Europe Against Cancer European Network for Cervical Cancer Screening
Grant Agreement n° SPC.2002475 (16 Dec 2002 - 15 Dec 2003) To co-ordinate and link quality assurance, monitoring and new
Scientific Institute technologies in cervical cancer screening in the EU of Public Health
SPACE-TIME VARIATION OF CERVICAL CANCER MORTALITY IN BELGIUM USING AN HIERARCHICAL BAYESIAN MODEL
(BELGIUM, 1969-1994)
Marc ARBYN1,2 Francis CAPET2 Arnošt KOMÁREK3 Emmanuel LESAFFRE3
1. European Network for Cervical Cancer Screening 2. Scientific Institute of Public Health, Unit of Epidemiology 3. Centre for Biostatistics, Faculty of Medicine Catholic University of Leuven, Kapucijnenvoer, B-3000 Leuven
IPH/EPI REPORTS Nr. 2003 –024 Epidemiology, December 2003; Brussels (Belgium) Scientific Institute of Public Heatlh, SIPH/EPI REPORTS N 2003 - 024 Depotnumber: D/2003/2505/024
Europe Against Cancer European Network for Cervical Cancer Screening Grant Agreement n° SPC.2002475 (16 Dec 2002 - 15 Dec 2003) To co-ordinate and link quality assurance, monitoring and new technologies in cervical cancer screening in the EU
SPACE-TIME VARIATION OF CERVICAL CANCER MORTALITY IN BELGIUM USING AN HIERARCHICAL BAYESIAN MODEL
(BELGIUM, 1969-1994)
Marc ARBYN1,2 Francis CAPET2 Arnošt KOMÁREK3 Emmanuel LESAFFRE3
1. European Network for Cervical Cancer Screening 2. Scientific Institute of Public Health, Unit of Epidemiology 3. Centre for Biostatistics, Faculty of Medicine Catholic University of Leuven, Kapucijnenvoer, B-3000 Leuven
Scientific Institute of Public Health J. Wytsmanstr. 14 1050 Brussels
Belgium Tel: + 32 2 642 50 21 Fax: +32 2 642 54 10 e-mail: [email protected] http://www.iph.fgov.be/epidemio/
IPH/EPI REPORTS Nr. 2003 – 024
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 2
1. Table of contents
1. TABLE OF CONTENTS...... 3
2. ABSTRACT...... 5 2.1. BACKGROUND ...... 5 2.2. MATERIAL AND METHODS ...... 5 2.3. RESULTS...... 5 2.4. DISCUSSION...... 5 3. KEY WORDS...... 6
4. INTRODUCTION...... 6
5. MATERIALS ...... 7
6. METHODS ...... 7 6.1. INTRODUCTION TO BAYESIAN MODELS ...... 7 6.2. SPACE-COHORT MODEL (SC)...... 8 6.2.1. Log-linear Poisson age-cohort model ...... 8 6.2.2. Bayesian Space-Cohort mode (SC)...... 9 6.2.3. Prior distributions for the SC model...... 10 6.3. SPACE-PERIOD MODEL (SP)...... 11 6.3.1. Log-linear Poisson age-period model ...... 11 6.3.2. Bayesian Space-Period model (SP) ...... 12 6.4. CARTOGRAPHIC DISPLAY STYLE...... 14 7. RESULTS ...... 15 7.1. SPACE-COHORT MODEL ...... 15 7.1.1. Log-linear Poisson age-cohort model ...... 15 7.1.2. Bayesian SC model ...... 15 7.1.3. Cohort effects estimated from the SC model...... 16 7.1.4. Space effects estimated from the SC model ...... 18 7.1.5. Local cohort effects (S*C model)...... 20 7.1.6. Contrast between local raw SCMRs and fitted cohort effects estimated from the S*C model ...... 23 7.2. SPACE-PERIOD MODEL...... 26 7.2.1. Loglinear Poisson age-period model...... 26 7.2.2. Bayesian SP model...... 26 7.2.3. Period effects estimated from the SP model ...... 27 7.2.4. Space effects estimated from the SP model...... 29 7.2.5. Local period effects (S*P model)...... 30 7.2.6. Contrast between local raw SMRs and fitted period effects estimated from the S*P model...... 33 8. DISCUSSION ...... 37
9. LIST OF ABBREVIATIONS ...... 42
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 3 10. BIBLIOGRAPHY...... 43
11. ANNEXES ...... 46 11.1. WINBUGS PROGRAMMES, INITIALS, DATA ...... 46 11.2. LOCATION OF FILES ...... 62 11.3. ALTERNATIVE CARTOGRAPHIC DISPLAY FOR THE SPATIAL DISTRIBUTION OF DISTRICT EFFECTS...... 63 11.4. SELECTED OUTPUT ...... 65 11.4.1. Space-cohort model without interactions ...... 65 11.4.2. Space-cohort model with interactions ...... 70 11.4.3. Space-period model without interactions ...... 75 11.4.4. Space-period model with interactions ...... 80
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2. Abstract
2.1. Background The variation of cervical cancer mortality in Belgium over time and place was analysed in separate previous studies. In this report we study changes over time and district (arrondissement) simultaneously using Bayesian hierarchical models.
2.2. Material and methods Data on mortality from cervical cancer, coded as 180, according to the 8th and 9th ICD classification, in the 43 Belgian districts, were obtained from the National Institute of Statistics. Variation over two time dimensions (cohort and period) and over place (districts), after controlling for age, was modelled using Bayesian models, following the approach described by Lagazo et al, 2001. Spatial components were defined as a Gaussian convolution of structured and unstructured components (CAR models). The Deviance Information Criterion (DIC) was used to compare models with and without space*time interactions.
2.3. Results Mortality changed substantially over successive cohorts. In global, mortality was higher than average in the older cohorts, women born before the 1920s, and lower than average in the younger cohorts born later. Two upward peaks could be discerned for the 1890-1899 and the 1915-1924 cohort. Since cohort 1930-39, mortality did not further decline. Even a discrete increase could be discerned in the youngest cohorts. In spite of the complex change over cohorts, mortality declined rather monotonously over calendar periods. The spatial effects relative to the average mortality varied between 0.73 and 1.52. Increased mortality was found in 5 neighbouring districts in the mid-west of Belgium: Charleroi, Soignies, Thuin, Namur and Philippeville, and further in the districts of Antwerp, Gent and Eeklo in the North of Belgium. A lower mortality risk (relative risk between 0.6 and 0.8) was observed in four districts: Tunhout, Tielt and the two Flemish-Brabant districts. The DIC was marginally smaller for models including interactions. Nevertheless their impact was practically ignorable.
2.4. Discussion Hierarchical Bayesian modelling yielded similar common time trend effects as obtained from previous ACP-Poisson regression models. But this time spatial effects could be modelled simultaneously, where ACP models would get into problems because of redundancy and over-saturation. Robust district and cohort effects could be estimated that were not influenced by space*time interaction. The risk of mortality is higher than average in districts were large agglomerations are localised, probably because of increased transmission in cities of the sexually transmittable HPV-virus, which is the main etiological risk factor for cervical cancer. Care must be taken with the interpretation of cervical cancer mortality trends, since an important part of uterine cancer deaths are certified as not otherwise specified (NOS). Nevertheless, preliminary correlation analyses did not indicate a substantial link between the local and periodic fitted relative risk and the proportion of NOS uterine
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 5 cancers. We can therefore conclude that the estimated spatial and time effects were probably not seriously biased by this certification problem.
3. Key Words
Age-cohort-period modelling, spatial analysis, space-time models, Bayesian models, geographical epidemiology, trend analysis, cervical cancer, mortality.
4. Introduction
In earlier publications, we analysed the temporal variation of mortality form cervical cancer in Belgium, from 1954 to 1994; and in the Flemish Region between 1969 and 1997 [Vyslouzylova, 1997; Arbyn, 2000a; 2002a; 2002b]. Spatial variation of mortality from breast cancer and uterine cancers at municipality level in the Flemish Region between 1993 and 1998 was explored recently for the Flemish Region [Arbyn, 2000].
In the current statistical research report we will explore Bayesian hierarchical models to analyse the simultaneous time and trend variation of cervical cancer mortality in Belgium between 1969 and 1994. The considered geographical subdivision is the district (“arrondissement”).
This statistical work was conducted within the frame work of the Europe Against Cancer Programme, in particular the European Network for Cervical Cancer Screening.
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5. Materials
Cervical cancer deaths were considered for females in the 43 districts (or “arrondissements”) of Belgium from 1969 to 1994. Deaths and corresponding population for each region were cross-classified by 14 age classes (20-24, ..., 80-84, 85 or more) and six calendar periods (1969, 1970-74, ... , 1990-94). Periods were defined as in previous trend analyses concerning the whole of Belgium or the Flemish Region [Vyslouzilova, 1997; Arbyn, 2000a; 2002a; 2002b]. This resulted in a first one-year period encompassing only 1969. All other periods encompass five calendar years. Age groups are 5-years wide except the last one, which is open-ended. Only the cause of death coded as 180 in the 8th (applied from 1969 to 1978) and the 9th ICD classification (applied from 1979 to 1994) are considered. Mortality rates and number of deaths can be found in Table 1. Data on number of deaths and female population were obtained from the National Institute of Statistics.
Table 1. Mortality from cervical cancer in Belgium by period and by age group between 1969 and 1994: rates per 100 000 women-years and number of deaths (between brackets).
Age Periods groups 1969 1970-1974 1975-1979 1980-1984 1985-1989 1990-1994 20-24 0.000 (0) 0.000 (0) 0.107 (2) 0.103 (2) 0.000 (0) 0.057 (1) 25-29 0.354 (1) 0.260 (4) 0.056 (1) 0.270 (5) 0.515 (10) 0.154 (3) 30-34 0.000 (0) 0.564 (8) 0.586 (9) 1.076 (19) 0.544 (10) 1.176 (23) 35-39 1.595 (5) 1.764 (27) 1.972 (28) 1.912 (29) 1.887 (33) 2.175 (40) 40-44 4.612 (15) 4.609 (75) 3.204 (49) 2.774 (39) 3.731 (56) 2.985 (52) 45-49 11.799 (39) 6.533 (107) 5.169 (83) 4.978 (75) 5.482 (76) 4.838 (72) 50-54 14.763 (34) 11.610 (152) 8.043 (129) 6.233 (98) 4.739 (70) 6.298 (86) 55-59 12.421 (35) 10.373 (133) 10.207 (128) 8.407 (131) 6.012 (92) 5.051 (73) 60-64 16.062 (46) 11.579 (164) 11.387 (140) 10.487 (126) 10.694 (160) 5.480 (81) 65-69 17.895 (48) 13.843 (185) 12.069 (158) 14.607 (168) 13.824 (156) 7.773 (110) 70-74 15.181 (33) 15.845 (176) 17.288 (201) 16.129 (189) 12.678 (131) 11.131 (115) 75-79 24.944 (37) 19.640 (151) 19.330 (167) 20.220 (191) 16.798 (163) 10.230 (90) 80-84 20.338 (17) 19.007 (82) 23.922 (119) 24.176 (144) 18.006 (121) 13.717 (99) 85+ 27.909 (13) 21.650 (53) 23.423 (69) 34.147 (120) 27.006 (123) 18.459 (101)
6. Methods
6.1. Introduction to Bayesian models
We have used the approach described by Lagazio et al. [2001] for modeling the space-time variation of the mortality rates. Four Bayesian models have been fitted – space-period and space-cohort models with and without interactions. Models with and without interactions were compared using the Deviance Information Criterion (DIC) [Spiegelhalter et al., 2002], being a generalization of the Akaike’s Information Criterion in the Bayesian framework. Lower values of DIC indicate a better model. DIC is defined as
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 7 DIC = D + (D − D(θ )) = D(θ ) + 2(D − D(θ )), where • D is the posterior expectation of the deviance and summarizes the fit of the model, • D (θ ) is the deviance evaluated at the posterior expectations of parameters.
The term D − D(θ ) represents the effective number of parameters.
Posterior distributions of the parameters of interest have been obtained using Gibbs sampling in WinBUGS [Spiegelhalter, 2000]. For a general introduction in the use of WinBUGS for spatial modeling, we refer to Lawson et al [2003]. For models without interactions, we have run two chains with a burn-in of 50 000 iterations and additional 50 000 iterations with 1:10 thinning to obtain a sample of 10 000 (2x 5 000) values. For models with interactions, we have run two chains with a burn-in of 100 000 iterations and additional 50 000 iterations with 1:10 thinning to obtain a sample of 10 000 values. Gelman-Rubin [1992], Geweke [1992] have been used to check for convergence. Diagnostic plots for the four models can be found in the technical report.
6.2. Space-cohort model (SC)
6.2.1. Log-linear Poisson age-cohort model Age-specific reference rates for the whole of Belgium were calculated using an age- cohort (AC) model [Clayton, 1987; Arbyn, 2002a] on the aggregated data of Table 1. The cohort is computed as period - age where period is the beginning of the concerned period and age the minimal age of the age group. The AC model fitted is a log-linear model for the observed counts with person-years at risk as an offset term, no constant term, 14 dummies for age groups and 18 dummies for cohort effect. There were 19 cohorts defined in this model — labeled as 1880, 1885, ... , 1970. The average of all cohorts was used as the reference. Detailed description of the AC model can be found below.
We use the following notation. • DTlj ... number of deaths in the lth age group of the jth cohort; • Tlj... number of person-years in the lth age group of the jth cohort. The AC model is given by • DTlj ~ Poisson(µlj ), 19 • log(µ / T )= age + cohort , with .cohort = 0 . lj lj λ l λ j ∑λ j j=1
The goodness of fit is assessed by the deviance (DEV), which is based on the ratio between the likelihoods (L) of the current and the saturated model (DEV = -2Ln (L{modeli} / L{saturated model}) [Nelder, 1972]. This log likelihood ratio statistic provides an overall measure of the adequacy of the model. It follows approximately a chi-square distribution whose number of degrees of freedom equals the amount of observations less the number of parameters included in the model [Frome, 1983]. Age-
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 8 specific reference rates were obtained by exponentiating the estimated coefficients for the age variables, i.e. exp( age ) was used as the age-specific reference rate for the lth λ l age group. The expected cases for each cohort and calendar period in each region can then be calculated by applying the age-specific reference rates to the cohort- (or period-) specific person-years of each region (cohort and period determine uniquely the age). The procedure is described more in detail below.
Notation. • DTijk ... observed number of deaths in the ith region, the jth cohort in the kth period; • ETijk ... expected number of deaths in the ith region, the jth cohort in the kth period; • Tijk ... number of person-years in the ith region and the jth cohort in the kth period;
Computation of the expected number of cases. age • ETijk = Tijk . exp ( ˆ ), λ l where agel = periodk - cohortj .
6.2.2. Bayesian Space-Cohort mode (SC) For the spatio-temporal analysis we have considered all nineteen birth cohorts (denoted as 1880, 1885, …, 1965, 1970). The data for the space-cohort (SC) model (observed and expected) cases were obtained by aggregating the data over the six periods. The following model is assumed.
