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BAYESIAN HIERARCHICAL MODELS FOR MAPPING LUNG CANCER MORTALITY IN ONTARIO

Marc-Erick Thériault

A thesis submitted in conformity with the requirements for the degree of Master of Science

Graduate Department of Comrnunity Healt h University of Toronto

Copyright @ 2000 by Marc-Erick Thériault

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Abstract

B-WESIAX HIERARCHIC,AL MODELS FOR ILIAPPIXG LUNG C-WCER

MORTALITY IN ONT-4FUO

Marc-Erick Thériault

Master of Science

Graduate Department of Community Health

University of Toronto

3000

Mapping rare disease incidence or mortality using maximum Likelihood estimates may

yield misleading results when regions have a low number of events or small sample size.

Sloreover. problems in mapping may arise wlie~i the number of events in an area is large

when frequentist pvalues are mapped. This thesis look at how Bayesian randoni effccts

estimates and postèrior probabilities might be used co offset the above problems tvliich

tend to visually dominate a map. Within the Bayesian framework. a spatially structured

mode1 is considered which allows prior information of spatially similar risks to be used.

The techniques are illustrated using lung cancer mortality data for Ontario. A simulation

study. based on the actual data, ?vas used to examine the effects of both population

size and fised relative risks on mean square error (including b i s ) and the ability to

distinguish high and low risk areas. Results show that mean square error is reduced in

Bayesian estimates at the cost of some bias. Maps of Bayesian posterior estimates and

Bayesian probabilities offer more consistency than their traditional counterpart.

Dedication

In memory of my Grandparents

Acknowledgements

The ultimate completion of this thesis depended on many different people. Even though

onIy a few are mentioned here, others were integral.

I'd first like to thank my supervisor Dr. Michael Escobar for his guidance, experience

and insight. I'm most thankful for his enthusiasm in taking on a topic that 1 really

wanted to pursue.

Many thanks go to a great group of statisticians that made up the cornmittee: Dr.

Peter Austin, Dr. George Tomlinson and Dr. James Stafford. They provided thorough

reviews and much input throughout the entire defense process.

Without the support of Dr. Paul Corey, who initiated the part-time program in

1995, this thesis would not have even gotten off the ground. 'Cheers' to a great Prograni

Direct or!

I'd also like to thank Dr. Toni Basinski. who got nie interested in the topic. Dr. Jack

Williams and Dr. David Naylor. for their support and use of the data. These Scientists

were essential to in? entry and continued success in the program.

Thanks to BH. LJ? DR, RF. TA, CS. RS, PO. HH. TKD and CT for al1 their help

and support. during the ups and dotvns. ovec the years.

Finallu, I'd like to thank my parents for their constant encouragement and uncondi-

tionsl support. Thanks to m. brothers and sister rvlio were always there for me. This

would not have been possible ttithout rny family.

Contents

1 Introduction

2 Background. Notation and Models 3

2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

. . . . . . . . . . 2.1.1 Frequentist Ma~i rnu rn Likelihood Mode1 (;LILE) 4

. . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hierarchical Bayesian models d a

'2.2.1 Gibbs Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Random Effects Models . . . . . . . . . . . . . . . . . . . . . . . . . . . S

. . . . . . . . . . . . . . . . . 2.3.1 Spatially Unstructured Mode1 (RE) 9

2.3.2 Spatially Structured Xlodel (CAR) . . . . . . . . . . . . . . . . . 10

. . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Estirnates and Probabilities 12

3 Preliminary Analysis 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 1lLE anaiysis 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 RE analysis 20

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 CAR analysis 23

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Final cornparison 32

4 Methods 36

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1 SimuiationStructure 36

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Cornparison of Estimates 39

. . . . . . . . . . . . . . . . . . . 4.3 Cornparison of Map Probability values 40

5 Simulation Results 41

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Estimates 41

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 M E 42

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Bias 46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Iterationmaps 50

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Map Probability values 62

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 ROC curves 62

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Iteration maps 66

6 Discussion 76

7 Conclusions 80

Appendices

A Reference map

B ANCOVA output 88

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.l >ISE output 85

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Bias output 91

. . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Area under ROC output 95

C Results for Estimates Analysis 98

. . . . . . . . . . . . . . . . . . . C.1 Tabular Results for Estimates Analysis 98

. . . . . . . . . . . . . . . . . . C.2 Graphical Results for Estimates Analysis 104

D Results for MP-value Analysis 110

. . . . . . . . . . . . . . . . . D.1 Graphiral Results for ROC curve Analysis 110

E Cornputer programs 116

E.1 BUGS prograrns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

E.L.1 REmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

E.1.2 CAR mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I l i

E.2 SAS Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

. . . . . . E.2.1 Simulate re1atii.e ïisk aad nin BUGS for Gibbs szmpler 117

. . . . . . . E.2.2 Calculate relative risk estirnates from Gibbs samples 125

List of Tables

Lung Cancer Mortality, Ontario 1996, Maximum likelihood estirnates of

relative risk (MLE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Surnrnary statistics for relative risk of Lung Cancer Mortality, Ontario

1996, Bayesian spatially unstructured model (RE) . . . . . . . . . . . . .

Sumrnary statistics for relative risk of Lung Cancer Mortality, Ontario

1996, Bayesian spatially structured model (CAR) . . . . . . . . . . . . .

SIean and variance estimates for models used in prciirninary nnalysis . .

..KCOVA results for USE . . . . . . . . . . . . . . . . . . . . . . . . . .

ANCOV.4 results for Bias . . . . . . . . . . . . . . . . . . . . . . . . . .

Frequency (and percentage) of model risk estimates assigned to 'High'

areas (ttl=49) and correct assignment to preassigned 'High' areas (ttl=9)

Ai'COVA results for area under the curvc . . . . . . . . . . . . . . . . .

Frequency (and percentage) of model MP-values assigned to 'High- areas

(ttl=49) and correct assignment to preassigned 'High' areas (tt l=9) . . .

Percentiles for model estimates . . - . . . . . . . - . . . . . . . . . . . .

List of Figures

Lung Cancer Mortality. Ontario 1996. Maximum likelihood estimates of

relative risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Lung Cancer Mortality. Ontario 1996. P-values based on two-tailed test

under Poisson assumption do. . = 1 . . . . . . . . . . . . . . . . . . . . . . 17 Boxplots and Lineplots comparing MLE vs . Bayesian " fixed" (Ieft) and

frequentist random effects vs . Bayesian RE (right) . . . . . . . . . . . . . 21

Histogram of l 0 ~ ( 0 ; \ ' ~ ~ ) . . . . . . . . . . . . . . . . . . . . . 22

Bayesian relative risk estimates of Lung Cancer llortalit y. Ontario 1996.

Spatially unstructured mode1 (RE) . . . . . . . . . . . . . . . . . . . . . 24

Bayesian 3IP-values. Spatially unstructured mode1 (RE) . . . . . . . . . 25

Bayesian relative risk est irnates of Lung Cancer Slortality. Ontario 1996 . Spatially structured model (CAR) . . . . . . . . . . . . . . . . . . . . . . 26

. Bayesian SIP.values Spatially st ructured mode1 (CAR) . . . . . . . . . . 2 1 Bosplots and Lineplots of relative risk estimates comparing \[LE vs . Bayesian

RE (left). Bayesian RE vs . Bayesian CAR (middle). Bayesian CAR vs .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MLE (right) 33

4.1 Simulation rnap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.1 Bosplots For 0 wlierc eo = i . 1. 3 (roms) and O0 = 1.5: 2.5.4.0 (columns) . 43 5.2 SISE vs Ba . eo IF analysis met hod . . . . . . . . . . . . . . . . . . . . . . 47

5.3 MSE vs Bo. eo by area assignment . . . . . . . . . . . . . . . . . . . . . . 48

5.4 Bias vs Bo. eo by analysis method . . . . . . . . . . . . . . . . . . . . . . 51

5.5 Bias vs Bo. eo by area assignrnent . . . . . . . . . . . . . . . . . . . . . . A

5.6 MLE maps for 0 where eo = and Bo = 1.5 . . . . . . . . . . . . . . . . .

5.7 RE maps for 8 where eo = ! and Bo = 1.5 . . . . . . . . . . . . . . . . . . A

5.8 CAR maps for B where eo = f anci 4 = 1.3 . . . . . . . . . . . . . . . . .

5.9 MLE rnaps for where eo = 1 and Bo = 1.5 . . . . . . . . . . . . . . . . .

5.10 RE maps for where e~ = 1 and Bo = 1.5 . . . . . . . . . . . . . . . . . .

5.11 CAR maps for 0 where eo = 1 and = 1.5 . . . . . . . . . . . .

5.12 ROC curves for e where eo = f , 1 ' 3 (rows) and Bo = l .5 ,2 .5 .4 .0 (columns) 5.13 Area under curve vs Bo, eo by analysis method . . . . . . . . . . . . . . .

5.14 MLE maps for MP-value where eo = f and $ = 1.5 . . . . . . . . . . . .

5.15 RE maps for MP-value where eo = $ and Bo = 1.5 . . . . . . . . . . . . .

5.16 CAR maps for NP-value wliere eo = f and $ = 1.5 . . . . . . . . . . . .

. 5 .K MLE niaps for MP-value where eo = 1 and Bo = 1.5 . . . . . . . . . . . .

. . . . . . . . . . . . . 5-18 RE maps for NP-value where eo = 1 and $ = 1.5

. . . . . . . . . . . . 5.19 CAR maps for MP-value where eo = 1 and do = 1.5

h.1 Ontario reference map of census divisions . . . . . . . . . . . . . . . . . .

C.1 Bosplots for eo = 4 = 1.5.2.5.4.0 . . . . . . . . . . . . . . . . . . . . .

