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Comparing multilevel and Bayesian spatial random effects survival models to assess geographical inequalities in colorectal cancer survival: a case study
International Journal of Health Geographics volume 13, Article number: 36 (2014)
Abstract
Background
Multilevel and spatial models are being increasingly used to obtain substantive information on arealevel inequalities in cancer survival. Multilevel models assume independent geographical areas, whereas spatial models explicitly incorporate geographical correlation, often via a conditional autoregressive prior. However the relative merits of these methods for large populationbased studies have not been explored. Using a casestudy approach, we report on the implications of using multilevel and spatial survival models to study geographical inequalities in allcause survival.
Methods
Multilevel discretetime and Bayesian spatial survival models were used to study geographical inequalities in allcause survival for a populationbased colorectal cancer cohort of 22,727 cases aged 20–84 years diagnosed during 1997–2007 from Queensland, Australia.
Results
Both approaches were viable on this large dataset, and produced similar estimates of the fixed effects. After adding arealevel covariates, the betweenarea variability in survival using multilevel discretetime models was no longer significant. Spatial inequalities in survival were also markedly reduced after adjusting for aggregated arealevel covariates. Only the multilevel approach however, provided an estimation of the contribution of geographical variation to the total variation in survival between individual patients.
Conclusions
With little difference observed between the two approaches in the estimation of fixed effects, multilevel models should be favored if there is a clear hierarchical data structure and measuring the independent impact of individual and arealevel effects on survival differences is of primary interest. Bayesian spatial analyses may be preferred if spatial correlation between areas is important and if the priority is to assess smallarea variations in survival and map spatial patterns. Both approaches can be readily fitted to geographically enabled survival data from international settings.
Background
The importance of understanding social inequalities in cancer survival is well recognized [1, 2], including the impacts of both residential area characteristics and individuallevel risk factors [3–6]. Much of the interest in the impact of arealevel effects on cancer outcomes has been driven by the emergence of statistical methods that are designed to model geographicallystructured data, including multilevel discretetime [7, 8] and more recently, Bayesian spatial [4, 5] survival models. Since practical usage of these terms can differ, in our context we define “multilevel” structure as having a clear hierarchical onetomany relationship between area and individuallevel variables [9].
Multilevel discretetime survival models [7, 8] are designed to account for the nested structure of individuals within geographical areas. They allow the simultaneous estimation of individual and arealevel effects by modelling complex sources of variation at different hierarchical levels [9, 10]. In this multilevel framework, observations from one geographical area are assumed to be statistically independent of those in another area, so any spatial associations between geographical areas are ignored [11].
In contrast, Bayesian spatial methods were developed to explicitly incorporate spatial associations between geographical areas while describing the geographical patterns across areas. Survival in this context can be modelled either at the individuallevel [12, 13] or by aggregating the unitrecord data across area and covariates of interest [4, 5]. Arealevel spatial effects are captured through modelled random terms for which an uncertainty distribution (the “prior” distribution) are specified [14]. This approach incorporates information from adjacent regions to help overcome data sparseness and account for betweenarea spatial associations. To date the widespread application of spatial models using large unit record datasets has been limited, since they usually require more advanced programming skills than typically required for standard statistical software packages and are computationally demanding [12, 13]. With a view to considering applications to large populationbased cancer registry data, we chose instead to estimate the spatial survival inequalities by fitting generalised linear spatial models [15] to aggregated data. These models can be readily implemented with freely available software packages [4, 5].
Since the multilevel discretetime and aggregated Bayesian spatial survival approaches require the researcher to ignore either the spatial or multilevel effects, respectively, the possibility that both effects may be simultaneously present in geographically structured data is overlooked. Hence multilevel models have been criticized in some instances for their inability to account for spatial dependencies of health outcomes [11, 16, 17]. The potential implications of adjusting solely for either multilevel or spatial effects in the context of geographical inequalities in health have however not been widely explored.
The literature on incorporating a spatial perspective into the multilevel setting is sparse and limited to smallscale studies [18–21]. For example multilevel membership models use additional random terms to model spatial clustering of neighboring areas with separate random terms used for each neighboring unit for each observation from a specific region [19, 22]. However the sheer number of random terms makes it computationally intractable to estimate these models when there are large cohorts covering a multitude of geographic areas. Alternatively the multilevel framework has been combined with regression approaches specifically designed to account for the spatial dependency of the arealevel residuals [20, 21]. These also have limitations, including their complexity, lack of statistical stability, computational demands and difficulties in the interpretation of resultant estimates, each of which pose conceptual and technical challenges to their widespread implementation. We were unable to find any literature on integrated multilevelspatial survival models. Nor, to our knowledge, have there been any studies that have used a casestudy approach to explore the implications of using multilevel discretetime and Bayesian spatial survival models on the same cohort.