Notation. • DT = 6 DT ... observed number of deaths in the ith region, the jth cohort; ij Σk =1 ijk
• ET = 6 ET ... expected number of deaths in the ith region, the jth cohort; ij Σk =1 ijk
• T = 6 T ... number of person-years in the ith region and the jth cohort. ij Σk =1 ijk
Model. DTij ~ Poisson(µij ), with µij = RRij · ETij , where i =1,... , 43 (regions), j =1,... , 19 (cohorts), ⎛ ⎞ ⎜ µ ij ⎟ log = log = , ⎜ ⎟ ()ij η ⎜ ⎟ RR ij ⎝ ET ij ⎠ where is a linear predictor, and RRij is a relative risk of the ith region and jth ηij ηij cohort. The estimated cohort effects (RRij=e ) are similar to the standardized cohort
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 9 mortality ratios (SCMR) described by Beral [1974]. The linear predictor is ηij specified in 2 ways. For the model without interactions: =α + struct + unstruct + cohort , ηij bi bi b j while for the model with interactions: =α + struct + unstruct + cohort + ac , ηij bi bi b j ccij
with ac = ac - ac - ac + ac . This centering is used to improve a convergence of ccij cij c1 j ci1 c11 the Markov Chain Monte Carlo simulation (MCMC).
The terms have the following meaning. • struct ... a structured spatial variability term; bi • unstruct ... an unstructured spatial variability term; bi • cohort ... the effect of the jth cohort; b j • ac ... an area- cohort interaction. cij
6.2.3. Prior distributions for the SC model
Let T struct = struct ,..., struct , b ()b1 b43 T unstruct = unstr ,..., unstr , b ()b1 b43 T cohort = cohort ,..., cohort , b ()b1 b19 T AC = AC ,..., AC . c ()c1,1 c43,19
The prior distributions for all effects are multivariate normals.
struct ~ N 0, −1 , b ( ()τ structκ struct ) unstruct ~ N 0, −1 , b ( ()τ unstr I 43 ) cohort ~ N 0, −1 , b ( ()τ cohortκ cohort ) AC ~ N 0, −1 , c ( (τ ACκ AC ) )
Precisely, both the structured spatial term bstruct and the cohort effect bcohort are assigned the Gaussian CAR prior distribution. For closer specification of matrices
κstruct, κcohort see Lagazio [2001]. Such priors are implemented in WinBUGS’ function car.normal(). Note that this function constraints the random effects to sum to zero. So that the following constraints are satisfied in the model.
43 19 struct = 0, cohort = 0. Σbi Σb j i=1 j=1
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 10 Specification of the matrix κAC can also be found in Lagazio [2001]. The prior for the interaction vector cAC is a Markov random field. The intercept term α is given a flat prior using WinBUGS function dflat(). Precision terms τstruct and τcohort are given a gamma prior Gamma(0.5,0.0005), Kelsall and Wakefield [1999]. The precision term τunstr is given a gamma prior Gamma(0.001,0.001). More notation is needed when describing a prior used for the interaction precision parameter τAC. Let AC denote c ij the prior conditional mean of AC given all other cAC terms (see equation (4) in cij Lagazio [2001]. Further let ni denote the number of neighbouring areas to the ith region. For na = 43 the number of districts in Belgium and nc =19 the number of cohorts involved in the model. Let nnij = ni if j =1 or j = nc and nnij =2ni otherwise. Let = x AC .(AC − AC ). The parameter τAC has then a gamma prior Gamma(rAC ξ ij nnij cij cij c ij + AC + AC Σi, j ξ ,lAC )with =1+ na nc , = 0.01+ ij . r 2 l 2
This prior distribution follows closely Lagazio [2001].
6.3. Space-period model (SP)
6.3.1. Log-linear Poisson age-period model Age-specific reference rates for the whole region (Belgium) were calculated using an age-period (AP) model [Clayton, 1987; Arbyn, 2002a] on the aggregated data of Table 1. The AP model fitted is a log-linear model for the observed counts with person-years at risk as an offset term, no constant term, 14 dummies for age groups and 5 dummies for period effect. There were 6 periods assumed in this model — labeled as 1969, 1970, 1975, 1980, 1985, 1990. The average of all periods was used as the reference. Detailed description of the (AP) model can be found below.
AP model Notation. • DTlk ... number of deaths in the lth age group of the kth period; • Tlk... number of person-years in the lth age group of the kth period. Model • DTlk ~ Poiss(µlk), 6 • log(µ / T )= age + period , whit .period = 0 . lk lk λ l λ k ∑λ k k =1 Age-specific reference rates were obtained by exponentiating the estimated coefficients for the age variables, i.e. exp ( age ) was used as the age-specific λl reference rate for the lth age group.
The expected cases for each calendar period and age group in each region were then calculated by applying the age-specific reference rates to the age- and period-specific person-years of ea h region. The procedure is described more in detail below. Notation.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 11 • DTilk... observed number of deaths in the ith region, the lth age group in the kth period; • ETilk ... expected number of deaths in the ith region, the lth age group in the kth period; • Tilk... number of person-years in the ith region and the lth age group in the kth period; Computation of the expected number of cases. age • ETilk = Tilk exp( ˆ ), λl
6.3.2. Bayesian Space-Period model (SP)
The data for the space-period (SP) model (observed and expected) cases were obtained by aggregating the whole data set over age groups. All available five-year period groups were used: 1969, 1970 (=1970-74), 1975 (=1975-79), 1980 (=1980-84), 1985 (=1985-89), 1990 (=1990-94). Note that the first period is based on observations from the only one year. The following model is assumed. Notation. DT = 14 DT observed number of deaths in the ith region, the kth period; ik Σl=1 ilk ET = 14 ET expected number of deaths in the ith region, the kth period; ik Σl=1 ilk T = 14 T number of person-years in the ith region and the kth period. ik Σl=1 ilk Model. DTik ~ Poiss(µik), with µik = RRik · ETik, where i=1,... , 43 (regions), k =1,... , 6 (periods), ⎛ µ ⎞ log ⎜ ik ⎟ = log()= , ⎜ ⎟ RRik ηik ⎝ EDT ik ⎠ where ηik is a linear predictor and RRik is a relative risk of the ith region and kth period. The linear predictor is specified in the following way.
model without interactions: = α + struct + unstr + period , ηik b1 b1 bk model with interactions: = α + struct + unstr + period + AP , ηik b1 b1 bk ccik with AP =AP − AP − AP + AP . ccik cik clk ccil cc11 This centering is used to improve a convergence of the MCMC.
The terms have the following meaning. • struct ... a structured spatial variability term; b1 • unstr ... an unstructured spatial variability term; b1 • period ... the effect of the kth period; bk • AP ... an area-period interaction. cik
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 12
Let struct struct struct = ( ,..., )T , b b1 b43 unstr = ( unstr ,..., unstr)T , b b1 b43
period = ( period ,..., period)T , b b1 b6 AP AP AP = ( ,..., )T . c c1,1 c43,6
The prior distributions for all effects are multivariate normals. struct ~ N(0,( )−1), b τ structκ struct unstr ~ (0,( )−1), b τ unstr I 43 period ~N(0,( )−1), b τ periodκ period AP ~ N,(0,( )−1), c τ APκ AP
Precisely, both structured spatial term bstruct and cohort effect bperiod are assigned the
Gaussian CAR prior distribution. For loser specification of matrices κstruct, κperiod see Lagazio [ 2001]. Such priors are implemented in WinBUGS function car.normal(). Note that this function constraints the random effects to sum to zero. So that the following constraints are satisfied in the model. 43 6 struct = 0, period = 0. Σbi Σbk i=1 k=1
Specification of the matrix κAP can also be found in Lagazio [2001]. The prior for the interaction vector CAP is a Markov random field. The intercept term α is given a flat prior using WinBUGS function dflat(). Precision terms τstruct and τperiod are given a gamma prior Gamma(0.5,0.0005). This prior distribution was suggested by Kelsall and Wakefield (1999) and improved a lot convergence of the resulting MCMC compared to commonly used prior Gamma(0.001,0.001). The precision term τunstr is given a gamma prior Gamma(0.001,0.001). More notation is needed when describing a prior used for the interaction precision parameter . Let AP denote the prior τAP c ik conditional mean of AP given all other AP terms (see equation (4) in Lagazio et al. cik c [2001]. Further let ni denote the number of neighbouring areas to the th region. Let na =43 be the number of assumed regions and np =6 the number of periods involved in the model. Let nnik = ni if k =1 or k = np and nnik =2 ni otherwise. Let = ξ ik . AP .( AP − AP). The parameter AP has then a gamma prior Gamma nnik cik cik c ik τ ( r AP ,l AP ) with
+ AP na np r =1+ , 2
AP ξ = 0.01+ Σi,k ik . l 2
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6.4. Cartographic display style
For display of fitted relative risks we categorised the median of the posterior distribution of the spatial effects into 7 categories as suggested by Knorr-Held and Raser [2000]. A log-scale was used to define the cut-offs between categories, in order to make the space of relative risks between 0 to 1 equal to the space of RRs ranging from 1 to infinity. A colour palette often used in relief maps was applied: 3 shadings of green (dark, intermediate and light) to indicate lower risk than the reference; yellow for the flexion zone indicating a risk around the reference; and further, orange, red and brown for risks that are progressively higher than the reference. The cut-off values are defined in Table 2. Table 2. Cut-offs for the categories and used shades to colour the districts according to the relative mortality risk. Shade From To Dark green ≥0.0 <0.5 Intermediate green ≥0.5 <0.6 Light green ≥0.6 <.8 Yellow ≥0.8 <1.2 Orange ≥1.2 <1.5 Red ≥1.5 <2.0 Brown ≥2.0 infinity
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7. Results
Over the period of 26 years, 6 406 women died from cervical cancer, which corresponds to a crude rate of 4.90/105 women-years (95% confidence interval 4.78- 5.02/105). This rate is at the same time the age-standardised rate since the overall age- specific cumulated women-years were used as reference population. How relative mortality risks vary by district and cohort is covered in subchapter 7.1, and by district and period in subchapter 7.2.
7.1. Space-cohort model
7.1.1. Log-linear Poisson age-cohort model
The deviance of fitted AC model was 105.6 on 52 degrees of freedom. Estimated coefficients age can be found in Table 3. λ l Table 3. Age- cohort log-linear model. Estimated effects of age, later on used to compute age-specific reference rates for the space-cohort Bayesian model.
Age group 20-24 25-29 30-34 35-39 40-44 45-49 50-54 age ˆ λ l -14.1399 -12.5029 -11.4755 -10.3700 -9.7859 -9.4284 -9.2794 age exp ( ˆ ) (per 105) λ l 0.07 0.37 1.04 3.14 5.62 8.04 9.33
Age group 55-59 60-64 65-69 70-74 75-79 80-84 85+ age ˆ λ l -9.3438 -9.2842 -9.2341 -9.1989 -9.1026 -9.0442 -8.8460 age exp ( ˆ ) (per 105) λ l 8.75 9.29 9.77 10.12 11.14 11.81 14.40
7.1.2. Bayesian SC model
For both models with and without interactions, a good convergence with respect to the parameters involved in the definition of the relative risk RRij was obtained. Also the cohort precision hyperparameter. τ cohort does not carry any problems. Slightly worse convergence was reached with respect to spatial and interaction precision parameters τ struct, τ unstr and τ AC. Nevertheless, this did not inhibit us from drawing reasonable inference with respect to the relative risk. MCMC summary of the scalar parameters can be found in Table 4.
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Table 4. Space-cohort Bayesian models. MCMC summary of scalar parameters.
SC model without interactions Parameter Mean SD MC error 2.5% Median 97.5% Sample size α -0.0381 0.0379 0.0007 -0.1131 -0.0380 0.0365 10000 struct τ 89.18 122.29 2.92 5.77 40.14 450.61 10000 unstr τ 42.92 86.11 2.20 13.48 27.28 167.12 10000 AC τ 43.16 18.22 0.25 15.95 40.36 86.56 10000
SC model with interactions Parameter Mean SD MC error 2.5% Median 97.5% Sample size α -0.0005 0.0500 0.0011 -0.0987 -0.0004 0.0967 10000 struct τ 750.96 1262.34 29.15 6.49 238.20 4380.07 10000 unstr τ 37.48 84.11 1.98 12.71 23.51 156.70 10000 cohort τ 45.38 16.60 0.28 16.33 42.19 92.65 10000 AC τ 29887.13 570.41 13.60 1994.0 2947.00 4206.07 10000
Table 5. Space-cohort model. Deviance information criterion and quantities used to compute it.
Quantity Model without interactions Model with interactions D 423.4 419.3 D(θ ) 400.2 395.6
DIC 446.6 443.0
7.1.3. Cohort effects estimated from the SC model
The estimated cohort effects (in exponentiated format,(RRj=exp(ηj)) are shown in Table 6 and plotted in Figure 1.
Table 6. Cohort effects and 95% credibility intervals, relative risk of mortality from cervical cancer in cohortj, compared to the Belgian average. Cohort effect Cohort Mid year Index (j) Median 0.025 0.975 .-1885 1880 1 1.63 1.20 2.24 1880-1889 1885 2 1.58 1.31 1.87 1885-1894 1890 3 1.67 1.46 1.89 1890-1899 1895 4 1.85 1.67 2.05 1895-1904 1900 5 1.72 1.56 1.88 1900-1909 1905 6 1.53 1.40 1.67 1905-1914 1910 7 1.33 1.21 1.45 1910-1919 1915 8 1.22 1.11 1.33 1915-1924 1920 9 1.18 1.08 1.29
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 16
Table 6. (continued) Cohort Mid year Index (j) Median 0.025 0.975 1920-1929 1925 10 0.90 0.82 0.98 1925-1934 1930 11 0.67 0.60 0.74 1930-1939 1935 12 0.58 0.51 0.64 1935-1944 1940 13 0.62 0.55 0.70 1940-1949 1945 14 0.62 0.54 0.70 1945-1954 1950 15 0.61 0.52 0.71 1950-1959 1955 16 0.69 0.58 0.82 1955-1964 1960 17 0.83 0.66 1.06 1960-1969 1965 18 0.75 0.54 1.02 1965-1974 1970 19 0.75 0.48 1.14
Belgium, 1969-1994; SC model
3
2
1 Cohort effect
.5
.3 0 5 10 15 20 Cohort
Figure 1. Cohort effects or relative risks of mortality from cervical cancer in Belgium between 1969- 1994,according to cohort (estimated from an SC model). In global, mortality was higher than average in the older cohorts (j ≤ 9, women born before 1924) and lower than the average in the younger cohorts (j>9, women born after 1920). The cohort effect increased until cohort C4 (j=4 corresponding to the cohort 1890- 1899) than it decreased substantially until C8. This decrease was interrupted in C9 (cohort 1915-1924). Mortality continued to decline further to a level significantly lower than the average until C12 (cohort 1930-39). From then mortality remained stable with a tendency to raise again in the youngest cohorts.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 17 7.1.4. Space effects estimated from the SC model The estimated area effects (in exponentiated format,(exp(ηi)) are shown in Table 7 and mapped in Figure 4.