C.2 Boxplots for eo = i. Bo = 1.5.2.5.4.0 . . . . . . . . . . . . . . . . . . . . . C.3 Boxp lo t s fo re~=1~Oo=1 .5~2 .~ .4 .0 . . . . . . . . . . . . . . . . . . . . .

C.4 Boxplots for eo = 2. 4 = 1.5.2.5.4.0 . . . . . . . . . . . . . . . . . . . . .

C.5 Boxplots for eo = 3. 60 = 1.572.57 4.0 . . . . . . . . . . . . . . . . . . . . .

D.1 ROC curves for eo = $.Bo = 1.5.2.5.4.0 . . . . . . . . . . . . . . . . . . .

D.2 ROC curvcs for eo = $.Bo = 1.3.2.5.4.0 . . . . . . . . . . . . . . . . . . . - D.3 ROC cirn-ris for eu = 1.19~ = 1.3.2.5.4.0 . . . . . . . . . . . . . . . . . . .

D.4 ROC curves foreo = 2 , & = 1.5,2.5,4.0 . . . . . . . . . . . . . . . . . . . 114

D.5 ROCcurvesfore~ =3:$=l .5 .2 .5 ,4 .0 . . . . . . . . . . . . . . . . . . . 115

Chapter 1

Introduction

blapping standardized rates and p-values are common approaclies to analyze grograpti-

ical variation in disease rnortality and incidence. Generally. tliese rnetliods attempt to

idcntify arcas which are truly differeiit from eacli otlier. This practice is used to providc

useful information sudi as generating etiology Iiypotheses. Differcnces iii rates tiave beeri

historically tested using frequentist methods. However. some of these methods are prone

to problems. Triie differences may be difficult to determine when areas with small nurn-

bers of subjects at risk have a greater chance to have a large estimated effect size due

to chance alone. Moreover, areas may appear significantly different since regions with

larger sample sizes give more power to detect 'deviant* rates. These areas tend to visiially

dominate a map. Therefore. smoothing or stabilizing rnethods are used to minimize in-

stability of estimates and probabilities. yet mnintaining geographic resolution. Generally

boundaries are drawn according to established definitions. Disease trends may or may

not follow these arbitrary boundaries. The prior belief of disease etiology or transmission

ma!? be incorporated in Bayesian rnetliods.

solved. Spatial statistics is an area mhich has benefited from the use o f tliese techniques.

The purpose of this study is to demonstrate to the reader tliat Bayesian random effect

estimates of relative risk and posterior probabilities are less prone to the problems en-

countered by traditional frequentist est imates and pvalues. hlso. random effects models

extend nicely to incorporate a spatial component which is readily feasible with current

Bayesian techniques and software. This study compares both tiequentist and Bayesian

methods along with both fixed and random effects (emphasis being on the random e f

fects model). The first Bayesian method assumes independent random effects that are

spatially unstructured. The second Bqesian method assumes conditional random efFects

based on regional adjacency This rnodel is called the conditional autorcgressivc model.

h simulation study is used to esamine mean square error (including bias) and a

rnethod's ability to distinguish between high and Inw risk areas wliile controlling risk

estimates and data size. Visiial iiispection of maps of estiniates and map probnbility

values ( NP-rilues) will also be iised to examine the diffcrenccs be twen t tic nict liods.

Kork by Bernardelli et al. (1995) [5] used a simulation stiidy to esiiniinc relat i~e risk

estimate mapping. The results indicate that the fully Bayesian approach is superior to

the conventional approach of mapping maximum likelihood estimates and p-values. This

thesis espands work of Bernardelli et al. [3]: by incorporating more extensive simulations

and addressing issues in mean square error and bias with both graphical and numerical

techniques to examine differences between methods.

This thesis is divided into seven chapters. The current chapter addresses the rationale

and aim of the thesis. The next chapter provides background. notation and introduces

the model information relevant for the remainder of the thesis. Chapter three provides

a case study outlining an initial assessrnent of the data and the clifferent rnodels. The

rnethocls of the simulation study are ciescribcd in cliapter four. The foUowing chapter

prcsciits tlic resiilts of t lic! siniiilat ion arid t liosc resiilts arc sytitliosizd iri t hc cliaptcr sis.

Coric.liisioris arc. c1r:iwti iii t liv f i ~ i i i l c h p r PI-.

Chapter 2

Background, Notation and Models

This chapter provides an introduction to the relevant background for this thesis. Tlie

first section introduces basic definitions and the frequentist estimate for relative risk.

The next section introduces Bayesian theory and current techniques in obtaining solu-

tions to Bayesian probleins. Random effects niodels. whose estimates ma? be consiciered

shnnhge estimators. are introduced in the following secticin. The raiidoni effects motlcl

is the basic framework for both the spatially structured and unstructured rnodels. The

next section introduces the spatially structured conditionally autoregressive model. That

section discusses how Bayesian methodology incorporates the spatial concept into a ran-

dom effects model. The final section section discusses how the Bayesian estimates and

probabilities are calculated in the thesis. Background litereture review. forma1 notation

and definition of the models rvill be made throughout the sections in this chapter.

2.1 Definitions

The Statistics Canada county definition was used for the maps in this tkicsis. It cic-

fines forty-nine contiguous arcas in the province of Ontario. Let ol~scrrcrl cotirits O =

(O1. . . . . O,,). n = 49 bc niut iiaily intlepcncl ~ l ) ~ ( . t < ' < [ cr)ilt ltS it l l

CHAPTER 2. BACKGROUND, NOTATION A N D MODELS 4

unknown area specific relative risks of niortality. In this thesis. the expected counts

are standardized to the overall age and geiider distribution of the data. Assuming the

average rate of the event of interest over i areas for j age groups and k genders is

where nijk is the total number a t risk. The expected number of events for an area is then

defined by

Therefore, assurning a non-contagious and rare outcorne. the likelihood of the relative

risk Bi is defined by

p(Oi 1 B i ) = e-'lE1 ( m ) O 1 O,!

In this thcsis. the overall rate f,k is obtained from the obsened data and not a standard

reference table. Event though the joint distribution of Oijh is multinomial~ Waller et al.

(1997) (referring to Xgresti (1990) pp.455-456) state that likelihood inference is unaffected

by conditioning on xi O i j k Therefore? it is common practice to retain the product Poisson likelihood. .Ml methods in this thesis rely on the Poissa(Bi Ei) likelihood. The

first method is frequentist and the final two are Bayesian.

2.1.1 Frequentist Maximum Likelihood Model (MLE)

The estimate that maxirnizes the likelihood (2.1) is

This niasimum likelihood est imate (LILE) is also known as the st antlardizcd mort ality

ratio (S.\ IR) (Bmslou- anci Da\- 1857 [il) for tlic it" area. This rstiniatr of risk is spatialIy

tiiistriictiircxl aiid is tlir orily frr~qiicrit~ist iiititho(1 ri.;iliiatcd i r i tlio siriiitliitioii stiiclc

r\ssurning that 0, are random Poissm(E,B,) variables. the map probability ualve

(MP-value) is defined as

This value is a cumulative probability and not a typical p - value. The definition above

is used to allow cornparisons with the Bayesian rnethods.

2.2 Hierarchical Bayesian models

Let O = (OL, . . . '0,) be random variables with joint density function po(O 1 8) where

8 = (O1, Oz, . . . ,O,) is a real-valued parameter lying in 8 = (8, x e2 x . + . x O,). Né assume 8 to be a set of open sets. Let E = ( E l . . . . . En) be constant. Let pe (0 ( u ) be a pnor

distribution function defined on 8 given real-valued hyperparameters & = ( I ! : ~ . . . . . c,).

The uncertainty in the value of the parameter 0 prior to ohserving O is clescribed by

a. Similarly, let p , (ci..) = (p,, (&,). . . . . pu,, ( l~r,)) be hyperprior distributions describing one's uncertainty in @, where ri, is a real-valued parameter lying in open set = (QI x

x - - - O - x ap). Using Bayes' Theorem. Ive define a hierarchical posterior dist7-ib~tion of B given O as

The calculation of the posterior distribution relies heavily on the calculation of the above

integrals.

Moreover. the posterior eqectation

or moments require more integral calculations.

Until recently. the calculation of the posterior distribution and posterior espectation

have been a major lirniting factor in the practical use of Bayesian techniques. One pas-

sible solution to calculating the posterior density is to use asymptotic methods. Normal

approximations such as the Bayes ian Central L i m i t Theorem method fail to pick up skew-

ness in the posterior. This may then iead to problematic konfiàence' intervais. Wide

the more complicated Laplace method (Tierney and Kadane 1986 [29]) yields more ac-

curate results, the posterior distribution must be unimodal. Both methods require a

large nurnber of data points. While both of tliese methods may be overcome. they can

be problematic in high dimensions. The expectation-mazimization (EM) algorithm of

Dempster et al. 1977 [ll] is a practical method to obtain the posterior mode. but not

the entire posterior distribution. The Gibbs sampler [12] allows one to sample values

from the full joint posterior distribution using a SIarkov Chain. With the auailability

and practicality of Gibbs sampling software. full' Bayesian analysis has I>ecomt. niore

popular. Therefore. only the fully Bayesian analysis will be assessed in this tliesis.

2.2.1 Gibbs Sarnpling

Consider the n+p dimensional problem where the n+p parameters are (81, . . . . &,W1?. . . , d p ) . Gisen the a~ailable data. the primary interest is the marginal posterior distribution of

each B i . Area specific surnrnary statistics such as the posterior mean or median (ie. pos-

terior espectation defined above) c m then be mapped. Thereforc. $yen the full joint

posterior distribution, the marginal densities can be readily computed as required. The

Gibbs sampler is described by the following iterative algorithm:

CHXPTER 2. BACKGROUND. NOTATION A N D ~ I O D E L S

(11 ( 1 ) ( 1 ) ( 1 ) (0) (01 ) Draw 8, p ( 0 , ( O. 0, . O? . . . . . On+ kvl . . . . . up . ( 1 ) ( 0 ) Draw 8'1') - p(ul 1 0.01?6)4". . . . .On . r2 . . . . . CF)).

