The normal density (2) is frequently used as mixture kernel and appropriate for our application. However, if necessary it allows easy extensions either in the mean structure or the variance-covariance structure. For one, one could allow component–specific variances leading to

. In addition, one could also think of giving up independence within area

*i* leading to a multivariate normal distribution with either structured or completely unstructured variance–covariance matrix

*Σ*. Furthermore, one could think of modelling component–specific variance–covariance matrices

*Σ*
_{
j
}. For two, instead of using a common slope model this could be generalized to component-specific slopes

*β*
_{
j
} leading

*f*(

*y*
_{
i
}|

*α*
_{
j
},

*β*
_{
j
},

*σ*
^{2}) or

. The E-step of the EM algorithm has to be changed appropriately, in the case of a common variance parameter

*σ*
^{2} leading to

which reflects the fact that there are component–specific slopes and

However, for our data constellation the proposed model (2) is appropriate as also Figure 1 suggests. A more rigorous analysis for this assumption requires fitting the model with random intercepts only and also the model allowing the slopes to be random as well. This has be done using a mixed model approach with a normal random effects assumption. The BIC-values associated with the model fits support the random-intercept only assumption.

Also, we have looked at the potential for curvature. This would correspond to an asymptotic change in life expectancy growth and relates to the discussion in Oeppen and Vaupel [1]. For males, the log-likelihood for the model with a quadratic term for year is -2*ℓ* = 2919.0, whereas the log-likelihood for the model without the quadratic terms is -2*ℓ* = 2920.2, leading to a likelihood ratio test statistic value of 1.2 with p-value 0.27, clearly not significant. For females, we have similar results.

A qualitatively different approach follows a conditional autoregressive model (CAR) which was originally suggested by Clayton and Kaldor [19] and more recently modified by Rasmussen [20]. In principle, the idea could be also utilized for spatial-temporal modelling as in this case and tries to implement a smoothing element by utilizing spatial information. The key element of CAR models is to model mean and covariance structure of the random effect (here the intercept in the temporal straight line model) by neighboring information. The ultimate goal is to reach a smooth map of the measure of interest (here level of life expectancy growth). This approach is quite meaningful, in particular, if the underlying process is likely to have a smooth characteristic. In our case, however, we were more interested in identifying a potential clustered structure in life expectancy growth for which we thought the likelihood based cluster approach is more appropriate. Hence we have not followed up on CAR models in this case.

In North Rhine-Westphalia (NRW), there is an apparent continuous rise in life expectancy at birth in men and women within the last twenty years. However, this pattern needs to be contemplated differentially. Our analysis shows that in North Rhine-Westphalia, life expectancy is predominantly higher in rural than in urban districts and differs considerably by region. Within the observed period from 1990 to 2010, levels of growths of life expectancy ranged from 70.3 to 73.7 years in men and from 77.3 to 80.2 in women. Life expectancy in the 54 districts was influenced by a latent categorical variable, which consists of seven categories or clusters. Each of the 54 districts is allocated into one of the seven clusters. This latent variable might be a surrogate variable for socio-economic factors. Life expectancy, as well as its counterpart mortality, strongly depends on factors like education, income, occupational status in addition to the factors sex and age. Most recent analyses of the European Prospective Investigation into Cancer and Nutrition (EPIC) showed that total mortality among men with highest education level is reduced by 43% compared to men with the lowest (hazard ratio (HR): 0.57, 95% confidence interval (CI) 0.52 – 0.61). Among women, the reduction was 29% (HR 0.71, 95% CI 0.64 – 0.78). In men, social inequalities were highly statistically significant for all causes of death examined. In women, the authors found a less strong, but statistically significant association with social inequalities for all causes of death except for cancer mortality and injuries (Gallo *et al.*[3]). For the region 29 (Gelsenkirchen), we found the lowest life expectancy for both, men and women. Socio-economic factors (see also *Health reporting unit at the NRW Centre for Health*[21]) support this finding and point to possible underlying causes of this result. For Gelsenkirchen, the lowest disposable income per inhabitant in NRW is documented (2009: 15,905 Euros / inhabitant; 80.8% of NRW average) as well as the highest rate of persons drawing unemployment benefits (12,189.7/100,000 inhabitants in 2009). In Gelsenkirchen, we observed the highest death rate per 100,000 inhabitants in 2010: 1,338.7 (Standardized Mortality Ratio (SMR): 1.17; NRW in total: 1.00). In 2009, only in Herne (47) and Dortmund (44) the proportion of smokers was higher (Gelsenkirchen: 31.4%; Herne: 35.0%; Dortmund: 32.3%). The opposite extreme of longest life expectancy for both sexes was found for two cities. Newborn girls and boys can expect the longest life in the regions 17 (Bonn) and 30 (Münster). In 2010, for Bonn and Münster the lowest SMRs of all NRW districts were documented (SMR: Bonn 0.83 / Münster 0.87). In contrast to Gelsenkirchen these cities have the lowest rate of persons receiving unemployment benefits (Bonn: 5,738.5/100,000; Münster: 5,090.3/100,000). Large universities are based in Bonn and Münster with thousands of students as city inhabitants. Therefore, the disposable income per inhabitant is above NRW average, but other regions show higher income rates. The proportion of smokers is relatively low in both cities (Bonn: 22.9%; Münster: 23.7%). In 2009, only in five rural districts the proportion of smokers was lower. Results for men and women differ slightly, as was reported for social inequalities in the EPIC cohort, too (Gallo *et al.*[3]). In men, besides Gelsenkirchen the cities Duisburg (2) and Oberhausen (7) are classified as regions with the lowest life expectancy of NRW. In women, it is only Gelsenkirchen.