Table 7. District effects and 95% credibility intervals, relative risk of mortality from cervical cancer in districti, compared to the Belgian average, estimated from the SC model. Relative risk of mortality District Province Code Index (i) Median 0.025 0.975 ANTWERPEN Antwerp 11 1 1.25 1.12 1.40 MECHELEN Antwerp 12 2 0.81 0.68 0.96 TURNHOUT Antwerp 13 3 0.73 0.61 0.87 BRUSSELS Brussels-Capital 21 4 1.06 0.95 1.18 HALLE-VILVOORDE Flemish-Brabant 23 5 0.75 0.65 0.87 LEUVEN Flemish-Brabant 24 6 0.76 0.65 0.88 NIVELLES Waloon-Brabant 25 7 0.96 0.81 1.12 BRUGGE W-Flanders 31 8 0.82 0.69 0.97 DIKSMUIDE W-Flanders 32 9 0.84 0.62 1.13 IEPER W-Flanders 33 10 0.97 0.77 1.21 KORTRIJK W-Flanders 34 11 0.85 0.72 1.01 OOSTENDE W-Flanders 35 12 1.04 0.85 1.27 ROESELARE W-Flanders 36 13 0.97 0.79 1.20 TIELT W-Flanders 37 14 0.77 0.58 1.00 VEURNE W-Flanders 38 15 0.98 0.73 1.28 AALST E-Flanders 41 16 1.08 0.92 1.27 DENDERMONDE E-Flanders 42 17 1.11 0.99 1.32 EEKLO E-Flanders 43 18 1.44 1.15 1.79 GENT E-Flanders 44 19 1.44 1.27 1.62 OUDENAARDE E-Flanders 45 20 0.88 0.70 1.08 SINT-NIKLAAS E-Flanders 46 21 0.98 0.81 1.17 ATH Hainaut 51 22 0.99 0.77 1.24 CHARLEROI Hainaut 52 23 1.52 1.35 1.72 MONS Hainaut 53 24 0.87 0.73 1.02 MOUSCRON Hainaut 54 25 0.90 0.69 1.16 SOIGNIES Hainaut 55 26 1.48 1.26 1.75 THUIN Hainaut 56 27 1.50 1.26 1.78 TOURNAI Hainaut 57 28 0.85 0.69 1.03 HUY Liège 61 29 1.05 0.84 1.31 LIEGE Liège 62 30 1.13 1.00 1.27 VERVIERS Liège 63 31 0.99 0.83 1.16 WAREMME Liège 64 32 0.96 0.74 1.22 HASSELT Limburg 71 33 1.01 0.86 1.19 MAASEIK Limburg 72 34 0.81 0.64 1.02 TONGEREN Limburg 73 35 0.93 0.75 1.13 ARLON Luxembourg 81 36 1.00 0.74 1.32 BASTOGNE Luxembourg 82 37 0.94 0.69 1.27 MARCHE-EN- FAMENNE Luxembourg 83 38 0.91 0.67 1.21 NEUFCHATEAU Luxembourg 84 39 0.93 0.70 1.21 VIRTON Luxembourg 85 40 1.01 0.74 1.35 DINANT Namur 91 41 1.01 0.80 1.27 NAMUR Namur 92 42 1.22 1.05 1.42 PHILIPPEVILLE Namur 93 43 1.41 1.10 1.80
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 18 Space-cohort model
11 13 43 46 35 31 72
12 38 44 42 32 37 71 36
41 24 33 45 23 21 73 34 54
25 64 57 51 62 55
61 53 52 92 63
83 91 56 93 82 Mortality from cervical cancer Distribution of relative risks (RRi) 84 0 to 0.5 (0) 0.5 to 0.6 (0) 0.6 to 0.8 (4) 0.8 to 1.2 (31) 81 1.2 to 1.5 (7) 85 1.5 to 2 (1) 2 to 3 (0) sc2.wor
Figure 2. Geographical distribution of area effects, relative risk of mortality from cervical cancer by districti, compared to the Belgian average. Areas are labeled with the NIS-district code (see Table 7). District-effects are estimated from the SC model.
raw SMR
11 13 43 46 35 31 72
12 38 44 42 32 37 71 36
41 24 33 45 23 21 73 34 54
25 64 57 51 62 55
61 53 52 92 63
83 91 56 93 82 Mortality from cervical cancer Distribution of raw SMRs 0 to 0.5 (0) 84 0.5 to 0.6 (0) 0.6 to 0.8 (13) 0.8 to 1.2 (24) 81 1.2 to 1.5 (6) 1.5 to 2 (0) 85 2 to 3 (0) rawsmr2.wor
Figure 3. Geographical distribution of raw SMRs, using the global all period mortality as reference rate.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 19 The highest mortality from cervical cancer is observed in district 52 (Charleroi), with an estimated risk that is 52% (CI: 35-72%) higher than the average. The estimated risk is between 20 to 50% higher in 4 other districts of the provinces of Hainaut and Namur (Soignies (code=55), Thuin (56), Namur (92) and Philippeville(93)) and also in the districts of Eeklo (43), Gent (44) and Antwerp (11). The estimated relative risk is lower than average in 4 districts: Turnhout (13), Halle- Vilvoorde (23), Leuven (24) and Tielt (37).The distribution of relative risks estimated from an SC model should be compared with the distribution of raw SMRs (Figure 3). These raw SMRs were computed using the overall age-specific rate as reference. For a discussion of the differences between raw SMR and estimated relative risks (RRi), we refer to the discussion. The geographical distribution of raw SMRs and estimated area effects is also
displayed using an alternative color palette definition, respectively in Figure 12 and in Figure 13 (see page 63).
7.1.5. Local cohort effects (S*C model)
The local cohort effects estimated for each district from the S*C model are plotted in Figure 4. The shape of all cohort effects curves are nearly identical indicating that the impact of the interactions can be ignored.
Figure 4. District specific cohort effects: relative change in mortality from cervical cancer by cohort Belgium, districts 1-9, 1969-1994, S*C model
Antwerpen Mechelen Turnhout 3 2 1 .5 .3
Brussels Vilvoorde Leuven 3 2 1 .5 .3 Cohort effect Nivelles Brugge Disksmuide 3 2 1 .5 .3 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Cohort
for each separate district, estimated from a space-cohort model, including cohort*district interactions.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 20 Belgium, districts 10-18, 1969-1994, S*C model
Ieper Kortrijk Oostende 3 2 1
.5 .3
Roeselare Tielt Veurne 3 2 1 .5 .3
Cohort effect Aalst Dendermonde Eeklo
3 2
1 .5 .3 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
Cohort
Belgium, districts 19-27, 1969-1994, S*C model
Gent Oudenaarde St-Niklaas 3 2 1 .5 .3
Ath Charleroi Mons
3 2
1 .5 .3 Cohort effect Mouscron Soignies Thuin
3 2 1
.5 .3 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
Cohort
Figure 4. (continued).
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 21 Belgium, districts 28-36, 1969-1994, S*C model
Tournai Huy Liège 3 2 1 .5 .3 Verviers Waremmes Hasselt
3 2
1
.5 .3 Cohort effect Maaseik Tongeren Arlon 3 2 1 .5 .3 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
Cohort
Belgium, districts 37-43, 1969-1994, S*C model
Bastogne Marche-en-Famenne Neufchâteau
3 2
1 .5 .3 Virton Dinant Namur
3 2 1
.5 .3
Cohort effect 0 5 10 15 20 0 5 10 15 20 Phillipeville 3 2 1 .5 .3 0 5 10 15 20
Cohort
Figure 4. (continued).
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 22 7.1.6. Contrast between local raw SCMRsa and fitted cohort effects estimated from the S*C model
The contrasts between the raw standardised cohort mortality ratios and the fitted cohort effects are illustrated in Figure 5. In the figures the raw SCMRsa are drawn in green dotted line, whereas the fitted cohort effects, estimated from the S*C model are drawn in a full red line.
Belgium, districts 1-9, 1969-1994, raw & fitted SCMR (S*C model)
Fitted cohort effect Raw SCMR
Antwerpen Mechelen Turnhout 10 3 1 .3 .1
Brussels Vilvoorde Leuven 10 3 1 .3 .1 Cohort effect Nivelles Brugge Disksmuide 10 3 1 .3 .1 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Cohort
Figure 5. Raw standardised cohort mortality ratio and fitted cohort effects, by district, estimated from an S*C model (cervical cancer mortality, Belgium, 1969-94).
a SCMR: standardised cohort mortality ratio.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 23 Belgium, districts 10-18, 1969-1994, raw & fitted SCMR (S*C model)
Fitted cohort effect Raw SCMR
Ieper Kortrijk Oostende
10 3 1 .3 .1
Roeselare Tielt Veurne 10 3 1 .3 .1 Cohort effect Aalst Dendermonde Eeklo 10 3 1 .3 .1 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Cohort
Belgium, districts 19-27, 1969-1994, raw & fitted SCMR (S*C model)
Fitted cohort effect Raw SCMR
Gent Oudenaarde St-Niklaas
10 3 1 .3 .1
Ath Charleroi Mons
10 3 1 .3 .1
Cohort effect Mouscron Soignies Thuin
10 3 1 .3 .1 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Cohort
Figure 5. (continued).
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 24
Belgium, districts 28-36, 1969-1994, raw & fitted SCMR (S*C model)
Fitted cohort effect Raw SCMR
Tournai Huy Liège 10 3 1 .3 .1 Verviers Waremmes Hasselt 10 3 1 .3 .1 Cohort effect Maaseik Tongeren Arlon 10 3 1 .3 .1 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Cohort
Belgium, districts 37-43, 1969-1994, raw & fitted SCMR (S*C model)
Fitted cohort effect Raw SCMR
Bastogne Marche-en-Famenne Neufchâteau 10 3 1 .3 .1 Virton Dinant Namur
10 3 1 .3 .1 Cohort effect 0 5 10 15 20 0 5 10 15 20 Phillipeville
10 3 1 .3 .1 0 5 10 15 20 Cohort
Figure 5. (continued).
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 25
7.2. Space-period model
7.2.1. Loglinear Poisson age-period model
The deviance of the fitted AP model was 175.3 on 65 degrees of freedom. Estimated coefficients age can be found in Table 8. λl
Table 8. Age-period log-linear model. Estimated effects of age, later on used to compute age-specific reference rates for the space-period model.
Age group 20-24 25-29 30-34 35-39 40-44 45-49 50-54 age ˆ λl -14.4401 -12.8301 -11.7082 -10.8073 -10.2174 -9.7503 -9.4624 age Exp( ˆ ) (per 105) λl 0.04 0.27 0.82 2.03 3.65 5.83 7.77
Age group 55-59 60-64 65-69 70-74 75-79 80-84 85+ age ˆ λl -9.3903 -9.1627 -8.9529 -8.7959 -8.6148 -8.4873 -8.2255 age Exp( ˆ ) (per 105) λl 8.35 10.49 12.94 15.14 18.14 20.61 26.78
7.2.2. Bayesian SP model
For both models with and without interactions, a good convergence with respect to the parameters involved in the definition of the relative risk RRik was obtained. The space-period model shows also quite good convergence of all precision parameters τ period, τ struct, τ unstr, τ AP. MCMC summary of the scalar parameters can be found in Table 9. Table 10 shows quantities used to compute the deviance information criterion for both models. The presence of interactions seems again to be important.
Table 9. Space-period Bayesian models. MCMC summary of scalar parameters.
(SC) model WITHOUT interactions Parameter Mean SD MC error 2.5% Median 97.5% Sample size α -0.0643 0.0351 0.0007 -0.1357 -0.0637 0.0045 10000 struct τ 84.96 129.01 3.08 5.21 33.44 470.1 10000 unstr τ 52.05 112.47 2.64 13.88 28.57 244.3 10000 period τ 47.51 30.51 0.43 9.10 40.86 124.1 10000
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 26 Table 9. (continued). (SC) model WITH interactions α -0.0293 0.0430 0.0009 -0.1138 -0.0300 0.0567 10000 struct τ 795.77 1298.67 31.27 6.77 250.40 4624.00 10000 unstr τ 35.83 88.66 2.29 12.49 23.34 120.1 10000 period τ 53.56 36.12 0.51 9.14 45.25 146.4 10000 AP τ 2092.6 498.57 10.53 1208.0 2055.5 3162.0 10000
Table 10. Space-cohort model. Deviance information criterion and quantities used to compute it.
Quantity Model without interactions Model with interactions D 276.1 256.7 D(θ ) 267.5 247.5
DIC 295.5 287.5
7.2.3. Period effects estimated from the SP model
The estimated period effects (in exponentiated format,(RRk=exp(ηk)) are shown in Table 11 and plotted in Figure 6.
Table 11. Period effects and 95% credibility intervals, relative risk of mortality from cervical cancer in periodk, compared to the Belgian average.
Period effect Period First year Index (k) Median 0.025 0.975 1969 1969 1 1.33 1.19 1.48 1970-74 1970 2 1.10 1.05 1.16 1975-79 1975 3 1.04 0.99 1.09 1980-84 1980 4 1.04 0.99 1.10 1985-89 1985 5 0.90 0.86 0.95 1990-94 1990 6 0.70 0.66 0.74
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 27
Belgium, 1969-1994; SP model
2
1.5
1 Period effect
.67
.5 1 2 3 4 5 6 Period
Figure 6. Period effects or relative risks of mortality from cervical cancer in Belgium between 1969- 1994, according to period (estimated from an SP model).
