triple-goal estimates for disease mapping

STATISTICS IN MEDICINEStatist. Med. 2000; 19:2295–2308

Triple-goal estimates for disease mapping

Wei Shen1;∗;† and Thomas A. Louis2

1Eli Lilly and Company; Lilly Corporate Center; Indianapolis; Indiana 46285; U.S.A.2Division of Biostatistics; School of Public Health; University of Minnesota; 420 Delaware Street; Box 303;

Minneapolis; Minnesota 55455; U.S.A.

SUMMARY

Maps of regional morbidity and mortality rates play an important role in assessing environmental equity.They provide e�ective tools for identifying areas with potentially elevated risk, determining spatial trend,and formulating and validating aetiological hypotheses about disease. Bayes and empirical Bayes methodsproduce stable small-area rate estimates that retain geographic and demographic resolution. The beauty of theBayesian approach lies in its ability to structure complicated models, inferential goals and analyses. Threeinferential goals are relevant to disease mapping and risk assessment: (i) computing accurate estimates ofdisease rates in small geographic areas; (ii) estimating the distribution of disease rates over the region; (iii)ranking the disease rates so that environmental investigation can be prioritized. No single set of estimates cansimultaneously optimize these three goals, and Shen and Louis propose a set of estimates that perform well onall three goals. These are optimal for estimating the distribution of rates and for ranking, and maintain a highaccuracy in estimating area-speci�c rates. However, the Shen=Louis method is sensitive to choice of priors.To address this issue we introduce a robusti�ed version of the method based on a smoothed non-parametricestimate of the prior. We evaluate the performance of this method through a simulation study, and illustrateit using a data set of county-speci�c lung cancer rates in Ohio. Copyright ? 2000 John Wiley & Sons, Ltd.

1. INTRODUCTION

Detection and assessment of the health e�ects of environmental exposures is of increasing in-terest among statisticians, epidemiologists and public health professionals. Environmental equity,including assessment of di�erential exposure and health e�ects of environmental agents withinvarious demographic subgroups, is an issue of increasing importance. Maps of regional morbidityand mortality rates play an important role in assessing environmental equity. They provide centraltools for identifying areas with potentially elevated risk, determining spatial trend, and formulatingand validating aetiological hypotheses about disease. Bayes and empirical Bayes methods producestable small-area rate estimates that retain geographic and demographic resolution, providing anattractive approach [1–7].

∗ Correspondence to: Wei Shen, Eli Lilly and Company, Lilly Corporate Center, Indianapolis, Indiana 46285, U.S.A.† E-mail: [email protected]

Copyright ? 2000 John Wiley & Sons, Ltd.

2296 W. SHEN AND T. A. LOUIS

The beauty of the Bayesian approach is its ability to structure complicated models, inferentialgoals and analyses. The prior and likelihood produce the full joint posterior distribution which gen-erates all inferences. Applications have burgeoned now that Markov chain Monte Carlo (MCMC)methods enable analysis of relevant models [8]. To take full advantage of the structure, methodsshould be linked to an inferential goal via a loss function.Three inferential goals are relevant to disease mapping and risk assessment: (i) computing

accurate estimates of disease rates in small geographic areas; (ii) estimating the distribution ofdisease rates over the region; (iii) ranking the disease rates so that environmental investigationcan be prioritized. No single set of estimates can simultaneously optimize these three goals, soShen and Louis [9] propose a set of ‘triple-goal’ estimates that perform well on all three goals. Theestimates are optimal for estimating the distribution of rates and for ranking. They maintain a highaccuracy in estimating area-speci�c rates. However, the method is sensitive to the prior. To addressthis issue, we introduce and evaluate a robusti�ed version based on a smoothed non-parametricestimate of the prior.The remainder of the paper is organized as follows. Section 2 introduces the model, inferential

goals, and structures the triple-goal estimates using the Bayesian approach. Section 3 discusses theissue of robustness with respect to choice of priors, and presents ways to improve the robustness.Section 4 presents a simulation study to evaluate the e�ciency and robustness of the triple-goalestimates. Section 5 provides an example using lung cancer mortality data from Ohio in 1968.Section 6 summarizes the results, discusses extension of the methods, and presents opportunitiesand challenges for future research.

