Hierarchical modelling of small area and hospital variation in short-term prognosis after acute...

STATISTICS IN MEDICINEStatist. Med. 2004; 23:2599–2621 (DOI: 10.1002/sim.1843)

Hierarchical modelling of small area and hospital variationin short-term prognosis after acute myocardial infarctionA longitudinal study of 35- to 74-year-old men in Denmark

between 1978 and 1997

SHren Rasmussen∗;†

National Institute of Public Health; Copenhagen; Denmark

SUMMARY

Models for analysis of trends in hospital and small area variation in case fatality after acute myocardialinfarction are presented. The data are from administrative registries in Denmark. Hierarchical modellingin a logistic regression with a Bayesian approach is used. Model selection is undertaken using thedeviance and the Bayesian information criteria. There is a modest trend for hospital variation in case-fatality rates that coincides with the introduction of new treatment strategies. This hospital variation isconsiderably larger than the variation at the area level. There is no trend for variation of the case-fatalityrates at the area level. Unstructured random e�ects slightly outperform spatially correlated random e�ectsat the area level. Somewhat high correlations over time within hospitals and within areas were detectedfor the case-fatality rates. Heavy-tailed distributions (T-distributions) could be an alternative for therandom e�ect distribution in data from administrative registries and compete in the model selectionwith the normal distribution in this study. Copyright ? 2004 John Wiley & Sons, Ltd.

KEY WORDS: acute myocardial infarction; covariance pattern; deviance information criterion; hierar-chical modelling; Markov chain Monte Carlo methods; spatial models

1. INTRODUCTION

In the last decade, great progress has been made in treating acute myocardial infarction (AMI)in western Europe and North America, as shown by several clinical trials. The lessons of thesestudies have gradually been implemented successfully at the population level, and treatmentfor AMI has therefore changed considerably. In the late 1980s, e�cient low-tech treatmentwith thrombolysis and aspirin was introduced. During the 1990s, an even more successfulhigh-tech approach was introduced, also at the population level, with percutaneous coronary

∗Correspondence to: SHren Rasmussen, National Institute of Public Health, SvanemHllevej 25, DK-2100,Copenhagen, Denmark.

†E-mail: [email protected]

Contract=grant sponsor: Danish Medical Research Council; contract=grant number: 22-00-0548

Received May 2003Copyright ? 2004 John Wiley & Sons, Ltd. Accepted January 2004

2600 S. RASMUSSEN

intervention (PCI) during the acute phase of treatment: initially in centres with catheterizationfacilities but later also in general hospitals. The medical treatment has also evolved to bemore complex, with a wide range of di�erent regimens in the acute and subacute phases ofAMI. At the same time, interest has been growing in guaranteeing the optimal quality ofhospital treatment for all patients, with an increasing number of clinical databases establishedto ensure this goal. Hospitals or treatment centres have been controversially compared usingleague tables, especially in the United States and United Kingdom [1, 2].Progress in computational power has increased the applications of Bayesian hierarchical

modelling. This paper presents a case study using these models in analysing prognosis afterAMI between units at di�erent levels with data from administrative registries. The healthcare services have increasingly used administrative data to compare incidence and prognosisafter AMI and Markov chain Monte Carlo (MCMC) methods allow the often massive databe exploited. The hierarchical models can successively increase in complexity, incorporatingrandom e�ects at di�erent levels and also letting these random e�ects vary across time trendsto investigate whether di�erences between units at the same level change over time. Thesemodels can further include spatial random e�ects to capture unmeasured spatially correlatedcovariates. The models expand further through the option of choosing di�erent priors andhyperpriors to stretch the �exibility of the models applied.This study used Bayesian hierarchical modelling to determine whether the improved and

more complex treatment of AMI patients has in�uenced the trends in the variation in short-term prognosis between hospitals and small areas. We also consider how much variation canbe attributed to hospitals and residential areas and thereby investigate what is most in�uentialfor the patient’s chances of survival. Another question relates to how the patient’s residencein�uences survival after AMI by considering spatial modelling, in which neighbouring areastend to be more similar because of such factors as the socioeconomic contexts in these areas.The correlations within hospitals and within areas over time are considered because theyin�uence the trends in variation and provide an intrinsic interpretation for the same units.

2. DATA

Data for the period 1978 to 1997 were extracted from two administrative registries: TheDanish National Hospital registry and Danish National Registry of Causes of Death. Theunique civil registration number, assigned to every permanent resident of Denmark identi�esa speci�c patient. An AMI case was de�ned as either admission with AMI as the primary orsecondary diagnosis. AMI was coded as 410 (ICD-8) or I21 + I22 (ICD-10) from the 8th and10th revision of the International Classi�cation of Diseases (ICD). For each patient, the �rstevent was recorded; later events were recorded only if they occurred more than 28 days afteradmission for the previous AMI. This eliminated problems with transfer between hospitals andreadmissions. The outcome measure was case-fatality (death from all causes) rate within 1–28days, for the patients who survived the �rst day after the event. This de�nition of case fatalitymeans that all these patients have been admitted to hospital, and the information is thereforederived from the hospital registries, excluding patients dying of AMI prior to admission.Further, all patients were de�nitely alive when admitted, eliminating problems associatedwith di�erences in admission procedures in for example emergency rooms. Madsen et al.[3] evaluated the validity of the data from the administrative registries for the diagnosis of

Copyright ? 2004 John Wiley & Sons, Ltd. Statist. Med. 2004; 23:2599–2621

HIERARCHICAL MODELLING OF SMALL AREA AND HOSPITAL VARIATION 2601

AMI. These results show that 95 per cent of the AMI admissions were con�rmed as beingan actual AMI. Furthermore Abildstrom et al. [4] have reported an improved prognosis at thepopulation level based on the data. Events among men 35–74 years of age residing in Denmarkwere analysed. The data were aggregated into 10 time periods, where each period covers 2years: (1978–1979; : : : ; 1996–1997). The events where a patient has residence at the remoteisland Bornholm were excluded from the data (about 1 per cent of the events). Denmarks’s270 municipalities not including the six municipalities from Bornholm, de�ned the area levelin the analysis. Only records from hospitals with more than 10 AMI admissions per year,indicating that acute referral and treatment were present at the hospital, were included inthe analysis. Data were furthermore broken down into eight 5-year age bands (35–39, 40–44; : : : ; 70–74). Empty cells de�ned within hospital, area and time period were excluded, sothat the data consist of 30 218 records covering 19 423 deaths from 125 598 events. Table Igives descriptive information on this AMI population. We further de�ne catchment zones asthe combination of the hospital that treats the patient and the residence of the patient. Thereare 270 municipalities and 85 hospitals, over the whole period, giving a theoretical maximumnumber of 22 950 catchment zones. But only 3825 catchment zones were not empty for eventsand therefore included in the analysis. This de�nition of a catchment zone is partly adapted inpart from Congdon and Best [5]. We use these zones to look for potential interaction betweenarea and hospital.