The mortality from cervical cancer was significantly higher than average in the first two periods. In the 3rd and 4th period mortality did not differ from the average, whereas in the last two periods mortality dropt significantly under the average line.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 28 7.2.4. Space effects estimated from the SP model Table 12. District effects and 95% credibility intervals, relative risk of mortality from cervical cancer in districti, compared to the Belgian average, estimated from the SP model. Relative risk of mortality District Province Code Index (i) Median 0.025 0.975 ANTWERPEN Antwerp 11 1 1.25 1.13 1.39 MECHELEN Antwerp 12 2 0.81 0.69 0.95 TURNHOUT Antwerp 13 3 0.73 0.61 0.86 BRUSSELS Brussels-Capital 21 4 1.06 0.97 1.17 HALLE-VILVOORDE Flemish-Brabant 23 5 0.75 0.65 0.86 LEUVEN Flemish-Brabant 24 6 0.76 0.65 0.88 NIVELLES Waloon-Brabant 25 7 0.96 0.82 1.12 BRUGGE W-Flanders 31 8 0.82 0.69 0.97 DIKSMUIDE W-Flanders 32 9 0.85 0.62 1.13 IEPER W-Flanders 33 10 0.97 0.77 1.21 KORTRIJK W-Flanders 34 11 0.85 0.72 1.00 OOSTENDE W-Flanders 35 12 1.04 0.86 1.26 ROESELARE W-Flanders 36 13 0.97 0.79 1.19 TIELT W-Flanders 37 14 0.77 0.59 1.00 VEURNE W-Flanders 38 15 0.98 0.73 1.28 AALST E-Flanders 41 16 1.08 0.93 1.26 DENDERMONDE E-Flanders 42 17 1.11 0.92 1.31 EEKLO E-Flanders 43 18 1.44 1.15 1.79 GENT E-Flanders 44 19 1.43 1.28 1.60 OUDENAARDE E-Flanders 45 20 0.88 0.70 1.08 SINT-NIKLAAS E-Flanders 46 21 0.98 0.82 1.17 ATH Hainaut 51 22 0.99 0.78 1.25 CHARLEROI Hainaut 52 23 1.52 1.36 1.70 MONS Hainaut 53 24 0.87 0.73 1.02 MOUSCRON Hainaut 54 25 0.90 0.69 1.15 SOIGNIES Hainaut 55 26 1.48 1.26 1.73 THUIN Hainaut 56 27 1.50 1.26 1.78 TOURNAI Hainaut 57 28 0.85 0.69 1.03 HUY Liège 61 29 1.06 0.84 1.30 LIEGE Liège 62 30 1.13 1.01 1.26 VERVIERS Liège 63 31 0.99 0.84 1.16 WAREMME Liège 64 32 0.96 0.75 1.22 HASSELT Limburg 71 33 1.01 0.86 1.19 MAASEIK Limburg 72 34 0.81 0.64 1.01 TONGEREN Limburg 73 35 0.93 0.75 1.13 ARLON Luxembourg 81 36 1.00 0.73 1.33 BASTOGNE Luxembourg 82 37 0.94 0.68 1.27 MARCHE-EN- FAMENNE Luxembourg 83 38 0.91 0.68 1.22 NEUFCHATEAU Luxembourg 84 39 0.93 0.70 1.21 VIRTON Luxembourg 85 40 1.01 0.74 1.35 DINANT Namur 91 41 1.01 0.80 1.26 NAMUR Namur 92 42 1.22 1.06 1.41 PHILIPPEVILLE Namur 93 43 1.41 1.10 1.80
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 29 Space-period model
11 13 43 46 35 31 72
12 38 44 42 32 37 71 36
41 24 33 45 23 21 73 34 54
25 64 57 51 62 55
61 53 52 92 63
83 91 56 93 82 Mortality from cervical cancer Distribution of relative risks (RRi) 0 to 0.5 (0) 84 0.5 to 0.6 (0) 0.6 to 0.8 (4) 0.8 to 1.2 (31) 81 1.2 to 1.5 (7) 1.5 to 2 (1) 85 2 to 3 (0) sp2.wor
Figure 7. Geographical distribution of area effects, relative risk of mortality from cervical cancer by districti, compared to the Belgian average. Areas are labeled with the NIS-district code (see Table 7). District-effects are estimated from the SP model.
The map in Figure 7 is identical to the one derived from the SC-model. Using an alternative cartographic display ( Figure 14) yield map that is hardly different. The values of the area effects shown in Table 12 (SP-model) are very similar to the ones in Table 7 (SC-model) as well. We can conclude that the SP and SC models result in the same relative district estimates.
7.2.5. Local period effects (S*P model)
The local period effects estimated for each district from the S*P model are plotted in. Figure 8. The shape of the period effect curves is nearly identical for all districts indicating that the impact of the interactions is negligible.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 30
Belgium, districts 1-9, 1969-1994, S*P model
Antwerpen Mechelen Turnhout 3 2
1 .5 .3
Brussels Vilvoorde Leuven 3 2 1 .5 .3 Period effect Nivelles Brugge Disksmuide 3 2 1 .5 .3 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
Period
Belgium, districts 10-18, 1969-1994, S*P model
Ieper Kortrijk Oostende 3 2
1
.5 .3
Roeselare Tielt Veurne
3 2 1 .5 .3 Period effect Aalst Dendermonde Eeklo 3 2 1 .5 .3 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
Period
Figure 8. District specific period effects: relative change in mortality from cervical cancer by cohort for each separate district, estimated from a space-period model, including period*district interactions.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 31 Belgium, districts 19-27, 1969-1994, S*P model
Gent Oudenaarde St-Niklaas 3 2 1 .5 .3 Ath Charleroi Mons 3 2 1 .5 .3
Period effect Mouscron Soignies Thuin 3 2 1 .5 .3 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 Period Belgium, districts 28-36, 1969-1994, S*P model
Tournai Huy Liège 3 2 1 .5 .3
Verviers Waremmes Hasselt 3 2 1 .5 .3 Period effect Maaseik Tongeren Arlon 3 2 1 .5 .3 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
Period
Figure 8. (continued).
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 32 Belgium, districts 37-43, 1969-1994, S*P model
Bastogne Marche-en-Famenne Neufchâteau 3 2 1 .5 .3
Virton Dinant Namur 3 2 1 .5 .3
Period effect 1 2 3 4 5 6 1 2 3 4 5 6 Phillipeville 3 2 1 .5 .3 1 2 3 4 5 6 Period
Figure 8. (continued).
7.2.6. Contrast between local raw SMRs and fitted period effects estimated from the S*P model
The contrasts between the raw SMR and the fitted period effects are illustrated in Figure 9. In the figures the raw SMRsa are drawn in green dotted line, whereas the fitted period effects, estimated from the S*P model are drawn in a full red line.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 33 Belgium, districts 10-18, 1969-1994, raw & fitted SMR (S*P model)
Fitted period effect Raw SMR
Ieper Kortrijk Oostende
3 2 1 .5 .3
Roeselare Tielt Veurne 3 2 1 .5 .3 Period effect Aalst Dendermonde Eeklo
3 2 1 .5 .3 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 Period
Belgium, districts 1-9, 1969-1994, raw & fitted SMR (S*P model)
Fitted period effect Raw SMR
Antwerpen Mechelen Turnhout 3 2 1 .5 .3
Brussels Vilvoorde Leuven 3 2 1 .5 .3
Period effect Nivelles Brugge Disksmuide 3 2 1 .5 .3 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 Period
Figure 9. Raw SMRs and fitted period effects, by district, estimated from an S*P model (cervical cancer mortality, Belgium, 1969-94).
Belgium, districts 19-27, 1969-1994, raw & fitted SMR (S*P model)
SpaceTimeBayModCvxBelgium Fitted period3CoverMenIPH effect.doc 13/11/1999 - 14:01 Raw SMR 34 Gent Oudenaarde St-Niklaas 3 2
Error! Reference source not found.. (continued).
Figure 9. (continued).
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 35
Belgium, districts 37-43, 1969-1994, raw & fitted SMR (S*P model)
Fitted period effect Raw SMR
Bastogne Marche-en-Famenne Neufchâteau 3 2 1 .5 .3
Virton Dinant Namur 3 2 1 .5 .3
Period effect 1 2 3 4 5 6 1 2 3 4 5 6 Phillipeville 3 2 1 .5 .3 1 2 3 4 5 6 Period
Figure 9. (continued).
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 36
8. Discussion
Age-cohort-period-space modeling In this space-time trend analysis we modeled variation in mortality from cervical cancer, certified as such, between 1969 and 1994, over the 43 Belgian districts, over periods or cohorts after controlling for age. We did not consider the simultaneous inclusion of period and cohort in order to avoid irresolvable identification problems, which should render the model extremely complex when also the spatial dimension is taken into account [Clayton, 1989b]. We did not separate linear (drift) from non- linear time trend effects. For a thorough discussion on the attribution of linear trend into period and/or cohort component in the frame work of log-linear Age-Period- Cohort modeling, we refer to our previous work [Arbyn, 2002a]. By applying smoothing methods we obtained more robust maps, where some spurious red or green spots were wiped away. Below we will interpret the main results and comment on some factors that could have influenced them.
Complex changes by cohort In spite of monotonously decreasing period effects, strong and complex cohort effects could be observed. The common component of the cohort trend was very strong and hardly modified by local influences, since time*place interactions were negligible. These cohort effects are related to changes in exposure of the female population to risk factors. The main etiologic agent for cervical cancer is infection with sexually transmittable human papilloma virusses [zur Hausen, 1994; Bosch, 1995; Munoz, 1997; Walboomers, 1999; Bosch, 2002]. The variation in mortality by cohort reflects changes in sexual behaviour over time of the successive generations. The upward peaks in the cohorts C4 and C9 were determined by women who were in their young adulthood during the world wars when prevalence of sexually transmitted diseases was high [Beral, 1974]. The interruption in the declining trend observed after C12 can be attributed to the increased sexual freedom and subsequent risk of HPV infection after the sixties. We refer to previous publications for a more thorough discussion of the possible explanations of cohort effects [Arbyn, 2002a; 2002b].
Screening effects Setting up a successful screening programme that reaches quickly all age groups of the target population can yield a clear abrupt period effect. This was not observed for Belgium, where screening coverage rose gradually since the late nineteen sixties and not homogenously over all targeted age groups [De Schryver, 1989; Arbyn, 2002a; 2002b]. This means that some part of the screening effect is captured by the cohort effect. The horizontal to slightly upward trend of the cohort effect curve after cohort C12 (born between 130-39) cohort can be explained as the net effect of screening in the context of increased primary risk. For instance, in England & Wales, the slope of the cohort effect involving the same cohorts was steeply rising, indicating low quality of screening in the seventies and early eighties [Patnick, 2000; Swerdlow, 2001]. The absence of such an import rising cohort effect in Belgium, might be a consequence of screening that was not or such a poor quality as in England & Wales and/or down- staging [Arbyn, 2002a; 2002b]. Critical reports on the low quality of screening in E & W prompted health authorities to reorganise screening by installing a nation wide call-recall system, incentives for general practitioners and stringent quality assurance procedures [Patnick, 2000]. These measures resulted quickly in a substantial decline
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 37 of mortality in all age groups in the target population [Quin, 1999; Sasieni, 1995; 1999; 2000]. The substantial drop in mortality, observed in the nineties in England, did not occur in Belgium, because no serious efforts to organise screening more efficiently were undertaken [Arbyn, 2000b; 2002c]. We must regret the delay in publication of mortality statistics impeding us to study the most recent years. It might be most interesting to verify if the slightly increasing decline observable in the period effect (Figure 6) of the last 2 periods could be intensified after 1995.
Smoothing of local contrasts Due to modeling, certain local contrasts were faded: for instance several districts colored in green in the raw SMR map (Figure 5) became yellow in the SC-model map due to neighbourship with yellow or orange districts. This was the case for Mechelen (NIScode=12), Brugge (31), Diksmuide (32), Maaseik (72) and Marche-en Famenne (83). Antwerp was shaded in orange in the SC map (district effect=1.25) while yellow in the raw SMR map (SMR=1.19). This was due to the fact that mortality in older cohorts was unlikely low (Figure 4). By modeling these cohort-effects wer upgraded to the Belgian mean, yielding a higher district effect in spite of the neighboring green and yellow areas.
Agglomeration effect Cervical cancer is more incident in areas with higher prevalence of HPV infections, such as urbanized centers where, in general transmission of sexually transmitted diseases is more intense. Boon linked data from Dutch cytological screening registers with demograhic data and found that the risk HPV-associated cytological lesions and cervical intraepithelial lesions increased with population density [Boon, 2003]. Figure 14 seems to confirm the relation between risk for cervical cancer and population density. The districts containing the agglomerations of Antwerp (NIS- code=11), Gent (44), Charleroi (52), Liège (62) and Namur (92) all have fitted relative risk that are higher than 1.2. Brussels (21), shows an SMR that is higher but not significantly higher than 1. This relation can be contaminated by the fact that districts containing agglomerations do contain non-urbanised municipalities as well. Nevertheless, spatial analysis of cervical cancer mortality (ICD-8 and ICD-9=180) in the Flemish Region in the period 1993-1998 at the level of the municipality confirmed the agglomeration effect more precisely [Arbyn, 2003]. This phenomenon might prompt health authorities to target campaigns first towards urbanized areas.
Impact of uterine cancer not otherwise specified The use of modeling of cervical cancer mortality is compromised because of a well- known certification problem: part of causes of deaths from cancer of the cervix and corpus uteri are declared as not otherwise specified uterine cancer (NOS). The proportion of all uterine cancer deaths without specification in Belgium was not constant over time [Arbyn, 2000a; 2002a]. The question raises if spatial variation of the cervix cancer SMR can be attributed to differences in the proportion of NOS. We therefore studied the relation between the SMRs and proportion of NOS. Death certificates were processed until the beginning of the 1990 by the provincial administrations. Later the communities took over. In Figure 10 the proportions of deaths due to cervix cancer, corpus and uterus NOS cancer are plotted by time period and province. Changes in the proportion of NOS uterine cancer are reflected in essentially coincide in the proportion corpus cancer. Nevertheless substantial changes
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 38 in the proportion of cervical cancer can be noticed. In previous work, we tried to correct for this certification problem at the level of the Flemish Region and the whole of Belgium. We concluded that the shape of the period- and cohort effect did not change substantially after correction for the NOS problem. Nevertheless, the slope of corrected cohort- & period effect did change and was more steeply declining that the equivalent effects without the correction.
pnos pcvx pcrp
Antwerpen Brabant W-Vlaanderen .8 .6 .4 .2 0
O-Vlaanderen Hainaut Liège .8 .6 .4 .2 0
Namur Luxembourg Namur .8 .6 .4 .2 0 1965 1975 1985 1965 1975 1985 1965 1975 1985 period
Figure 10. Proportion of uterine cancers death certified as respectively of cervical, corpus and NOS origin, variation over time period and province.
We studied the correlation between the raw SMRs the fitted relative risks and the proportion of NOS (Table 13). In global, SMRs decreased in districts where the % of NOS was higher. But, the ST models faded out this linear relation. The correlation coefficient was only -0.07, and the slope of the regression line was not significantly different from unity.
Table 13. Relation between the raw SMR or the fitted RR and the proportion of NOS. Corrleation coefficient Slope 95% CI SMR -0.2729 -0.843 -1.211 -0.476 Fitted RR -0.0784 -0.115 -0.296 0.066
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 39 Relation between fitted RR and proportion of NOS 2 52 56 55 43 44 93
52 56 55 92 43 52 44 52 11 1.5 56 55 56 55 93 44 43 44 43 62 93 93 4241 61 21 35 52 71 85 55 5692 8111 91 4344 46 33 11 11633651 Fitted RR 1
31 82 72 12 84921273 72 242383 45 37 37 24 13 5424 42 41 343213 57 235323 62 13 35 311221 72 61 8546 718137 24 51 91 63 38 363313 842373 25 8264 45 5483 3112 32 34 72 53 57 .5 37 24 2313 0 .5 1 Proportion of NOS
Figure 11. Scatter plot and regression line of fitted relative risks derived from the S*P model against the proportion of uterine cancer deaths coded as NOS. Red points marked with their NIS code are derived from districts colored red or orange in Figure 7, green NIS-codes are derived from districts colored in green. The not marked dots come from districts colored in yellow.
Figure 11 shows the weakly negative relation between the fitted RR and the % of NOS. It allows distinction between the districts by color used in Figure 7. No clear relation can be noticed for the red districts. This observation allows to conclude that the spatial effect was not just an artifact due to certification. Nevertheless this conclusion requires further analysis.