These findings support results of a socio-spatial cluster analysis conducted in 2007 by Strohmeier *et al.*[2] which was mentioned already in the introduction. Based on social indicators six clusters were established for NRW, which classified the 54 districts into six types which were dubbed as follows: poverty pole, family zone, cities dominated by administrative and service units, rising regions / suburban counties, heterogeneous cities, heterogeneous rural districts. Like in our analysis, the poverty pole (representing areas which are in several ways socially deprived) included the cities Gelsenkirchen, Duisburg, and Oberhausen, but also the cities Dortmund and Herne which are all located in the Ruhr area.

In relation to the NRW health indicators the authors found a significantly lower male and female average life expectancy in the poverty pole. In our analyses also more cities, especially of the Ruhr area, are categorized into the clusters of lower life expectancy. The Ruhr area is an urbanized, high density area comprising 11 cities and 4 counties with about 5 million inhabitants, formerly characterized by heavy industry and now undergoing a structural change towards e.g. information technology and health care industry. An additional underlying cause for lower life expectancy in this area might still be environmental. The Heinz Nixdorf RECALL study (Fuks *et al.*[22]), which included 4,291 participants from the Ruhr cities Bochum (43), Essen (3) and Mülheim a.d. Ruhr (6), recently confirmed that residential proximity to high road traffic (≤ 50m) and road traffic noise exposure (24h mean noise (Lden) > 65 dB) have a tendency toward higher blood pressure and an elevated prevalence of hypertension. Data from this study also showed that a reduction in distance between residence and major roads by half was associated with a 7.0% (95% CI 0.1 – 14.4%) higher coronary artery calcification (CAC) (Hoffmann *et al.*[23]).

In a subgroup of the RECALL study population, participants residing in Essen (n=1,641) and Mülheim (n=1,742) for which digitized information on inner city roads was available, prevalence of coronary heart disease at high traffic exposure showed significantly elevated OR=1.85 (95% CI 1.21 – 2.84, adjusted for cardiovascular risk factors and background air pollution) (Hoffmann *et al.*[24]). Further analysis showed a stronger effect for men (OR=2.33, 95% CI 1.44 – 3.78), which might account for the difference among men and women in our cluster analysis. Another analysis of the RECALL data investigated if the association of road traffic exposure and subclinical cardiovascular disease might be modified by socio-economic characteristics of individuals or neighborhoods. Participants with low socio-economic status (SES) and simultaneous exposure to high road traffic had highest levels of CAC (Dragano *et al.*[25]). The prevalence of high CAC was 23.9% in higher-educated men with low traffic exposure but 37.7% in lower-educated men with high road traffic exposure (women: 22.0% vs. 28.1%).

The cluster analysis of life expectancy once more stresses the differences between urban and rural regions in North Rhine-Westphalia. The latent component categorizing the 54 districts into seven categories can be interpreted as a surrogate comprising several underlying factors. The results point to districts where an accumulation of problems has negative impact on health. For males, only three cities are classified into the lowest cluster category, with 5.4% of the total NRW population living there. For women, only Gelsenkirchen is classified into this cluster. Given the emerging insight into possible underlying causes, chances for these cities to improve their outcome may come into closer reach.