3. METHOD

In the present paper we propose Bayesian hierarchical models and multi-level models usingMCMC methods, which allows unmeasured variables to be modelled as random e�ects atboth the hospital and area level at the same time and thereby their respective in�uence onthe short-term prognosis can be compared. In addition we model potential variation over timeand likely spatial dependence between areas. Fitting the model is demanding because, forexample patients from the same area can be treated at di�erent hospitals and thus formalnesting does not exist between areas and hospitals, resulting in a cross-classi�ed model [6].We would also anticipate a strong correlation between consecutive measurements for the levelof interest, because we would expect, for example, the outcome within a hospital to be nearlyidentical close in time. It is important to address this correlation, which in�uences the trendsin variation.The Bayesian approach through MCMC methods also o�ers some �exibility in choice of

priors for random e�ects. This has been traditionally a normal distribution but given that thesource is administrative data a more heavy-tailed distribution could be appropriate becausethese data potentially can produce outliers through coding bias such as an autonomous hospitalregistration procedure that con�icts with the registration of other hospitals by using otherdiagnosis classi�cations for an actual AMI and thereby in�uence the outcome. One would tryto partly eliminate these kind of outliers in advance by carefully deciding, which measuresor de�nitions to use. Furthermore estimation with MCMC methods is less biased for binaryresponse models compared to quasi-likelihood methods [7].We use the WinBUGS [8] programme to analyse data. This programme supports Bayesian

inference by MCMC simulation obtained by updating parameters using Gibbs sampling [9],which successively samples from the conditional distribution of each parameter given all the


2602 S. RASMUSSEN

TableI.Characteristicofthestudypopulation.

Period

1978=79

1980=81

1982=83

1984=85

1986=87

1988=89

1990=91

1992=93

1994=95

1996=97

Cases

13979

13929

14202

13829

13518

12711

12109

11735

10528

9,058

Deaths

2621

2611

2477

2395

2313

2001

1571

1425

1127

882

Case-fatalityrateinpercent

1919

1717

1716

1312

1110

Numberofhospitals

7977

7676

7473

7171

7069

Rangefornumberof

p10

4454

6051

5952

5759

4136

cases(hospitals)

p25

7881

9477

9285

8679

6864

median

177

181

187

182

183

174

171

165

150

131

p75

238

245

249

257

239

221

213

213

208

176

p90

343

337

346

342

347

344

314

316

308

270

Rangeforcase-fatality

p10

1313

1112

1210

88

75

rate(hospitals)inpercent

p25

1615

1315

1413

1110

88

median

1918

1717

1715

1312

1010

p75

2121

2019

2018

1514

1312

p90

2325

2322

2219

1716

1515

Rangefornumberof

p10

1112

1212

1312

1111

109

cases(areas)

p25

1616

1717

1717

1616

1413

median

2624

2627

2625

2623

2120

p75

4547

4847

4946

4345

3733

p90

107

103

110

111

104

9894

9086

72

Rangeforcase-fatality

p10

98

67

65

55

00

rates(areas)inpercent

p25

1313

1112

1210

87

64

median

1817

1617

1615

1212

119

p75

2222

2121

2120

1715

1413

p90

2729

2626

2725

2219

2120



others in the model. Browne et al. [6] outline these full conditionals, considering some of themodels proposed in this study.We propose an approach for model selection with three steps in which one has to take

into account the intensive amount of computation to address the problems of interest. We�rst identify the best-�tting �xed e�ect model using common maximum likelihood estimation.Then we use families of random intercept and slope models within this �xed e�ect modelto select candidate models for further evaluation. We hereby screen models to determinewhether there simultaneously are trends according to hospital and area random e�ects. Thelast step involves other type of models in which we examine di�erent covariance patterns forthe random e�ects.

3.1. First step: �xed e�ects model

Using logistic regression with time t as a quantitative covariate, we found that the following�xed model, where pit is the case-fatality rate within 1–28 days, was the most parsimoniousacceptable:

log(

pit1− pit

)= �0 + �1iagei + �2iagei(t − �t) + �3(t − �t) + �4(t − �t)2

i=1; : : : ; 8 (1)

Age is included in the model as a categorical covariate with eight age groups (35–39,40–44; : : : ; 70–74). The estimates of the parameters in model (1) are presented in Table II.Di�erent combinations of random e�ects can then be evaluated for area and hospital withinthe same �xed e�ect model (1). Thus (1) becomes an equivalent to the usual sex and agestandardization in analysis of spatial Poisson models for incidence and event data [10, 11], butwithout reducing the complexity in terms of data aggregation. This has some computationaldisadvantages.

3.2. Second step: unstructured random e�ects according to area and hospital

We assume that the area random e�ects ua and hospital random e�ects hj independently follownormal distributions with zero mean and unknown variance. Thus,

ua ∼ N(0; �2u); and hj ∼ N(0; �2h) (2)

respectively. These e�ects may describe the extent to which individuals from a given areaor a given hospital have a tendency towards death or survival above what can be explainedby known individual characteristics: that is, in this study, age and time period. A multi-levelmodel with both area random e�ects and hospital random e�ects is speci�ed as

log(

pitaj1− pitaj

)= �0 + �1iagei + �2iagei(t − �t) + �3(t − �t) + �4(t − �t)2 + ua + hj

i=1; : : : ; 8; a=1; : : : ; 270 j=1; : : : ; 85 (3)


2604 S. RASMUSSEN

Table II. Estimates from the �xed e�ect model using Bayesian logistic regression; 10 000 iterationsused for calculations.

Parameter Mean 95% CI

�4 −0:010 −0:012 −0:008�3 −0:081 −0:091 −0:070�2(65−69) −0:004 −0:019 0.011�2(60−64) −0:012 −0:030 0.004�2(55−59) −0:027 −0:048 −0:007�2(50−54) −0:052 −0:079 −0:026�2(45−49) −0:083 −0:117 −0:047�2(40−44) −0:041 −0:094 0.010�2(35−39) −0:002 −0:079 0.073�2(70−74) 0�1(65−69) −0:291 −0:334 −0:248�1(60−64) −0:656 −0:703 −0:607�1(55−59) −0:972 −1:028 −0:917�1(50−54) −1:391 −1:464 −1:391�2(45−49) −1:620 −1:723 −1:520�1(40−44) −1:698 −1:836 −1:562�1(35−39) −1:639 −1:856 −1:431�1(70−74) 0�0 −1:101 −1:134 −1:068D 26 805PD 17DIC 26 822

95% CI: 95 per cent credible interval. Mean centring of time period.

where the �rst level is patients in their respective subgroups de�ned by age and time period,the second level is area and the third level is hospital.