Cartographic display style A good map facilitates communication with the interested public. “A map … just as a graph of an epidemic curve is more eloquent than a list of monthly incidence figures …” [Boelaert, 1998]. Map contrasts depend on the used cartographic display style in particular on the scale of the chosen mapped indicator, the number and delineation of the categories and the choice of the color shade palette [Arbyn, 2000c]. We used a fixed, data-independent system to define the cut-offs. Often in cancer atlases data-dependent systems are used based on quantiles, or arbitrarily chosen percentiles. This yield maps that by prior definition always contain a given number of green or red spots, that attract attention of the reader. This yields nice but also often misleading maps. Our display style allows more intuitively interpretable maps. If spatial variability is limited, more oligo-chromic maps are obtained. We used the same cut-offs (Table 2) as in previous work (Arbyn, 2003). The ranges were quite large, yielding a certain lack of “power” but more robustness. We did not incorporate the credibility intervals to distinguish between significant an non-significant differences. This is really required anymore to deal with since the Bayesian smoothing from the CAR model shrinks RRs from areas with less deaths more towards the RRs in neighbouring districts, whereas the RRs in areas with more deaths are more resistant to shrinkage. Nevertheless, for comparison
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 40 we plotted also maps using narrower ranges and incorporating the credibility intervals (Figure 13 and Figure 14). This was useful to explore certain hypotheses such as the agglomeration effect. Probably this last display style provides to much detail in the extreme values (red and dark green spots).
Further work In this first report, we tried to modelise spatial and time trend effects simultaneously. For the future we want to work further and plan the following activities: • Assess the goodness of fit of Bayesian spatial models by formal residual analysis. • Compare with CAR models including adjacency matrices with general smoothing methods. • Correction of cervical cancer mortality for the part of NOS cancer deaths are probably from cervical origin, see our previous work on the level of Belgium and the Flemish Region. • Development of nested hierarchical models applied on data aggregated on the municipality level, with municipality within districts, provinces, regions and finally the whole country as respective hierarchical levels. • Inclusion of covariate information (socio-economic, demographic, screening, …) aggregated at the same geographical levels • Modeling of international data.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 41
9. List of abbreviations
AC: age-cohort model. ACP: age-cohort-period model. AIC: Akaike’s Information Criterion. AP: age-period model. BUGS: Bayesian inference Using Gibbs Sampling CAR model: conditional auto-regressive model. CI: Bayesian credibility interval (posterior 2.5 and 97.5 percentiles of the posterior distribution); 95% confidence interval. DEV: deviance. DIC: Deviance Information Criterion. DT : number of deaths. EDT: expected number of deaths. HPV: human papilloma virus. ICD: International Classification of Diseases. L: likelihood. MC error: Monte-Carlo error. MCMC simulation: Markow-chain Monte Carlo simulation. NIS: National Institute of Statistics. NOS: not otherwise specified. RR: relative risk. SC model: Space-Cohort model. SCMR: standardised cohort mortality ratio. SD: standard deviation. SMR: standardised mortality ratio. SP model: Space-Period model. T: number of person-years at risk.
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10. Bibliography
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Arbyn M, Van Oyen H. Cervical cancer screening in Belgium. Eur J Cancer 2000b;36: 2191-2197.
Arbyn M, Ghafariyan M, Lesaffre M. Mapping of epidemiological indicators: practical problems, theoretical concepts. In: Methods in Public Health: Spatial Epidemiology. Conference organized by the Belgian Association of Public Health, the Belgian Statistical Society, and the Quetelet Society. Leuven, 24th of November, 2000c.
Arbyn M, Van Oyen H, Sartor F, Tibaldi F, Molenberghs G. Description of the influence of age, period and cohort effects on cervical cancer mortality by loglinear Poisson models (Belgium, 1955-94). Arch Public Health 2002a; 60:73-100.
Arbyn M, Geys H. Trend of cervical cancer mortality in Belgium (1954-94): tentative solution for the certification problem of not specified uterine cancer. Int J Cancer 2002b; 102: 649-654.
Arbyn M, Temmerman M. Belgian Parliament calls for organised cervical cancer screening and HPV research throughout Europe. Lancet Oncol 2002c;3:74.
Arbyn, M., Geys, H., Faes, C., Molenberghs, G. Spatial variation of mortality from uterine and breast cancer in the Flemish Region (1993-1998). IPH/EPI-REPORTS Nr. 2001-016 (D/2001/2505/33), October 2003. Brussels.
Beral V. Cancer of the cervix: a sexually transmitted infection? Lancet. 1974; May 25:1037-1040.
Boon ME, Van Ravenswaay Claasen HH, Van Westering RP, Kok LP. Urbanisation and the incidence of squamous and glandular epithelium of the cervix. Cancer 2003; 99: 4-8.
Boon ME, van Ravenswaay Claasen HH, Kok LP. Urbanization and baseline prevalence of genital infections including Candida, Trichomonas, and human papillomavirus and of a disturbed vaginal ecology as established in the Dutch cervical screening program. Am J Obstet Gynecol 2002;187:365-9.
Bosch FX, Manos MM, Munoz N, Sherman M, Janssen AM, Peto J, Schifmann MH, Moreno V, Kurman R, Shah KV. Prevalence of human papillomavirus in cervical cancer: a worldwide perspective. JNCI 1995; 87:796-802.
Bosch FX, Lorincz A, Munoz N, Meijer CJ, Shah KV. The causal relation between human papillomavirus and cervical cancer. J Clin Pathol 2002; 55: 244-265.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 43 Clayton D,.Schifflers E. Models for temporal variation in cancer rates : I : Age-period and age-cohort models. Stat Med 1987;6:449-67.
Clayton D,.Schifflers E. Models for temporal variation in cancer rates. II : Age- period-cohort models. Stat Med 1987;6:469-81.
De Schryver A. Does screening for cervical cancer affect incidence and mortality trends? The Belgian experience. Eur J Cancer Clin Oncol 1989;25:395-9.
Frome EL. The analysis of rates using Poisson regression models. Biometrics. 1983;39:665-674.
Gelman A, Rubin DB. Inference from iterative simulation using multiple sequences. Statistical Science 1992; 7: 457–511.
Geweke J. Evaluating the accuracy of sampling-based approaches to calculating posterior moments. In Bayesian Statistics, 4, (ed. Bernardo JM., Berger JO, Dawid AP, Smith AFM.), 1992; Oxford: Clarendon Press.
Kelsall, J. E., and Wakefield, J. C. (1999). Discussion of “Bayesian models for spatially correlated disease and exposure data”, by Best et al. In Bayesian Statistics, 6, (ed. Bernardo, J. M., Berger, J. O., Dawid, A. P., and Smith, A. F. M.) Oxford: Oxford University Press, p. 151.
Knorr-Held L, Rasser G. Bayesian detection of clusters and discontinuities in disease maps. Biometrics 2000, 56: 13–21.
Lagazio C,Dreassi E, Biggeri A. A hierarchical Bayesian model for space-time variation of disease risk. Statistical Modelling 2001; 1: 17–29.
Lawson AB, Browne WJ, Vidal Rodeiro CL. Disease mapping with WinBUGS and MLwiN. West Sussex PO19 8SQ: John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, 2003.
Munoz N. Human papillomavirus and cervical cancer: epidemiological evidence. In: Franco E, Monsonego J, eds. New Developments in Cervical Cancer Screening and Prevention. Oxford: Blackwell Science, 1997.
Nelder JA, Wedderburn RWM. Generalized linear models. J R Statist Soc A. 1972; 135: 370-384.
Patnick J. Cervical cancer screening in England. Eur J Cancer 2000;36:2205-2208.
Quinn M, Babb P, Jones J. Effect of screening on incidence and mortality from cancer of the cervix in England: Evaluation based on routinely collected statistics. BMJ 1999;318:904-908.
Spiegehalter DJ, Best NG, Carlin BP. Bayesian deviance, the effective number of parameters, and the comparison of arbitrarily complex models. Research Report 1998; 98-009, Division of Biostatistics, University of Minnesota.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 44 Sasieni P, Cuzick J, Farmery E. Accelerated decline in cervical cancer mortality in England and Wales. Lancet 1995;346:1566-1567.
Sasieni P, Adams J. Effect of screening on cervical cancer mortality in England and Wales: analysis of trends with an age period cohort model. BMJ 1999;318:1244- 1245.
Sasieni PD, Adams J. Analysis of cervical cancer mortality and incidence data from England and Wales: evidence of a beneficial effect of screening. J Royal Stat Soc A 2000;163:191-209.
Spiegelhalter, DJ, Best NG, Carlin BR, van der Linde A. Bayesian measures of model complexity and fit. J Royal Stat Soc B 2002; 64: 583-616.
Spiegelhalter D, Thomas A, Best N. WINBUGS: Bayesian Inference Using Gibs Sampling. MRC Biostatistics Unit, Institute of Public Health, Cambridge & Department of Epidemiology and Public Health, Imperial College School of Medicine, London, 2000, (http://www.mrc-bsu.cam.ac.uk/bugs).
Swerdlow A, dos Santos LR, Doll R. Cancer incidence and mortality in England and Wales: trends and risk factors. Oxford: Oxford Universisty Press, 2001.
Vyslouzilova S, Arbyn M, Van Oyen H, Drieskens S, Quataert P. Cervical cancer mortality in Belgium, 1955-1989. A descriptive study. Eur J Cancer 1997;33:1841-5.
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11. Annexes
11.1. WINBUGS programmes, initials, data
11.1.1. Space-cohort model without interactions
WinBUGS program: model { for (i in 1:sumNumNeigha) {weia[i] <- 1} for (j in 1:sumNumNeighc) {weic[j] <- 1}
### PRIORS for structural area and cohort effects ### ------area.struct[1:na] ~ car.normal(mapa[],weia[],numa[],tau.area.struct) cohort[1:nc] ~ car.normal(mapc[],weic[],numc[],tau.cohort)
### PRIOR for unstructural area effect ### and MODEL specifiction ### ------for (i in 1:na) { area.unstr[i] ~ dnorm(0.00,tau.area.unstr) for (j in 1:nc) { observe[i,j] ~ dpois(lambda[i,j]) log(lambda[i,j]) <- log(expect[i,j])+alpha+area.struct[i]+area.unstr[i]+cohort[j] rr[i,j] <- lambda[i,j]/expect[i,j] dev.residual[i,j] <- observe[i,j]*(log(observe[i,j]+0.01)-log(lambda[i,j]))+(lambda[i,j]-observe[i,j]) } }
### PRIORS for remaining parameters ### ------alpha ~ dflat() tau.area.struct ~ dgamma(0.5,0.005) tau.cohort ~ dgamma(0.5,0.0005) tau.area.unstr ~ dgamma(0.001,0.001) sigma.area.struct <- 1/tau.area.struct sigma.cohort <- 1/tau.cohort sigma.area.unstr <- 1/tau.area.unstr
### Deviance ### ------deviance <- sum(dev.residual[1:na,1:nc])
### Space effect ### ------for (i in 1:na) {space[i] <- area.struct[i] + area.unstr[i]}
### Relative risks for each cohort ### to be used by GeoBUGS ### ------for (i in 1:na) {rrc1[i] <- rr[i,1]} for (i in 1:na) {rrc2[i] <- rr[i,2]} for (i in 1:na) {rrc3[i] <- rr[i,3]} for (i in 1:na) {rrc4[i] <- rr[i,4]} for (i in 1:na) {rrc5[i] <- rr[i,5]}
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 46 for (i in 1:na) {rrc6[i] <- rr[i,6]} for (i in 1:na) {rrc7[i] <- rr[i,7]} for (i in 1:na) {rrc8[i] <- rr[i,8]} for (i in 1:na) {rrc9[i] <- rr[i,9]} for (i in 1:na) {rrc10[i] <- rr[i,10]} for (i in 1:na) {rrc11[i] <- rr[i,11]} for (i in 1:na) {rrc12[i] <- rr[i,12]} for (i in 1:na) {rrc13[i] <- rr[i,13]} for (i in 1:na) {rrc14[i] <- rr[i,14]} for (i in 1:na) {rrc15[i] <- rr[i,15]} for (i in 1:na) {rrc16[i] <- rr[i,16]} for (i in 1:na) {rrc17[i] <- rr[i,17]} for (i in 1:na) {rrc18[i] <- rr[i,18]} for (i in 1:na) {rrc19[i] <- rr[i,19]} }