2. INFERENTIAL GOALS AND METHODS

Consider the basic two-stage hierarchical model for disease mapping:

�kiid∼ G

Yk | �k indep∼ Poisson(nk�k); k =1; 2; : : : ; K(1)

where Yk is the observed number of events, nk is the population at risk, and �k is the true rate inarea k. Our inferential goals are:

Goal 1: Produce an e�ective estimate of disease rate for each geographic area. This ‘estimateof centre’ is the usual goal for most statistical analyses, Bayesian or otherwise. Traditionalestimates let each area stand on its own and use the area-speci�c maximum likelihood estimate(MLE): �k =Yk=nk . Many researchers have shown that the MLE can be improved upon bytaking advantage of the two-stage structure (borrowing information) [4; 6]. The Bayes advantageholds even when parameters of the prior must be estimated (for K¿4), producing an empiricalBayes (EB) procedure, or the prior parameters are themselves given hyperpriors. With ak theestimates of the �k , under squared error loss (SEL=K−1∑ [ak − �k ]2), the posterior means(PM) �pmk = �k =E(�k |Yk) are optimal. This posterior mean shrinks the traditional MLE towardsthe overall mean by an amount that depends on the relation between the prior variation andsampling variation. For example, if G is gamma(�; �), E(�)= ��= �; V (�)= �2=�, then the

Copyright ? 2000 John Wiley & Sons, Ltd. Statist. Med. 2000; 19:2295–2308

TRIPLE-GOAL ESTIMATES FOR DISEASE MAPPING 2297

posterior distribution of �k is gamma(Yk+�; (nk+�=�)−1), and the posterior mean and varianceof �k are

�pmk =Yk + �nk + �=�

=(1− sk)(Yknk

)+ sk�

V (�k |Yk) = Yk + �(nk + �=�)2

where sk = �=(�nk + �) represents the amount of shrinkage from the raw rate (MLE= Yk=nk)to the state-wide rate (prior mean = �). For counties with a smaller population, the shrinkagee�ect and variance reduction are greater.Goal 2: Produce an e�ective estimate of the histogram of the true underlying rates. For thisgoal, there is no explicit attention to producing good estimates for individual geographic areas.Rather, the goal is, for example, to estimate the number of areas with true rates in excess ofa threshold or to estimate and compare the distribution of disease rates in two states or othercollection of areas. To structure this goal, let the empirical distribution function (EDF) of the�s be

GK (t)=1K

K∑k=1I{�k6t}

where I(:) is the indicator function. With A(t) the estimated EDF, for integrated squared errorloss (ISEL(A;GK)=

∫[A(t)−GK (t)]2 dt), Shen and Louis [9] show that the posterior expected

EDF �GK (t)=E[GK (t) |Y] is optimal. Furthermore, with the added constraint that A(:) is adiscrete distribution with at most K mass points, the ISEL-minimizer (GK) has mass 1

K at

Uj = �GK−1(2j − 12K

); j=1; : : : ; K

Goal 3: Rank the rates, or estimate the extremes. Ranking disease rates is important in identi-fying areas with unusually high rates, and in prioritizing public health investigations orinterventions. Let Rk be the true rank in area k, then Rk =

∑Kj=1I{�k¿�j}: If ties do not occur, the

smallest � has rank 1 and so on. With Tk the estimated rank of �k , for squared error loss on theranks (SELR=K−1∑ [Tk − Rk ]2), the posterior expected ranks �Rk =E[Rk |Y ] =

