Slides[1]

Modelling and Implementation of spatio-temporal disease mapping (STDM) Antonio L´ opez-Qu´ılez

Grup d’Estad´ıstica espacial i temporal en Epidemiologia i Medi Ambient

(GEeitEma – UV)

Facundo Mu˜ noz

Departament d’Estad´ıstica i Investigaci´ o Operativa

Modelling and Implementation of STDM

STDM Workshop 2009

1 / 26

So we need to implement a STDM model...

... but WHICH ONE? Multiple alternatives and proposals → no BEST Selection criteria: automatic fit versatile fast Accurate!

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

2 / 26

So we need to implement a STDM model...

... but WHICH ONE? Multiple alternatives and proposals → no BEST Selection criteria: automatic fit versatile fast Accurate!

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

2 / 26

So we need to implement a STDM model...

... but WHICH ONE? Multiple alternatives and proposals → no BEST Selection criteria: automatic fit versatile fast Accurate!

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

2 / 26

Framework for STDMs

Four stages: 1 2 3 4

Probabilistic model for observations Components of the linear predictor Structures of the effects Inference methodology

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

3 / 26

Framework: Probabilistic model for observations

Poisson, Binomial or Negative Binomial distribution? Relative or absolute risk? External or internal standardization of the population? All depend on the dataset, and do not condition the spatio-temporal modelling, so... As appropriate

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

4 / 26

Framework: Probabilistic model for observations

Poisson, Binomial or Negative Binomial distribution? Relative or absolute risk? External or internal standardization of the population? All depend on the dataset, and do not condition the spatio-temporal modelling, so... As appropriate

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

4 / 26

Framework: Probabilistic model for observations

Poisson, Binomial or Negative Binomial distribution? Relative or absolute risk? External or internal standardization of the population? All depend on the dataset, and do not condition the spatio-temporal modelling, so... As appropriate

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

4 / 26

Framework: Components of the linear predictor

ηijk = g(rijk ) = Int.+Ck + Si + Tj + CSik + CTjk + STij + CSTijk +εijk | {z } | {z } main effects

interaction terms

Intercept: Base risk Main effects: Additional risk of belonging to group k, living in region i and period j respectively. Interaction terms: Contribution to the risk due to a combination of the effects. Extra variability term: Overall effect of other minor factors.

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

5 / 26

Framework: Components of the linear predictor

ηijk = g(rijk ) = Int.+Ck + Si + Tj + CSik + CTjk + STij + CSTijk +εijk | {z } | {z } main effects

interaction terms

Intercept: Base risk Main effects: Additional risk of belonging to group k, living in region i and period j respectively. Interaction terms: Contribution to the risk due to a combination of the effects. Extra variability term: Overall effect of other minor factors.

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

5 / 26

Framework: Components of the linear predictor

ηijk = g(rijk ) = Int.+Ck + Si + Tj + CSik + CTjk + STij + CSTijk +εijk | {z } | {z } main effects

interaction terms

Intercept: Base risk Main effects: Additional risk of belonging to group k, living in region i and period j respectively. Interaction terms: Contribution to the risk due to a combination of the effects. Extra variability term: Overall effect of other minor factors.

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

5 / 26

Framework: Components of the linear predictor

ηijk = g(rijk ) = Int.+Ck + Si + Tj + CSik + CTjk + STij + CSTijk +εijk | {z } | {z } main effects

interaction terms

Intercept: Base risk Main effects: Additional risk of belonging to group k, living in region i and period j respectively. Interaction terms: Contribution to the risk due to a combination of the effects. Extra variability term: Overall effect of other minor factors.

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

5 / 26

Framework: Components of the linear predictor

ηijk = g(rijk ) = Int.+Ck + Si + Tj + CSik + CTjk + STij + CSTijk +εijk | {z } | {z } main effects

interaction terms

Intercept: Base risk Main effects: Additional risk of belonging to group k, living in region i and period j respectively. Interaction terms: Contribution to the risk due to a combination of the effects. Extra variability term: Overall effect of other minor factors.