This describes one iteration in the Gibbs sampier. After an initial burn-in of B

iterations, where the sampier reacbes convergence, arid a irl fur tiier i trra

for each of the n + p parameters are obtained. Therefore, using the a above,

(O(,? O:'\'). . . . . op, y * . , &y 1) -b ri (el. 02,. . . .en' krl . . . . . C;J

lgori t hm defined

- p . . . . 0 . . . . vP 1 O ) as J I -+ cc.

are Monte Car10 estimates (Pnr (B i ( O). 1 O)) of marginal posterior distributions

The above algorithm requires the calculation of full conditional distributions in which

each of the n + p parameters are drawn from. The full joint posterior distribution can be

sampled more easily when one specifies conjiigate distributions for po(O 1 8) . po(O ( dr)

and p&) [15. 161. The fi111 conditional distribution are not reqiiired by B W S software

[28] to sample the full posterior distribution. Generally, one needs only to set up the like-

lihood and the prier distribution in the BUGS prograrn. Given samples frorn the full joint

posterior distribution. estimates of the marginal posterior distributions nere esarnined.

Tliere arc niaiiy tecliniqiics iised to cictermiric convergence of the G i h sanipling cliaiii.

111 t h tliesis. only visiial inspcctiori of trace plots [vas prrforrncd.

CHAPTER 3 - BACKGROUND. NOTATION AND MODELS 8

The Bayesian met hods discussed above assume observations to be condit ionally inde-

pendent given parameters that incorporate spatial dependence or independence via a

prior distribution. This approach borrows strength from al1 the data to allow unsta-

ble risk estimates to be shrunk toward overall or local means. Tlie Bayesian methods

incorporate the use of random effects models.

2.3 Random Effects Models

Often, with a combinat ion of rare disease and areas witli siiiall population. the observed

number of events is greater then espected under Poisson inference 131. The extra-Poisson

variation is the variation in escess of that captured by thc likeliliood aime. This ariscs

from an area's heterogeneity in the rate of disease risk. It is reasonable to think tliat each

area has its own mortality rate which depends on other non-rneasurable factors. Random

effects models can captures both Poisson and estra-Poisson nriation. thereby obtaining

estimates which better reflect the true heterogeneity of the relative risk [3.4. 5.91. Tliese

estimates of random effects models can be referred as shrinkage estimators. The extra-

Poisson variation is rnadelled t hrough a prier distribut ion wi t h dispersion hyperpararneter

O*. This parameter represents the variation in true relative risks. Both Bayesian models

discussed in t liis t hesis allow the component of estra-Poisson variabili ty to be modelled.

The major differencc between fully Bayesian and empirical Bayes analysis is the way this

parameter is estimated. The fully Bayesian rnethod allows the iincertainty abolit o2 to

be carried over to the estimates of the relative risks [SI. Even with this difference, the

relative risk estimates O btained by bot h ernpirical Bayes and full y Bayesian techniques

have been demonstrated to yield similar results [4. 721.

hltlioiigh cstablishecl for some time in analysis of variance. rnncloni effects niodels have

siirgrcl in the fields of niiiltilel-el inocielling and lorigitiiclinal cL;itii atialysis. SAS pro~icles

PROC MIXED [ZG] ii~id Splits providcs lme [?O] for riiodrliing raticlor~i ~ ~ f i w s h o a r iiiodrls.

As part of tlie preliniinary nnalysis. 1 provide a brief esa~iiination of tlie frequentist risk

estimates for ranclom effects iising the S.AS macro G L I M M I X (iised for geiieralized linear

models). but do not apply it to the simulation study.

Consider the following model

where p is the overall mean and ci is the iLh area specific random effect. Typicall- the

assumption of random versus fixed effects is based on whether the analyst believes the

effects are sampled from some population. Although the counties in a province may

not be thought of as a sarnple from a population of counties, the use of random effects

models. in tliis thesis. is used to obtain smoothed estimates which rnay othenvise be

considered 'unstable or 'noisy' when modelling through a fixed inodel. In fi~ct. the

m ~ ~ i m u r n likelihood estirnate 2.2 is the solution to the fixed effects model.

Within the Bayesian framework. the random effects can be modelled througli a lcvel

of hierarclv via a prior distribution. In ttiis thesis. the ci term is inodcl!ed with a

Normal distribution. The Bormal distribution is easily extended to adjust for area level

covariates by adding fixed effects with a #xi term. Recall the data esamined here has

been standardized for age and sex. The simulations in this thesis do not apply any

covariates. The two Bayesian rnodels differ in the way the random effects are modelled.

The different priors on the random effects will refiect the analyst's prior belief about

the kind of geographical variation expected in relative risk. The first Bayesian mode1

assumes a spatially unstruct ured model while tlie second mode1 is spatially st ruc t ured.

2.3.1 Spatially Unstruct ured Mode1 (RE)

The spatially unstructured or heterogeneity random effects model (RE) for log relative

risk (Claytori and Kaldor 1987 [SI) is defincd Iiierarcliiçally as follows:

This mode! d l ~ w s the area specific random effect hi to be modelled about a Normal

(CCh, rh) distribution where ph may be interpreted as the overall mean p in (2.5) and rh

(= &), called the precision, is the overall variability of random effects. The fully Bayesian uh

method allow one to investigate the uncertainty about both ph and T~ by adding another

level to the hierarchy as seen in the last two lines of (2.6) with the use of hyperprior

distributions. In this thesis. values of hyperparameters &. & 06 and ,8,h are chosen such that ph and rh are modelled with diffuse or non-inlonative prior distributions. The

use of hyperprior distributions highlights the major advantage over frequentist random

effects models. The ability to capture uncertainty in ph and rh highlights the advantage

over empirical Bayes methods. Both models exploit the use of random effects models.

The random effects are 'shrunk' towards the overall mean with the amount of shrinkage

depending on both prior (and hyperprior) distributions and the amount of data observed

j through the likelihood).

The full conditional distributions can be found in Mollie 1996 (221. For the random

effect hi, given all other random effects h-i, the conditional distribution for the spatially

unstructured model is defined by

(hi - 4)' P(hi ( h - i , ~ , ~ E ) a exp Oihi- Eiexp{hi) - 2'7;

2.3.2 Spatially Structured Model (CAR)

Frequestist methods have been established for some time in the area of spatial statistics

(Ripley 1981 [24]: Riple~- 1988 [25] and Cressie 1993 [101). An autoregrcssi~e xnoclcl to

acc:ount for regional iltljacc11c:y. iutroducecl b>- Bcsag 1974 ['I]. is tliscussc~rl in i d l of t lw

CHAPTER 3 . BACKGROUND, NOTATION A N D ~ ~ O D E L S 11

aboce rcferences. k i n g empirical Bayes methods. Clayton aiid Kaidor 1987 [9] apply the

model in mapping log relative risks. Their model assumed tliat the conditional variance

of an area effect, given its neighbouring areas, to be constant. This may not be strictly

appropriate for maps with varying number of neighbours. Besag et al. 1991 [3] proposed

a model whereby the conditional variance for an area i given al1 other area log relative -7 risk is inversely proportional the number of areas adjacent to it. 1 nis moàei continues

to be extended and applied, in the Bayesian context [[4? 5, 22, 23, 301.

If i is an area of interest, we define w, = 1 if area i is adjacent to area j and

wi, = O othenvise. The conditionally autoregressive model (CAR) requireç that the joint

distribution of 10g(0~ ) , given Zog(0,) and hyperparameter oz. is Normally dist ributed wit h

mhere j belongs to the set of neighbours of i. c, is tlie i lh area specific rnndom effect.

1 u!ii = x:=, w,, and Et = , xy~, w,, c, The inverse variance-covariance mat rix is de fined by

Equivalent 1- in terms of precision rC (where T = 5 ) the inverse variance-covariance matris is defined by

The Bayesian CAR mode1 is thercfore defined hierarchically as

CHAPTER 2 . BACKGROUND. NOTATION AND ~ ~ O D E L S

where ni is the number of adjacent counties to county i, j belongs to the set of neighbours

of i and ui+ = xy,, wij- The random effects are conditional on the average of the adjacent area random effects. Therefore, the effects are 'shrunk' towards local means

where, again. the amount of shrinkage depends on priors and the amoiint of data. Note.

the area specific variation (r:) is the proportion of overall variation of conditional random

effects as required by (2.7).

For the random effect ci, given al1 other random effects c+ the conditional distribu-

tion for the spatially structured model is defined by

U& (c* - ci)- P(ci 1 c-,;O.O:) N esp Oit* - E,esp {ci) - 3 'L

- o c

2.4 Estimates and Probabilities

When convergence has been attained from the Gibbs sampler, area specific relative risk

can be estirnated by the sarnple median (8:-4'ES) or sample mean of the log relative risks

defined by

where bl is the number of samples for the ith area. Meng 1957 [21] proposed the use of

posterior.probabilities from Bayesian randorn effects models as a itirthod to combat the

miiltipic coiriparisoris prol~lcm. Bernarcielli et al. 1992. 1995 [A. 51 iipplied this concept

CHAPTER 2. BACKGROUND. NOTATION AND ~ I O D E L S

If the sample global rnean is defined as

the Bayesian MP-value is defined as

This may be defined as the posterior probability of an area specific relative nsk being

greater than the global mean [4, 5, 211. The MP-value can then be estimated by

where indicator function I ( . ) = 1 if espression is tme. I ( . ) = O otherwise.