WinBUGS data: list( na = 43, nc = 19, sumNumNeigha = 204, sumNumNeighc = 36, observe = structure(.Data = c(0, 1, 12, 38, 67, 99, 86, 83, 96, 80, 53, 33, 32, 27, 18, 9, 7, 0, 0, 0, 0, 6, 9, 13, 13, 22, 16, 13, 12, 6, 11, 8, 4, 2, 2, 1, 0, 0, 0, 1, 2, 9, 7, 13, 12, 17, 7, 13, 11, 6, 7, 3, 1, 3, 0, 0, 0, 1, 18, 24, 62, 87, 97, 89, 97, 85, 75, 55, 45, 34, 16, 14, 3, 2, 0, 0, 0, 1, 6, 8, 25, 25, 30, 19, 20, 27, 16, 14, 12, 6, 5, 1, 0, 0, 0, 0, 1, 2, 12, 19, 16, 26, 22, 19, 16, 10, 8, 8, 7, 6, 3, 1, 0, 0, 0, 2, 4, 9, 21, 21, 17, 5, 20, 20, 10, 10, 4, 5, 4, 4, 0, 0, 0, 0, 0, 3, 8, 14, 12, 22, 11, 12, 17, 4, 8, 4, 1, 2, 2, 0, 0, 0, 0, 0, 1, 1, 1, 4, 3, 1, 1, 2, 1, 2, 0, 1, 2, 0, 0, 0, 0, 0, 1, 5, 4, 7, 10, 6, 6, 7, 6, 4, 2, 2, 1, 0, 0, 0, 0, 0, 0, 2, 5, 12, 9, 18, 12, 11, 12, 19, 12, 5, 3, 3, 4, 0, 1, 0, 0, 1, 1, 1, 4, 10, 14, 13, 8, 12, 11, 4, 4, 5, 3, 0, 0, 1, 0, 0, 0, 1, 0, 3, 7, 15, 11, 10, 8, 5, 5, 5, 3, 3, 0, 0, 1, 0, 0, 0, 0, 2, 1, 2, 6, 3, 1, 6, 2, 2, 0, 1, 2, 0, 3, 0, 0, 0, 0, 0, 2, 3, 0, 7, 5, 1, 7, 2, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 8, 13, 24, 31, 15, 15, 19, 13, 8, 8, 8, 2, 0, 3, 0, 0, 0, 0, 3, 4, 8, 21, 11, 18, 14, 10, 10, 4, 8, 2, 1, 1, 0, 0, 0, 2, 1, 2, 5, 6, 8, 13, 12, 13, 3, 6, 3, 0, 2, 1, 0, 1, 0, 0, 0, 5, 16, 26, 46, 51, 67, 52, 50, 55, 30, 15, 14, 12, 7, 3, 3, 0, 0, 0, 0, 3, 6, 4, 11, 6, 5, 11, 2, 7, 2, 3, 2, 1, 0, 0, 0, 0, 0, 1, 1, 5, 10, 16, 11, 13, 18, 12, 8, 2, 7, 5, 2, 0, 2, 0, 0, 0, 1, 1, 4, 3, 12, 4, 4, 4, 8, 5, 3, 0, 0, 0, 1, 1, 0, 0, 1, 6, 19, 42, 60, 57, 53, 49, 45, 39, 27, 13, 11, 7, 8, 5, 1, 2, 0, 1, 2, 5, 9, 17, 19, 16, 14, 15, 15, 13, 7, 2, 3, 3, 1, 0, 0, 1, 0, 0, 1, 2, 3, 9, 10, 6, 2, 5, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 2, 7, 14, 14, 22, 21, 19, 20, 18, 13, 7, 6, 0, 0, 2, 1, 1, 0, 1, 1, 4, 13, 11, 30, 18, 14, 15, 11, 10, 5, 5, 2, 0, 0, 0, 0, 0, 0, 4, 2, 8, 9, 12, 15, 8, 5, 4, 7, 6, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 7, 7, 6, 9, 6, 3, 6, 5, 5, 4, 2, 0, 0, 0, 0, 2, 5, 20, 42, 56, 57, 54, 58, 45, 43, 19, 20, 12, 15, 8, 5, 4, 0, 0, 0, 1, 5, 5, 25, 20, 26, 11, 23, 19, 9, 1, 3, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 3, 8, 8, 3, 6, 4, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 2, 6, 10, 18, 19, 15, 17, 15, 16, 13, 9, 7, 4, 0, 0, 0, 0, 0, 0, 1, 5, 4, 5, 6, 8, 3, 6, 2, 3, 5, 2, 2, 3, 0, 0, 0, 1, 0, 4, 5, 7, 10, 4, 2, 9, 12, 6, 6, 2, 2, 3, 1, 2, 0, 0, 0, 0, 2, 1, 4, 3, 4, 2, 3, 3, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0,
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 47 1, 0, 2, 0, 4, 2, 4, 2, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 3, 5, 2, 4, 2, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 4, 4, 1, 6, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 3, 3, 0, 3, 1, 4, 5, 6, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 10, 4, 8, 11, 5, 4, 5, 2, 0, 1, 0, 1, 1, 0, 0, 0, 1, 4, 11, 28, 27, 25, 25, 23, 23, 10, 7, 5, 7, 2, 1, 1, 0, 0, 0, 1, 2, 9, 5, 10, 3, 4, 7, 3, 4, 1, 3, 1, 0, 1, 1, 0, 0), .Dim = c(43,19)), expect = structure(.Data = c(0.5823, 4.0294, 10.2954, 20.6203, 35.9246, 53.0779, 58.7962, 56.8755, 58.1054, 68.5416, 62.4863, 50.7876, 36.5982, 26.3818, 16.1367, 7.7636, 2.6587, 0.7941, 0.1136, 0.1792, 1.1943, 2.7591, 5.4793, 10.1106, 15.6258, 17.8022, 17.1197, 17.3049, 20.662, 19.1205, 15.9816, 11.4925, 8.1616, 5.1722, 2.5067, 0.8683, 0.2607, 0.0369, 0.1075, 0.7244, 1.7671, 3.8289, 7.2826, 12.2685, 15.6085, 15.8195, 16.2694, 20.6074, 20.9694, 18.481, 14.18, 10.2825, 6.7226, 3.3633, 1.2121, 0.359, 0.0526, 1.1348, 7.6225, 17.984, 33.3635, 53.1903, 71.6165, 75.6371, 70.1332, 71.4026, 81.343, 68.2725, 52.0693, 37.205, 27.075, 17.174, 8.2044, 2.8703, 0.8952, 0.1256, 0.2813, 1.8722, 4.4821, 9.3716, 16.6311, 24.6156, 27.7826, 26.96, 28.908, 37.1465, 35.6749, 30.0096, 21.7862, 15.9591, 10.0357, 4.6946, 1.576, 0.4449, 0.0645, 0.2358, 1.5813, 3.8691, 7.8705, 13.8708, 20.3821, 23.5522, 22.7572, 23.5773, 28.9851, 27.3169, 22.4425, 16.5601, 12.3018, 7.9651, 3.8653, 1.3061, 0.386, 0.0527, 0.1804, 1.2329, 3.0264, 5.8128, 9.8123, 14.0574, 15.2901, 14.4256, 15.6468, 19.7308, 18.1297, 15.0212, 11.8261, 9.7457, 6.5449, 2.8579, 0.9008, 0.2468, 0.0393, 0.148, 0.9989, 2.4482, 5.101, 9.0624, 13.7231, 15.578, 14.6454, 14.0772, 17.3882, 16.472, 14.0928, 10.5444, 7.4375, 4.6024, 2.1847, 0.7485, 0.2247, 0.0336, 0.0276, 0.189, 0.4434, 0.9045, 1.5767, 2.3956, 2.8055, 2.4625, 2.582, 3.4356, 3.1665, 2.4055, 1.7793, 1.172, 0.7197, 0.3592, 0.1272, 0.0399, 0.0061, 0.0595, 0.3904, 0.9712, 2.1725, 3.835, 5.5735, 6.2436, 5.9375, 6.139, 7.7446, 6.9168, 5.2916, 3.831, 2.6913, 1.659, 0.7861, 0.2739, 0.0839, 0.0131, 0.126, 0.8868, 2.3105, 5.1275, 9.0757, 13.6667, 15.2945, 14.259, 14.4063, 18.7561, 17.7895, 14.7039, 10.8109, 7.7196, 4.6612, 2.2373, 0.7715, 0.2421, 0.0361, 0.0759, 0.5373, 1.3942, 2.9368, 5.2625, 7.9556, 9.087, 8.3885, 8.1927, 10.0075, 9.1868, 7.6336, 5.459, 3.744, 2.2857, 1.1019, 0.3715, 0.1126, 0.0168, 0.0682, 0.4561, 1.1524, 2.4951, 4.663, 7.116, 7.8775, 7.1293, 7.2094, 9.4725, 8.941, 7.8149, 5.7004, 3.8948, 2.2822, 1.0789, 0.3698, 0.1168, 0.0182, 0.0462, 0.3129, 0.7259, 1.5561, 2.889, 4.3033, 4.6503, 4.3912, 4.1128, 5.4801, 5.3847, 4.4813, 3.2665, 2.2639, 1.3494, 0.6722, 0.2388, 0.0744, 0.011, 0.0361, 0.2367, 0.5821, 1.1707, 2.0035, 2.9311, 3.2074, 3.0346, 3.112, 3.8032, 3.4102, 2.7415, 1.9814, 1.3475, 0.8236, 0.3982, 0.1434, 0.0435, 0.0066, 0.1542, 1.0111, 2.4925, 5.2042, 9.3161, 14.0691, 15.8409, 14.6353, 14.8347, 18.6571, 17.8031, 14.8007, 10.1786, 7.0943, 4.4573, 2.1618, 0.7587, 0.2277, 0.0335, 0.0963, 0.6207, 1.5157, 3.2252, 5.8163, 8.9438, 10.339, 9.8486, 9.8164, 12.1634, 11.3187, 9.8091, 7.0062, 5.0144, 3.2869, 1.5539, 0.5212, 0.1585, 0.0238, 0.0514, 0.345, 0.8474, 1.7526, 3.1627, 4.8003, 5.1151, 4.6824, 4.4132, 5.4142, 5.1833, 4.4022, 3.1201, 2.1535, 1.3626, 0.623, 0.2101, 0.0649, 0.0103, 0.3572, 2.4412, 6.077, 11.939, 19.6339, 28.6333, 31.0872, 28.5432, 28.4157, 33.7611, 31.366, 26.5026, 19.2614, 13.7205, 8.5051, 4.0348, 1.4079, 0.4378, 0.0632, 0.0947, 0.6091, 1.4222, 3.0165, 5.1144, 7.2483, 7.7763, 7.1354, 6.8103, 8.164, 7.5623, 6.1748, 4.1135, 2.9169, 1.8587, 0.8717, 0.3019, 0.0914, 0.0137, 0.0975, 0.6551, 1.672, 3.3965, 6.283, 10.2052, 11.9957, 11.3954, 11.843, 13.5866, 12.5677, 11.1035,
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 48 8.3597, 6.2811, 3.9263, 1.7752, 0.6091, 0.1858, 0.0292, 0.0746, 0.4686, 1.0891, 2.1194, 3.4002, 4.6356, 5.0525, 4.8964, 5.1829, 6.0783, 5.2197, 3.7941, 2.6418, 1.9714, 1.3866, 0.6387, 0.2091, 0.0604, 0.0096, 0.3324, 2.2293, 5.5908, 11.0957, 18.8635, 26.5705, 27.72, 25.3689, 28.6011, 35.6441, 30.422, 22.1851, 15.3316, 11.2743, 8.1147, 3.715, 1.2124, 0.3559, 0.0561, 0.209, 1.3894, 3.4682, 6.6876, 10.8463, 15.2735, 17.0522, 15.4415, 17.1908, 21.0638, 17.493, 12.5207, 8.5162, 6.7777, 4.9196, 2.2928, 0.7273, 0.2056, 0.0327, 0.0511, 0.3476, 0.888, 1.9368, 3.3416, 4.7143, 4.7974, 4.1322, 4.2931, 5.4704, 4.7982, 3.7054, 2.4937, 1.9775, 1.3441, 0.5965, 0.185, 0.0535, 0.0091, 0.126, 0.8321, 2.0763, 4.0662, 6.8781, 9.7759, 10.7126, 9.8674, 10.4265, 12.59, 10.8863, 8.2689, 5.7563, 4.506, 3.1792, 1.4546, 0.4804, 0.1401, 0.0214, 0.1036, 0.6909, 1.7101, 3.3307, 5.7576, 8.041, 8.7362, 8.008, 8.6916, 10.8248, 9.3266, 7.0139, 5.036, 3.7165, 2.6463, 1.227, 0.4072, 0.1159, 0.0177, 0.148, 0.9702, 2.3018, 4.2673, 6.8334, 9.4366, 9.8941, 8.611, 9.0674, 11.0614, 9.5631, 7.1016, 4.648, 3.6027, 2.5686, 1.1511, 0.383, 0.1103, 0.0183, 0.0821, 0.5486, 1.2761, 2.3545, 3.972, 5.6188, 6.0505, 5.5645, 5.8164, 7.1664, 6.1749, 4.5659, 3.2888, 2.3405, 1.7382, 0.822, 0.2698, 0.0766, 0.0115, 0.5105, 3.4859, 8.4742, 16.0217, 26.4391, 37.2502, 39.9648, 36.8374, 39.9711, 48.4447, 43.3874, 32.5975, 22.8328, 16.2519, 11.117, 5.05, 1.663, 0.4899, 0.0743, 0.187, 1.2448, 3.0114, 5.6858, 9.3037, 13.1387, 15.2786, 14.5621, 14.973, 17.8089, 15.5462, 12.31, 9.4384, 6.4119, 4.2858, 2.0905, 0.722, 0.2171, 0.0329, 0.0556, 0.3476, 0.8072, 1.5748, 2.7004, 3.9061, 4.1651, 3.8907, 4.0961, 4.7453, 4.2357, 3.1757, 2.1951, 1.6055, 1.2063, 0.5558, 0.1799, 0.0498, 0.0071, 0.0979, 0.6651, 1.6323, 3.5519, 6.5975, 10.6439, 13.084, 13.708, 15.4601, 20.9657, 20.8174, 18.546, 14.3254, 9.8809, 6.7476, 3.3941, 1.1655, 0.338, 0.0504, 0.0428, 0.2952, 0.7281, 1.5155, 2.8956, 4.7678, 6.1478, 6.5543, 6.9157, 9.6464, 10.3607, 9.641, 7.6051, 5.2708, 3.4992, 1.8413, 0.6631, 0.2007, 0.0282, 0.0691, 0.4525, 1.0751, 2.1734, 4.064, 6.4549, 7.7238, 7.9668, 8.4444, 10.6335, 10.3691, 9.1285, 7.0131, 4.9928, 3.3406, 1.6213, 0.5659, 0.171, 0.0252, 0.0288, 0.2001, 0.4758, 0.8839, 1.61, 2.3543, 2.6422, 2.6253, 2.7577, 3.5488, 3.2763, 2.4986, 1.8266, 1.2495, 0.8612, 0.4099, 0.1442, 0.0449, 0.0066, 0.0199, 0.1319, 0.3686, 0.6942, 1.198, 1.9239, 2.2375, 2.2487, 2.2315, 2.4637, 2.2162, 1.7444, 1.1694, 0.7926, 0.5754, 0.294, 0.1049, 0.0335, 0.0051, 0.0291, 0.2077, 0.4977, 0.9112, 1.6526, 2.2334, 2.6088, 2.6519, 2.7188, 3.1893, 2.7225, 2.0808, 1.4311, 1.0277, 0.7408, 0.3727, 0.1312, 0.0396, 0.0061, 0.042, 0.2814, 0.6638, 1.2191, 1.9787, 3.0379, 3.3564, 3.2556, 3.3009, 3.785, 3.276, 2.4254, 1.7204, 1.2073, 0.851, 0.4235, 0.1445, 0.0429, 0.007, 0.0419, 0.2663, 0.5847, 1.1155, 1.8896, 2.6415, 2.8128, 2.6057, 2.5428, 3.1566, 2.8813, 2.1846, 1.5084, 0.936, 0.758, 0.374, 0.13, 0.0378, 0.0056, 0.0681, 0.4813, 1.1076, 2.0375, 3.5093, 5.0387, 5.4059, 5.4608, 5.6736, 6.686, 5.6709, 4.4672, 3.1266, 2.1354, 1.5507, 0.7362, 0.258, 0.078, 0.012, 0.201, 1.324, 3.1065, 6.0362, 10.2258, 14.3315, 15.6488, 14.6231, 15.7221, 19.1441, 16.9376, 12.9452, 9.2689, 6.7802, 5.0881, 2.3616, 0.7796, 0.2267, 0.0345, 0.0452, 0.3024, 0.6927, 1.3144, 2.1883, 3.0283, 3.4213, 3.2466, 3.5367, 4.2362, 3.7911, 2.8691, 2.042, 1.505, 1.0833, 0.472, 0.1605, 0.0491, 0.0074), .Dim = c(43,19)), numa = c(3, 6, 5, 2, 7, 7, 6, 6, 5, 5, 7, 3, 5, 4, 3, 5, 5, 2, 8, 6, 4, 4, 5, 3, 3, 8, 4, 4, 6, 4, 5, 7, 5, 3, 5, 3, 4, 5, 5, 2, 5, 6, 4),
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 49 mapa = c( 21, 3, 2, 21, 17, 6, 5, 3, 1, 34, 33, 6, 2, 1, 6, 5, 26, 17, 16, 7, 6, 4, 2, 33, 32, 7, 5, 4, 3, 2, 42, 32, 26, 23, 6, 5, 19, 18, 14, 13, 12, 9, 15, 13, 12, 10, 8, 25, 15, 13, 11, 9, 28, 25, 20, 19, 14, 13, 10, 15, 9, 8, 14, 11, 10, 9, 8, 19, 13, 11, 8, 12, 10, 9, 26, 20, 19, 17, 5, 21, 19, 16, 5, 2, 19, 8, 21, 20, 18, 17, 16, 14, 11, 8, 28, 26, 22, 19, 16, 11, 19, 17, 2, 1, 28, 26, 24, 20, 43, 42, 27, 26, 7, 27, 26, 22, 28, 11, 10, 27, 24, 23, 22, 20, 16, 7, 5, 43, 26, 24, 23, 25, 22, 20, 11, 42, 41, 38, 32, 31, 30, 35, 32, 31, 29, 38, 37, 35, 30, 29, 42, 35, 33, 30, 29, 7, 6, 35, 34, 32, 6, 3, 35, 33, 3, 34, 33, 32, 31, 30, 40, 39, 37, 39, 38, 36, 31, 41, 39, 37, 31, 29, 41, 40, 38, 37, 36, 39, 36, 43, 42, 39, 38, 29, 43, 41, 32, 29, 23, 7, 42, 41, 27, 23), numc=c(1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1), mapc=c( 2, 1,3, 2,4, 3,5, 4,6, 5,7, 6,8, 7,9, 8,10, 9,11, 10,12, 11,13, 12,14, 13,15,
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 50 14,16, 15,17, 16,18, 17,19, 18) )
WinBUGS initials: ## Chain 1: list( alpha=0, tau.area.struct=1, tau.area.unstr=1, tau.cohort=1, area.struct=c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0), area.unstr=c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0), cohort=c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0) )
## Chain 2: list( alpha=0, tau.area.struct=0.5, tau.area.unstr=1.5, tau.cohort=1, area.struct=c( 0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5, 0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5, 0.5,-0.5,0.5), area.unstr=c( -0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5, -0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5, -0.5,0.5,-0.5), cohort=c( -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2) )
11.1.2. Space-cohort model with interactions