∑Kj=1 Pr[�k¿

�j |Y] are optimal. These �Rk are shrunken towards the mid-rank (K + 1)=2, thus compressing‘percentiles’ towards the 50th. The �Rk are usually not integers which can be attractive, sinceinteger ranks can over-represent distance and under-represent uncertainty. For example, ranks( �Rk) of 1, 1.1, 3.9 indicate similarity between the �rst two areas, and considerable di�erencebetween the third and the �rst two. Ranking the �Rk produces integer ranks Rk = rank( �Rk).Goal 4: Triple-goal estimates. In many policy settings, communication and credibility will beenhanced by reporting a single set of estimates with good performance for all three inferentialgoals. However, no single set of estimates can simultaneously optimize all three goals. Theposterior means (�k) are the optimal estimates for area-speci�c rates. However, the EDF of theposterior means is always under-dispersed [10; 11]. In addition, ranking the posterior means toobtain the estimated ranks is inappropriate [9; 12]. This leads to the search for estimates thatcan compromise among goals 1, 2 and 3. The goal here is three-fold: to have a set of estimatesthat produce (i) good area-speci�c rates; (ii) a good EDF of the rates; and (iii) good ranks.



Louis [11] and Ghosh [10] present a constrained Bayes (CB) estimator that produces area-speci�c estimates with a histogram that has the appropriate centre and spread. Devine et al.[11] investigate the use of these compromise rules on spatially aligned data. The CB estimates(a1; : : : ; ak) minimize posterior expected SEL subject to the constraints:

1K∑ak =

1K∑�k

1K − 1

∑(ak − �a)2 =

1K∑�k +

1K − 1

∑(�k − ��)2

where �k =var(�k |Yk), �a=K−1∑ak and ��=K−1∑�k . The constraints require that the �rst twomoments of the estimates match those of the true parameters (under posterior expectation). TheCB estimates have a closed-form representation:

�cbk = �: +

[1 +

1K

∑�k

1K−1

∑(�k − �:)2

]12

(�k − �:)

Since the term in square brackets is greater than 1, the CB estimates are more spread out thanare the posterior means. The ranks produced by the CB estimates are always the same as thoseproduced by the posterior means. The CB approach works well for exchangeable Gaussian samplingmodels, less well for other models.Shen and Louis [9] propose a more general approach (called GR). The GR approach proceeds

as follows:

1. Minimize the ISEL for estimating GK using GK (goal 2 – EDF).2. Minimize the SEL for estimating the ranks using Rk (goal 3 – ranks).3. Estimate �k by letting �

grk = URk ; k =1; : : : ; K (goal 1 – area-speci�c estimates).

Note that �gr’s are optimal for estimating GK and the ranks. Shen and Louis [9] evaluate the GRapproach by mathematical analyses and simulations, and �nd it better than CB and PM in a varietyof models. GR always produces an accurate EDF, equivalent or better ranks than either PM or CB,and has very little loss in area-speci�c estimation accuracy compared to CB and PM. Conlon andLouis [14] apply this approach in a case study of lip cancer incidence in Scotland using spatialcorrelation models for estimating area-speci�c rates, the distribution of rates, and ranks.

3. ROBUSTNESS

Bayesian procedures can be very non-robust to prior misspeci�cation. In the standard asymptoticanalyses, the precision of each area estimate increases (possibly by integration over time) andthe in uence of the prior decreases. More relevant to disease mapping are asymptotics basedon increasing the number of areas either by broadening coverage or re�ning partitions. In spatialstatistics, these asymptotics are referred to as increasing domain and in�ll asymptotics, respectively[15]. For either of these asymptotics, if the assumed prior is incorrect (either with wrong parametersor in the wrong parametric family), the EDF estimate ( �GK) will not converge to the true priordistribution.



Empirical Bayes (or the hierarchical Bayes approach) is more robust to prior misspeci�cationthan is selecting a single prior, but as Morris [16] shows, parametric EB su�ers from lack ofrobustness when the true prior is not from the assumed parametric family. For estimation ofarea-speci�c rates, when the assumed two-stage hierarchical model is valid, EB produces e�cientestimates for area-speci�c rates and performs better than the MLEs. However, if the model doesnot hold (for example, the prior is incorrect) performance is good ‘on average’ over all areas(the Stein e�ect), but can have poor performance for an outlying rate.Broadening the parametric family of possible priors improves robustness, leading to consideration