In our case study we apply the two analytical methods of multilevel discretetime and Bayesian spatial survival models to a populationbased cohort of colorectal cancer (CRC) patients and examine the relative merits of the two approaches. Our focus is not on statistically comparing the estimates obtained [23], but rather comparing the interpretation of the output generated by the two approaches, and to discuss the differing data transformations, model assumptions and parameterization required for both approaches, in addition to specific software and computing considerations. Given the increasing popularity of both methods to assess geographical disparities in health, and the increasing interest in using large, populationbased administrative datasets to examine these disparities, this case study aimed to improve understanding of the relative merits of the two approaches.
Methods
Approval for this study was obtained from the University of Queensland Social and Behavioral Sciences Ethical Review Committee and Queensland Health.
Study cohort
Histologically verified cases of invasive CRC (ICDO3: C18C20, C21.8) among individuals aged 20–84 years diagnosed between January 1, 1997 and December 31, 2007 with complete address information who survived for at least one day after diagnosis (n = 22,727) was acquired from the statewide populationbased Queensland Cancer Registry [24]. Information extracted from pathology forms was used to obtain the American Joint Committee on Cancer categories [25] for the stage at diagnosis and surgical margins as described previously [3].
Residential address at diagnosis was geocoded and assigned to a Statistical Local Area (SLA) [3]. SLAs are administrative units that cover the whole state with no gaps or overlaps, and are typically responsible for local infrastructure and thus deemed to be socioeconomically relevant to their residents. There were 478 SLAs in Queensland in 2006 with a median population of 5,810 (range 7: 77,523) and median area of 14 km^{2} (range 0.3:106,188). Geographic remoteness at CRC diagnosis was classified according to the 2006 Australian Standard Geographical Classification Remoteness Index [26] and arealevel disadvantage measured by the Index of Relative Socioeconomic Advantage and Disadvantage [27].
Survival data
Patients were followed for allcause mortality status until 31^{st} December 2010 with annual matching to the Registrar of Births, Deaths and Marriages and the (Australian) National Death Index [24]. Survival was calculated in years from date of diagnosis to death or the study end point. Survival times were truncated at 5 years of follow up to allow efficient computation of the complex survival models and to be consistent with previous studies [3, 4].
Statistical analysis
Multilevel analysis was carried out with MLwiN version 2.26 [28] (University of Bristol, UK) that requires a onceoff purchase while spatial modelling was performed using the freely accessible WinBUGS version 1.4 [29]. Both packages carry out Markov chain Monte Carlo (MCMC) estimations, provide diagnostic tests and plots to visually assess convergence of resulting chains and allow specification of random effects, and can be interfaced with Stata (Statacorp LP, TX, USA), as well as R [30]. However WinBUGS allows greater flexibility in the number of chains and choice of priors specified by the user whereas MCMC models in MLwiN can only be fitted with default priors and run on a single chain.
Multilevel discretetime survival
While continuous time approaches are most commonly used for survival analysis there are several advantages to discretetime models, especially in multilevel settings with large public health data sets [3, 8]. Fitting multilevel survival models requires an initial data restructuring in which a record is created for each time point that an individual survives. Generating such an expanded persontime dataset using months or days rather than years would increase the original dataset by 12 to 300fold. Given the size of our initial cohort and available computational resources it was not feasible to implement this additional expansion. Hence multilevel discretetime survival models that employ years as the time variable considerably reduce both the size of the expanded dataset and the computational demands for the subsequent survival analysis. In addition, parameter estimates from multilevel discretetime and continuoustime Cox survival models have been shown to be comparable in a number of studies [3, 8]. Thus discretetime survival models are preferred in the multilevel framework [3, 8].
Multilevel discretetime survival models were fitted to an expanded personperiod file containing a sequence of binary responses for each individual from each year [8]. The data file specifically incorporates censoring into the analysis, in that a censored individual will have a sequence of zero’s for each year whereas one who dies has a value of one for the year of death and zero for previous years (Appendix 1). Multilevel discretetime survival models estimate the unexplained variation within and betweenSLAs with the residuals for different areas assumed to be independent of each other. The hazard function from the multilevel discretetime survival model describes the conditional probability of death in interval t given they were still alive in the previous interval [8]. When modeling the hazard with the logit link; the exponentiated coefficients are interpreted as odds ratios (OR).
Multilevel discretetime survival model specification
The discretetime hazard function (h_{ tij }) for followup interval t and individual i in the j^{th} SLA is defined as the probability of a death (e_{ tij }) occurring during the followup interval t, given that no death has occurred in a previous year, i.e.:
which is the standard response probability for a binary variable. Therefore multilevel discretetime survival models are essentially logistic regression models with the response variable being the binary indicator e_{ tij } in the personperiod file. We fitted a multilevel randomeffects logistic model which was specified as:
where β_{ 0 } is an intercept for the j^{th} SLA that varies randomly across the SLAs, x_{ tij } is a vector of covariates with coefficient β which represents the effect of covariates on the hazard at follow up interval t for individuals in the reference (baseline) category of each variable (the baseline hazard), ml _u_{ j } is the random effect for each SLA j which is normally distributed with mean 0 and variance ml _σ^{2}_{ u } and f(t) is a function of followup interval used to model the baseline hazard on a logistic scale. A dummy variable was used for each time period: i.e. baseline logit hazard takes form f(t) = f 1D 1 + f 2D 2 + … + f 5D 5 [8]. Finally the model assumes constant hazards over each followup interval [8].
Bayesian spatial survival
We modified a previously described [4] Bayesian spatial Poisson model to analyse five year allcause survival. This model is specified as:
where d_{ mtj } is the observed number of deaths among the CRC cohort in the m^{th} stratum [across all included covariates], t^{th} followup interval and j^{th} SLA. The value d_{ mtj } has a Poisson distribution with mean μ_{ mtj }, y_{ mtj } is persontime at risk, α_{ t } is a timevarying intercept, and β_{ m } represents the coefficients of the vector of covariates x, spat _u_{ j } is the unexplained spatial variation in the modeled count of deaths for each area j and spat _v_{ j } is the unexplained nonspatial variation in the modeled deaths [31]. The total variation is:
where spat_σ^{2}_{ u(m) } is the marginal variance for the spatial effect and spat_σ^{2}_{ v } is the variance for the nonspatial effect.
The input data are aggregated by each combination of individual and arealevel covariates at the SLA level. A Poisson distribution is assumed for the modeled outcome (the observed mortality count in each stratum), while aggregated survival time is included as an offset variable in the model. This is a piecewise exponential model, where the followup time is divided into distinct intervals and the hazard is assumed constant across each interval. If the time intervals are split at the occurrence of each event (death), the Poisson survival model is equivalent to the Cox proportional hazards model [32, 33]. We selected annual time intervals due to the size of the dataset. Similar to the multilevel discretetime model, for each individual and time interval, death (the response) is defined as 1 if the individual dies within that interval and 0 otherwise. However unlike the multilevel discretetime model deaths are then aggregated across each stratum prior to being modelled. This Bayesian spatial model includes separate terms for the spatially correlated (spat _u_{ j }) and the spatially uncorrelated unexplained variation (spat _v_{ j }), where j is the SLA. The spatial term depends upon geographical location and implies that neighboring areas influence each other more than nonneighbors [34] whereas the spat_v_{ j } term accounts for variation which is independent of geographical location.
Estimation of the survival models
Multilevel discretetime
Models were estimated with MCMC simulations [22] in MLwiN 2.26 [28] (University of Bristol, UK) interfaced with Stata 12.0 (StataCorp, Texas) [35]. Default noninformative uniform priors (Appendix 1) were used for the fixed parameters and an inverse gamma distribution for the betweenarea variance. Parameter estimates were obtained from 80,000 iterations after discarding an initial 40,000 iterations. The underlying hazard was described by a dummy variable for each year [8]. A three step modeling strategy was adopted as described previously [3]. Truncating survival times to five years allowed efficient MCMC estimation of the multilevel discretetime models after expansion for this relatively large dataset.
Bayesian spatial
Models were fitted with the MCMC algorithm within WinBUGS 1.4 software [29] interfaced with Stata 12.0 [36]. After a burnin period of 250,000 iterations a further 100,000 iterations were monitored. Models were developed systematically: first we fitted a null model with only random effects, then we added all the individual covariates before including area disadvantage and remoteness, first separately and then together for the final fully adjusted model.
The spatial variance (spatu_{ j }) was modeled with an intrinsic conditional autoregressive (CAR) prior [31] with the neighboring SLAs primarily defined based on common borders, as previously described [37]. Diffuse normal priors were chosen for the intercept and regression coefficients. Model specification was completed by assigning weakly informative hyperpriors to the two precision (inverse variance) parameters. Prior distributions and the associated sensitivity analyses are further described in Appendix 2. Model inferences were relatively insensitive to the choice of hyperpriors (Additional file 1).
Model practicalities
Since single chain MCMC simulations were the only option within MLwiN [22], they were used for all analyses.
MCMC chain convergence (for both approaches) was assessed by visual inspection of the trace, density and autocorrelation plots of the posterior distributions for monitored parameters. Default diagnostic tests in MLwiN [22, 38] were used for multilevel discretetime models and the Geweke test (p <0.01 criteria for nonconvergence) for Bayesian spatial models [39]. Model residuals (both approaches) were also graphically examined for goodnessof fit.
Model fit within the set of multilevel discretetime or Bayesian spatial models was evaluated using the Bayesian deviance information criterion (DIC) [40] with smaller DIC values (≤7) indicating improved fit. As DIC values are sensitive to the underlying data structure [41], these were not used for comparisons between approaches but rather for comparing models within each approach.
Parameter estimates from multilevel and spatial Poisson models are presented as odds ratios (OR) and relative risks (RR), respectively, with 95% credible intervals (CrI).
Random effects
Multilevel discretetime
The median odds ratio (MOR) [10] that expresses arealevel variance from multilevel models on the odds ratio scale was used to quantify arealevel survival variation (Appendix 1). The value of MOR is always ≥ 1 with larger values indicating greater geographical variation. The intraclass correlation coefficient (ICC) is often used to quantify the contribution of the arealevel variance to the total variance in multilevel linear models. However, the use of such measures in the context of logistic regression is questionable and not recommended in standard multilevel literature due to problems in their computation and interpretation [10, 42–44]. Alternative measures include the median odds ratio (MOR) [10] that expresses arealevel variance from multilevel models on the odds ratio scale. This was used to quantify arealevel survival variation in the present study.