3.2.1. Random slope regression models. Because we are interested in examining variationover time, we introduce interaction between time and the random e�ects. These models aresometimes denoted as random slope models: that is each random e�ect has it own interceptand slope. For brevity, we present the models for the hospital random e�ects such that

(hj1

hj2

)∼ MVN

(0;

[�21h

�12h �22h

])(4)

where the model is, for example:

log(

pitj1− pitj

)= �0 + �1iagei + �2iagei(t − �t) + �3(t − �t) + �4(t − �t)2 + hj1 + hj2(t − �t)

i=1; : : : ; 8 j=1; : : : ; 85 (5)



Looking at the quadratic e�ects, we can expand the covariance matrix even further so that:

hj1

hj2

hj3

∼ MVN

0;

�21h

�12h �22h

�13h �23h �3h

(6)

where the model now is, for example:

log(

pitj1− pitj

)= �0 + �1iagei + �2iagei(t − �t) + �3(t − �t) + �4(t − �t)2

+hj1 + hj2(t − �t) + hj3(t − �t)2

i=1; : : : ; 8 j=1; : : : ; 85 (7)

Thus, these models will predict hospital performance in case-fatality rates compared with thenational average adjusted for trend and age. In the same fashion, we can introduce interactionbetween time and area random e�ects ua. The parameterization in (5) and (7) means that theintercept and slopes are examined at time: t − �t.

3.2.2. Variance and covariance over time. Random slope models give rise to potential vari-ance heterogeneity in time within area and say hospitals. The random slope model in (5)induces a polynomial for the variance in the linear predictor:

Var(hj1 + hj2(t − t0)) = �21h + 2�12h(t − t0) + �22h(t − t0)2 (8)

Correspondingly, a random slope model with time squared from (7) induces a fourth orderpolynomial:

Var(hj1 + hj2(t − t0) + hj3(t − t0)2) = �21h + 2�12h(t − t0) + �22h(t − t0)2 + 2�13h(t − t0)2

+2�23h(t − t0)3 + �23h(t − t0)4

These models also allow for correlation in time within the same area and hospital, which isa very reasonable assumption, for example, from (5), where s �= t

Cov(hj1 + hj2(t − t0); hj1 + hj2(s− t0)) = �21h + �12h((t − t0) + (s− t0))

+�22h(t − t0)(s− t0) (9)

The correlation over time depends on the spacing and position. As seen in (9) the modelfrom (5) is parsimonious using three parameters for modelling a covariance matrix of varioussizes. To add more complexity to the random slope model, an extra error term can be added


2606 S. RASMUSSEN

to (5) such as rjt , where rjt ∼ N(0; �2e), that is, a random interaction between time and thelevel. This changes (8) to

Var(hj1 + hj2(t − t0) + rjt)=�21h + 2�12h(t − t0) + �22h(t − t0)2 + �2e (10)

This extra term resembles a residual in a linear regression. It can be interpreted as a residualon a ‘macro’ level in this setting.

3.2.3. T -distributed random e�ects. An alternative to the normal distribution assumption isthe T-distribution with a heavy tail, for example, with four degrees of freedom. This distribu-tion is not so in�uenced by possible outliers as the normal distribution. Using the multivariateT-distribution, we re-express (4) as(

hj1

hj2

)∼ MVT

(0;

[�21h

�12h �22h

]; 4

)(11)

3.3. Spatially correlated area random e�ects

For discrete spatial distribution at the area level, we use the model proposed in Reference[12], where

�i = vi + ui

vi =∑i �=jzijv∗j where zij=

1ni

when i ↔ j otherwise zij=0

v∗j ∼ N(0; �2s ); ui ∼ N(0; �2u)

(12)

and ni is the number of neighbours to area i. The symbol ↔ indicates that i is a neighbour toj. In this manner, each area is a member of a higher-level unit that contains its neighbours:that is, shares a common boundary. Thus, spatial variance is modelled as a special case ofmultiple membership [6]. Langford et al. [12] consider other weighting schemes in (12), buta priori there is no general knowledge about these choices in this setting. Langford et al. [12]operate with a covariance term between the unstructured area random e�ects and the spatialarea random e�ects such that in (12), cov(v∗i ; ui)=�s; u. Note that the variance of the spatiale�ects depends on the number of neighbours:

Var(�i)=1ni�2s + �

2u (13)

The mode or the average of neighbours is usually used in determining the relative importanceof the spatial and the unstructured variance components in (13). The average of neighbourswas 4.50 in this study. Using (12) and cov(v∗i ; ui)=�s; u, the covariance between area i and jcan be expressed as

Cov(�i; �j)=�s; u(zij + zji) +∑

k �=j; k �=izikzjk�2s (14)

such that the covariance depends on the number of common neighbours, as seen in the lastterm in (14) and the �rst term disappears in (14), if i does not border j [12].



3.4. Model selection using the deviance information criterion

Parsimonious models are selected by using the newly proposed deviance information criterion(DIC) [13]. This criterion penalizes the mean posterior deviance D with a term measuringmodel complexity, the so-called ‘e�ective number of parameters’ PD. In the binomial model,D is de�ned as the usual binomial deviance [14]:

D=2∑iyi log

(yinipi

)+ (ni − yi) log

(ni − yini − nipi

)(15)

The term yi is the number of deaths, ni is the denominator (the number of cases) and pi isthe probability of death. In general, DIC is de�ned as

DIC=D(�) + 2PD=D+ PD=D+ (D −D(�))=2D −D(�) (16)

where � are the parameters in the model [13]. We use the mean parameterization versionof the DIC, using the posterior means of pi for calculating D(�). Spiegelhalter et al. [13]outlined other choices. Models with smaller values of DIC denote better �ts. The e�ectivenumbers of parameters are often less than the total number of parameters in the model due to‘borrowing of strengths’ across levels in hierarchical models. A disadvantage of using the DICis that reliable methods of �nding standard error of the measure do not exist [15], althoughthe standard error for D is easily obtained. This gives an indication of how much samplingerror there is when DIC is compared for di�erent models. In principle, determining whichmodel to choose is not possible using exclusively DIC, where models with a di�erence of1–2 comparing DIC are candidates for the best-�tting model, whereas models with di�erencesbetween 3 and 7 cannot be entirely disquali�ed [13].