WinBUGS program: model { for (i in 1:sumNumNeigha) {weia[i] <- 1} for (j in 1:sumNumNeighc) {weic[j] <- 1}
### PRIORS for structural area and cohort effects ### ------area.struct[1:na] ~ car.normal(mapa[],weia[],numa[],tau.area.struct) cohort[1:nc] ~ car.normal(mapc[],weic[],numc[],tau.cohort)
### PRIOR for unstructural area effect
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 51 ### and MODEL specifiction ### ------for (i in 1:na) { area.unstr[i] ~ dnorm(0.00,tau.area.unstr) for (j in 1:nc) { observe[i,j] ~ dpois(lambda[i,j]) log(lambda[i,j]) <- log(expect[i,j])+alpha+area.struct[i]+area.unstr[i]+cohort[j]+eff.int.ac[i,j] rr[i,j] <- lambda[i,j]/expect[i,j] dev.residual[i,j] <- observe[i,j]*(log(observe[i,j]+0.01)-log(lambda[i,j]))+(lambda[i,j]-observe[i,j]) } }
### AREA-COHORT INTERACTIONS ### ------for (i in 1:na) {
### parameters for the prior of psi[i,1] ### ------int.ac.bar[i,1] <- int.ac[i,2]+mean(adj.int.ac[(off[i]+1):off[i+1],1])- mean(adj.int.ac[(off[i]+1):off[i+1],2]) nadj.int.ac[i,1] <- 1.0*numa[i]
### parameters for the prior of psi[i,j], j=2,...nc-1 ### ------for (j in 2:(nc-1)){ int.ac.bar[i,j] <- (int.ac[i,j-1]+int.ac[i,j+1])/2+ mean(adj.int.ac[off[i]+1:off[i+1],j])- (mean(adj.int.ac[off[i]+1:off[i+1],j-1])+mean(adj.int.ac[(off[i]+1):off[i+1],j+1]))/2 nadj.int.ac[i,j] <- 2.0*numa[i] }
### parameters for the prior of psi[i,nc] ### ------int.ac.bar[i,nc] <- int.ac[i,nc-1]+mean(adj.int.ac[(off[i]+1):off[i+1],nc])- mean(adj.int.ac[(off[i]+1):off[i+1],nc-1]) nadj.int.ac[i,nc] <- 1.0*numa[i];
### precision parameter of the prior for psi[i,j] ### ------for (j in 1:nc) {int.ac.prec[i,j] <- nadj.int.ac[i,j]*tau.int.ac}
### create eff.int.ac[i,j] such that sum(eff.int.ac[,j])=0 and sum(eff.int.ac[i,])=0 ### use it then in a model specification ### and assign a prior to interaction terms ### ------for (j in 1:nc){ int.ac[i,j] ~ dnorm(int.ac.bar[i,j],int.ac.prec[i,j]) eff.int.ac[i,j] <- int.ac[i,j]-int.ac[i,1]-int.ac[1,j]+int.ac[1,1]
### an obscure way to define later on a prior for tau.int.ac ### ------tau.int.ac.like[i,j] <- nadj.int.ac[i,j] * int.ac[i,j] *(int.ac[i,j] - int.ac.bar[i,j]) } }
### compute adj.int.ac ### adj.int.ac[off[i]+1:off[i+1],j] are interactions comming from ### all neighbours of the ith area and the jth cohort ### ** it is used when computing int.ac.bar[i,j] ### ------
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 52 for (l in 1:adj) {for (j in 1:nc) {adj.int.ac[l,j] <- int.ac[mapa[l],j]}}
### PRIORS for remaining parameters ### ------alpha ~ dflat() tau.area.struct ~ dgamma(0.5,0.0005) tau.cohort ~ dgamma(0.5,0.0005) tau.area.unstr ~ dgamma(1.0E-3,1.0E-3) r.int.ac <- 1 + (na)*(nc)/2 l.int.ac <- 0.01 + sum(tau.int.ac.like[,])/2 tau.int.ac ~ dgamma(r.int.ac,l.int.ac) sigma.area.struct <- 1/tau.area.struct sigma.cohort <- 1/tau.cohort sigma.area.unstr <- 1/tau.area.unstr sigma.int.ac <- 1/tau.int.ac
### Deviance ### ------deviance <- sum(dev.residual[1:na,1:nc])
### Space effect ### ------for (i in 1:na) {space[i] <- area.struct[i] + area.unstr[i]}
### Relative risks for each cohort ### to be used by GeoBUGS ### ------for (i in 1:na) {rrc1[i] <- rr[i,1]} for (i in 1:na) {rrc2[i] <- rr[i,2]} for (i in 1:na) {rrc3[i] <- rr[i,3]} for (i in 1:na) {rrc4[i] <- rr[i,4]} for (i in 1:na) {rrc5[i] <- rr[i,5]} for (i in 1:na) {rrc6[i] <- rr[i,6]} for (i in 1:na) {rrc7[i] <- rr[i,7]} for (i in 1:na) {rrc8[i] <- rr[i,8]} for (i in 1:na) {rrc9[i] <- rr[i,9]} for (i in 1:na) {rrc10[i] <- rr[i,10]} for (i in 1:na) {rrc11[i] <- rr[i,11]} for (i in 1:na) {rrc12[i] <- rr[i,12]} for (i in 1:na) {rrc13[i] <- rr[i,13]} for (i in 1:na) {rrc14[i] <- rr[i,14]} for (i in 1:na) {rrc15[i] <- rr[i,15]} for (i in 1:na) {rrc16[i] <- rr[i,16]} for (i in 1:na) {rrc17[i] <- rr[i,17]} for (i in 1:na) {rrc18[i] <- rr[i,18]} for (i in 1:na) {rrc19[i] <- rr[i,19]} }
WinBUGS data: - same as for the model without interactions with additionally off=c(0, 3, 9, 14, 16, 23, 30, 36, 42, 47, 52, 59, 62, 67, 71, 74, 79, 84, 86, 94, 100, 104, 108, 113, 116, 119, 127, 131, 135, 141, 145, 150, 157, 162, 165, 170, 173, 177, 182, 187, 189, 194, 200, 204)
WinBUGS initials:
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 53 ## Chain 1: list( alpha=0, tau.area.struct=1, tau.area.unstr=1, tau.cohort=1, tau.int.ac=1, area.struct=c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0), area.unstr=c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0), cohort=c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), int.ac= structure(.Data= c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), .Dim=c(43,19)) )
## Chain 2: list( alpha=0, tau.area.struct=0.5, tau.area.unstr=1.5, tau.cohort=1, tau.int.ac=1, area.struct=c( 0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5, 0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5, 0.5,-0.5,0.5), area.unstr=c( -0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5, -0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5, -0.5,0.5,-0.5), cohort=c(
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11.1.3. Space-period model without interactions
WinBUGS program: model { for (i in 1:sumNumNeigha) {weia[i] <- 1} for (k in 1:sumNumNeighp) {weip[k] <- 1} area.struct[1:na] ~ car.normal(mapa[],weia[],numa[],tau.area.struct) period[1:np] ~ car.normal(mapp[],weip[],nump[],tau.period)
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 55 for (i in 1:na) { area.unstr[i] ~ dnorm(0.00,tau.area.unstr) for (k in 1:np) { observe[i,k] ~ dpois(lambda[i,k]) log(lambda[i,k]) <- log(expect[i,k])+alpha+area.struct[i]+area.unstr[i]+period[k] rr[i,k] <- lambda[i,k]/expect[i,k] dev.residual[i,k] <- observe[i,k]*(log(observe[i,k]+0.01)-log(lambda[i,k]))+(lambda[i,k]- observe[i,k]) } } alpha ~ dflat() tau.area.struct ~ dgamma(0.5,0.005) tau.period ~ dgamma(0.5,0.0005) tau.area.unstr ~ dgamma(1.0E-3,1.0E-3) sigma.area.struct <- 1/tau.area.struct sigma.period <- 1/tau.period sigma.area.unstr <- 1/tau.area.unstr
### Deviance ### ------deviance <- sum(dev.residual[1:na,1:np])
### Space effect ### ------for (i in 1:na) {space[i] <- area.struct[i] + area.unstr[i]}
### Relative risks for each period ### ------for (i in 1:na) {rrp1[i] <- rr[i,1]} for (i in 1:na) {rrp2[i] <- rr[i,2]} for (i in 1:na) {rrp3[i] <- rr[i,3]} for (i in 1:na) {rrp4[i] <- rr[i,4]} for (i in 1:na) {rrp5[i] <- rr[i,5]} for (i in 1:na) {rrp6[i] <- rr[i,6]} }
WinBUGS data: list( na = 43, np = 6, sumNumNeigha = 204, sumNumNeighp = 10, observe = structure(.Data= c(32, 79, 132, 182, 166, 150, 5, 18, 32, 32, 24, 27, 8, 22, 6, 28, 27, 21, 35, 193, 199, 143, 130, 104, 4, 46, 43, 37, 40, 45, 10, 29, 27, 38, 32, 40, 7, 33, 31, 28, 36, 21, 4, 24, 31, 27, 21, 13, 1, 1, 6, 4, 1, 7, 8, 15, 23, 7, 3, 5, 9, 21, 33, 29, 17, 19, 2, 16, 33, 19, 12, 10, 2, 15, 18, 19, 13, 10, 1, 4, 8, 11, 4, 3, 0, 7, 8, 7, 5, 5, 7, 19, 33, 40, 30, 42, 9, 21, 19, 15, 30, 21, 10, 19, 8, 17, 8, 16, 18, 104, 82, 93, 90, 65, 3, 8, 6, 18, 18, 10, 5, 22, 18, 27, 28, 13, 5, 16, 6, 15, 5, 4, 23, 114, 100, 123, 45, 40, 13, 30, 29, 36, 22, 13, 4, 17, 7, 7, 5, 2, 5, 31, 39, 42, 28, 22, 5, 38, 24, 44, 13, 16, 7, 25, 22, 10, 13, 4, 5, 17, 11, 10, 14, 8, 27, 121, 100, 68, 101, 48, 6, 31, 27, 19, 51, 16, 0, 12, 7, 7, 8, 5, 15, 31, 20, 25, 38, 25, 3, 17, 6, 9, 7, 13, 6, 13, 12, 19, 12, 14, 2, 5, 7, 5, 6,
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 56 3, 1, 3, 2, 3, 9, 1, 2, 5, 5, 3, 4, 3, 1, 8, 5, 5, 5, 5, 2, 6, 6, 3, 5, 7, 1, 9, 11, 12, 19, 4, 8, 41, 28, 42, 48, 33, 2, 11, 13, 8, 8, 13), .Dim = c(43,6)), expect = structure(.Data= c(22.8449, 115.8083, 121.2598, 125.4805, 129.9196, 133.994, 6.5693, 33.2191, 34.9401, 37.7003, 39.698, 41.9972, 5.5199, 28.1545, 30.8342, 34.7708, 38.8775, 43.4439, 32.5014, 162.91, 161.6121, 158.7109, 157.4227, 149.9758, 10.7927, 55.1702, 59.5326, 63.4676, 67.7687, 72.6516, 8.8395, 44.9318, 47.7112, 51.4639, 55.2079, 59.4683, 5.9858, 30.5174, 33.0084, 35.2698, 37.7507, 41.3541, 5.5795, 28.4189, 30.2547, 32.5561, 35.0047, 37.6573, 1.0479, 5.2507, 5.3974, 5.8473, 6.0798, 6.4744, 2.432, 12.2364, 12.6343, 13.3297, 13.8149, 14.4511, 5.7979, 29.4408, 31.1461, 33.0145, 34.8187, 36.8612, 3.1673, 16.0587, 17.0388, 18.5431, 19.8543, 21.5748, 3.0169, 15.2919, 15.7636, 16.5465, 17.6436, 18.6784, 1.6809, 8.521, 9.3337, 10.4006, 10.8285, 11.5502, 1.2051, 6.109, 6.319, 6.6944, 7.3106, 8.2225, 6.0924, 30.8269, 32.0183, 33.2518, 34.6105, 36.3437, 3.899, 19.6793, 20.4739, 21.835, 22.8915, 24.1798, 1.9913, 10.079, 10.1922, 10.2936, 10.7395, 11.2458, 12.2413, 61.7351, 63.8022, 65.6801, 67.5185, 69.2835, 3.0099, 15.2014, 15.6769, 16.0582, 16.3629, 16.8677, 4.368, 22.1116, 23.3314, 24.8144, 26.3009, 27.9931, 2.1073, 10.5537, 10.843, 11.0619, 11.0442, 11.3781, 11.8998, 59.7853, 60.5252, 60.6118, 61.0107, 61.5668, 7.1259, 35.8071, 36.0702, 35.701, 35.6991, 36.2546, 1.9414, 9.7793, 9.9853, 10.1287, 10.2023, 10.3737, 4.2977, 21.612, 22.1128, 22.6164, 23.1177, 23.8209, 3.5261, 17.7511, 18.3705, 18.8481, 19.3124, 20.3081, 4.2026, 21.0583, 20.8882, 20.4744, 20.4601, 20.8534, 2.5968, 13.0543, 12.8724, 12.3525, 12.5962, 13.1348, 16.9782, 85.3781, 86.5638, 86.223, 86.5795, 87.6385, 6.1694, 30.8911, 31.2526, 31.9662, 32.7628, 34.273, 1.6935, 8.5094, 8.6273, 8.7557, 8.8829, 9.2902, 4.9817, 25.6865, 28.9582, 32.6535, 36.4175, 41.1588, 2.2719, 11.7733, 13.4978, 15.553, 17.7077, 20.3809, 2.9471, 15.0098, 16.2705, 17.8461, 19.4481, 21.4773, 1.0886, 5.5278, 5.7756, 5.856, 6.0251, 6.3597, 0.8277, 4.1595, 4.3756, 4.5874, 4.6237, 4.8679, 1.028, 5.1721, 5.2935, 5.5686, 5.8319, 6.1946, 1.3288, 6.675, 6.7076, 6.8116, 7.0494, 7.5041, 1.1015, 5.5668, 5.7692, 6.0627, 6.1564, 6.3505, 2.1863, 10.9866, 11.3336, 11.8988, 12.3294, 13.2098, 6.2138, 31.324, 32.7834, 34.4958, 35.6514, 37.2797, 1.4327, 7.2207, 7.2923, 7.3104, 7.4371, 8.0847), .Dim = c(43,6)), numa = c(3, 6, 5, 2, 7, 7, 6, 6, 5, 5, 7, 3, 5, 4, 3, 5, 5, 2, 8, 6, 4, 4, 5, 3, 3, 8, 4, 4, 6, 4, 5, 7, 5, 3, 5, 3, 4, 5, 5, 2, 5, 6, 4), mapa = c( 21, 3, 2, 21, 17, 6, 5, 3, 1, 34, 33, 6, 2, 1, 6, 5, 26, 17, 16, 7, 6, 4, 2, 33, 32, 7, 5, 4, 3, 2, 42, 32, 26, 23, 6, 5, 19, 18, 14, 13, 12, 9, 15, 13, 12, 10, 8, 25, 15, 13, 11, 9, 28, 25, 20, 19, 14, 13, 10,
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 57 15, 9, 8, 14, 11, 10, 9, 8, 19, 13, 11, 8, 12, 10, 9, 26, 20, 19, 17, 5, 21, 19, 16, 5, 2, 19, 8, 21, 20, 18, 17, 16, 14, 11, 8, 28, 26, 22, 19, 16, 11, 19, 17, 2, 1, 28, 26, 24, 20, 43, 42, 27, 26, 7, 27, 26, 22, 28, 11, 10, 27, 24, 23, 22, 20, 16, 7, 5, 43, 26, 24, 23, 25, 22, 20, 11, 42, 41, 38, 32, 31, 30, 35, 32, 31, 29, 38, 37, 35, 30, 29, 42, 35, 33, 30, 29, 7, 6, 35, 34, 32, 6, 3, 35, 33, 3, 34, 33, 32, 31, 30, 40, 39, 37, 39, 38, 36, 31, 41, 39, 37, 31, 29, 41, 40, 38, 37, 36, 39, 36, 43, 42, 39, 38, 29, 43, 41, 32, 29, 23, 7, 42, 41, 27, 23), nump=c(1,2,2,2,2,1), mapp=c( 2, 1,3, 2,4, 3,5, 4,6, 5) )
WinBUGS initials: ## Chain 1: list( alpha=0, tau.area.struct=1, tau.area.unstr=1, tau.period=1, area.struct=c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0), area.unstr=c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0), period=c( 0,0,0,0,0,0))
## Chain 2: list(
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 58 alpha=0, tau.area.struct=0.5, tau.area.unstr=1.5, tau.period=1, area.struct=c( 0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5, 0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5, 0.5,-0.5,0.5), area.unstr=c( -0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5, -0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5, -0.5,0.5,-0.5), period=c( -0.2,0.2,-0.2,-0.2,0.2,-0.2))
