## 3. Mortality decline over time

I start with an analysis of death trends over time for single years of age partitioned into ten-year age groups. Because the age structures of the male and female populations are different I need to account for such differences analyzing death rates over such broad age categories. This can be accomplished by using the following intensity regression model [Note 1].

 AGE + YEAR*SEX (1)

In this model, estimates of the AGE effect will correspond to the standard age-specific schedule of death rates, and the interaction term will produce estimates of the relative trends in death rates, separately for males and females. The later is closely related to trends in age-standardized mortality rates except for the fact that we don't need to select a standard population because it is estimated indirectly by regression procedure. The estimation procedure uses a maximum likelihood method known in statistical literature [i.e. Kleinbaum et al. 1998] as the Poisson regression model. Note that all independent variables (YEAR, AGE, SEX) are factors, that is, one parameter is estimated for each factor level. All necessary constraints are also taken into account in order to provide model identifiability.

First, to capture the general pattern in mortality decline I fitted Model (1) to all ages. The female population in the year 1921 was selected as a reference group, which means that the AGE effect is close to the female death rates observed in that year. This estimated schedule of death rates, shown in Figure 1a, appears similar to those of other countries: a high peak of infant mortality, a bump of mortality in the 20s and then an exponential increase until the highest ages. The attentive reader will also have noticed "steps" of death rates at ages 40, 50, 60, 65, 70, and 80 - these are due to age heaping at these ages.

The interaction term of the fit (YEAR*SEX) is plotted in Figure 1b on the logarithmic scale. Every point on this graph can be interpreted as a ratio of the standard mortality schedule (Figure 1a) to a mortality schedule observed in a particular year and population. For example, the estimate for the male population in 1921 (first point on the blue curve) is 1.09, which means that male mortality in 1921 was on average 9% higher than female mortality in the same year. If we look at the last points reached by both curves in 1997, for females the estimate is 0.25 and for males 0.40. Thus, the level of female mortality in 1997 is about 25% and that of male mortality in 1997 is about 40% of the female mortality level in 1921.

As can be seen in Figure 1b, until the mid-1930s the excess of male mortality is rather small and the rate of decline is comparable to that of the female population. From this time onwards, female mortality started to decline faster than male mortality, the level of which nearly stagnated in the mid-1950s. Such differences in mortality trends led to the emergence of a gap between male and female death rates, which peaked in the mid-1970s. In later years the decline in female mortality slowed, however, while male death rates decreased at a higher rate. In most recent years the death rates for males and females are converging, thereby reducing the gap which had emerged before.

It would be an oversimplification to expect that Model (1) will fit the entire data set well, but it conveniently summarizes the sex differentials in mortality decline over time in one graph - much the same as it is summarized by life expectancy at birth. An analysis of goodness of fit, which is not included here, shows that the model, indeed, does not fit the data very well [Note 2], but nevertheless the results are included in the report because they are helpful for revealing the general trends.

The trends in individual age groups can be quite different from the average shown in Figure 1. To explore the dynamics of death rates more closely I fitted Model (1) separately for 10-year age groups and plotted the interaction term in a similar fashion in Figure 2. The AGE effects are not shown in this figure since AGE has been used only as a control variable.

We can see that the most striking reductions in mortality occurred in infancy (age 0) and childhood (1-9 years of age), where death rates declined markedly and remarkably uniformly for both sexes, thus making the sex differential in mortality at these ages very small at the end of the 1990s. For females, the estimate in 1997 at age 0 is about 0.04 which means that the death rate at this age fell by a factor of 25 since 1920. In the most recent year death rates continued to decline at a similar rate as before, even though they had already reached exceptionally low levels until 1990.

Trends at young and young adult ages (10-19, 20-29, 30-39, and 40-49 years of age) show a pattern quite different from that found in other age groups. Until the mid-1950s death rates fell extremely rapidly both for males and females, but then they stagnated to a large extent and even increased for males at ages close to 20 in the 1970s and at ages 30-39 in the 1990s. Starting with the year 1980 the mortality decline for males and females accelerated again and death rates resumed their decline at a rate comparable with that before 1950. Another interesting feature of these graphs is that female mortality exceeded male mortality in the 1930s (Figures 2d and 2e). This excess of female mortality is not found on any of the other graphs. Males suffer from higher mortality throughout the whole period and age range analyzed.

At ages 50-59 and 60-69 female mortality declined more or less at a constant rate over the whole period of observation. This is indicated by almost linear downward trends in the female death rates (Figures 2g and 2h). In the male population at the same ages, trends in death rates have been quite different. Until 1970 virtually no changes in male mortality are to be observed. The death rates stayed at an almost constant level. It was only after 1970 that they started to decline. The rate of decline, however, was higher than the rate of decline observed in the female population during the period 1970-1997. This has led to a convergence of the levels of male and female death rates in the most recent years.

At older ages (Figure 2i and 2j) death rates decline more or less uniformly for both sexes. The level of mortality was almost the same for males and females in the 1920s, but due to the higher rate of mortality decline in the female population, the curves diverged over time. At the end of the 1970s mortality decline in the male population at ages 70-79 accelerated appreciably and the gap between the sexes was reduced in this age group.

In summary, the analysis conducted here demonstrates usefulness of the proposed statistical model for exploring age-specific mortality trends. It also helps to reveal details in the trends and get a deeper understanding of mortality developments. By using more rigid models aiming at capturing general trends in age-specific mortality, however, some important details can be lost. For example, the recent application of the Lee-Carter model [Lee and Carter 1992] by Tuljapurkar to mortality data for G7 countries led to the conclusion that "In every country over this period [Note 3], mortality at each age has declined exponentially at a roughly constant rate" [Tuljapurkar et al. 2000]. If this would be the case the curves shown in Figure 2 would be well approximated by a linear function for the years 1950 onwards. However, it is obvious that the pattern of mortality decline is much more complicated, especially for males. At ages 50-70, for example, it is clear that male death rates remained on a constant level until the late 1970s and only then started to decline. It seems to be the case that the Lee-Carter model summarizes information in matrix of death rates in some important ways but it doesn't provide a good fit to the data.

 Sex differentials in survival in the Canadian population, 1921-1997: A descriptive analysis with focus on age-specific structure Kirill Andreev © 2000 Max-Planck-Gesellschaft ISSN 1435-9871 http://www.demographic-research.org/Volumes/Vol3/12