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

5 / 26

Framework: Components of the linear predictor A toy example:

ηijk = Intercept

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

6 / 26

Framework: Components of the linear predictor A toy example:

ηijk = Intercept + Si

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

6 / 26

Framework: Components of the linear predictor A toy example:

ηijk = Intercept + Tj

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

6 / 26

Framework: Components of the linear predictor A toy example:

ηijk = Intercept + Si + Tj

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

6 / 26

Framework: Components of the linear predictor A toy example:

ηijk = Intercept + Si + Tj + STij

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

6 / 26

Framework: Components of the linear predictor A toy example:

ηijk = Int. + Ck + Si + Tj + CSik + CTjk + STij + CSTijk + εijk

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

6 / 26

Framework: Structures of the effects

Lack of data ; Additional hypothesis: Neighbouring regions are more likely to have similar risk levels risk levels are expected to evolve smoothly on time

STRUCTURE Borrowing strength Many alternatives for implementing structure in practice

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

7 / 26

Framework: Structures of the effects

Lack of data ; Additional hypothesis: Neighbouring regions are more likely to have similar risk levels risk levels are expected to evolve smoothly on time

STRUCTURE Borrowing strength Many alternatives for implementing structure in practice

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

7 / 26

Framework: Structures of the effects

Lack of data ; Additional hypothesis: Neighbouring regions are more likely to have similar risk levels risk levels are expected to evolve smoothly on time

STRUCTURE Borrowing strength Many alternatives for implementing structure in practice

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

7 / 26

Framework: Structures of the effects

Lack of data ; Additional hypothesis: Neighbouring regions are more likely to have similar risk levels risk levels are expected to evolve smoothly on time

STRUCTURE Borrowing strength Many alternatives for implementing structure in practice

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

7 / 26

Framework: Structures of the effects

Lack of data ; Additional hypothesis: Neighbouring regions are more likely to have similar risk levels risk levels are expected to evolve smoothly on time

STRUCTURE Borrowing strength Many alternatives for implementing structure in practice

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

7 / 26

Framework: Structures of the effects

Lack of data ; Additional hypothesis: Neighbouring regions are more likely to have similar risk levels risk levels are expected to evolve smoothly on time

STRUCTURE Borrowing strength Many alternatives for implementing structure in practice

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

7 / 26

Framework: Structures of the effects

Spatial effect Si : Random effects Heterogeneity effect Clustering (CAR) effect BYM specification

Two-dimensional splines Penalized splines

Combination of Penalized splines and CAR models (new!) Mar´ıa Durb´ an, Dae-Jin Lee P-spline mixed-models for spatio-temporal data Today 15:30hs

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

8 / 26

Framework: Structures of the effects

Spatial effect Si : Random effects Heterogeneity effect Clustering (CAR) effect BYM specification

Two-dimensional splines Penalized splines

Combination of Penalized splines and CAR models (new!) Mar´ıa Durb´ an, Dae-Jin Lee P-spline mixed-models for spatio-temporal data Today 15:30hs

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

8 / 26

Framework: Structures of the effects

Spatial effect Si : Random effects Heterogeneity effect Clustering (CAR) effect BYM specification

Two-dimensional splines Penalized splines

Combination of Penalized splines and CAR models (new!) Mar´ıa Durb´ an, Dae-Jin Lee P-spline mixed-models for spatio-temporal data Today 15:30hs

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

8 / 26

Framework: Structures of the effects

Temporal effect Tj : Parametric models linear, quadratic, ...

First or second order Random Walks (RW) Autoregressive processes (AR) Splines

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

9 / 26

Framework: Structures of the effects

Spatio-temporal interaction STij : Remember: STij only accounts for individual differences from Int. + Si + Tj Yet another classification (Following Knorr-Held (2000)): Type Type Type Type

(GEeitEma – UV)

I: Independence II: Temporal structure III: Spatial structure IV: Spatio-temporal structure

Modelling and Implementation of STDM

STDM Workshop 2009

10 / 26

Framework: Structures of the effects

Spatio-temporal interaction STij : Remember: STij only accounts for individual differences from Int. + Si + Tj Yet another classification (Following Knorr-Held (2000)): Type Type Type Type

(GEeitEma – UV)

I: Independence II: Temporal structure III: Spatial structure IV: Spatio-temporal structure

Modelling and Implementation of STDM

STDM Workshop 2009

10 / 26

Framework: Structures of the effects Spatio-temporal interaction STij : Type I Unstructured set of values for each combination of region and period