Similarly, in terms of the median (ë), the MP-value may be defined by

whicli is then estimated by

After completing M Gibbs cycles on n area specific relative risk estirnates. the global

median (or mean of log relative risk) is obtained from the n x M Gibbs samples. Each

sarnple value is tested against the global median. The estimated MP-value represents

the proportion of times the area specific relative risk is greater than the overall median

relative risk. Therefore. a relative risk much larger than the global rnedian tends to yield

a large proportion. The use of the global mean or median is arbitrary and up to the

analyst to decide which constant to use for the calculation of the posterior probabilities.

Although another method is to compare each area specific effect against the mean (or

median) within each Gibbs cycle. Bernarclclli et al. 1995 [5] used the global rnean. For

simplicity and robustness. both estimates and SIP-valiics werc bascd on nicdiaiis in tliis

Chapter 3

Preliminary Analysis

This chaptcr introduces the preliminary analysis used to examine cancer mortality data

for Ontario. Maps of estimates and probabilities will also be presented throughout this

chapter: for the models of specific interest. The first section looks at the traditional

maximum likelihood analysis. This section will provide a comparison 114th an analogous

Ba1:esian model. The second section will look a t the unstructured random effccts model.

Although the Bayesian model is of primary interest, this section provide a comparison

with a frequentist random effects model. The next section incorporates the spatially

structured autoregressive model to the data. The final section provides surnmary of

estimates for both overall mean and variance. A visual comparison of the estimates

from one model from each of the previous sections. These three models will be further

esaniined in the simulation studv of the following chapters.

3.1 MLE analysis

Lung cancer mortality data was estracted from the Ontario Cancer Registry database.

h coliort aas identified as tliose persons. age 50 years and older. witli a diagnosis of

lung cancer in 1996 and dyirig by the end of the sanie year. The ohscrvcd and cspected

niirithcr of clmtlis. 11asrd on '2.1. for cadi coiirity apprar in Tal~le 3.1. Tlic traclitional

inaximum likelihood estimates (MLE defined by 2.2) and p-values based on a trvo-tailed

test against uni@

also appeax in Table 3.1. -4 reference map is provided in Appendix A as Figure A.1. This

maps labels the geographic region with the county identifier (CD) in Table 3.1. There is

an average of 4.4 (median of 4) neighbous for each county. Both Essex County (37) and

Sudbury Regional Municipality (53) have only one adjacent neighbour and Wellington

County (23) has nine neighbours.

The largest relative risk estirnate is 2.21 in Prince Edward County (13) with p-value

c 0.001. The smallest relative risk estimate of 0.66 belongs to Perth County (31) with

p-value = 0.07. The number of observed deaths Vary from a high of 789 in the Toronto

area (20) to a low of 5 in Manitoulin District (5 1). These two counties have risk estimates

that are very similar, but only Toronto has a statistically significant result ( p d u e <

0.001). A rnap of the relative risk estimates appear as Figure 3.1. Estimates larger tlian

1.30 appear as red areas. Lighter shades of red reflect areas of gradually lower risk.

Similarly, areas whose estimates were smaller than 0.70 appear as blue. Lighter shade

of blue reflect the gradua1 increase in relative risk. The white areas indicate regions

where the relative risk estimate falls between 0.90 and 1.10. The legend contains the

distribution of counties. Recall there are 49 counties. There are 8 counties, scattered

throughout Ontario. with risk estimates larger then 1.30. There appears to be 5 distinct

clusters of 'high' risk counties. One cluster with 3 counties. another with 2 and the rest

consist of only 1 area. Note counties 57, 56, 54, 48 and 49 are relatively high risk counties

that are adjacent to Sudbury District (52) which is relatively low risk. Sudbury District

(52) district encornpasses Sudbury Regional Municipality (53) which is relatively higb

ri&. There does not seem to be an\- apparent continuity of the relative risk estimates in

the map.

Thcrr arc 11 çourities ni th p-wlticis lcss thari 0.05. Eight of tliosr coiintics han! risk

Figure 3.1: Lung Cancer Mortaüty, Ontario 1996, Mzuimum likelihood estimates of

relative risk

Figure 3.2: Lung Cancer Mortaüty, Ontario 1996, P-values based on two-tded test under

Poisson assumption Bor = 1

Table 3.1 : Lung Cancer Mortalit. Ontario 1996. Maxi-

mum likelihood estimates of relative risk (MLE)

CD County name Population O bserved Expected SMR P-value

STOEtMONT

PRESCOTT AND RUSSELL UNITED COUNTIES

C)TTPPWP--CA-PXLFI"I'Y REGION.4L MUNICWAC'ITY

LEEDS AND GRENVILLE UNITED COUNTIES

LANARK COUNTY

FRONTENAC COUNTY

LENNOX AND XDDWGTON COUNTY

HASTLIGS C O W T Y

PRINCE EDWARD COUNTY

NORTHUMBERLXXD COUIJTY

PETERBOROUGH COUKTY

\TCTOFU COUSTY

DURHAM REGIOSXL >IUIiICIPALITY

YORK REGIOXAL MUXICIPALITY

TORONTO M ETROPOLITAN i\llU?iiCIPALITY

PEEL REGIONAL MUNICIPALITY

DUFFERIB COUKTY

WELLINGTON COUNTY

HALTON REGIOXAL MUNICIPALITY

H,AhlILTOS-iVEST1VORTH REGIONAL MU&.

NIAGARA REGIOSAL blUNICIPALITY

HALDNAND-NORFOLK REGIONAL MUN-

BR4XT C O i D T l -

W.4TERLOO REGION AL MUNICPALITY

PERTH COUXTY

OXFORD COC'STY

ELGIX COUSTl-

KEST COC'XTY

ESSES COCSTl*

LA5iBTOS COVXTY

-

CD County name Population Observed Expected SMR P-value

MIDDLESEX COUXTY

HCRON COUXTI*

BRUCE COUNTY

GREY COUXTY

SIMCOE C O U N N

>iIUSKOKA DlSTRlCT MU'IICIPALITY

HALIBURTON COUNTY

RENFREW COUNTY

NIPISSING DlSTRICT

PARRY SOUND DISTRICT

kIXNITOULIX DISTRICT

SUDBURY DISTRICT

SUDBURY REGIONXL MUNICIPALITY

TIblISKXhIING DISTRICT

COCHRANE DISTMCT

ALGOMA DISTRICT

THUNDER B.\Y DISTRICT

RAINY RIVER DISTRICT

KENOR4 DISTRICT

TOTAL 2959380 3921 3921.0

estimates greater than 1 and the remaining 3 have risk estimates less than one. A map

of pvalues appears as Figure 3.2. The legend attempts to combine both statistically

significant results (via pvalues) with risk estimate information (where 'High' has MLE

> 1 and 'Low' has MLE < 1). The significantly high counties tend to correspond to the

map based on estimates. One county in south-western Ontario, Essex county, has pvaliie

= 0.025, but a relative risk cstimatc of 1.20. This couoty, therefore, did not appear as

a high rate county on the estimates map. Conversely, the county of Muskoka appears

'high' in the estirnates map with relative risk of 1.36 and pvalue = 0.067. Therefore,

this county did not appear on the p-wlue map. Discontinuities appear in this map. as

did the estimates map.

For comparison only, a Bayesian model was fit to this model. This Bayesian 'fised'

efFect model used separate diffuse Normal distributions on each effect. No MP-values were

calculated. The 11 counties with p-values < 0.05 in the MLE analysis have a 2.5-97.3

percentile interval that does not capture 1.0.

Referring to the ieft-most plot Figure 3.3 (the riglit-most will be discussed in the nest

section), the MLE and Bayesian relative risk estimates are shown for comparison. The

boxplots (not traditional bosplots) indicate the 2.5, 25, 50, 75 and 97.5 percentiles. The

Iineplots join estimates between the two methods for the same counties. A reference line

at 1.0 was added to the plot representing no increase or decreased risk of rnortality.

Except for a couple of counties, below the the reference line, which appear to diverge

away from 1, the estimates are very similar.

3.2 RE analysis

Recall the assumption that the county specific observed number of cleaths follorvs a Pois-

son distribution. The saniplc variance to sample niean ratio of courity specific deaths is

quite liirgf! ( l72.2). S inw t lir k t è i ii[>[>c;ir to c x h i l ~ i t est ra-Poissoii iiiriat iori (owrdispcr-

I

O'Z

Figure 3.4: Histogram of log(8yLE)

sion), i t is reasonable to mode1 using random effects models. A Iiistogram of log relative

risks appears as Figure 3.1. The log maximum likelihood estimates appear Normally

distributed with an overall mean of 0.0730 (p=0.019). median of 0.0796 and standard

deviation of 0.2272. The county specific relative risks were then estimated with the

Bayesian random effects mode1 (RE) from (2.6). The Gibbs sampler cvas run with a

burn-in of 200 and a further 1000 samples in the chain were kept. Al1 the chains appear

to converge within 50 iterations. The summary statistics of the Gibbs samples for each

county appear in Table 3.2. T h e county specific median appears to be slightly less or

equal to the mean. The MP-value (2.12) was also calculated. The global median SMR

from al1 the county specific Gibbs sarnples was 1.053. There are 4 counties with a 2.5

percentile which are greater than 1.0. Those counties each have an MP-value greater

than 92%. There are 3 counties with a 97.5 percentile lower than 1.0. Each of those

counties have an MP-value close to 0%. These T counties \vit h a range falling outside of

1.0 are al1 captured in the SILE analysis with a p-value < 0.05. In the LILE analysis.

the 4 remaining counties with ii pvaliic less than 0.05 with 1.0 falling bettveen ttic 2.5

and 9ï.-j percriitilc. in tlic RE ariitlysis (coiititirs 12. 54. .X and 59). tiad hlP-\-aliies froni

0.783 to 0.908. -4s indicatetl by tlie IIP-value. these counties have relative risk esrimates

larger t han 1 .O.