of a fully non-parametric approach. To this end, Laird [17; 18] derives a non-parametric maximumlikelihood (NPML) estimate of the prior and uses it in EB analysis. For estimation under the SEL,posterior means produced by the NPML prior are competitive with those from parametric priors,even when the assumed parametric model is correct [19]. However, for more general inferencesthe NPML estimate is somewhat unsatisfactory due to its being discrete, having too narrow asupport, and frequently being under-dispersed. These features make it especially unattractive forestimating posterior tail areas and other features not captured by the posterior mean and variance(for example, GK).Smoothing the NPML estimate has the potential of improving performance, especially for small

sample sizes, but computational di�culties have impeded implementation. Proposed remedies tothese problems include using a non-parametric hyperprior (for example, Dirichlet process) [20],using a mixture of Gaussians [21], directly smoothing the estimated prior [22; 23], and adding‘pseudo-data’ to the data set and then computing the NPML estimate.Laird and Louis [24] propose an alternative, called smoothing by roughening (SBR), based on

the EM algorithm for �nding the NPML estimate. The SBR process starts with a smooth estimateof the prior G(0) (for example, one may start with a parametric estimate of the prior), and employsthe recursion:

G(�+1)(t;Y) = EG(�) [GK (t) |Y]

=1K

K∑k=1PG(�) (�k6t |Y)

The recursion is an EM algorithm, and at each step produces a posterior expected GK . If iteratedto convergence, it will produce the NPML estimate so long as the support of G(0) includes thesupport of the NPML. However, if G(0) is continuous, convergence is very, very slow. Terminatingthe EM after a relatively small number of iterations produces a smoothed non-parametric estimate(indeed, by ‘roughening’ G(0)). Increasing the number of iterations as sample size increases allowsthe SBR estimate to adapt to the data and move toward the NPML estimate.Shen and Louis [19] propose an accurate and e�cient computing algorithm to implement the

SBR method. They show that the SBR estimate of the prior is robust to prior misspeci�cation, andis exible to choices of initial estimate and stopping iteration. Empirical Bayes inferences based onthe SBR estimate of the prior are robust and e�cient with respect to a variety of inferential goals.

4. SIMULATIONS

Performance of the PM, CB and GR approaches with respect to all three inferential goals un-der a known prior have been evaluated by Shen and Louis [9] and Devine et al. [13]. Here



Table I. Performance of the parametric and SBR empirical Bayes approaches in triple-goalsestimation: simulation results.

Prior Goal Measure MLE Parametric SBRPM CB GR PM CB GR

Mixture of gammas �’s SEL 134 91 95 107 80 82 86GK ISEL 289 406 283 353 307 266 216Rs SELR 57 54 54 54 54 54 53

Gamma �’s SEL 121 45 50 51 45 54 53GK ISEL 461 474 266 216 396 294 233Rs SELR 151 141 141 141 141 141 141

SEL: squared error loss, reported as 10000 times the actual value.ISEL: integrated squared error loss, reported as 100000 times the actual value.SELR: squared error loss for the ranks.

we investigate EB performance (e�ciency and robustness) of these approaches by a simulationstudy.Data are generated from the two-stage model (1), where K =50, nk =5+2(k−1); k =1; : : : ; K

and G=0:5×gamma(�1; �1) + 0:5×gamma(�2; �2). We evaluate methods under two prior cases:1. Gamma: (�1; �1)= (�2; �2)= (20; 0:02).2. Mixture of gammas: (�1; �1)= (20; 0:01), and (�2; �2)= (30; 0:02).

For each case, 100 samples are generated.Four approaches are compared: MLE; PM; CB, and GR. Other than the MLE, all other ap-

proaches require estimation of the prior. Two prior estimates are considered: parametric and SBR.The parametric approach assumes a conjugate prior gamma(�; �). The prior parameters are esti-mated by the method of moments [25]. An alternative approach is the maximum likelihood methodbased on the marginal distribution, and is computationally expensive. Since K =50 is large, thedi�erence between these two parametric approaches is small. For SBR, we start with a gammadistribution with larger mean and variance than that of the true prior and stop at the 20th iteration.The discrete computing algorithm [19] is used to calculate G(�), where the continuous G(0) isapproximated by a discrete distribution with 200 equally spaced grid points on a wide enoughinterval. The NPML approach is excluded from this analysis, since it is ine�ective for estima-tion of GK and related goals. Comparison of the approaches uses the loss functions presentedin Section 2.Table I presents results from the simulation study. For the Poisson=gamma model, SBR produces