Bayesian spatial
The relative contribution of the spatial component to the total variance was calculated using the spatial fraction [5] (Appendix 2). If the spatial fraction is close to 1 the spatial effect dominates, otherwise if close to 0 the unstructured component dominates [5]. This measure allows quantification of the extent to which the unexplained variation is associated with geographical location.
Differences between approaches
Table 1 summarizes the main features and differences between the multilevel discretetime and Bayesian spatial approaches used in this case study. The assumptions, underlying concepts and interpretation of arealevel effects for the two approaches are compared and contrasted in Table 2.
Results
Study population
The final cohort had a median age at diagnosis of 68 years and median followup time of 5.0 years with unadjusted 5year allcause survival of 58.1% (95% CI: 57–58) (Table 3). All covariates in Table 3 had significant bivariate associations with survival outcomes (log rank test: 0.001 ≤ p <0.003).
Statistical analysis
The fullyadjusted maineffects multilevel discretetime model (Model 5; Additional file 2) had the smallest DIC value indicating it had the best fit to the data and so was the preferred multilevel model. Similarly, the bestfitting model for the Bayesian spatial analysis (out of Models 7–13) was Model 11 (Additional file 3) which simultaneously adjusted for all aggregated individual and arealevel covariates and included both random effects. Based on the MCMC diagnostic tools, all monitored parameters converged for both multilevel discretetime and Bayesian spatial models. No problems with model fit were detected on visual inspection of model residuals for both approaches.
Full adjustment for all considered covariates substantially reduced the residual geographical variation in survival for both approaches (Tables 4 and 5). The final multilevel discretetime model had for example a nonsignificant arealevel effect (p = 0.118) with an associated MOR of 1.07 (Table 4). For the spatial analysis the final smoothed RR estimates for allcause deaths ranged from 0.86 to 1.20 (median 0.99) with CrIs that generally overlapped the average value of 1.00 (Additional file 4). This illustrates that much of the geographical variability in survival was accounted for by the included covariates. Only 55% of the variance in the fully adjusted Bayesian spatial model was spatially structured, and the estimated spatial fraction had a wide 95% CrI (35–73; Table 5). As the spatial fraction is the ratio of the marginal spatial structured variance to the sum of the variance of both marginal spatial structured and unstructured random effects, a value close to the midpoint of 0.5 suggests that neither the spatial or the unstructured effect is dominant.
The observed patterns for the main effect parameter estimates generated from the two modeling approaches were broadly similar (Table 6), although the CrI of the multilevel estimates were generally equal to or wider than those for the Bayesian spatial model. As expected, within those categories with large numbers of deaths (e.g. Stage IV cancers), there were large differences in the OR and RRs estimates due to the violation of the rare disease assumption when using ORs to estimate RRs.
Discussion
To our knowledge this is the first report of a casestudy approach to explore the implications of using multilevel discretetime [7, 8] and Bayesian spatial survival models [4, 5] for the same populationbased cohort. These complex models were estimated using MCMC methods to reduce estimation bias for multilevel discretetime models [7] and produce more reliable smallarea estimates for spatial analyses [14].
Through a systematic comparison of the two approaches this study highlights important differences between the multilevel and spatial perspectives in analyzing cancer survival including model specification, underlying concepts, assumptions regarding modeleffects and interpretation of arealevel random effects in the context of populationbased data that typically cover numerous geographical areas and have long term followup.
While the fixed estimates from the two approaches cannot be compared directly [23], we found that adjusting for within or between area clustering had only a minimal impact on the broad patterns for the fixed estimates. For example, people from remote areas had poorer allcause survival than those from major cities for both approaches. This is consistent with a recent simulation study that found fixed effects were similar for multilevel and spatial methods [17].
A key feature of the multilevel approach is its ability to relate the estimated geographical variation to the total survival differences between individual patients. A number of additional parameters have also been developed for multilevel logistic regression, such as the MOR, which uses the estimated arealevel random effect to quantify the median variability in survival between two randomly selected patients from two different areas with identical individuallevel characteristics [10, 44]. However there is a lack of well accepted and robust measures for reporting the magnitude and impact of smallarea variation in survival from Bayesian spatial models in a meaningful manner [45]. Tango’s MEET [46] is a global clustering test that has been previously used to formally evaluate the significance of the modelled spatial variation in Bayesian spatial survival results [4, 37], but computational difficulties with the large number of variables in our models precluded this approach here.
An important strength of the Bayesian spatial models adopted for this case study is their ability to account for spatial associations while borrowing information from neighboring areas to enable stable smallarea estimates. Using aggregated spatial models potentially also allows greater flexibility in incorporating more years whereas the multilevel model requires curtailing the data. Moreover, the Bayesian spatial model, unlike the multilevel discretetime model, can be easily modified to conduct relative survival analyses [4, 5], the preferred approach when reporting populationbased cancer survival estimates [47].