The estimates from the random effects model are mapped in Figure 3.5 using the

same range defined in the map of maximum likelihood estimates in Figure 3.1. There

appears to be much shrinkage to the overall mean. There is only one county (13) with a

relative risk estimate greater than 1.3. and ün1y 3 counties with an cstimntc less than 0.9

(19,20,21). With 30 of 49 counties in the 0.90-1.09 range, it is very difficult to make any

comments about spatial continuity in risk estimates. The map of MP-values, in Figure

3.6, appears spatially discontinuous and similar to the MLE pvalue map. Figure 3.2.

Generally, the lowest risk counties are in the southern part of the province.

A frequentist model mas fit to this model for comparisons with the Bayes estimates

only. It will not be examined beyond this section. The mode1 rvas fit with the GLIMMIX

macro in SAS. The model revealed one significantly high and two significantly loir. randorn

effects for counties 01. 19 and 20 respectively. The second plot of Figure 3 .3 displays

both the frequentist and Bayesian relative risk estimates for cornparison. The relative

estimates appear very similar with slightly more shrinkage occurring in the frequentist

estimates.

3.3 CAR analysis

The Bayesian spatially structured random effects model (CAR) from (2.8) was also mod-

elled. The sumrnary statistics of the Gibbs run for each county appear in Table 3.3. The

MP-values in the table is based on a global median of 1.077. There are 7 counties with

a 2.3 percentile is greater than 1.0. The 4 counties identified. in the RE analysis with

the sanie critcria. are captured here. Those counties each have ari NP-ralue greater tlian

89%. TTlicre are 3 countics mitli a 97.5 perceritilc loncr ttian 1.0. Tlicsci coiirities are

the saine as tliosc iclcntificd i ~ i tlicl RE ;itiiilysis. Tlir cstiniattls arc riiiippctl iri Figiirc

Figure 3.5: Bayesian relative risk estimates of Lung Cancer Mortaüty, Ontario 1996,

Spatially nnstructured mode1 (RE)

Figure 3.6: Bayesian MP-dues, Spatially unstnictured mode1 (RE)

Figure 3.7: Bayesian relative risk estimates of Lung Cancer Mortality, Ontario 1996,

Spatially stnictured mode1 (CAR)

Figure 3.8: Bayesian MP-dues, Spatially structurecl mode1 (CAR)

Table 3.2: Surnmary statistics for relative risk of Lung

Cancer L l ~ r t a l i t y ~ Ontario 1996. Bayesian spatially un-

structured mode1 (RE)

CD Mean STD 2.5% 97.5% Median M P - d u e

CD Mean STD 2.5% 97.5% Median MP-value

Table 3.3: Summary statistics for relative risk of Lung

Cancer blortality, Ontario 1996, Bayesian spatially stmc-

tured mode1 (CAR)

CD Mean STD 2.5% 97.5% Median MP-value

CD Mean STD 2.5% 97.5% Median MP-value

3.7 with the range consistent with the previous two maps (Figures 3.1 and 3.5). There

appeae to be more variation as cornpared to the RE model. The amount of shrinkage

appears to be les : but there also appears to be local smoothing with more continuity

between adjacent counties. Recall the counties around Sudbury District. The relatively

low county (52), in the MLE analysis, has been locally smoothed to relatively high. There

are 21 coünties in the 0.30-1.09 range cornparcd t o 13 and 30 for estimztec of the MLE

and RE models, respectively. The MP-value value map for this model appears in Figure

3.8. There appears to be sorne spatial continuity, but not as much as the estimates map.

The counties in the south have very low probabilities. Some discontinuity appears in the

south-east part of the province. The Sudbury District county relative risk estirnate is

smoothed slightly upwards.

3.4 Final cornparison

Estimates of overall mean and variance appear in Table 3.4. The values are on the

log scale. There is mked evidence that the overall mean is different from zero. The

intervals for the Frequentist empirical and random effects do not capture zero, while the

interval for the Frequentist fked mode1 and the Bayesian spatially unstructured random

effects include zero. The frequentist randorn effects models has smaller variance than the

Bayesian RE model. Since this is a fully Bayesian model. the larger variance rnay be due

to uncertainty carried over from the hyperprior distributions. As rnentioned earlier. an

advantage of a fully Bayesian analysis allows one to obtain an interval for the dispersion

parameter. The variance for the Bayesian CAR model is larger then the Bayesian RE

rnodel. This is due to the spatial constraints placed on the random effects.

For tlic rernainder of t his thesis. frequentist SILE. Ba-esian RE (spatially unstruc-

turccl) and Bayesian CAR (spatially structtired) niodcls are consitlered. Figure 3.9 dis-

plays t lie I>os/lirie plots ancl provide a coniparat ive \-i(?w o f t lie ii11ioiirit aiid dire

Table 3.4: SIean and variance estimates for models used

in preliminary analysis

Frequentist Bayesian

1 ModeÏ 1 Parameter 1 Estirnate %%CI 1 Estirnate 2.5%-97.5% 1

! Sample std std ) 0.2272 - 1 - - 1

shrinkage for both the RE and CAR models. The arnount of spread in the risk estimates

coincide with the values in Table 3.4. There was much shrinkage in the estirnates from

the MLE model to both the RE and CAR models. This [vas also observed in the maps.

The distribution of the RE model estimates appears symmetric while the MLE and CAR

estimates appear skewed.

The leftmost graph displays how the BILE estimates were slirunk to an overall mean

of the the Bayesian RE model. Larger relative risk estimates are 'pulled' downward while

smaller estimates have been 'pulled' upwards. The lines corresponding to the 2 largest

MLE risk estimates have been shrunk dramatically by the RE model. These 2 counties.

13 and 59: have counts of 29 and 16 respectively York Regional Municipality (19). which

has the second to smallest MLE risk estimate and a count of 120. was not greatly affected

by the RE model. Perth County (31) ivith a cocint of 19 (the lon?est risk estimate). has

been shrunk upwards to cross several lines including York's line. ..\fso, note that several

counties go from having a risk l e s than one to gea te r than one.

The middle graph shows that the CAR model display more variation than the RE

model. The added heterogeneity is due to the spatial constraints place on the randorn

effects. The estimates do not appear to have the sanie ranks. Therefore. tliere is sonie

stirinkige occurririg ttiat is not toiviisds a single overall tricari. like t h RE riio(k4.

Fixed effects

&dom effects

p

p

0.1665 (-0.2422,0.5752)

0.0582 (0.0004,0.1160)

- - 0.05605 (-0.0032?0. 1151)

The third graph compares estimates between the CAR and hILE niodels. Most of the

lines have been shriink from the MLE inodel to the CAR model. -4 few ?VILE model risk

estimates greater than one became larger under the CAR model. A couple of low risk

counties in the 'VILE model have been 'pulled' upwards by quite a large margin. Recall

Sudbury District (52) in the MLE and CAR maps (figures 3.1 and 3.7) which was greatly

3nccted 5' its neighbouring countier.

Chapter 4

Methods

Based on the preliminary results of the previous chapter, this chapter outline the meth-

ods used for a simulation study to examine mean squared error, bias and the ability

to distinguish between high and low risk areas. The first section describes the overall

structure of the simulatioris including the developmeiit of a simulation map. The nest

section introduces the definitions of mean sqvared error and bias to allow for coniparisons

between the different methods. The final section describes the use of receiuer operator

curves (ROC) for cornparing MP-values for different methods.

4.1 Simulation Structure

Recall the two frequentist based maps in Figures 3.1 and 3.2. Using the resiilts from these

two maps. a simulation map was created based on the the union of the dark red areas

in these maps. This approach identified 'hot spots' or areas with either large estimates

(which rnay be unstable) or significantly high risk areas (which rnay be susceptible to

large number of events). The simulation map appears as Figure 4.1. Red areas represent

areas which are açsigned as 'higti'. Those areas adjacent to the 'hi&' areas arc coloured

grey and labeled as 'boundarf areas. .\II otlier areas are assigtied -loiv0 clesignatcd by

a white coloiir oti tlic! niap. Tlicrc are nirie .higli' areas. sistecti %oiincl;iry' arcas and

Figure 4.1: Simulation map

tmenty-four 'low' areas. Most of the 'low' areas appear as one large cluster in south-west

region of Ontario.

The MLE, RE and CAR models were examined under different scenarios. Each sce-

nario simulated the observed number of events 0' = (O;, . . . ,O&): from a P&SSO~(E ;B~~)

where Ei = eo x Ei, i = 1,. . . ,49. The Ei7s are the original data while eo is a constant

multiplier for the expected counts. The multiplier eiiher iriflatts ur deilatrs tlir aiiiu uiit

of data for each area. In this thesis, eo = f , h, 1,2? 3. The 8 o i ' ~ are relative risks assigned to an area based on the category of 'high', 'boundary' or 'low'. In this thesis. Bo is as-

signed the values 1.5. 2.5 or 4 for 'high' areas. Al1 other areas ('boundary' and 'lowl) are

assigned Bo = 1. Therefore, each scenario or experiment is a function of both one OF three

assigned relative risks, Bo, and one of five data size multipliers, eo, resulting in a total of

fifteen experiments. Each scenario was simulated 100 times. Within each simulated iter-

ation, the Gibbs sampler \vas run with a 'burn-in' of B = 300 and M = 1.000 iterations

to ensure coiirergence to the posterior distribution of the Bayesian models. After visttal

inspection of sample area specific traceplots, convergence \vas commonlp attained within

50 iterations. Convergence and diagnostics will not be addressed further in this thesis.

