Multilevel discrete-time | Bayesian spatial | |
---|---|---|

Assumptions
| ||

Data structure | Data is hierarchically structured with individuals nested within geographical areas. | Data is assumed to be spatially structured at the aggregated level. |

Individuals | Individuals (level 1) living in the same area (level 2) are assumed to be correlated | No individual-level data is retained |

Hazard | Constant hazards over each follow-up interval. | Constant hazards over each follow-up interval. |

Area-level effects | Area-level random effect is constant and normally distributed. Area-level random effects for different geographical areas are independent of each other; hence any spatial associations between neighboring areas are ignored. | Area-level random effect is not assumed to be constant; rather it depends on the spatial relationship between areas with the assumption that the mean outcome between two neighboring areas is more similar than that between two more distant areas. |

Modelled outcome | These are essentially logistic regression models with the outcome variable being a binary indicator that gives the probability of a death occurring in a follow-up interval given that no death has occurred in the previous year. | A Poisson distribution is assumed for the modeled outcome (i.e. observed mortality count) in each aggregated stratum. However the usual assumption for a Poisson model, that the variance equals the mean, is relaxed since additional random effect parameters are included. |

Underlying concepts
| ||

Baseline hazard | The baseline hazard is modelled on the logistic scale as a function of the follow-up interval. | The baseline hazard is not specifically defined as this is a semi-parametric model. |

Censoring | The censoring information is included. A censored individual has a sequence of zero’s for each year whereas a person who dies has a value of one for the year of death and zero for previous years. | The censoring information is included. A censored individual has a sequence of zero’s for each year whereas a person who dies has a value of one for the year of death and zero for previous years. However deaths are then aggregated acrosseach stratum. |

Equivalence to Cox model | Multilevel logistic regression with expanded dataset is a good approximation to the Cox proportional hazard model [8]. | The Poisson survival model is a good approximation to the Cox proportional hazards model [32, 33]. |

Spatial smoothing | No spatial smoothing is incorporated | Models borrow information from adjacent regions (termed ‘spatial smoothing’) to help overcome data sparseness, allow shrinkage towards overall risk, produce more robust estimates and account for between-area spatial associations [49]. |

Spatial structure | An individual’s probability of death is statistically dependent on their area of residence at diagnosis. Spatial proximity to other areas is not considered. | The spatial structure is encoded into the prior distribution specified for the random effects and requires the definition of relationships between spatially close SLAs [31]. The variable is assumed to be normally distributed relative to the neighbourhood mean. |

Levels of variance | The total variance is partitioned at different levels: between individuals living in the same area (individual-level) and that between two different areas (area-level). | The overall variance cannot be decomposed over different analytical levels. However the 2 random effects at the area-level allow the variance to be partitioned into spatially structured and unstructured variance. |

Interpretation of the area-level random effects
| ||

Number | One type | Two types |

Nature | Area-level random effects disregard any spatial correlation that may be present in the data and ignore the specific effect of location. | The spatially correlated area-level random effect assumes similarity between neighboring areas and quantifies the residual variation that is associated with geographical location. The uncorrelated or unstructured area-level random effect assumes independence between areas and allows for area-level variation that is not spatially correlated. |