Implementation: iid

Usually white noise: STij ∼ N (0, σ 2 )

Models: Non-persistent local circumstances causing a slight increase or decrease in the rates in a specific region-period

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

11 / 26

Framework: Structures of the effects Spatio-temporal interaction STij : Type I Unstructured set of values for each combination of region and period

Implementation: iid

Usually white noise: STij ∼ N (0, σ 2 )

Models: Non-persistent local circumstances causing a slight increase or decrease in the rates in a specific region-period

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

11 / 26

Framework: Structures of the effects Spatio-temporal interaction STij : Type I Unstructured set of values for each combination of region and period

Implementation: iid

Usually white noise: STij ∼ N (0, σ 2 )

Models: Non-persistent local circumstances causing a slight increase or decrease in the rates in a specific region-period

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

11 / 26

Framework: Structures of the effects Spatio-temporal interaction STij : Type II Region-specific temporal evolution structures independent of that in the neighbouring regions

Implementation: As many forms as the temporal main effect (Parametric, Splines, RW, AR, ...)

Models: Risk determinants affecting specific regions and inducing deviations from the global trend

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

12 / 26

Framework: Structures of the effects Spatio-temporal interaction STij : Type II Region-specific temporal evolution structures independent of that in the neighbouring regions

Implementation: As many forms as the temporal main effect (Parametric, Splines, RW, AR, ...)

Models: Risk determinants affecting specific regions and inducing deviations from the global trend

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

12 / 26

Framework: Structures of the effects Spatio-temporal interaction STij : Type II Region-specific temporal evolution structures independent of that in the neighbouring regions

Implementation: As many forms as the temporal main effect (Parametric, Splines, RW, AR, ...)

Models: Risk determinants affecting specific regions and inducing deviations from the global trend

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

12 / 26

Framework: Structures of the effects Spatio-temporal interaction STij : Type III Period-specific spatial structures independent of adjacent periods

Implementation: Typically CAR

Models: Risk determinants affecting an area larger than the region size, but not persistent in time

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

13 / 26

Framework: Structures of the effects Spatio-temporal interaction STij : Type III Period-specific spatial structures independent of adjacent periods

Implementation: Typically CAR

Models: Risk determinants affecting an area larger than the region size, but not persistent in time

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

13 / 26

Framework: Structures of the effects Spatio-temporal interaction STij : Type III Period-specific spatial structures independent of adjacent periods

Implementation: Typically CAR

Models: Risk determinants affecting an area larger than the region size, but not persistent in time

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

13 / 26

Framework: Structures of the effects Spatio-temporal interaction STij : Type IV Full spatio-temporal structure

Implementation: Parametric function of time, with spatially-structured coefficients Random effect with given structure matrix Non-parametric smoothing function (splines) with spatially structured coefficients

Models: Risk determinants exceeding the limits of one or more regions, and that persist for more than one period of time (GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

14 / 26

Framework: Structures of the effects Spatio-temporal interaction STij : Type IV Full spatio-temporal structure

Implementation: Parametric function of time, with spatially-structured coefficients Random effect with given structure matrix Non-parametric smoothing function (splines) with spatially structured coefficients

Models: Risk determinants exceeding the limits of one or more regions, and that persist for more than one period of time (GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

14 / 26

Framework: Structures of the effects Spatio-temporal interaction STij : Type IV Full spatio-temporal structure

Implementation: Parametric function of time, with spatially-structured coefficients Random effect with given structure matrix Non-parametric smoothing function (splines) with spatially structured coefficients

Models: Risk determinants exceeding the limits of one or more regions, and that persist for more than one period of time (GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

14 / 26

Framework: Structures of the effects

Spatio-temporal interaction STij : The four types are suited to modelling different kinds of phenomena Versatility ; Overmodelling! Usual practice: first explore, then choose the adequate interaction type and use that not automatic

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

15 / 26

Framework: Structures of the effects

Spatio-temporal interaction STij : The four types are suited to modelling different kinds of phenomena Versatility ; Overmodelling! Usual practice: first explore, then choose the adequate interaction type and use that not automatic

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

15 / 26

Framework: Structures of the effects

Spatio-temporal interaction STij : The four types are suited to modelling different kinds of phenomena Versatility ; Overmodelling! Usual practice: first explore, then choose the adequate interaction type and use that not automatic