11.1.4. Space-period model with interactions
WinBUGS program: model { for (i in 1:sumNumNeigha) {weia[i] <- 1} for (j in 1:sumNumNeighp) {weip[j] <- 1}
### PRIORS for structural area and cohort effects ### ------area.struct[1:na] ~ car.normal(mapa[],weia[],numa[],tau.area.struct) period[1:np] ~ car.normal(mapp[],weip[],nump[],tau.period)
### PRIOR for unstructural area effect ### and MODEL specifiction ### ------for (i in 1:na) { area.unstr[i] ~ dnorm(0.00,tau.area.unstr) for (j in 1:np) { observe[i,j] ~ dpois(lambda[i,j]) log(lambda[i,j]) <- log(expect[i,j])+alpha+area.struct[i]+area.unstr[i]+period[j]+eff.int.ap[i,j] rr[i,j] <- lambda[i,j]/expect[i,j] dev.residual[i,j] <- observe[i,j]*(log(observe[i,j]+0.01)-log(lambda[i,j]))+(lambda[i,j]-observe[i,j]) } }
### AREA-PERIOD INTERACTIONS ### ------for (i in 1:na) {
### parameters for the prior of psi[i,1] ### ------int.ap.bar[i,1] <- int.ap[i,2]+mean(adj.int.ap[(off[i]+1):off[i+1],1])- mean(adj.int.ap[(off[i]+1):off[i+1],2]) nadj.int.ap[i,1] <- 1.0*numa[i]
### parameters for the prior of psi[i,j], j=2,...nc-1 ### ------for (j in 2:(np-1)){ int.ap.bar[i,j] <- (int.ap[i,j-1]+int.ap[i,j+1])/2+ mean(adj.int.ap[off[i]+1:off[i+1],j])- (mean(adj.int.ap[off[i]+1:off[i+1],j-1])+mean(adj.int.ap[(off[i]+1):off[i+1],j+1]))/2 nadj.int.ap[i,j] <- 2.0*numa[i] }
### parameters for the prior of psi[i,np]
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 59 ### ------int.ap.bar[i,np] <- int.ap[i,np-1]+mean(adj.int.ap[(off[i]+1):off[i+1],np])- mean(adj.int.ap[(off[i]+1):off[i+1],np-1]) nadj.int.ap[i,np] <- 1.0*numa[i];
### precision parameter of the prior for psi[i,j] ### ------for (j in 1:np) {int.ap.prec[i,j] <- nadj.int.ap[i,j]*tau.int.ap}
### create eff.int.ap[i,j] such that eff.int.ap[i,1]=0 and eff.int.ap[1,j]=0 ### use it then in a model specification ### and assign a prior to interaction terms ### ------for (j in 1:np){ int.ap[i,j] ~ dnorm(int.ap.bar[i,j],int.ap.prec[i,j]) eff.int.ap[i,j] <- int.ap[i,j]-int.ap[i,1]-int.ap[1,j]+int.ap[1,1]
### an obscure way to define later on a prior for tau.int.ac ### ------tau.int.ap.like[i,j] <- nadj.int.ap[i,j] * int.ap[i,j] *(int.ap[i,j] - int.ap.bar[i,j]) } }
### compute adj.int.ap ### adj.int.ap[off[i]+1:off[i+1],j] are interactions comming from ### all neighbours of the ith area and the jth period ### ** it is used when computing int.ap.bar[i,j] ### ------for (l in 1:adj) {for (j in 1:np) {adj.int.ap[l,j] <- int.ap[mapa[l],j]}}
### PRIORS for remaining parameters ### ------alpha ~ dflat() tau.area.struct ~ dgamma(0.5,0.0005) tau.period ~ dgamma(0.5,0.0005) tau.area.unstr ~ dgamma(1.0E-3,1.0E-3) r.int.ap <- 1 + (na)*(np)/2 l.int.ap <- 0.01 + sum(tau.int.ap.like[,])/2 tau.int.ap ~ dgamma(r.int.ap,l.int.ap) sigma.area.struct <- 1/tau.area.struct sigma.period <- 1/tau.period sigma.area.unstr <- 1/tau.area.unstr sigma.int.ap <- 1/tau.int.ap
### Deviance ### ------deviance <- sum(dev.residual[1:na,1:np])
### Space effect ### ------for (i in 1:na) {space[i] <- area.struct[i] + area.unstr[i]}
### Relative risks for each cohort ### to be used by GeoBUGS ### ------for (i in 1:na) {rrp1[i] <- rr[i,1]} for (i in 1:na) {rrp2[i] <- rr[i,2]}
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 60 for (i in 1:na) {rrp3[i] <- rr[i,3]} for (i in 1:na) {rrp4[i] <- rr[i,4]} for (i in 1:na) {rrp5[i] <- rr[i,5]} for (i in 1:na) {rrp6[i] <- rr[i,6]} }
WinBUGS data: - same as for the model without interactions with additionally off=c(0, 3, 9, 14, 16, 23, 30, 36, 42, 47, 52, 59, 62, 67, 71, 74, 79, 84, 86, 94, 100, 104, 108, 113, 116, 119, 127, 131, 135, 141, 145, 150, 157, 162, 165, 170, 173, 177, 182, 187, 189, 194, 200, 204)
WinBUGS initials: ## Chain 1: list( alpha=0, tau.area.struct=1, tau.area.unstr=1, tau.period=1, tau.int.ap=1, area.struct=c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0), area.unstr=c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0), period=c( 0,0,0,0,0,0), int.ap= structure(.Data= c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), .Dim=c(43,6)) )
## Chain 2: list( alpha=0, tau.area.struct=0.5, tau.area.unstr=1.5, tau.period=1, tau.int.ap=1, area.struct=c( 0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5, 0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5, 0.5,-0.5,0.5), area.unstr=c( -0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5, -0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5, -0.5,0.5,-0.5), period=c(
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 61 -0.2,0.2,-0.2,-0.2,0.2,-0.2), int.ap= structure(.Data= c( -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2, -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2, -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2, -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2, -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2, -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2, -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2, -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2, -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2, -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2, -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2, -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2, -0.2,0.2,-0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2,0.2,-0.2), .Dim=c(43,6)) )
11.2. Location of files
Text report file: w:\word\SpaceTimeBayeModCvxBelgium3.doc
Data file: w:\arnost\
Statistical processing of WINBUGS output (produced with Stata, version 7). *.do files in w:\arnost\Oct03\
Tabulation of WINBUGS output (produced with EXCEL) w:\arnost\Oct03\SP.XLS, SC.XLS, SPINT.XLS, SCINT.XLS
Maps: (produced with MapInfo, version 10). Figure 2: w:\arnost\Oct03\sc2.wor Figure 3: w:\arnost\Oct03\rawsmr2.wor Figure 7: w:\arnost\Oct03\sp2.wor Figure 12: w:\arnost\Oct03\rawsmr1.wor Figure 13: w:\arnost\Oct03\sc1.wor Figure 14: w:\arnost\Oct03\sp1.wor
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 62
11.3. Alternative cartographic display for the spatial distribution of district effects.
In Figure 12 to Figure 14, we use an alternative definition of the color palette to shade districts, using a combination of the statistical significance and the value of the relative spatial effect. The cut-offs for the extreme shades are less extreme than in the initial maps (Figure 3, Figure 4 and Figure 7). The display style below provides more contrasts in particular in densely populated areas with a higher absolute number of deaths. Therefore we can observe a higher number of districts in green and orange or red. This is for instance the case for Brugge and Mechelen (in green) and Liège (orange). These districts show district effects which are significantly different from unity but which are in value not so extreme.
A small difference can be observed between the SC-model map (Figure 13) and the SP-model map (Figure 14): Tielt (district 37) is yellow in the first while light-green in the second. This was only due to rounding. The maps with modeled district effects shows one more orange spot (Liège, code=62) and one more light green spot (Mechelen, code=12) in comparison with the previous maps.
raw SMR
11 13 43 46 35 31 72
12 38 44 42 32 37 71 36
41 24 33 45 23 21 73 34 54
25 64 57 51 62 55
61 53 52 92 63
83 91 56 93 82
Mortality from cervical cancer
Distributon of relative risk (R) 84 Upper CI<0.9 (5) Upper CI between 0-1 (5) R<1 but CI includes 1 (20) R>1 but CI includes 1 (5) 81 Lower CI between 1-1.1 (1) 85 Lower CI >1.1 (7) rawsmr2.wor
Figure 12. Geographical distribution of raw SMRs, using the global all period mortality as reference rate.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 63 Space-cohort model
11 13 43 46 35 31 72
12 38 44 42 32 37 71 36 41 24 33 45 21 34 23 73 54
25 64 57 51 62 55
61 53 52 92 63
83 91 56 93 82 Mortality from cervical cancer Distributon of relative risk (Ri) 84 Upper CI<0.9 (3) Upper CI between 0-1 (2) R<1 but CI includes 1 (21) R>1 but CI includes 1 (8) 81 Lower CI between 1-1.1 (2) 85 Lower CI >1.1 (7) sc1.wor
Figure 13. Geographical distribution of area effects, relative risk of mortality from cervical cancer by districti, compared to the Belgian average, estimated from the SC model. Districts are labeled with their NIS code (see Table 7).
Space-period model
11 13 43 46 35 31 72
12 38 44 42 32 37 71 36
41 24 33 45 23 21 73 34 54
25 64 57 51 62 55
61 53 52 92 63
83 91 56 93 82
Mortality from cervical cancer Distributon of relative risk (R) 84 Upper CI<0.9 (3) Upper CI between 0-1 (3) R<1 but CI includes 1 (20) 81 R>1 but CI includes 1 (8) Lower CI between 1-1.1 (2) 85 Lower CI >1.1 (7) sp1.wor
Figure 14. Geographical distribution of area effects, relative risk of mortality from cervical cancer by districti, compared to the Belgian average, estimated from the SP model. Districts are labeled with the NIS code (see Table 7).
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 64
11.4. Selected output
In this section, we show a selected output from the Bayesian analysis. Specifically, the output concerning the precision parameters (τ’s) for all four considered models is given.
11.4.1. Space-cohort model without interactions
Figure 15. Posterior densities for the precision parameters.
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Figure 166. Traces of sampled values for the precision parameters.
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Figure 177. Autocorrelation plots for the precision parameters.
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Figure 188. Gelman-Rubin convergence plots for the precision parameters.
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Figure 19. Geweke convergence plots for the precision parameters.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 69 11.4.2. Space-cohort model with interactions
Figure 20. Posterior densities for the precision parameters.
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Figure 21. Traces of sampled values for the precision parameters.
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Figure 22. Autocorrelation plots for the precision parameters.
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Figure 23. Gelman-Rubin convergence plots for the precision parameters.
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Figure 24. Geweke convergence plots for the precision parameters.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 74 11.4.3. Space-period model without interactions
Figure 25. Posterior densities for the precision parameters.
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Figure 26. Traces of sampled values for the precision parameters.
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Figure 27. Autocorrelation plots for the precision parameters.
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Figure 28. Gelman-Rubin convergence plots for the precision parameters.
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Figure 29. Geweke convergence plots for the precision parameters.
SpaceTimeBayModCvxBelgium3CoverMenIPH.doc 13/11/1999 - 14:01 79 11.4.4. Space-period model with interactions
Figure 30. Posterior densities for the precision parameters.
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Figure 31. Traces of sampled values for the precision parameters.
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Figure 32. Autocorrelation plots for the precision parameters.
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Figure 33. Gelman-Rubin convergence plots for the precision parameters.
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Figure 34. Geweke convergence plots for the precision parameters.
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