How to estimate mortality trends from grouped vital statistics

Background

Mortality data at the population level are often aggregated in age classes, for example 5-year age groups with an open-ended interval for the elderly aged 85+. Capturing detailed age-specific mortality patterns and mortality time trends from such coarsely grouped data can be problematic at older ages, especially where open-ended intervals are used.

Methods

We illustrate the penalized composite link model (PCLM) for ungrouping to model cancer mortality surfaces. Smooth age-specific distributions from data grouped in age classes of adjacent calendar years were estimated by constructing a two-dimensional regression, based on B-splines, and maximizing a penalized likelihood. We show the applicability of the proposed model, analysing age-at-death distributions from cancers of all sites in Denmark from 1980 to 2014. Data were retrieved from the Danish Cancer Society and the Human Mortality Database.

Results

The main trends captured by PCLM are: (i) a decrease in cancer mortality rates after the 1990s for ages 50–75; (ii) a decrease in cancer mortality in later cohorts for young ages, especially, and very advanced ages. Comparing the raw data by single year of age, with the PCLM-ungrouped distributions, we clearly illustrate that the model fits the data with a high level of accuracy.

Conclusions

The PCLM produces detailed smooth mortality surfaces from death counts observed in coarse age groups with modest assumptions, that is Poisson distributed counts and smoothness of the estimated distribution. Hence, the method has great potential for use within epidemiological research when information is to be gained from aggregated data, because it avoids strict assumptions about the actual distributional shape.

Efficient estimation of smooth distributions from coarsely grouped data

Ungrouping binned data can be desirable for many reasons: Bins can be too coarse to allow for accurate analysis; comparisons can be hindered when different grouping approaches are used in different histograms; and the last interval is often wide and open-ended and, thus, covers a lot of information in the tail area. Age group-specific disease incidence rates and abridged life tables are examples of binned data. We propose a versatile method for ungrouping histograms that assumes that only the underlying distribution is smooth. Because of this modest assumption, the approach is suitable for most applications. The method is based on the composite link model, with a penalty added to ensure the smoothness of the target distribution. Estimates are obtained by maximizing a penalized likelihood. This maximization is performed efficiently by a version of the iteratively reweighted least-squares algorithm. Optimal values of the smoothing parameter are chosen by minimizing Akaike's Information Criterion. We demonstrate the performance of this method in a simulation study and provide several examples that illustrate the approach. Wide, open-ended intervals can be handled properly. The method can be extended to the estimation of rates when both the event counts and the exposures to risk are grouped.