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

15 / 26

Framework: Structures of the effects

Spatio-temporal interaction STij : The four types are suited to modelling different kinds of phenomena Versatility ; Overmodelling! Usual practice: first explore, then choose the adequate interaction type and use that not automatic

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

15 / 26

Framework: Inference methodology

Bayesian approach + + + −

Unified general approach Greatest modelling flexibility Detailed posterior information on every parameter Specification of priors

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

16 / 26

Framework: Inference methodology

Bayesian approach + + + −

Unified general approach Greatest modelling flexibility Detailed posterior information on every parameter Specification of priors

MCMC − − − −

(GEeitEma – UV)

Very slow! (hours, days, ...) Technical aspects (convergence, mixing, ...) not automatic not suitable for a fast and automatic service

Modelling and Implementation of STDM

STDM Workshop 2009

16 / 26

Framework: Inference methodology

Bayesian approach + + + −

Unified general approach Greatest modelling flexibility Detailed posterior information on every parameter Specification of priors

INLA (Rue et al. (2009)) + Fast + Straightforward − Restricted to latent GMRF + Leonhard Held, Birgit Schr¨ odle Spatio-temporal disease mapping using INLA Today 15:30hs

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

16 / 26

Framework: Inference methodology

Frequentist approach Empirical Bayes techniques + Fast + Feasible in many situations + Much work in progress - Not straightforward (reparameterization, develop specific numerical algorithms, ...) - Variability in point estimates often understated

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

17 / 26

STDMs in the literature Classified according to the structure of the temporal evolution of the estimated risk for each region (Tj + STij ) a Parametric b Temporally independent c Smooth temporal evolution

(a)

(GEeitEma – UV)

(b)

Modelling and Implementation of STDM

(c)

STDM Workshop 2009

18 / 26

STDMs in the literature Parametric models Tj + STij is a linear or quadratic function of time coefficients may or may not be spatially structured + Information is shared in time (and possibly in space) + Straightforward formulation − Inappropriate for long periods of time: too restrictive! References: Bernardinelli et al. (1995) Sun et al. (2000) Assun¸c˜ ao, Reis and Oliveira (2001)

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

19 / 26

STDMs in the literature Temporally independent models Tj + STij is a sequence of spatial models + Information is shared in space + Temporal evolution is not restricted to any specific shape − No sharing of information in time − High dimensional models ; identifiability and overmodelling problems References: Waller et al. (1997) Xia and Carlin (1998) Nobre, Schmidt and Lopes (2005)

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

20 / 26

STDMs in the literature Smooth temporal evolution models Tj + STij is a smooth function of time + Temporal evolution is not restricted to any specific shape + Information is shared in time − Many alternatives for STij ; model selection criteria References: Knorr-Held (2000) Macnab and Dean (2001, 2002) Lagazio, Biggeri and Dreassi (2003) Schmid and Held (2004) Richardson, Abellan and Best (2006) Mart´ınez-Beneito, L´ opez-Qu´ılez and Botella-Rocamora (2008)

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

21 / 26

Discussion

Additional information Versatility ; few additional information Almost everything relies on spatio-temporal structure Flexible models

Standardizing Age as a covariate ; Powerful! Age-specific count data is needed Age standardization ; more versatile

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

22 / 26

Discussion

Additional information Versatility ; few additional information Almost everything relies on spatio-temporal structure Flexible models

Standardizing Age as a covariate ; Powerful! Age-specific count data is needed Age standardization ; more versatile

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

22 / 26

Discussion

Additional information Versatility ; few additional information Almost everything relies on spatio-temporal structure Flexible models

Standardizing Age as a covariate ; Powerful! Age-specific count data is needed Age standardization ; more versatile

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

22 / 26

Discussion

Additional information Versatility ; few additional information Almost everything relies on spatio-temporal structure Flexible models

Standardizing Age as a covariate ; Powerful! Age-specific count data is needed Age standardization ; more versatile

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

22 / 26

Discussion

Additional information Versatility ; few additional information Almost everything relies on spatio-temporal structure Flexible models

Standardizing Age as a covariate ; Powerful! Age-specific count data is needed Age standardization ; more versatile

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

22 / 26

Discussion

Additional information Versatility ; few additional information Almost everything relies on spatio-temporal structure Flexible models

