# Model for excess deaths (survival)

The model for the excess death rate is a piecewise Poisson Bayesian spatial relative survival model, closely based on that first described by Fairley et al. (2008), and since applied by others (Cramb et al. 2011, Cramb et al. 2012, Saez et al. 2012). The first stage is the likelihood model for the observed number of deaths \(d_{kji}\) in the \(k^{\text{th}}\) age group, the \(j^{\text{th}}\) follow up interval and the \(i^{\text{th}}\) area:

$$d_{kji} \sim \text{Poisson}\left(\mu_{kji}\right).$$

The second stage comprises

$$\text{log} \left(\mu_{kji}-d_{kji}^*\right)=\text{log}(y_{kji})+\beta_{0j}+\boldsymbol{x\beta}_k+R_i$$

where \(y_{kji}\) is person-time at risk, \(d_{kji}^*\) is the expected number of deaths based on population mortality, \(\beta_{0j}\) is the intercept (which varied by follow-up year), \(\beta_k\) is the coefficient of the predictor variable vector \(\boldsymbol{x}\) (representing the broad age groups, sex and cancer type), and \(R_i\) is the spatial component modelled with the Leroux CAR prior (Leroux et al. 2000).

The key output used in the maps from this model is the excess hazard ratio (EHR), which is calculated as \(\text{exp}(R_i)\). The model was run separately for males, females and persons. The covariates included were to prevent bias in the estimates due to differing age structures (4 age groups, representing ages 15-54, 55-64, 65-74 and 74-89 years), sex (only for persons, separated into males and females), or cancer types between areas (only for all cancers and head & neck cancers).

The input data for the excess death models are the observed number of deaths \(d_{kji}\), the expected population-level number of deaths \(d_{kji}^*\), and the person-time at-risk \(y_{kji}\).

As for the national summary, to calculate these estimates, relative survival methods were used. Patients who were still alive at 31st December 2014 were considered censored. Survival calculations only included people aged between 15 and 89 years at diagnosis. Patients whose cancer diagnosis was based on death certificate or autopsy only have also been excluded, as well as those with a survival time of zero days or a date of diagnosis after date of death.

The period approach and life table method were used to calculate observed survival, which also generates observed deaths (\(d_{kji}\)). This approach involves dividing the time after diagnosis into a series of discrete time intervals. Five annual intervals were used, and comparisons against models using quarterly intervals for the first year showed no differences in the small-area estimates. Person-time at risk (\(y_{kji}\)) was also obtained from this, as it reflects the sum of the patients alive within each discrete time interval.

Expected deaths \(d_{kji}^*\) (based on Australian unit record file mortality data obtained from the Registries of Births, Deaths and Marriages) was calculated using the Ederer II method (Ederer et al. 1961) for each 5-year age group (15-19,20-24,…,85-89 years), aggregated calendar year (2006-2010, 2011-2014), and groups of SA2s that were considered to be neighbours to enhance stability of the population mortality estimates.

## References

Cramb SM, Mengersen KL, Baade PD 2011. Atlas of Cancer in Queensland: geographical variation in incidence and survival, 1998 to 2007. Brisbane Viertel Centre for Research in Cancer Control, Cancer Council Queensland.

Cramb SM, Mengersen KL, Turrell G, Baade PD. Spatial inequalities in colorectal and breast cancer survival: premature deaths and associated factors. Health Place. 2012; 18(6):1412-1421.

Ederer F, Axtell LM, Cutler SJ. The relative survival rate: a statistical methodology. NCI Monographs. 1961; 6:101-121.

Fairley L, Forman D, West R, Manda S. Spatial variation in prostate cancer survival in the Northern and Yorkshire region of England using Bayesian relative survival smoothing. British Journal of Cancer. 2008; 99(11):1786-1793.

Leroux BG, Lei X, Breslow N. Estimation of disease rates in small areas: a new mixed model for spatial dependence. 2000. 135-178. In Halloran ME, Berry D (Eds). Statistical models in epidemiology, the environment and clinical trials. New York: Springer.

Registries of Births, Deaths and Marriages, the Coroners and the National Coronial Information System. Unit record mortality data for Australia by state of usual residence, 2006–2015. Brisbane, Australian Coordinating Registry, 2017.

Saez M, Barceló MA, Martos C, Saurina C, Marcos-Gragera R, Renart G, Ocaña-Riola R, Feja C, Alcalá T. Spatial variability in relative survival from female breast cancer. Journal of the Royal Statistical Society: Series A (Statistics in Society). 2012; 175(1):107-134.