Modelling Count Data: Outline• Characteristics of count data and the Poisson distribution • Applying the Poisson: Flying bomb strikes in South London• Deaths by horse-kick: as a single-level model Poisson model fitted in MLwiN

• Overdispersion: types and consequences, the unconstrained Poisson, the Negative Binomial

•Taking stock: 4 distributions for modeling counts

•Number of extramarital affairs: the incidence rate ratio (IRR); handling categorical & continuous predictors; comparing model with DIC•Titanic survivor data; taking account of exposure, the offset

•Multilevel Poisson and NBD models; estimation and VPC•Applications: HIV in India and Teenage employment in Glasgow•Spatial models: Lip cancer in Scotland; respiratory cancer in Ohio

2

Some characteristics of count data

Very common in the social sciencesNumber of children; Number of marriagesNumber of arrests; Number of traffic accidentsNumber of flows; Number of deaths

Counts have particular characteristics•Integers & cannot be negative•Often positively skewed; a ‘floor’ of zero In practice often rare events which peak at 1,2 or 3 and rare at higher values

Modelled by •Logit regression models the log odds of an underlying propensity of an outcome;•Poisson regression models the log of the underlying rate of

occurrence of a count.

3

The Poisson distribution results if the underlying number of random events per unit time or space have a constant mean () rate of occurrence, and each event is independent Simeon-Denis Poisson (1838) Research on the Probability of Judgments in Criminal and Civil Matters

Applying the Poisson: Flying bomb strikes in South LondonKey research question: falling at random or under a guidance systemIf random independent events should be distributed spatially as a Poisson

distributionDivide south London into 576 equally sized small areas (0.24km2)Count the number of bombs in each area and compare to a PoissonMean rate = = [229(0) + 211(1) + 93(2) + 35(3) + 7(4)

+1(5)]/576= 0.929 hits per unit square

0 1 2 3 4 5+

Observed 229 211 93 35 7 1

Poisson 227.5 211.3 98.15 30.39 7.06 1.54

Very close fit; concluded random

Theoretical Poisson distribution I

4

Theoretical Poisson distribution II

Probability mass function(PMF) for 3 different mean occurrences

Mean: 10

Mean: 4Mean: 1

0 10 20 30

0.0

0.1

0.2

0.3

0.4

conts

Pro

bPoisson PMF for different means

When mean =1; very positively skewedAs mean occurrence increases (more common event), distribution approaches Gaussian; So use Poisson for ‘rareish’ events; mean below 10Fundamental property of the Poisson: mean = varianceSimulated 10,000 observations according to PoissonMean Variance Skewness0.993 0.98 1.004.001 4.07 0.5010.03 10.38 0.33Variance is not a freely estimated parameter as in Gaussian

5

0 1 2 3 4 5+

0

50

100

Deaths

Fre

quency

Bortkewicz L von.(1898) The Law of Small Numbers, LeipzigNo of soldiers killed annually by horse-kicks in Prussian cavalry; 10

corps over 20 years (occurrences per unit time)The full data 200 corps years of observations

As a frequency distribution (grouped data)Deaths 0 1 2 3 4 5+Frequency 109 65 22 3 1 0

Mean Variance Number of obs0.61 0.611 200

Interpretation: mean rate of 0.61 deaths per cohort year (ie rare)Mean equals variance, therefore a Poisson distribution

Death by Horse-kick I: the data

1 1 0 0 2 0 2 0 1 1 2 0

0 0 2 0 0 2 0 1 1 1 1 ……

6

Death by Horse-kick II: as a Poisson

With a mean (and therefore a variance) of 0.61 Deaths 0 1 2 3 4 5+Frequency 109 65 22 3 1 0Theory 109 66 20 4 1 1

Again :The Poisson results if underlying number of random events per unit time or space have a constant mean () rate of occurrence, and each event is independent

Formula for Poisson PMF 2, ,1,0 ,!

)(

xx

exPMF

x

e is base of the natural logarithm (2.7182) is the mean (shape parameter): the average number of events in a

given time interval x! is the factorial of x

EG: mean rate of 0.6 accidents per corps year; what is probability of getting 3 accidents in a corps in a year?

0226.01*2*3

6.0)3(

36.0

e

PMF

7

Horse-kick III: as a single level Poisson model

1)var(

....)log(

)(~

)(~

0

5.0^

0

1100

....

00

1100

i

ii

iii

xx

i

iiii

ii

e

z

xx

zey

Poissony

e ii

Modeling on the loge scale, cannot make prediction of a negative count on

the raw scale Level-1 variance is constrained to be an exact Poisson, (variance =mean)

General form of the single-level model

Observed count is distributed as an underlying Poisson with a mean rate of occurrence of ;

That is as an underlying mean and level-1 random term of z0 (the Poisson ‘weight’)

Mean rate is related to predictors non-linearly as an exponential relationship

Model loge to get a linear model (log link)

The Poisson weight is the square root of estimated underlying count, re-estimated at each iteration

Variance of level-1 residuals constrained to 1,

8

Horse-kick IV: Null single-level Poisson model in MLwiN

The raw ungrouped counts are modeled with a log link and a variance constrained to be equal to the mean