results comparable to that of the parametric approach. For the Poisson=mixture of gammas model,SBR improves signi�cantly over the parametric approach. Of the three approaches, GR seemsto bene�t the most. For example, its SEL and ISEL reductions from the parametric prior to theSBR prior are the largest in the Poisson=mixture of gammas model. Of the three inferential goals,estimation of GK is in uenced the most by choice of the prior. On the other hand, ranks are nearlyunchanged by choice of prior estimate.Although GR pays no particular attention to reducing �-speci�c SEL, its SEL is only slightly

larger than that of PM and CB [9]. However, GR has a big advantage in estimating the GK ifthe estimated prior is valid. Figure 1 shows scaled histogram estimates under the parametric andSBR priors for the Poisson=mixture of gammas model. The scaled histograms are compared tothe true prior density. PM, CB and GR based on the parametric prior all have the wrong shape.



Figure 1. Histogram estimates under the parametric and SBR priors: simulation results from thePoisson=mixture of gammas model. The histograms are compared to the true prior density. The MLE isover-dispersed. The parametric approach is inferior to the SBR. However, only the GR based on the SBR

has the right spread and shape.

PM=SBR is under-dispersed. CB=SBR has the right spread, but a sub-optimal shape. Only GR=SBRhas the right spread and shape. In this set of simulations, even though the underlying model isnon-exchangeable, PM, CB and GR all produce similar ranks.MLE is a robust approach with respect to all goals. Its high variability causes it to be inferior

to all EB estimates in estimating the �s and the ranks, regardless of the choice of prior. TheGR=SBR approach dominates the MLE in all three inferential goals in this set of simulations.

5. MAPPING OHIO LUNG CANCER RATES

The data we analyse, originally studied by Devine, include lung cancer death counts Yk andcounty population counts nk (in 1000s), as observed in each of the K = 88 counties in Ohio from1968 to 1988 [26]. These data were originally taken from a public use data tape [27] containingage-speci�c death counts by underlying cause and population estimates for every county in theUnited States. Several researchers [7; 13; 28–30] have studied the spatial and=or temporal patterns ofthe Ohio lung cancer death rates using Bayesian hierarchical models. For illustration, we will use thewhite male data observed in 1968. Our goals are two-fold: (i) to investigate the in uence of theprior estimates on EB estimation; (ii) to compare the MLE, the PM, CB and GR estimates withrespect to various estimation goals.We again use model (1) in Section 2. The Poisson sampling distribution is a reasonable choice

to model death counts. The independence assumption is partially supported by the analysis of



Figure 2. Parametric and non-parametric estimates of the prior. The SBR estimate of the prior is smooth, andprovides a good compromise between the NPML and the parametric estimates. Ohio lung cancer mortality

data, white males, 1968.

Waller et al. [7]. They found very little spatial pattern of the underlying county rates in 1968.However, a temporal trend (from 1968 to 1988) of increasing spatial correlation was observed.

5.1. In uence of the prior

We start with estimation of the prior. Figure 2 displays the parametric estimate of the priorbased on the Poisson=gamma model, the NPML estimate, and the SBR estimate by starting at theparametric prior estimate and terminating after 20 iterations. From Figure 2, the NPML estimateof the prior is discrete with only four mass points, even though there are K =88 counties. Ofcourse, a posterior distribution based on the NPML prior is also discrete, with exactly the samemass points as the NPML prior, making them ine�ective for posterior analyses involving the wholeposterior distributions (for example, tails). The parametric and SBR priors are roughly the samefor the middle part of the distribution, however they di�er substantially in the tails.Figure 3 compares the rate estimates based on the parametric and SBR priors for three dif-

ferent approaches: PM; CB, and GR. The parametric and SBR estimates are similar for ratesclose to the prior means, and di�er for both high and low rates. The di�erences between theparametric and SBR estimates are the largest for GR, supporting the �nding that the GR esti-mates are more sensitive to the choice of prior than are PM and CB. Although the area-speci�cestimates are somewhat di�erent (in particular in the lower and upper extremes), the ranks turnout to be similar between the parametric and SBR approaches (di�erences in rank are no greaterthan 4).