The simulations can be simplified with the following pseudocode:

do over SCENARIOS eo x 80 (eo = i, 4,1,2,3; Bo = 1.5,2.5,4) do ITERATION = 1 to 100 f o r each county t

Er = eo x Ei

simulate 01 -- Poisson(OoiE~)

calculate &y LE = Or/E: do over BAYESIAN methods (RE,CAR)

do b= 1 t o B (Gibbs 'burn-in')

discard B ! ~ )

end

do m= 1 t o M (Gibbs samples)

store 01'") end

calculate ~f-"""" = median(0j;"). . . . .O(" ' ) 1 'If )

end BAYESIAN

end ITERATION

end SCENARIOS

where i = 1,. . . ,49.

The SCENARIO and ITERATION steps were done using SAS (~6.12: Windows95). BUGS

[28] (vO.6O:DOS) was used to perform the Gibbs sampling for the BAYESIAN çteps.

4.2 Cornparison of Estimates

The mean squared error (MSE) of relative risk estirnates. 0 = (0,. . . . y &). pooled in

each of the 'high7, 'boundary' or 'low' areas was used to esarnine cornparisons between

methods. If the true value of the parameter is Bo = (Oo,i,. . . . 0oeq9) the XlSE is defined

to be

where 6 is equai to some statistic. It can be shown that

where 13iase(9) = E ~ ( & ) - Bo

With k e d parameter go, the &ISE is estimated under each scenario' for each 'highr,

'boundarf and .lo+ nreas. by

where na is the number of areas in either the 'high7, 'boundary' or 'low' group. Similarly

the bias of ê is estimated by

For cpantitim comparisons. anaiysis of covariance (;\'\'CO\--\) was used to csamine

the effefects of estiniatc sim and data sizc on ILSE aiid bias n-hik coritrolliiig for hot11

method of analysis (.VILE. RE and CAR) and area type (high. boundary and low). Since

shrinkage can occur either 'upwards' or -downwards', the absolute value of the bias is

taken. Due to normality violations, the response was transformed using a log trans-

formation, for both MSE and bias. Graphical cornpaxisons disptayed boxplots of the

distribution of estimates for each of the methods. Maps of estimates h m simulation

iterations under select scenarios were also used to examine the differences.

4.3 Cornparison of Map Probability values

To examine MP-values, receiver operator cames (ROC) were used to determines a method's

ability to distinguish between 'high' and non-high ('boundary' and 'Iow') areas. The Y

avis represents the proportion of 'liigh' areas greater than a cut-off value c (true positive

rate). Similady. the X axis represents the proportion of non-high areas greater than the

same cut-off point (false positive rate). Therefore. a method mit h high distinguishing

ability results in a curve which is close to both the Y=l and S=O lines. The ROC pro-

vide graphical cornparison only. The area under the ROC quantifies the distinguishing

ability. A value closer to 1 indicates high discriminating ability while values closer to .5

indicate low discriminating ability. The area under the ROC was calculated using the

trapezoid rule. r\NCOVA was used to esarnine the differences in methods of analysis on

the area under the ROC curve. The response ivas transformed using the logit transforma-

tion defined as l o g ( - ) due to normaiiv violations. Maps of MP-values from selected

scenarios and simulation iteration ri-ere used to esamine the difference in rncthods.

Chapter 5

Simulation Result s

This chapter presents the results of the simulation study described in the previous c h a p

ter. Note that logarithmic transformations were used for BISE and IBias( while the logit

of the area under the ROC was used to stabilize variance in the analysis of covariance

(ANCOVA). Interactions (and contrasts )are implemented to examine differences in rates

of increase or decrease. The complete output of rcsults appear in Appendis C. Each scc-

tion provides the results numerically and graphically. The first section examines ttie

relative risk estimates with subsections focusing on MSE, bias and iteration maps. The

final section examines the MP-values with subsections focusing on ROC and iteration

maps.

5.1 Estimates

The numerical summaries (percentiles, estimated MSE and estimated bias) for the pooled

relative risk estimates by both assignment area and analysis type for each eo x O0 com-

bination appear in Appendix C. For gaphical cornparison. boxplots displaying the 2.5,

25' 50 (median). 75 and 97.5 percentiles by each combination also appear in Appendis

B. Eacli graph plots the relative risk estimates pooled for eacli of the 'liigh' (on top).

%oiinclary' (in ttie niiddle) and 'loiv' (on the bottom) arcas by cadi a~iiil>-sis nictliotl

MLE (on left). RE (in the middle) and CAR (on right). Figure 5.1 displays the boxplots

in a matrix for 8 where e~ = f y 1.3 (roms) and $ = 1.5.2-5.4.0 (columns). The leftmost

cohmn identifies the scenarios where Bo = 1.5, the middle column for 80 = 2.5 and the

nghtmost column for Bo = 4.0. The top row identifies the scenarios where eo = i, the middle row for eo = 1 and bottom row for eo = 3. Note the boxplots in the 'high' section

s e not disp!ayed QI? the came $cale over fio. For hrevity: only a subset of scenarios were

displayed here.

5.1.1 MSE

Results of the ANCOVA appear in Table 5.1. The highest level represents the main

effects including some second order interaction t e m s of interest. Sublevel term provide

cornparisons made with contrasts. The pvalues are not adjusted for multiple conipar-

isons. Scatterplots of log(L1SE) venus Bo and eo with least squared lines by method of

analysis appears in Figure 5.2 and by assignment area in Figure 5.3.

Figures 5.1. 5.2 and 5.3 indicate a decrease in M E with increased eo and an increase

in kISE witli increased Bo. These results are confirmed statistically significant with

ANCOVA results in Table 5.1.

The average log(h1SE) is lowest for the Bayesian methods. RE (-1.553) and CAR

(- l.487), than the SILE mode1 ( - 1.302). These differences are significant alt hough there

is no evidence that the two Bayesian methods are significantly different. The scatterplots

appear to indicate that the different methods of analysis decrease at a different rate over

eo (pvalue=0.0023). There does not appear to be differences between the two Bayesian

methods. The two Bayesian methods do appear to be different from the MLE method.

The scatterplots show that the Bayesian methods increase at a significantly faster rate

over Bo (pvalue=O.OOOS), than the MLE method. There is evidence that the two Bayesian

methods are different tliaci the MLE nietliod and no evidence that there arc tliffcrcrices

hctween t lie two Baysiari met liods.

Figure 5.1: Bosplots for è where eo = i! 1.3 (rows) and Bo = 1.5' 2.3.4.0 (columns)

The mean log(MSE) is lowest for the 'low' and highest for the 'high' areas The means

are -1.805. -1.599, -0.939 for loml boundal and high areas respectively. These means

are significantly different (pwlue=0.0001). There is no evidence that the 'boundary'

area is different from the 'loiv' area (pvalue=0.1337). There is maginal evidence (p

value=0.0378) that the rates of decrease by area assignment over eo are difTerent. The

rate of decrea~e for the two Bayesian methods are not diflermt. There is no evidence that

the rates of increase of log(MSE) over Bo by assigned area are different (pvalue=0.1842).

The bottom left graph of figure 5.1 shows that both RE and CAR methods for Bo = 1.5

and eo = appear to eshibit less variation than the MLE method. Note the RE method

in the 'high' geographic regions for Bo = l.5,eo = $, 95% of the coverage does not

capture Bo. This difference is reduced as both Bo and ea increase. SIoreovero the RE

method displays less variation than the CAR method for low Bo and low eo.

Table 5.1: A'JCOVA results for MSE

Effect F Value P-value

80

eo

MLE, RE, CAR

MLE vs. RE, CAR

RE vs. CAR

High, Boundary, Low

High vs Boundary, Low

Boundary vs Low -- -

O0 xMLE. RC, CAR

MLE vs. RE, CAR

RE vs. CAR

O0 x High, Boundary, Low

High vs Boundary. Low

B o t i n d q vs. Low

eo x MLE. RE, CAR

MLE vs. RE, CAR

RE vs. CAR

eo x High, Boundary, Low

High vs Boundary, Low

Boundary vs. Low

5.1.2 Bias

Results of the ANCOVA for log(1 biasl) appear in Table 5.2. Scatterplots of log(l biasl)

versus eo and Bo with least squared lines by method of analysis appears in Figure 5.4 and

by assignment area in Figure 5.5.

The log(lbias() does not appear to increase or decrease over increasing O0 (gvalue=0.3183).

There is evicience that iogij bias j j ciecreases as eo increasts.

The average log((bias1) is -2.616, -1.554 and -1.362 for the MLE, CAR and RE methods

respectively. There is evidence that these means are different. In Figure 5.1. the bILE

method appears to be centered around $ witli very little or no b i s . Botli RE and CAR

rnethods appear to be biased downward for the 'high' areas and biased upward for the

'boundary' areas. Figure 5.1 also shows that there is very little bias among the 'loi\-*

areas across al1 scenarios regardless of method. The rates of decrease over eo by niethod

do not appcar to be different. There is eridence that the rates over O0 by method arc

different (p~alue=0.03%7). Tliere is some evidence that the IWO Bayesian rnethods rates

are different.

There is evidence that the log(lbias1) differs by area assignment. The average log(lbias1)

is -2.476, -1.775 and -1.280 for the 'low'. 'boundary' and 'high' areas respectively. How-

ever, the rates of decrease over eo and Bo by area assignrnent do not appear to be different.