There are also limitations to both approaches. Multilevel discretetime survival analysis requires an initial restructuring to the personperiod format so that standard binary response regression can be carried out [8]. Given the size of our primary dataset, the additional expansion required for analyzing survival outcomes over the entire time period or with shorter time intervals (e.g. days, months) rather than years was not possible under our computing specifications. This is a key limitation of MLwiN, which, as the most widely used software for multilevel modeling, may make this approach computationally infeasible [48]. Parameter estimates from continuous time survival models have however been shown to be comparable to those from multilevel discretetime survival models [3, 8, 49]. For the Bayesian spatial model, the estimates are based on data aggregated by geographical units; hence making inferences at the individual level are subject to the wellknown ecological bias [14, 50]. Both models when run under the computationally intensive MCMC were very timeconsuming. An alternative option could be to use the R package INLA (Integrated Nested Laplace Approximation) [51] to generate results instead. This method approximates fully Bayesian inference and generates within seconds or minutes rather than days, but is only available for selected models [52].
Given the differences between the two approaches, the choice of analytic methods will depend on the research questions of interest, data characteristics, and available computational resources. Multilevel models may be more appropriate if a clear hierarchical structure is apparent and the primary objective is to quantify the independent impact of individual and arealevel factors on survival differences while accounting for the clustering at the different analytical levels. Spatial analyses may however be preferred if the spatial correlation between areas has a theorized impact on the observed inequalities, or if the goal is to study geographical variation in cancer survival at the smallarea level and then create maps of the smoothed relative risk estimates to understand spatial patterns. Such maps can prove useful in identifying areas with lower survival (or elevated relative risk of mortality) relative to all other regions within the overall study area [4, 5] with the potential to guide targeted strategies for improving survival and allocating resources.
The approaches described in the current study are generalizable in terms of wider international settings, geographical units (i.e. not restricted to SLAs) and cancer sites that can be analysed. These models can be fitted to datasets from any populationbased or hospitalbased cancer registry provided that there is sufficient information to estimate survival and assign cases to a geographical unit. Finally these models can be readily extended to look at geographical inequalities in survival for other diseases and conditions than cancer.
Conclusions
As spatial models more accurately define the geographical composition with the study cohort by accounting for spatial proximity, perhaps the optimum approach would be to integrate these two approaches by combining the spatial structure and neighboring information with a multilevel survival model that retains the nested structure. Literature on incorporating a spatial perspective into the multilevel setting is comparatively rare [18–21]. Various conceptual and technical challenges have limited the easy implementation of multilevel spatial models in practice including their inherent complexity, computational demands and concerns about the statistical stability and interpretation of model estimates [19–21]. This may be a promising area for further research.
Appendix 1 Multilevel discretetime survival analysis
Multilevel discretetime survival models [8] were adopted to analyse geographical variations in five year allcause survival for individual patients. As described elsewhere [3], this approach requires an initial expansion of the dataset to allow survival models to be specified as multilevel binary response models.
Data expansion
We used the death or censoring time, r_{ ij }, and an indicator δ_{ ij } which was 0 if death had not occurred and 1 if death had occurred by fiveyears for each individual i in the j^{th} SLA in the original data, to create for each followup interval t (years) up to r_{ ij } a binary response e_{ tij } which was coded as:
Hence if an individual died during the third year after diagnosis their discrete responses were (e_{1ij}, e_{2ij}, e_{3ij}) = (0,0,1), while someone who was censored in the third year had response vector (0,0,0). This restructured dataset is often referred to as a personperiod file[8].
Priors
The intercept and fixed parameters were assigned diffuse uniform priors (mean 0, variance 1.0). A weakly informative hyperprior of Gamma (0.1, 1000) was used for the precision ml _τ_{ u } (inverse variance) on the arealevel random effect ml _u_{ j. } These are the default prior distributions in MLwiN [22]. Given the large number of area level units (478 SLAs) inferences are unlikely to be sensitive to the choice of prior distributions for the arealevel variance [53].
Median odds ratios
The median odds ratio (MOR) was calculated as described previously [10]:
where Ζ_{0.75} is the 75^{th} percentile of the normal distribution and ml _σ^{2}_{ u } is the estimated arealevel variance from the MCMC simulations. A 95% credible interval for the MOR was generated from the posterior distribution of the variance [43].
Appendix 2 Bayesian spatial survival analysis
Priors
An exchangeable normal prior spat_v_{ j } ~ N(0,spat_σ^{2}_{ v }) was specified for the nonspatial random effect where spat _σ^{2}_{ v } is the variance. The spatial dependence (spat _u_{ j }) across SLAs was estimated using an intrinsic conditional autoregressive (CAR) prior [31] defined as:
where ω_{ jk } =1 if j, k are adjacent SLAs and 0 otherwise and spat_σ^{2}_{ u } is the variance for the spatial effect. Neighbors were defined using an adjacency matrix as described previously [37]. Diffuse normal priors were used for the intercept and fixed effects and weakly informative Gamma hyperpriors for the precision parameters spat _τ_{ u } and spat_τ_{ v }.
Sensitivity analyses were conducted by specifying three different Gamma (Γ) distributions for spat _τ_{ u } and spat_τ_{ v } and two uniform (Unif) priors for the standard deviation (spat _σ_{ u }, spat _ σ_{ v }) [4]:

