##### 2.2 Methods
For Austria I estimated further life expectancy at age 50 by calculating the average of the exact ages at death; for Denmark further life expectancy at age 50 was calculated on the basis of life tables that were corrected for left truncation. This was achieved by calculating occurrence and exposure matrices which take into account an individual’s age on 1 April 1968. For example, a person who was 70 at the beginning of the study and who died at age 80 enters the exposures for ages 70 to 80 but is not included in the exposures for ages 50 to 69. When calculating the life tables, I estimated the central age-specific death rate based on the occurrence-exposure matrix for two-year age-groups. The corresponding life table mortality rate is derived by the Greville Method [9] .

For both countries, results are presented as the difference between the mean age at death of people born in a specific week and the average mean age at death in the study period.

If people born in a specific month experience a higher mortality risk than others, the distribution of birth dates of the total population changes with age [20]. To test this for Austria I compared the monthly distribution of the number of births in the years 1880 to 1907 with the distribution of birthdays among those who died at ages 90–99 in the years 1988 to1996. I also calculated for five-year age-groups the proportion of women in the 1981 census born in winter. For Denmark I compared the distribution of birthdays of the survivors of the cohorts born between 1899 and 1908 at ages 60–69 and 90–99.

To test whether the seasonal distribution of deaths explains the differences in life span, the distribution of deaths was "uniformized" in the Austrian data set. The dates of death were redistributed such that over the year they follow a uniform distribution, while their rank order remains unchanged. In other words, the weeks of death d(t) are sorted that

d(1)<=...<=d(t)<=...<= d(n), with n equals the total number of deaths. Under a uniform distribution deaths will be observed in each week. Then the k-th death is reassigned to week d(j), with , the integer part of the expression inside the brackets (e.g. if the total number of deaths n equals 1040 then in each week i=20 persons should have died. Then the 65th person who has died is reassigned to week 4.) Average age at death according to week of birth is then recalculated based on d(j). This redistribution slightly increases the life span of people who died at the beginning of the year while it reduces the life span of those who died at the end of the year. If the seasonal pattern in life span is in fact caused by the seasonal distribution of deaths alone, the differences in life span should then disappear.

The longitudinal nature of the Danish data set allows me to model , the force of mortality at age x, directly. The general mathematical specification of the model is

 {1}

where is a function of the baseline hazard ; the value of the covariate and the regression coefficient that measures the effect of the covariate i on the force of mortality.

In this model the baseline hazard is piecewise constant. The model assumes that the age-specific death rate for five-year age groups j follows a step function and that the death rate within the five-year age groups is constant.

Thus,

 {2}

with j = 50-54, 55-59, 60-64, 65-69, 70-74, 75-79, 80-84, 85-89, 90+

All other covariates are sets of binary variables that represent the levels of categorical covariates. The following covariates are included:

• month of birth [MOB]. This variable divides the year into thirteen periods, each consisting of four weeks. Week one starts at day one, week two at day eight, and so forth;
• time lived since the last birthday [TSLB]. This covariate divides the year into three periods: (1) the twelve weeks immediately following an individual’s birthday (the week of birthday is defined as week one), (2) week 13 to week 40 (28 intermediary weeks) and (3) the twelve weeks before an individual’s next birthday (week 41 to week 52). This variable is defined for each year of an individual’s life starting with 1 April 1968;
• current month, which is measured in 13 four-week periods;
• period: the years are divided into the six groups 1968-1974, 1975-1979, 1980-1984, 1985-1989, 1990-1994, 1995-1998.
• birth cohort: 1860-1879, 1880-1889, 1890-1899, 1900-1909, 1910-1918.

The first and the last covariate are independent of time. The other three covariates are time-varying. The second and the third variable account for two factors: first, for the possibility of a ‘birthday-effect’ and second, for the seasonality in mortality. If they are both included at the same time in one model, the model does not converge; probably this is due to the high correlation between the two variables [Note 1] .

Period and cohort effects cannot be included simultaneously in a model that accounts for age effects. Thus, four models were estimated. Model 1a includes sex, period, month of birth and current month; Model 1b uses the same covariates but adjusts for cohort factors instead of period factors. Model 2a includes sex, period and the second-order interaction term "time period since last birthday" x "month of birth"; Model 2b is similar to the latter model but adjusts for cohort factors. The two last models include a full set of second-order interaction terms between "month of birth" and "time period since last birthday" in order to account for the seasonal differences in the risk of mortality. The interaction is modelled by 38 dummy variables, each representing a [MOB, TSLB] pair, with the (born in week 1-4, 1-12 weeks since last birthday) cell omitted as the reference group. The survival models were estimated with the program Rocanova [25].

 Longevity and Month of Birth: Evidence from Austria and Denmark Gabrielle Doblhammer © 1999 - 2000 Max-Planck-Gesellschaft ISSN 1435-9871 http://www.demographic-research.org/Volumes/Vol1/3