3.5. Alternative measures of �t

An alternative to DIC could be the Bayesian information criterion (BIC) also known asSchwarz’s information [16, 17], for example, calculated as

BIC=D+ q log(N − n) (17)

where q is the number of covariance parameters or variance components, n is the number of�xed e�ects and N is the number of observations, such that in our case N � n. Hereby thenumber of covariance parameters, such as q=2 in (3), will penalize the model instead of thenumber of e�ective parameters. Models with smaller values of BIC denote better �ts. Giventhe de�nition of BIC and the relatively big sample in our case, we would expect heavilyparameterized models to be more penalized by BIC than DIC. We only use DIC in the �rstand second step of the model selection procedure and return to the BIC when discussingcandidate models for the best �t.

3.6. Third step: covariance pattern models

To consider other choices for modelling variation at the area and especially the hospital level,we analysed four di�erent models with a 10× 10 covariance matrix, corresponding to thenumber of time points. A large number of covariance patterns are available for use in mixedmodels, but we inspect only the most common patterns.


2608 S. RASMUSSEN

3.6.1. Unstructured covariance matrix. A model where the covariance matrix is unstruc-tured is

hit ∼ MVN

0;

�21 : : : �1;10

.... . .

...

�10;1 · · · �210

t=1; : : : ; 10 (18)

This model is heavily parameterized with 10 ∗ (10− 1)=2 + 10=55 covariance parameters tobe estimated.

3.6.2. Autoregressive model. A �rst-order autoregressive model with a fading correlation intime such that correlation is highest between consecutive measurements and smaller far awayin time controlled by the parameter � is

hit ∼ MVN

0; �

2

1 �1 : : : �10

�1. . .

...

:

�10 �9 · · · 1

t=1; : : : ; 10 (19)

This can be implemented in WinBUGS by using the fact that also the precision matrix in (19)has a simple form, see Reference [18, p. 193]. This model has two covariance parameters �2

and �.

3.6.3. Compound symmetry. A model where the covariance matrix has equal o�-diagonalelements (an equi-correlation model) is

hit ∼ MVN

0; �2

1 : : : �

.... . .

...

� · · · 1

t=1; : : : ; 10 (20)

This can be implemented in WinBUGS, imposing a positive correlation, by setting

hit = h∗i + h

∗it ; h∗

i ∼ N(0; �2hosp); h∗it ∼ N(0; �2hosp× time) (21)

and then �=�2hosp=(�2hosp+�

2hosp× time) and �

2 =�2hosp+�2hosp× time in (20). This uses the obvious

analogy with the intraclass correlation in for example one-way random e�ects ANOVA. Thismodel has two covariance parameters, �2hosp and �

2hosp× time.

3.6.4. No covariance model. This model is speci�ed such that each 10 periods have di�erenthospital random e�ects, and without covariance between these e�ects over time, that is themodel mentioned above, where the o�-diagonal entries in the covariance matrix in (18) is setto zero. This model has 10 variance components or covariance parameters.



3.6.5. Additional random e�ects. These four models also potentially include area as an un-structured random e�ect and area as a spatial random e�ect. Other models where the normaldistributions is exchanged with heavy-tailed T-distributions are straightforward in (19), (20)and (21).

3.7. Speci�cation of priors and hyperpriors

Hyperpriors on the inverse variance components were speci�ed to be vague using Gammapriors Ga(0.001,0.001) with a mean 0:001=0:001=1 and a variance 0:001=0:0012 =1000. For�xed e�ects, almost �at priors were speci�ed using the normal distribution with a largevariance e.g. 10,000. Hierarchical centring and mean centring of �xed e�ects for improvingconvergence was used [9]. For the inverse of the covariance matrices �−1

n in (4), (6), (11)and (18), we assumed a Wishart prior with n degrees of freedom, which represent a highprior uncertainty, and an inverse- scale matrix R, a n× n matrix with elements 0.01 on thediagonal and 0 on the o�-diagonal. This prior was also used when introducing the covarianceterm �s; u in (12). The correlation coe�cient � was given a positive uniform prior U (0; 1) in(19). We used a burn-in on 5000 iterations followed by 5000 iterations using two chains withdispersed starting values. Convergence of the chains was checked using the Gelman-Rubinconvergence statistic [19], implemented in the WinBUGS Software.Other alternatives for proper hyperpriors on the variance components are possible [19, 20],

where a natural simple candidate is a positive uniform prior U (0; k) with a suitable chosenk, which is furthermore easy to implement in WinBUGS. Instead we chose the ‘traditional’gamma and Wishart priors for the inverse variance components because they are conjugate inthe full conditionals for these parameters and, in addition, they give relatively high weight forthe variance components close to zero [20], assuming small entries in the inverse-scale matrixfor the Wishart distribution. This is reasonable for our kind of analysis where the outcomeis mortality and the parameters of interest represent di�erences between areas and hospitalsand because the measure is particular constructed to behave conservatively in the sense thatit potentially reduces these di�erences compared with a measure that includes the day of theevent.

4. RESULTS

The crude case-fatality rates on the population level have improved considerably, decreasingfrom 19 to 10 per cent in the period (Table I). The lower mortality can also be observed bylooking at the case-fatality rates within both areas and hospitals, where di�erent percentilesshow a downward trend. In addition, the crude number of cases re�ects the decreasing inci-dence of AMI.

4.1. First step of model selection

The �xed e�ects analysis in Table II shows that the trends in case fatality have been especiallybene�cial to men 40–59 years old and also that there has been an accelerating downwardtrend in case fatality for all the age groups. Figure 1 presents the �tted case-fatality ratescorresponding to the model in (1). This model has a DIC of 26 822 and 17 e�ective parameters(Table II), which corresponds to the true number of parameters in the model [13]. The DIC,


2610 S. RASMUSSEN

0%

5%

10%

15%

20%

25%

30%

35%

78/79 80/81 82/83 84/85 86/87 88/89 90/91 92/93 94/95 96/97

Year

Cas

e fa

talit

y ra

te

35-39

40-44

45-49

50-54

55-59

60-64

65-69

70-74

Age

Figure 1. Predicted case-fatality rates from the �xed e�ect model: mean calculated by 10 000 iterations.

considering our nearly negligible priors, approximately equals the Akaike information criterion(AIC) for this model, when using maximum likelihood estimation [13]. We use DIC de�nedas (16) to discriminate between di�erent models in step 2 and step 3 in the model selectionprocedure.