1.
spat _ τ _{ u } ~ Γ(0.1, 100), spat _ τ _{ v } ~ Γ(0.1, 100)

2.
spat _ τ _{ u } ~ Γ(0.5, 1000), spat _ τ _{ v } ~ Γ(0.5, 1000)

3.
spat _ τ _{ u } ~ Γ(0.1, 10), spat _ τ _{ v } ~ Γ(0.001, 1000)

4.
spat _ σ _{ u } ~ Unif(0, 10), spat _ σ _{ v } ~ Unif(0, 10)

5.
spat _ σ _{ u } ~ Unif(0, 1000), spat _ σ _{ v } ~ Unif(0, 1000)
Priors 1 to 2 had means and variances on the precisions of (10, 1000); (500, 500000); and for Prior 3, spat _τ_{ u } had (1, 10), while spat_τ_{ v } had (1, 1000), respectively. Priors 4 and 5 had means and variances on the standard deviations of (5, 8.3) and (500, 83333.3).
Models were compared in in terms of DIC statistics [40], cumulative distribution plots of deviance [54], summary measures of the posterior distribution of monitored parameters and convergence diagnostics.
Spatial fraction
If spat_σ^{2}_{ u(m) } is the marginal variance for the spatial effect and spat_σ^{2}_{ v } is the variance for the nonspatial effect then the spatial fraction (Ψ) [5] is:
Abbreviations
 CRC:

