmed that the model fits well for deaths in infancy and it also fits for under-five mortality. In this paper several statistical characteristics of the new model are derived and presented. Other advantages of using the new model are: (1) the model itself is a specific statistical function, a probability density function (pdf); in statistical analysis, pdf plays a wider and more significant role than a simple mathematical formula; (2) with the use of pdf as a formula, continuous representation of mortality becomes possible. The new age pattern of mortality for birth to pre-teen ages is tested with data sets from Australia, Japan and Sweden. Later entire age-range are fitted with the same single year data of mortality and parameters of model are estimated. For the comparison purposes H-P model was also fitted to the same data sources. The above analysis and comparison of observed and fitted probabilities of death suggest the following. As far as the pattern of errors is concerned the new model is similar to that of Heligman and Pollard along the no-error line. The concept of a no-error line is somewhat arbitrary however, as the observed data possess certain random errors due to undefined random causes. When random fluctuations occur, the models fit the data well, with a succession of small positive and negative errors. A fear that the use of a graduating formula might change the observed age pattern is unwarranted as shown especially in the application of the models to Swedish data. Where the purpose is to produce models that can be used to project future mortality, graduation of random fluctuations is desirable. On the basis of the trials in this paper it can be concluded that new mortality model can be used to graduate the declining first part of human mortality and the new model (modified Heligman-Pollard) can be used to graduate overall mortality. Several applications of the new mortality model are to be named as a. To remove awkward irregularities and inconsistencies; b. To make results more precise; c. To construct a life table; d. To aid inferences from incomplete data; e. To facilitate comparison; f. To aid forecasting.