4.2. Second step of model selection

Initial analysis in which hospital variation varied in time generally had the best �t comparedwith models where hospital variation was believed to be constant as in (2). This led tomodels in which hospital variation varied over time. Initially, models where both spatialvariance and unstructured variance at the area level had random slopes with time were inferiorcompared with models where hospital variation has a random slope with time as in (5) or (7),and with either unstructured area variation or spatial area variation without random slopes.Table III presents the results for di�erent random slopes models for the hospital randome�ects called model 2A to model 2F. All models except model 2A has areas as randome�ects either unstructured or spatial (model 2D). The best-�tting model in terms of DIC isthe model (10), with an extra residual (model 2E) with a DIC of 26 549 and PD=196, andarea as unstructured random e�ects, such that the extra parameters induced by the macroresidual bene�t the model’s �t and still maintain the parsimony. We conclude from step 2that there exists correlation within hospitals perhaps a trend in hospital variation, and thereseem to be no trends for the area random e�ects. We compared the spatial distribution withand without the covariance parameter in (12) and no di�erence in terms of DIC were seen;the correlation was only slightly positive corrs; u=0:21 (95%CI − 0:43 − 0:75) such that we



TableIII.Standarddeviationoftherandom

e�ectsforrandom

slopemodels:meancalculatedby10000iterationswith95%CI.

Model

2A2B

2C2D

2E2F

Randompart

hosp+hosp

∗ time

unstr+hosp+

unstr+hosp+

spat+hosp+

unstr+hosp+hosp

∗ time+

unstr+hosp+

hosp

∗ time

hosp

∗ time+hosp

∗ time∗time

hosp

∗ time

hosp

†time

hosp

∗ time

Unstructured

� u=0:05

� u=0:05

� u=0:05

� u=0:05

� u=0:05

e�ects

(0.02–0.08)

(0.02–0.09)

(0.02–0.09)

(0.02–0.08)

(0.02–0.09)

Spatiale�ects

� s=0:06

(0.02–0.14)

Hospitale�ects

� 1h=0:14

� 1h=0:15

� 1h=0:14(0:11–0:18)

� 1h=0:14

� 1h=0:14

T� 1h=0:12

(0.12–0.17)

(0.12–0.18)

corr12h=

−0:24(−0:62–0:19)

(0.11–0.18)

(0.11–0.18)

(0.10–0.16)

corr12h=

−0:25

corr12h=

−0:26

� 2h=0:02(0:01–0:03)

corr12h=

−0:25

corr12h=

−0:23

Tcorr12h=

−0:20

(−0.60–0.15)

(−0.62–0.15)

corr13h=

−0:15(−0:49–0:24)

(−0.61–0.17)

(−0.62–0.21)

(−0.55–0.21)

� 2h=0:02

� 2h=0:02

� 3h=0:010(0:007–0:013)

� 2h=0:02

� 2h=0:02

T� 2h=0:017

(0.01–0.03)

(0.01–0.03)

corr23h=0:17(−0:29–0:55)

(0.01–0.03)

(0.01–0.03)

(0.012–0.024)

� ht=0:09

(0.05–0.12)

D(std.dev)

26468(17.0)

26444(23.1)

26409(25.4)

26440(22.8)

26353(41.7)

26441(23.1)

PD

101

123

151

128

196

126

DIC

26569

26567

26560

26568

26549

26567

Numberof

34

75

54

covariance

parameters(q)

BIC

26503

26491

26491

26499

26412

26488

∗ Indicatesrandom

slope.

† Indicatesrandom

interaction.

Pre�xTrefertoaparameterfrom

aT-distributionwithfourdegreesoffreedom.


2612 S. RASMUSSEN

choose to continue with the most parsimonious model where �s; u is set to zero. Unstructuredrandom e�ects slightly outperform the spatial distribution in terms of DIC. For convenience,we continue in the model selection with area distributed as unstructured random e�ects, whencomparing models in step 3 and return to the spatial distributed random e�ects, when thebest model is selected. The negative correlation in the random slopes models (Table III) e.g.corr12h= − 0:26 for model 2B means that there is a decrease in the hospital variation overtime using (8) almost until the end of the study period.

4.2.1. Random e�ects distributed as heavy-tailed T-distributions. We considered the modelwith random slopes with time for hospitals as multivariate T-distributed with four degrees offreedom (11) and normally distributed area random e�ects (model 2F). This did not improvethe �t, with a DIC of 26 567 with PD=126, but equals the DIC for model 2B (Table III).

4.3. Third step of model selection

We consider the models with four di�erent covariance patterns presented in Section 3.6 con-sidering the hospital variation together with area as an unstructured random e�ect. The re-sults from these models are presented in Table IV. The last model in Table IV is a modelwhere area also has compound covariance pattern (model 3G). The best �t is observed withthe compound symmetry model (model 3C), which has a DIC of 26 551 with PD=213. Thesame di�erence in DIC seen in step 2 for the random slope models is also seen when theunstructured area random e�ects are replaced with the spatial area random e�ects comparingDIC for model 3C and model 3F. Considering models from the normal distribution familywith regard to the random e�ects, we conclude that two candidates for the best �t are therandom slope models with the extra residual term (model 2E) from step 2 and among thecovariance pattern models, the model with the compound covariance pattern (model 3C) witha DIC of 26 549 and 26 551, respectively. The compound covariance model (model 3D) witha heavy-tailed T-distribution did not improve the �t, but it is also a candidate model for thebest �t together with model 3G.

4.4. Comparing models with BIC

Tables III and IV also present the results for BIC. As expected model 3A and model 3E withmore covariance parameters than the other models do not �t very well using BIC. The best�tting model using BIC is the compound symmetry model using the T-distribution (model3D) with a BIC of 26 330 + 3× log(125 598 − 17)=26 365 using (17) and the compoundsymmetry model for both area and hospital (model 3G) with a BIC of 26 364.

4.5. Interaction between area and hospital as random e�ects

Using the de�nition of catchment zones we analysed a model in which this term was includedcombined with the random e�ects in model 2B. But this model has a DIC of 26 594 withPD=157 (estimates not presented), a DIC larger than, for example, model 2B. We choosetherefore not to add more complexity to this model.


TableIV.Variancecomponentsforcovariancepatternmodels:meancalculatedby10,000iterationswith95%CI.

Model

3A3B

3C3D

3E3F

3G

CovarianceUnstructured

Auto-regression

Compound

Compound

Nocovariance

Compound

Compound

patternlevel

Hosp.

Hosp.

Hosp.

T-distributionHosp.

Hosp.

Hosp.