Figure 3. Comparison of the parametric and SBR estimates of the county-speci�c rates. Compared to PMand CB, the GR estimates are in uenced the most by the prior estimates, particularly in the tails. Ohio lung

cancer mortality data, white males, 1968.

Table II. Comparison of the parametric and SBR estimates of lung cancer death rates for selected counties.

County Y N MLE SE Parametric EB SBR EB RankPM CB GR PM CB GR GR=SBR

Cuyahoga 435 677 0.642 0.031 0.631 0.734 0.686 0.633 0.731 0.651 87De�ance 3 18 0.166 0.096 0.373 0.323 0.318 0.358 0.302 0.275 5Geauga 4 30 0.132 0.066 0.324 0.245 0.266 0.296 0.205 0.232 2Greene 17 58 0.294 0.071 0.365 0.309 0.282 0.359 0.303 0.248 3Hamilton 248 375 0.662 0.042 0.641 0.749 0.734 0.634 0.733 0.662 88Lucas 128 205 0.623 0.055 0.594 0.675 0.663 0.610 0.695 0.646 86Mercer 3 17 0.172 0.099 0.377 0.329 0.330 0.363 0.310 0.308 8Portage 11 59 0.186 0.056 0.302 0.209 0.238 0.274 0.172 0.209 1Washington 6 27 0.222 0.091 0.368 0.315 0.294 0.355 0.297 0.262 4

Y , number of deaths; N , population at risk (in thousands); SE, standard error; Rates, number of deaths per thousand; Rank,the lowest rate receives the smallest rank (1), the highest rate receives the largest rank (88).

Table II presents rate estimates for nine counties; most are in uenced by choice of the prior.Hamilton county (which includes Cincinnati), ranked the highest in Ohio by all methods exceptthe MLE, has the largest di�erence between its GR estimates. This di�erence is directly relatedto the di�erence in the right tail of the two priors. The parametric prior has a longer right tail,thus forcing the largest GR estimate to further right than the SBR prior. Cuyahoga county (whichincludes Cleveland) provides a similar example. On the contrary, the CB estimates for Hamiltonand Cuyahoga are less a�ected because both priors have similar mean and variance. Given that theobserved rates are the most stable in Hamilton and Cuyahoga due to the large population sizes,the GR=SBR estimates appears to be a better choice. For counties with lower rates, the e�ect ofthe SBR prior is to bring the PM, CB and GR rates closer to the MLEs. This may bene�t for



Figure 4. Maps of Ohio lung cancer mortality in white males in 1968. The map based on the MLEs is highlyvariable, while the map based on the PM rates shrinks too much to the state-wide mean. The map based on

the GR rates provides a good compromise.

counties like Portage (south east of Cleveland), whose PM=parametric estimate of rate (0.302)maybe too far (¿2 standard errors) away from the MLE (0.186).

5.2. Comparison among the MLE, PM, CB and GR

For PM, CB and GR, the results will be described for the SBR prior. The relative performanceof PM, CB and GR is similar for the parametric prior.Figure 4 compares the raw and smoothed lung cancer maps in Ohio. The raw disease map

appears highly variable, as the rates in neighbouring counties range from the lowest to the highest.In the raw map the observed rates are below 0.25 (per 1000) in 13 counties, and above 0.65in 14 counties. A further look of the 13 counties with observed rates below 0.25 reveals that11 of them have fewer than 5 deaths. Similarly, of the 14 counties with observed rates above0.65, 10 of them have populations size below 20 000. The small values of the death count andpopulation size can cause the raw rates to be highly unstable and inaccurate, such that additionor deletion of one or two deaths can have a dramatic e�ect on the observed rates. Using the PMrates, raw rates are smoothed toward the state-wide mean, producing a more stable disease map.In the smoothed PM map, all rates are in the interval (0.25, 0.65). The smoothed map based onthe GR estimates (similar for CB) provides a good compromise between the maps using the rawrates and the posterior means. In all three disease maps, no obvious spatial patterns of lung cancermortality are apparent.Although the PM estimates successfully stabilize the raw rates, for estimating the histogram they