Figure 5.2: MSE vs O0,etl by analysis method

Figure 5.3: MSE vs Bo, ea by area assigrnent

Table 5.2: ANCOVA resuhs for Bias

E ffec t F Value P-value -

00

eo

MLE, RE, CAR

MLE vs. RE, CAR

RE vs. CAR

High, Boundary, Low

High vs Boundary, Low

Boundary vs Low

x MLE. RE. CAR

MLE vs. RE, CAR

RE vs. CAR

x High. Boundary, Low

High vs B o u n d q , Low

Boundary vs. Low

eo x MLE. RE. CAR

MLE vs. RE, CAR

RE vs. CAR

eo x High, Boundary, Low

High vs Boundary, Low

Boundary vs. Low

5.1.3 Iteration maps

bfaps of estimates from four simulated iterations are provided to allow for coniparisons

and consistency between the methods. These maps were randomly selected. The maps

are also examined to determine concordance with the simulation map of Figure 4.1:

focussing on gradation in risk over areas. This is proposed for a visual comparison o n l .

Relative risk estimates based on MLE, RE and CAR mocieis for four iterations uricler

scenario eo = f and O0 = 1.5 are mapped in Figures 5.6-5.8. Iteration maps for the

scenario that closely resembles the actual data, eo = 1 and Bo = 1.5. appears in Figure

3.9-5.11. The ranges of stiading categories tend to be different as the distributions of

the relative risk estimates are quite different under the different models. The nurnber of

regions in each range can be viewed in the legend of individual maps. .A table is provided

with counts of assigned 'red' areas (out of 49 counties) and coiints of correct assignment

of 'high' areas (out of 9). The ranges are fixcd. within scenario. to allow for an 'objective'

comparison.

The first set of maps in Figure 5.6 displays some consistency with the simulation map.

The bright red areas indicates areas with relative risk estimates greater than 1.30. The

large central regions and the south-western tip appear as 'high' areas. Spurious 'high'

risk regions appear in the south-west. initially set to predominantly 'low' areas.

The nest set of maps in Figure 5.7 illustrate modelling the snme simiilatrd data under

the RE assumption. The bright red areas indicates areas with relative risk estimates

p a t e r than 1.05. The different scale is iiscd to acconimodate the arnount of slirinkage

that occurred in the estimates, using the random effects model. This set of maps look

more variable compared to the MLE rnethod. Uçing the same range reveals a slightly

different number of counties in each. AI1 maps capture the south-west tip as 'high'. The

large central areas also tend to bc -Iiigh'. -411 rnaps revwl tlic similar spurioiis resiilts

seen in tlic prc~ioiis set of tniips. Tlicrc appcars to Iw niorr contiiiiiity I~c!t~vc~ri *higli'.

-Iio~i~i

Figure 5.4: Bias vs &, eo by analysis method

0.0 0.5 1 .O 1.5 2 0 2 5 3.0

Figure 5.5: Bias vs Bo, eo by area assignment

Figure 5.8 displays the CAR maps wliich appear smoother witli more continuity

between areas than the previous sets of maps. The dark red areas indentifies areas with

relative risk estimates greater than 1.15. The maps appear similar to the RE maps

but more spatially smoothed and have higher concordance to the simulation map. The

spurious results in the south-west r e $ m appearc; tn h~ s m o o t h ~ d out in the first and third

maps. The north-west area (Rainy River District59). originally set as % i g l ~ ' ~ appears to

have lower risk compared to other 'high' risk areas.

The relative risk estiniates based on MLE. RE and CAR models for four itcratiotis

under scenario Bo = 1.5 and ea = 1 are mapped in Figures Z.9-5.11. The SILE maps

in Figure 5.9 use the same range as the previous scenario. There appears to be more

consistency with the simulation map with regards to 'high' areas. There are still sonie

spurious results in the south-west region but not as manu in the previous scenario.

The RE maps in Figure 5.10 have somc concordance with the simulation map. The'

also appear to be much tess variable than the same method under the previous scenario.

The RE maps are noticeably smoother compared to the 'VILE maps. The north-aest

county in the rnaps appear less consistent with 'high' areas. -411 maps capture the south-

west tip as 'high'. The large central areas also tend to be 'high' along with the south-

central Ontario areas. The spurious results seen in the SILE set of maps. has been

smoothed out somewhat but still appear in the maps. As in the previous scenario. there

also appears to be more continuity between 'high'. 'boondarf and 'low' areas. Cnlike

the previous scenario, there appears to be higher concordance with the simulation rnap.

Figure 3.11 displays the CAR maps which maintain the same range a s the RE maps

for cornparison. The maps appear to have high concordance with the simulation map.

Spatial continiiiiy is esemplified in tliis set of maps. The sptirious resiilts in the south-

wcst area appcar to be snioothcd out niore iii thc secoiid arid foiirtli rnaps. Ttic riorth-

iwst ami iipprars to corrcdy cirriioristratc~ 'liigli' risk ivliirli \vas iiot t h ciisc in th

Figure 5.6: MLE maps for 8 where eo = f and go = 1.5

Figure 5.7: RE maps for 6 where e,, = ; and q = 1.5

Figure 5.8: CAR maps for 8 where eo = and Bo = 1.5

previous scenario. The sinal1 county of Sudbury Regional Sliitiicipality (53) wliich is

totally surrounded by Sudbury District ( 5 2 ) appears more coiisistent with the simulation

map than the RE maps.

nu m m - ) gzK;i 1

Figure 5.10: RE maps for 6 where eo = 1 and Bo = 1.3

Figure 5.11: CAR maps for 4 where Q = 1 and Bo = 1.5

Table 5.3 is a table of counts after assigning the same range for colour coding to al1

estimates over each method. within each of the two above scenarios. The counts are

tabulated for each of the iteration maps esamined above using the ranges defined in the

RE model in the previous section. An overall proportion is dso provided summarizing

al1 100 iterations. Under the first scenario, each of the methods appear to have similar

proportions. In the second scenario, it appears that more areas in the MLE and CAR

models, were assigned, than the RE model. The MLE and CAR methods pick up more

correct 'high' assignments. Comparing the RE and the CAR models, there is a 52%

(12.5 to 19.0) increase in the proportion of areas assigned 'high' with a 35.1% (60.4 to

81.6) increase in proportion of correct -high' assignments. Even though. overall. the CAR

mode1 assigns 28.4% (19.0 to 24.1) mare 'higli' risk estimates than the CAR niodel. tliere

is only a 10.5% (81.6 to 90.2) gain in picking up the correct 'hight risk areas.

Table 5.3: Frequency (and percentage) of model risk es-

timates assigned to 'High' areas (tt1=49) and correct as-

signment to preassigned %gh' areas (ttl=9)

MLE RE C .AR eo Iteration High Correct High Correct High Correct

Overall ( 4 . ) (83.6) (49.8) (74.7) (48.2) (81.3)

1 1.5 1 3 (26.5) 8 (88.9) 5 (10.2) 5 (55.6) 9 (1 .4 ) 7 (77.5)

5.2 Map Probability values

5.2.1 ROC curves

ROC curves were used to determine the analysis method's ability to discern between

'high' vs 'non-high' ('boundary' plus 'low') risk areas. ROC curves were used for graphical

comparisons. Only a subset of scenarios were displayed here. AI1 ROC curves appear in

Appendix LI. Figure 5.12 displays the ROC in a matrix. The rom (eo = f ,1' 3) represent

increasing eo up on the Y axis, while the columns (do = 1.5.2.5,4.0) represent increasing

Bo dong the X axis. The solid line represents the MLE method while the dotted and

daslied lines represent the RE and CAR methods, respectively.

For nunierical cornparison, the area under the ROC curves w r e calculatcd using

trapezoid rule and AXCOVA was used on data for al1 sceriarios. The resiilts of the

ANCOVA appcar in Table 5.4. Scatterplots of logit of the area iincler ROC versils eo and

$ with least squared lincs by methoci of analysis appears in Figure 3.13.

The curves in Figure 5.12 do not appear to be very differcnt within eacli scenario.

The CAR lines appears to fa11 consistently outside the RE lines. ünder scenario 4 = 1.5

and eo = f , the CAR line starts outside then crosses to the inside of the MLE line. then

cuts outside again. The RE line appears to run parallel with the CAR line, crossing

outside the MLE line at about 50% Non-high. 90% High.

.Al1 lines get further away from the diagonal as both $ and eo increase. There is also

statistical evidence that the overall discerning ability increases with increasing eo and Oo.

The mean logit of area under the curves was 1.819, 2.196 and 2.247 for the MLE,

CAR and RE methods respectively. This difference is not statistically significant (p-

value=0.7654).

Tliere is no evidence tliat the rate of increase is over eo is differcnt for cadi of tlic

met tiorls.

Figure 0.12: ROC curves for 0 where eo = f ' 1.3 (rows) and $ = 1.5.2.5.4.0 (colurnns)

CHAPTER 5. SIMULATION RESULTS

the MLE mcthod, there was not enough evidence iising .-\XCOV.l ((p=0.1957).

Table 5.4: AXCOV-A results for area under the curve

Effect F Value P-value

80 67.93 0.0001

eo 22.91 0.0001

MLE, RE, CAR 0.27 0.7654

MLE vs. RE, CAR 0.52 0.4767

RE vs. CAR 0.02 0.8839

e0 x MLE, RE? CAR 1-71 0.1957 MLE vs. RE, CAR 3.33 0.0763

RE vs. CAR 0.08 0.7753

eo x MLE, RE. CAR 0.09 0.9103

MLE vs. RE, CAR 0.19 0.6678

RE vs. CAR 0.00 0.9739

5.2.2 Iteration maps

Maps of MP-values €rom four simulated iterations are provided under scenarios Bo = 1.5

and eo = and Bo = 1.5 and eo = 1.

The MP-value maps in Figures 5145.19 coincide with the previous maps of estirnates

Figures 5.6-5.9.