Standardizing Age as a covariate ; Powerful! Age-specific count data is needed Age standardization ; more versatile

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

22 / 26

Discussion

Interactions More interaction terms ; more flexibility Many parameters, unidentifiability and overmodelling problems Keep it simple

Model selection Common practice: trying alternative models Not automatic

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

23 / 26

Discussion

Interactions More interaction terms ; more flexibility Many parameters, unidentifiability and overmodelling problems Keep it simple

Model selection Common practice: trying alternative models Not automatic

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

23 / 26

Discussion

Interactions More interaction terms ; more flexibility Many parameters, unidentifiability and overmodelling problems Keep it simple

Model selection Common practice: trying alternative models Not automatic

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

23 / 26

Discussion

Interactions More interaction terms ; more flexibility Many parameters, unidentifiability and overmodelling problems Keep it simple

Model selection Common practice: trying alternative models Not automatic

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

23 / 26

Discussion

Interactions More interaction terms ; more flexibility Many parameters, unidentifiability and overmodelling problems Keep it simple

Model selection Common practice: trying alternative models Not automatic

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

23 / 26

Discussion

Implementation Specificity of the model complex models ; specific situations simpler models ; more versatility Complexity and scope of the inference method simpler models ; more inference tools available & more likely to work smoothly without fine tuning (automatic fit) Speed Inference method Accuracy EB estimation ; typically yields consistent and nearly unbiased point estimates, but often understates their variability MCMC estimation ; careful diagnosis of simulation outcomes

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

24 / 26

Discussion

Implementation Specificity of the model complex models ; specific situations simpler models ; more versatility Complexity and scope of the inference method simpler models ; more inference tools available & more likely to work smoothly without fine tuning (automatic fit) Speed Inference method Accuracy EB estimation ; typically yields consistent and nearly unbiased point estimates, but often understates their variability MCMC estimation ; careful diagnosis of simulation outcomes

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

24 / 26

Discussion

Implementation Specificity of the model complex models ; specific situations simpler models ; more versatility Complexity and scope of the inference method simpler models ; more inference tools available & more likely to work smoothly without fine tuning (automatic fit) Speed Inference method Accuracy EB estimation ; typically yields consistent and nearly unbiased point estimates, but often understates their variability MCMC estimation ; careful diagnosis of simulation outcomes

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

24 / 26

Discussion

Implementation Specificity of the model complex models ; specific situations simpler models ; more versatility Complexity and scope of the inference method simpler models ; more inference tools available & more likely to work smoothly without fine tuning (automatic fit) Speed Inference method Accuracy EB estimation ; typically yields consistent and nearly unbiased point estimates, but often understates their variability MCMC estimation ; careful diagnosis of simulation outcomes

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

24 / 26

Conclusions

Euroheis 2 project: STDM fast, automatic, user-oriented Framework for STDM: popular alternatives for each of the four stages Classification of models: structure of the temporal trends Discussion: summary of relevant aspects to be considered

We hope this provides all the necessary elements to take a well-based decision

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

25 / 26

Conclusions

Euroheis 2 project: STDM fast, automatic, user-oriented Framework for STDM: popular alternatives for each of the four stages Classification of models: structure of the temporal trends Discussion: summary of relevant aspects to be considered

We hope this provides all the necessary elements to take a well-based decision

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

25 / 26

Conclusions

Euroheis 2 project: STDM fast, automatic, user-oriented Framework for STDM: popular alternatives for each of the four stages Classification of models: structure of the temporal trends Discussion: summary of relevant aspects to be considered

We hope this provides all the necessary elements to take a well-based decision

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

25 / 26

Conclusions

Euroheis 2 project: STDM fast, automatic, user-oriented Framework for STDM: popular alternatives for each of the four stages Classification of models: structure of the temporal trends Discussion: summary of relevant aspects to be considered

We hope this provides all the necessary elements to take a well-based decision

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

25 / 26

Thank you

Modelling and Implementation of spatio-temporal disease mapping

Grup d’Estad´ıstica espacial i temporal en Epidemiologia i Medi Ambient

Departament d’Estad´ıstica i Investigaci´ o Operativa

http://www.euroheis.org/

(GEeitEma – UV)

Modelling and Implementation of STDM

STDM Workshop 2009

26 / 26