Hosp.andarea

Unstructured� u=0:05

� u=0:05

� u=0:05

� u=0:05

� u=0:07

� u=0:05

� u=0:06

�2 area=0:0022

orspatial

(0.02–0.08)

(0.02–0.08)

(0.02–0.08)

(0.02–0.08)

(0.04–0.11)

(0.02–0.08)

(0.03–0.10)

(0.0004–0.0060)

e�ects

� s=0:06

�2 area

×time=0:0021

(0.02–0.14)

(0.0004–0.0060)

� area=0:51(0.15–0.87)

Hospital

∗� h=0:17

�2 hosp=0:032� h=0:18

�2 hosp=0:020

T� h=0:14

T�2 hosp=0:012

†� h=0:17

�2 hosp=0:020

� h=0:18

�2 hosp=0:020

e�ects

(0.14–0.21)(0.021–0.043)(0.15–0.21)

(0.012–0.031)

(0.11–0.17)

(0.007–0.021)

(0.15–0.21)

(0.011–0.030)

(0.15–0.21)

(0.011–0.0031)

� hosp=0:84

�2 hosp

×time=0:012

T�2 hosp

×time=0:006

�2 hosp

×time=0:012

� hosp

×time2=0:011

(0.72–0.92)

(0.006–0.019)

(0.003–0.010)

(0.006–0.019)

(0.005–0.018)

� hosp=0:63

T� hosp=0:61

� hosp=0:62(0.43–0.78)

� hosp=0:66(0.46–0.83)

(0.46–0.80)

(0.41–0.79)

D(std.dev)26313(31.7)

26361(35.6)

26338(39.7)

26330(39.2)

26346(38.9)

26332(39.6)

26317(44.4)

P D254

193

213

221

273

220

234

DIC

26567

26554

26551

26551

26619

26552

26551

q‡56

33

311

34

BIC

26970

26396

26373

26365

26475

26367

26364

∗ RefertoTableV.

† RefertoFigure2.

‡ No.ofcovarianceparameters.

Pre�xTrefertoaparameterfrom

aT-distributionwith4degreesoffreedom.

2614 S. RASMUSSEN

4.6. Comparison of �xed e�ects

A small di�erence is expected when comparing the posterior means of the �xed e�ects inmodel (1) with the �xed e�ects in the random e�ect models. This di�erence partly dependentson the sizes of the variance components in a hierarchical logistic regression, because the �xede�ects in random e�ect models are interpreted as subject speci�c estimates [21]. For brevity,we choose not to present the results for the �xed e�ects for models presented in Tables IIIand IV.

4.7. Comparison of hospital and area variation

The case-fatality rates seem to vary at the hospital level. The standard deviation of the hospitalrandom e�ects is about 4-fold larger than the standard deviation of the area random e�ectsin Model 3C (Table III). The standard deviation of the unstructured random area e�ects isalmost similar for the di�erent covariance pattern models: �u ranges from 0.05 to 0.07 (TableIV). The covariance pattern models implying equal hospital variance over time are also quitealike with a standard deviation of 0.18 (95% CI 0.15–0.21) for the hospital random e�ectsin the compound symmetry and 0.17 (95% CI 0.14–0.21) in the �rst-order auto-regressionmodel. Using the formula exp(2× 1:96�h) this corresponds to a range of variation from 1.9-to 2.0-fold for all except the most extreme 5 per cent of the hospitals on the relative oddsscale. Figure 2 shows the variance heterogeneity for di�erent models at the hospital level.The e�ect of ignoring the correlation in successive measurements and thereby underestimatingthe variance is clearly seen in model 3E. This model exhibits a relative large di�erence inhospital variation in terms of standard deviation from 0.08 to 0.18. Figure 2 shows that theother models only induce slightly di�erences in hospital random e�ects variation.

4.8. Correlation within hospitals and municipalities

Table V presents the correlation matrices for the unstructured covariance pattern (model 3A),the auto-regression covariance pattern (model 3B) and the random slope model with themacro-residual (model 2E) together with the standard deviation of the hospital random e�ectsfor each year. The restrictions forced upon the elements in the auto-regression correlationmatrix give a quite high correlation within hospitals on consecutive measurements �1 = 0:84,but it is in line with the correlation matrix for model 2E using (9) and (10). The unstructuredcovariance matrix gives a vague idea of how the correlation pattern is distributed at the hospitallevel, but there is a trend towards smaller correlations at the end of the study period. Thecompound symmetry correlation matrix induces a constant correlation with �hosp = 0:63 using(21) (Table IV). Also the compound symmetry model using the heavy-tailed T-distributioninduces a correlation of T�hosp = 0:61 (Table IV). The spatial model for the area e�ects anda compound covariance pattern for hospitals induce a correlation of �hosp = 0:62 (model 3F)within hospitals. The correlation within hospitals was virtually unchanged in model 3G, with�hosp = 0:66 and �area = 0:51 (Table IV) for municipalities.

4.9. Check of convergence

In our analysis we saw negligible co-linearity (cross-chain correlation) between the area andhospital variance components. The cross-chain correlation between these components was typ-ical between −0:1 and 0.1 indicating that there appears to be enough information in data with



0

0.05

0.1

0.15

0.2

0.25

78/79 80/81 82/83 84/85 86/87 88/89 90/91 92/93 94/95 96/97

Year

Lo

g-o

dd

s

2C

3A

3C

3E

Model

Figure 2. Standard deviation of the random e�ects for selected models at the hospital level:mean calculated by 10 000 iterations.

regard to the number of areas nested into hospitals [6]. Assessing the Gelman-Rubin con-vergence statistic, it was a rule of thumb that convergence was achieved after approximately3000 iterations, such that summary statistics after 5000 iterations was a safe policy.

4.10. Random e�ects for hospitals

Figure 3 illustrates the random e�ects, for six chosen hospitals with di�erent numbers ofpatients admitted and trends, are illustrated for each period. We compare the random e�ectsfrom three models with the ratio between observed and expected deaths using the �xed e�ectmodel (1) with aggregation over hospitals and time. The high correlation induced by therandom slope and the covariance pattern models between consecutive measurements is seenin Figure 3 such that the random hospitals e�ects from these models are more alike anddependent of the preceding random e�ects compared to the more scattered predictions fromthe �xed e�ect model. Note that the hospital random e�ects are compared to the overallperformance with all hospitals, in the particular period, when adjusted for age, trend and arearandom e�ects. Thus hospital no. 1, for example, improved its performance throughout the


2616 S. RASMUSSEN

Table V. Correlation matrices and standard deviation for random e�ects at hospital level for model 2E,model 3A and model 3B: mean calculated by 10 000 iterations.

78=79 80=81 82=83 84=85 86=87 88=89 90=91 92=93 94=95 96=97 Std. dev (95%CI).

