Table 2 Assumptions, underlying concepts and interpretation of area-level effects: multilevel discrete-time and Bayesian spatial survival models

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.