Colorectal cancer
 CI:

Confidence interval
 CrI:

Credible interval
 CAR:

Conditional autoregressive
 DIC:

Deviance information criterion
 MCMC:

Markov chain Monte Carlo
 MOR:

Median odds ratio
 OR:

Odds ratio
 RR:

Relative risk
 SLA:

Statistical Local Area.
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Acknowledgments
This work was supported by a research grant from the (Australian) National Health and Medical Research Council (NHMRC) (ID561700). Associate Professor Peter Baade is supported by an NHMRC Career Development Fellowship (ID1005334) and Professor Gavin Turrell is supported by an NHMRC Senior Research Fellowship (ID 1003710).
The NHMRC is an external funding agency that provided funds to conduct this research. They had no input into the content or conclusions of this paper.
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The authors declare that they have no competing interests.
Authors’ contributions
PDB conceived the study. PD performed the analysis. PD, SMC and PDB drafted the manuscript. All authors contributed to, read and approved the final manuscript.
Electronic supplementary material
Example of sensitivity analysis for Bayesian spatial survival models.
Additional file 1: Kernel density plots for estimated relative risks (RR) of allcause death by arealevel remoteness: A: major cities; B: inner cities and C: remote from Bayesian spatial survival models with Gamma priors specified for the precision (inverse of variance). of 1: τ_{u} ~ Γ(0.1, 10), τ_{v} ~ Γ(0.1, 10); 2: τ_{u} ~ Γ(0.5, 1000), τ_{ v } ~ Γ(0.5, 1000); 3 : τ_{u} ~ Γ(0.1, 10) τ_{v} ~ Γ(0.001, 1000); or Uniform (Unif) priors on the standard deviation of : 4 : σ_{u} ~ Uniform(0, 10), σ_{ v } ~ Unif(0, 10) or 5 : σ_{ u } ~ Unif(0, 1000), σ_{ v } ~ Unif(0, 1000). (PDF 88 KB)
12942_2014_604_MOESM2_ESM.xlsx
Additional file 2:Model comparisons for multilevel discretetime survival models. Table S1 shows Bayesian deviance information criterion (DIC) values for different models with smaller values (difference of at least 7 units) indicating better modelfit. Models with difference of 3–5 units can be weakly distinguished. The pD values represents the effective number of parameters. (XLSX 12 KB)
12942_2014_604_MOESM3_ESM.xlsx
Additional file 3:Model comparisons for the Bayesian spatial survival models. Table S2 shows Bayesian deviance information criterion (DIC) values and pD values, which due to borrowing of strength from adjacent areas is often less than the total number of model parameters. Larger values indicate estimates have undergone less smoothing. (XLSX 12 KB)
12942_2014_604_MOESM4_ESM.pdf
Additional file 4:Median smoothed relative risk (RR) and credible intervals by statistical local areas (SLA). The median smoothed relative risk (RR) estimates from final Bayesian spatial model for allcause survival by statistical local areas (SLAs) in Queensland. The black line is the RR, grey lines are the 95% credible intervals (CrI) and the red horizontal line indicates the Queensland average. (PDF 52 KB)
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Dasgupta, P., Cramb, S.M., Aitken, J.F. et al. Comparing multilevel and Bayesian spatial random effects survival models to assess geographical inequalities in colorectal cancer survival: a case study. Int J Health Geogr 13, 36 (2014). https://doi.org/10.1186/1476072X1336
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DOI: https://doi.org/10.1186/1476072X1336
Keywords
 Bayesian
 Multilevel
 Colorectal cancer
 Epidemiology
 Allcause survival
 Spatial