Random slope model with an extra residual (model 2E)1.00 �1 = 0:21(0.17–0.25)0.80 1.00 �2 = 0:20(0.16–0.24)0.78 0.78 1.00 �3 = 0:19(0.16–0.22)0.75 0.76 0.78 1.00 �4 = 0:18(0.15–0.21)0.71 0.73 0.76 0.75 1.00 �5 = 0:17(0.14–0.21)0.65 0.68 0.74 0.72 0.74 1.00 �6 = 0:17(0.14–0.21)0.59 0.63 0.71 0.69 0.72 0.73 1.00 �7 = 0:17(0.13–0.21)0.52 0.57 0.66 0.65 0.68 0.71 0.73 1.00 �8 = 0:17(0.13–0.21)0.45 0.50 0.61 0.60 0.65 0.69 0.72 0.74 1.00 �9 = 0:17(0.13–0.21)0.38 0.43 0.55 0.55 0.60 0.65 0.70 0.73 0.76 1.00 �10 = 0:18(0.14–0.23)

Unstructured covariance pattern (model 3A)1.00 �1 = 0:21(0.15–0.27)0.69 1.00 �2 = 0:23(0.17–0.29)0.56 0.66 1.00 �3 = 0:20(0.14–0.27)0.58 0.64 0.50 1.00 �4 = 0:18(0.13–0.24)0.37 0.45 0.45 0.54 1.00 �5 = 0:17(0.13–0.24)0.48 0.44 0.34 0.44 0.35 1.00 �6 = 0:16(0.11–0.22)0.52 0.39 0.33 0.29 0.13 0.39 1.00 �7 = 0:18(0.13–0.26)0.43 0.51 0.41 0.45 0.35 0.43 0.32 1.00 �8 = 0:17(0.11–0.24)0.42 0.45 0.41 0.39 0.25 0.35 0.30 0.45 1.00 �9 = 0:15(0.10–0.22)0.53 0.55 0.44 0.40 0.29 0.47 0.39 0.48 0.41 1.00 �10 = 0:20(0.13–0.30)

Autoregression covariance pattern (model 3B)1.00 �1 = 0:17(0.14–0.21)0.84 1.00 �2 = 0:17(0.14–0.21)0.71 0.84 1.00 �3 = 0:17(0.14–0.21)0.60 0.71 0.84 1.00 �4 = 0:17(0.14–0.21)0.51 0.60 0.71 0.84 1.00 �5 = 0:17(0.14–0.21)0.43 0.51 0.60 0.71 0.84 1.00 �6 = 0:17(0.14–0.21)0.37 0.43 0.51 0.60 0.71 0.84 1.00 �7 = 0:17(0.14–0.21)0.32 0.37 0.43 0.51 0.60 0.71 0.84 1.00 �8 = 0:17(0.14–0.21)0.27 0.32 0.37 0.43 0.51 0.60 0.71 0.84 1.00 �9 = 0:17(0.14–0.21)0.24 0.27 0.32 0.37 0.43 0.51 0.60 0.71 0.84 1.00 �10 = 0:17(0.14–0.21)

whole period, and close to a level approximately resembling the overall performance, wherelog odds equals zero (Figure 3).

4.11. Sensitivity to choice of hyperpriors

We checked the posterior means for Model 2E, the best �tting random slope model in termsof DIC, using a positive uniform prior U (0; 100) as hyperprior for both �1h; �2h and �ht at thehospital level and �u at the area level. The correlation corr12h between �1h and �2h was assumedto be uniformly distributed U (−1; 1). The posterior means and 95 per cent credible intervalswere as follows: �1h=0:14 (95% CI 0.10–0.17), �2h=0:02 (95% CI 0.01–0.03), �ht =0:09(95% CI 0.06–0.12), �u=0:05 (95% CI 0.02–0.08) such that changing the hyperpriors hadvery little impact with the results except for the correlation which was slightly decreased



Hospital No. 1

Year78/79 82/83 86/87 90/91 94/95

Hospital No. 49

Year

log

odds

78/79 82/83 86/87 90/91 94/95-0.8

0.0

Hospital No. 3

Year78/79 82/83 86/87 90/91 94/95

Hospital No. 76

Year

log

odds

78/79 82/83 86/87 90/91 94/95

-0.1

0.2

log

odds

-0.1

0.2

log

odds

-0.1

0.2

log

odds

-0.1

0.3

log

odds

-0.1

0.3

Hospital No. 18

Year

78/79 82/83 86/87 90/91 94/95

Hospital No. 6

Year78/79 82/83 86/87 90/91 94/95

Model:Black square: fixed-effect model Black triangle: model 2E White triangle: model 3A Star: model 3C

Figure 3. Comparisons of trends for random e�ects within six selected hospitals: meancalculated by 10 000 iterations.

corr12h= −0:32 (95% CI –0.86–0.26) compared to the results for Model 2E in Table III. Thecorresponding mean posterior deviance and e�ective number of parameters were 26 352 and198, respectively. Using positive uniform priors U (0; 100) for �2hosp; �

2hosp× time at the hospital

level and �u at the area level in Model 3C, which is the best �tting covariance model interms of DIC, gave similar results compared to using the vague conjugate hyperpriors (TableIV). The posterior means and 95 per cent credible intervals for the correlation and standarddeviation for the random e�ects at the hospital and area level were: �hosp = 0:63 (95% CI 0.45–0.80), �h=0:18 (95% CI 0.15–0.21) and �u=0:05 (95% CI 0.02–0.09). The correspondingmean posterior deviance and the e�ective number of parameters were slightly di�erent, 26 332and 219, compared to the results for Model 3C, although DIC equals 26 551, which is thesame as for the vague conjugate priors (Table IV).

5. DISCUSSION

This study considered the use of Bayesian models in complex routinely collected sparse datafrom administrative registries in analysing case-fatality rates following AMI with the purpose


2618 S. RASMUSSEN

of evaluating whether unexplained variance is located at the hospital level and area level andwhether this changes over time. The models imply that the hospital variation is considerablelarger than the area variation. The best model in terms of DIC suggests a pattern in hospitalvariation with a downward trend at the beginning of the study period and an indication of anupward trend in the end. This model also indicates a considerable positive correlation withinhospitals over time, such that if a hospital ‘performs’ better than the average for one period,we would also expect it to ‘perform’ better than the average the next period and vice versa.The best model also suggests that these correlations decrease over time. Overall there is notrend in area variation and no convincing support for a spatial correlation between areas. Noneof the model selection criteria, DIC or BIC, can persuasively select between the competingmodels. They can exclude some models, and the choice of the ‘best’ model depends on thecriteria chosen. This is perhaps not surprising given the nature of the two criteria, in whichBIC tends to penalize models with a relative large number of parameters.We chose to use Bayesian modelling because the MCMC methods split the complexity