For scenario Bo = 1.5 and e~ = f . the MLE map has areas scattered and discontinuous

throughout Ontario. There is little consistency, even at 90% probability? with each of

the other maps and the simulation rnap. The south-western tip does appear in al1 the

maps as 'high'.

The RE maps in Figure 5.15 appear more continuous than the MLE rnaps. More of

the 'high' areas are captured with probability geater than 60%. Spurious resuits still

exist? especially, in the south-western region. The maps are very sirnilar to estimates

maps in Figure 5.7.

The CAR maps in Figure 5.16 maintain tlie same range as tlie RE niaps and also

appear very similar to estimates map in Figure 5.8. Alore areas are captured as 'Iiigh'

than the RE method. Again, there is some concordance with the simulation map but

some spurious results do exist.

The MLE MP-value maps of scenario Bo = 1.5 and eo = 1 in Figure 5.17 appear to

capture most of the 'high' areas. Some other areas are sporadically assigned to 'high'

but the maps appear to be similar to the simulation map. There also appears to higher

concordance under this scenario than the same method under the previous scenario.

The RE maps of MP-values in Figure 5.18 appear quite similar to the estimates

maps of Figure 5.10. These maps have high concordance with the simulation map. The

'high' areas are captured by this method and the maps are very similar to their hILE

counterpart maps. Sonic spurioiis rcsults still appear in the south-west arcas.

Figure 5.19 displays the maps of 1IP-valiies iindcr the CAR asstiiriptioii. Siriiiliir

to ttic cstirtiatcs tiiaps. tticse riiaps < l i ~ p l i i sortie coticordiiiicc* witli t h siiiiiiliitiori riiiq).

Figure 5.13: Area under curve vs Bo, eo by analysis method

Figure 5.14: MLE maps for MP-value where eo = f and Bo = 1.5

Figure 5.15: RE maps for MP-value whem eo = a and Bo = 1.5

Figure 5.16: CAR maps for MP-value where eo = a and Bo = 1.5

They also tend to smooth out some of the spurioiis results identified by the precious two

methods. Most of the 'high' areas are captured by this method and the last map is the

only one that captures al1 but one area a t 80% probability The CAR method picks up

more 'high' and 'low' areas as indicated by the red and white areas indicative of higher

discerning ability.

Table 5.5 is a table of counts after assigning the same range to al1 MP-values over

each method, within each of the two above scenarios. The first scenerio shows a 55.2%

(23.0 to 35.7) increase in the proportion of 'high' assignments from the RE to the C.4R

model. There was a 35.1% (56.2 to 73.9) increase in the proportion of correct assignments.

Comparing the CAR model to the iLI LE model? there was a 47.6% (33.7 to 52.7) increaçe

in 'high' assignment. with a 12.8% (75.9 to 85.6) increase in correct assignnients. In the

scenario that resembles the data, eo = 1 and Bo = 1.3. there was a 23.0% (17.8 to 21.9) in

'high' assignments with 13.3% (73.6 to 83.4) increase in correct assignrnents comparing

the RE and CAR models. Comparing the CAR to 'VILE models. there was a 58% (21.9

to 34.9) increase in ;tiigh7 assignments. but only a 10.5% (83.4 to 92.3) increase in correct

assignments.

Table 5.5: Frequency (and percentage) of mode1 MP-

values assigned to 'High' areas (ttl=49) and correct as-

signment to preassigned 'High' areas (ttl=9)

- -

!VI LE RE CAR

eo Bo Iteration High Correct High Correct High Correct

Figure 5.17: MLE maps for MP-value where eo = 1 and Bo = 1.5

-*

Figure 5.18: RE maps for MP-value where q = 1 and Bo = 1.5

Figure 5.19: CAR maps for MP-value where eo = 1 and 00 = 1.5

Chapter 6

Discussion

Using both numerical and graphical analyses, it was found that both 'VISE and bias

decreased while increasing the amount of data (eo). Moreover. increasing the assigned

'high' relative ri& estimate ( O 0 ) results in increasing bISE and bias neither increased nor

decreased. The area under the ROC increased tvith an increase in both data and nt111

relative risk.

The MSE for the Bayesian methods increased over null relative riskt but \vas consis-

tently lower than the MSE for the MLE method which remained relatively unchanged

(particularly in the ' b o u n d q ' and 'low' areas). Ail three methods appear to converge to

the same MSE for large assigned relative risk. Similarly, as data increased. the Bayesian

methods (with lower M E ) approached the same MSE.

The bias analysis indicated that the bias in the Bayesian methods nrere consistently

higher than the MLE method. This result was noticed over both increased data and nul1

relative risk. The rates of decrease over the amount of data were simiIar for the three

different methods. There tvas evidence that the rates were different over null relative risk.

Noreover. therc was evidenre that the rates for the Bayesian rnethods w r e different. witli

the CAR methotl increasing (similar to tlie MLE methocl) and ttir clecreasing for tlie RE

mctho(1.

The area under the ROC captured by the Bayesian methods was consistentl- larger

over both increasing 'high' relative risk esimate and data examined in this stiid. -41-

though the scatterplots (with the least squared lines) appeared to indicate steeper rise in

dope over increased null relative risk for the Bayesian methods, there was no statistical

evidence for this difference. The rates of increase over increased data multiplier were

relatively similar between al1 methods.

By increasing the null relative risk in 'high' areas, the magnitude of the difference in

assigned relative risk increases between 'high' and 'non-high' regions. This is an artifact

of the simulation study. The pooled estimates for boundary areas consistently lie between

the low and high areas. Smoothing pulls estimates of the boundary areas farther away

frorn the null relative risk assigned as one, to values greater tlian one. Sirnilarly. those

'high' areas are smoothed downcvards below the null relative risk (Oo = 1.5.2.5 or 4).

The RE method pulls estimates towards a global mean. whereas the CAR method pulls

estimates towards local means. The amount of pull or smoothing depends on a relative

weight determined bu the amount of data.

Recall the tight distributions for the RE method in scenario O0 = 1.5. eo = f . The

prior distribution on the random effects was a Normal distribution with relatively large

variance. Even though diffuse priors were used, there was a considerable eRect on the

estimates in scenarios with small amounts of data (or low e o ) The amount of data did

not affect the &ISE from the CAR method as much as it did for the RE method because

the variance was constrained by adjacent areas.

Fundamental to Bayesian methods is that with the increase in data, the MSE and

b i s becorne similar to the MLE method. That was shown here. The prior distribution

becornes 'less important' to the postenor distribution, since the likelihood dominates the

posterior as data increases [8, 131.

It was sliown tii21t the cstimatcs from tlic Bayesian nicthotls IKE bitiscd with srnaII

M E . The iisc of tlic iteration iiiaps iwrc iiiore ticlphil to detcrniiric corisistcricy with tlic!

simulation map. The maps became less variable, within rnethod. as both nul1 relative

nsk and data increased. The Bayesian methods revealed continuity in the maps which

were more similar to the simulation map than the MLE maps. The Bayesian random

effects models, through the use of a prior distribution on the effects, allowed one to

control the MSE while allowing spatial (or non-spatial) continuity. The maps of the first

scenario appeared quite variable. Although the prior distribution and hyperparameten

were fked over al1 scenarïos in the simulation study, the analysis require more thought

within each scenario in practice. The 'tweaking' under each scenano: may have lead to

more favourable results.

The south-western tip, Essex county (37). originally 'high', appeared as 'high' in most

of the iteration maps. There is a t least a 2.5 fold increase in the espected number of

deaths for Esses county than the nest largest count among 'high' areas.

Spurious results appeared in many of the maps. These were identified as regions in the

simulation where number of events was driven largely due to relatively small population

(E:) . Gelinan and Pricc (i41 state that maps of posterior means (dong with means and

p-values) are subject to systematic artifacts related to county populations.

The ability to distinguish between 'high' versus non-high areas appeared consistent

between the rnethods. The discerning ability of the Bayesian methods may have been

darnpened by combining 'boundary' and 'low' in the non-high group. Bernardelli et al.

(1995) [5] showed a marked advantage in the Bayesian methods with the ROC curves.

However, it is unclear whether the 'boundary' areas were included in the non-high areas.

As Bo increases, in the 'higli' regions only, the difference between 'high' and non-high

areas increase. Therefore, by design, al1 methods capture more area under the ROC

curve. The area under the ROC curves gets larger as eo increases. Recall as eo increases.

>ISE decreases. -4s this happens? the variance around the estimates decrease. Therefore.

the 'higli' region S IP-values incrcascs.

Tlic iteratioii niaps for tlic LIP-~iliics wcrc gcncrally sirriilar to tlic cstirri;it(! rtiaps.

The MLE maps had larger MP-values as eo increased. Similarly. the Bayesian methods

too' had a greater number of larger MP-values.

Gelman and Price [NI define marginal z-score as a score that measures the discrep

ancy of the county estimate with respect to its marginal distribution, averaging over

the unknown county parameter Br. The Bayesian MP-value behaves similarly depending

on county and global estimates which, in turn, only depend on the samples from the

marginal posterior distribution.

Since the MP-value is a function of the estimates, the iteration maps of MP-values

yield similar results to the estimates maps since Bayesian methods for estimates and

probabilities synthesize the raw effect and 'sample' size value. This was noticed in maps

presented in Beranardelli et al. (1992? 1995) [4, 51.

It is possible that. by design of the simulation study, the ?VILE method is favoured

over the spatial model. Even though the 'boundary? areas (those areas adjacent to 'high'

risk areas) are considered. they are always assigned a relative risk of one. As the assigned

'high' area risk increases. the 'boundary' areas result in relative risk estimates larger than

one. This may be unfair to the spatial model.

Chapter 7

Conclusions

Bayesian met hods exploit the use of random effects model to successfiill y reduce variance.

even thoiigh the estimates of relative risk are biased. Therc did not appear to be an

increase