overshrink the raw rates toward the state-wide mean. Figure 5 compares EDFs of various ratesagainst the SBR prior estimate. As expected, the EDF of the MLEs is substantially over-dispersed,and the EDF of the PM estimates is signi�cantly under-dispersed. The EDF of the CB estimates



Figure 5. Comparison of the distribution of the Ohio lung cancer mortality in white males in 1968. The MLEis over-dispersed, PM is over-shrunk. CB has the right spread, though inaccurate in the tail areas. GR has

the right spread and shape.

roughly matches the estimated prior, however, it does not work well for either tail. By design, theEDF of the GR estimates (which is essentially one more iteration of the EM algorithm) matches theSBR prior very closely. The GR approach is appropriate for identifying counties with excessivelyhigh rates. For example, the GR approach identi�es two counties (Hamilton and Cuyahoga) withtrue rates above 0.65, as compared to none by the PM approach, four by the CB approach, and14 by the MLE approach.Figure 6 compares the ranks produced by the MLE, PM and GR approaches. The county with

the lowest rate receives rank 1, and the county with the highest rate receives rank 88. Note thatPM and CB always produce identical ranks. The ranks produced by the PM approach are similarto the ranks produced by the GR approach. The largest di�erences between the GR and PM ranksis 5. The Spearman correlation between the PM and GR ranks is 0.999. However, there could belarge di�erences in ranks between the MLE and GR approaches, mainly due to the instability ofraw rates. For example, Morgan county was ranked the 6th lowest by the MLE approach, and the30th lowest by GR. There was only one lung cancer death in Morgan county, with a populationsize of 5300. Similarly, Vinton county was ranked the third highest by the MLE approach, andonly the 23rd highest by GR. There were only four lung cancer deaths in Vinton county, with apopulation size of 4800.



Figure 6. Comparison of county ranks of the Ohio lung cancer mortality in white males in 1968. PM andGR produces similar ranks, while the MLE ranks can be very di�erent due to its high variability.

6. DISCUSSION

Accurate disease maps play an important role in public health decision making. A disease mapbased on the area-speci�c MLEs su�ers from high variability, a particular problem for rare dis-eases with low incidence rates. ‘Borrowing information’ becomes an important strategy to reducevariation and to stabilize rates. All of PM, CB and GR are capable of stabilizing rates, however,the EDFs of PM or CB do not have the right shape, and the EDF of PM is always under-dispersed.The ranks based on PM or CB are not as good as those based on GR, in particular, if the pos-terior variances of the rank estimates are large. The GR approach provides the best single set ofestimates that are e�ective for multiple inferential goals including estimating area-speci�c rates,estimating the distribution of rates in the entire region, and ranking rates.PM, CB and GR are all sensitive to choice of priors. Unlike PM and CB, GR depends consid-

erably on the whole prior distribution, not merely the �rst two moments. Thus the GR estimatesmay bene�t the most from an improved prior. Our simulation and data analysis of Ohio lungcancer mortality show that the SBR method improves the robustness of the GR approach.In this paper, we focus on the two-stage independence model (that is, the area-speci�c rates are

assumed to be independent of each other). These model assumptions can be relaxed. Recent



advances in computational methods (MCMC) allow one to develop complicated models with manylevels in the hierarchy for broad distributional assumptions including spatial correlation and timetrend among the true disease rates. Devine et al. [13] have applied the CB approach to spa-tially aligned data. Conlon and Louis [14] have implemented the GR approach in hierarchicalmodels including spatial covariance, covariates and hyperprior information. Extensions of SBR orother robust methods to spatially correlated models are critical in order to realize the potentialof GR. However, generalization of the SBR method is not straightforward, and this requiresfurther research.

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