induced by the crossed and spatial random e�ects into the full conditionals, which makesthe calculations feasible considering the relative large number of observations. The MCMCmethods also perform better than the quasi-likelihood methods in the binomial distribution byentirely using the distribution assumptions for the random e�ects and by taking into accountthe uncertainty of the estimation of the variance components when assessing the precisionof the �xed and the random e�ects. The opportunity for using heavy-tailed distributions forthe random e�ects in a straightforward manner combined with the information criterion forselecting models is also a bene�t. Using vague priors restricts the prior information in�uencein the MCMC methods. Sensitivity analyses using uniform priors on the parameters of interestshowed little e�ects on the results compared to results using conjugate priors as the gammaand Wishart distribution.Bayesian modelling using MCMC methods is a powerful tool for building very com-

plex models, and the danger of over-parameterization and data dredging is therefore alwaysimminent. We therefore think the build-up from simple models to more complex modelsis important such that more advanced models always can be compared to simpler modelsto determine whether the results are consistent with previous �ndings. We also emphasisethat the presented models in terms of complexity match the goal they were intended toaccomplish.We interpret heterogeneity in variation over time for hospitals to be a result of the intro-

duction of thrombolysis and aspirin treatment for AMI. This treatment was introduced as astandard in late 1988, and with a swift propagation into Denmark’s hospitals so that it wasfully implemented within 1–2 years. Compared with the novel applied invasive procedures,such as PCI, this medical treatment is rather low-tech and thereby applicable on all hospitals.It was also very e�cacious in reducing the mortality in selected patients by about 40 percent in clinical trials [22]. It thereby supports the hypothesis that an e�cacious treatmentdispersed swiftly to all hospitals will increase the homogeneity in survival rates betweenhospitals. This could also reduce the correlation within hospitals, because a hospital, in thephase of implementing the treatment, is more sensitive to experiencing a change in its rankorder such that the ‘performance’ in consecutive years would be more scattered compared tothe average. The �nding that the smallest hospital variance heterogeneity co-indices wherethe thrombolysis and aspirin treatment is fully implemented in 1988–1991 further implies thehypothesis mentioned. Primary PCI was gradually introduced from the mid-1990s, however



mainly in �ve hospitals with catheterization facilities and a few close hospitals. Further PCIin unstable patients initially treated with thrombolysis was increasingly used during the 1990sand may also account for decreasing case fatality after AMI. This could account for someheterogeneity between hospitals during this period of time as a result of potential unequalgeographical access to these invasive procedures for AMI patients.This study indicates that most of the variation in the case-fatality rates is located at the

hospital level rather than at the area level. Some of this variation could be due to codingpractice, in that hospitals with higher case-fatality than average may classify di�erently thanhospitals below the average. Especially unstable angina pectoris was and is a competing di-agnosis to acute myocardial infarction with noticeably lower death rates. These classi�cationdi�erences might well have decreased over time, such that in the beginning of the studyperiod there were several competing diagnoses to AMI, which again might increase the het-erogeneity between hospitals. Bearing this in mind, the variation between hospitals could, toa certain extent, result from coding practice and not a systematically di�erence in the careand treatment they provide, although the hospital variation displays a heterogeneity associatedwith a potential treatment e�ect. Furthermore our control for case-mix (age and secular trend)is rather crude, and some of the di�erences between hospitals are therefore a consequence ofthe health of their respective patient population before they reach hospital and the severity ofthe AMI. However, robust estimation with a heavy-tailed T-distribution was not superior formost models, indicating that the measure used for case fatality showed no departure from thenormal assumption for the random e�ects and a diminishing proportion of signi�cant outliersthereby exist. This indicates, among other things, that the preliminary work with data includ-ing patients through the disease classi�cation system and excluding patients who die on the�rst day of the event and using a �xed follow-up on 28 days has been bene�cial in removingthe in�uence of potential outliers, especially among hospitals. Nevertheless, looking at thecompound symmetry model, we see that the heavy tailed T-distribution model competes withits normal distributed equivalent at the hospital level, favoured by BIC, but also a competitivemodel using DIC. We can therefore not quite abandon this distribution as an alternative to thevariation in hospital random e�ects. Finally, case-fatality rates did not vary within a hospitalfor patients residing in di�erent areas such that the lack of interaction between hospital andarea subsides.The relatively high correlation between consecutive measurements at the hospital level is

rewarding, because it also indicates the validity of the measure and de�nitions used. It couldbe a potential benchmark in analysing trends in such measures as performance, when usingadministrative registries in Denmark, such that a �nding of zero or negative correlation, withinhospitals or other units, could cause questions about the validity of the data and the de�nedmeasures of interest, if there is no obvious explanation for these �ndings [23].Some studies have demonstrated an association between socioeconomic attributes and case

fatality on the individual level [24] and the socioeconomic characteristics and case-fatality atthe area level [25, 26]. Assuming that these socioeconomic features exhibit spatial variationat municipality level, we would also expect a more pronounced level of spatial variation inour analysis. This is not the case although the model with spatial variation participates withthe model with unstructured variation at the area level.This reduced presence of spatial variation for small areas warranted further investigation.

Removing of the hospital random e�ects in (3) and ignoring time trends at the area levelshowed that the municipalities were noticeably spatially correlated using (12). The standard


2620 S. RASMUSSEN

deviations for the spatially distributed random e�ects were �s=0:14 (95% CI 0.08–0.19) and�u=0:07 (95% CI 0.04–0.09) with a correlation of corrs; u=0:60 (95% CI 0.08–0.87), suchthat the spatial and unstructured components were equally in�uential using (13). The numberof e�ective parameters for this model was 82 and DIC was 26 736. This was compared to theunstructured model where the standard deviation for the random e�ects was �u=0:10 (95%CI 0.08–0.13), the number of e�ective parameters was 97 with a DIC on 26 751. An apparentexplanation for this spatial clustering could be that the hospital catchment areas, which wouldalmost be a grouping of the municipalities depending on the hospital’s geographical location,were partially responsible for this spatial patterning, and including hospital random e�ectswould therefore weaken the signi�cance of this pattern considerably.

6. CONCLUSION

Using a Bayesian approach, this case study demonstrated the use of hierarchical modelling inthe study of trends in hospital area and small area variation in mortality after AMI consideringdi�erent priors and hyperpriors. Although models are favoured in which the hospital variationis not constant in time, this study also shows that the odds of dying is almost twice as highin the hospitals with the worst ‘performance’ as in those with best ‘performance’ during a20-year period. This hospital variation is considerable larger than the variation at the arealevel, and unstructured random e�ects slightly outperform spatially correlated random e�ectsat the area level.

ACKNOWLEDGEMENTS

I would like to thank Associate Professor Svend Kreiner, Deputy Director Mette Madsen and Se-nior Consultant Jan Kyst Madsen for valuable comments through the writing process. Also thanks toProfessor Robin Henderson for suggestions on the almost �nal manuscript and the three referees forhelpful comments. This study was supported by a grant from the Danish Medical Research Council(22-00-0548).

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