-0.494 is the mean rate of occurrence on the log scale

Exponentiate -0.494 to get the mean rate of 0.61

0.61 is interpreted as RATE; the number of events per unit time (or space), ie 0.61 horse-kick deaths per corps-year

9

So far; equi-dispersion, variances equal to the mean Overdispersion: variance > mean; long tail, eg LOS (common) Un-dispersion: variance < mean; data more alike than pure Poisson process; in multilevel possibility of missing level

Consequences of overdispersion Fixed part SE’s "point estimates are accurate but they are not as precise as

you think they are" In multilevel, mis-estimate higher-level random part Apparent and true overdispersion: thought experiment•: number of extra-marital affairs: men women with different means

Who?Mean Var Comment

Men&Women

0.55 0.76 Overdisp

Men 1.00 1.00 Poisson

Women 0.10 0.10 Poisson

apparent: mis-specified fixed part, not separated out distributions with different meanstrue: genuine stochastic property of more inherent variabilityin practice model fixed part as well as possible, and allow for overdisperion

Overdispersion I: Types and consequences

10

Overdispersion II: the unconstrained Poisson

Not significantly different from 1; No evidence that this is not a Poisson distribution

•Deaths by horse-kick

•estimate an over- dispersed Poisson

• allow the level-1 variance to be estimated

11

•Instead of fitting an overdispersed Poisson, could fit a NBD modelHandles long-tailed distributions; An explicit model in which variance is greater than the meanCan even have an over-dispersed NBD

vy

zz

xx

zezey

NBDy

iiii

iiii

iii

iiiiii

ii

/)|var(

;

....)log(

)(~

)(~

2

^

1

5.0^

0

1100

1000

•Same log-link but NBD has 2 parameters for the level-1 variance; that is quadratic level-1 variance, v is the overdispersion parameter

Overdispersion III: the Negative binomial

12

Overdispersion IV: the Negative binomial

Horse=kick analysis: Null single-level NBD model; essentially no change, v is estimated to be 0.00 (see with Stored model; Compared stored model)

Overdispersed negative binomial;No evidence of overdispersion; deaths are

independent

13

First Bernoulli and Binomial Bernoulli is a distribution for binary discrete events y is observed outcome; ie 1 or 0 E(y) = ; underlying propensity/probability for occurrence Var(y) = /1-

Mean: Variance: /1-

0.01 0.01/0.99 = 0.0099

0.5 0.5*0.5 = 0.2500

0.99 0.99/0.01 = 0.0099

Binomial is a distribution for discrete events out of a number of trials

y is observed outcome; n is the number of trials, E(y) = ; underlying propensity/ of occurrence Var(y) = [/(1- )]/n

Linking the Binomial and the Poisson: I

• Least variation when denominator is large (more reliable), and as underlying probability approaches 0 or 1

14

• Poisson is limit of a binomial process in which prob → 0, n→∞

• Poisson describes the probability that a random event will occur in a time or space interval when the probability is very small, the number of trials is very large, and each event is independent

• EG : The probability that any automobile is involved in an accident is very small, but there are very many cars on a road in a day so that the distribution (if each crash is independent) follows a Poisson count

• If non-independence of crashes (a ‘pile-up’), then over-dispersed Poisson/NBD, latter used for ‘contagious’ processes

• In practice, Poisson and NBD used for rare occurrences, less than 10 cases per interval,& hundreds or even thousands for

denominator/ trials [Clayton & Hills (1993)Statistical Models in Epidemiology OUP]

Linking the Binomial and the Poisson: II

15

If common rate of occurrence (mean >10) then use raw counts and Gaussian distribution (assess Normality assumption of the residuals)

If rare rate of occurrence, then use over-dispersed Poisson or NBD; the level-1 unconstrained variance estimate will allow assessment of departure from equi-dispersion; improved SE’s, but biased estimates if apparent overdispersion due to model mis-specification

Use the Binomial distribution if count is out of some total and the event is not rare; that is numerator and denominator of the same order

Taking stock: 4 distributions for counts

•Mean variance relations for 4 different distributions that could be used for counts

16

Affairs Cons Children Religious YearsMarried Age

0 1 NoKids Slightly 10-15 years 37.0

0 1 NoKids Somewhat < 4 years 27.0

3 1 NoKids Slightly < 4 years 27.0

0 1 WithKids Anti 6-10 years 32.0

3 1 WithKids Slightly < 4 years 27.0

0 1 WithKids Very 6-10 years 57.0

Single categorical predictor Children with NoKids, as the base

Modeling number of extra-marital affairs Single- level Poisson with Single categorical Predictor

Extract of raw data (601 individuals from Fair 1978)

Fair, R C(1978) A theory of extramarital affairs, Journal of Political Economy, 86(1), 45-61

Understanding customized predictions…………….

17

Log scale: NoKids: -0.092 WithKids-0.092+0.606= 0.514First use equation to get underyling log-number of events then exponeniate to get estimated count (since married)

As mean/median counts NoKids: expo(-0.092)= 0.91211 Withkids: expo(-0.092 + 0.606) = 1.6720

Those with children have a higher average rate of affairs (but have they been married longer?)

Single- level Poisson with Single categorical Predictor

Understanding customized predictions

18

Modeling number of extra marital affairs: the incidence rate ratio (IRR)

BUT can get this directly from the model by exponentiating the estimate for the contrasted category

expo(0.606) = 1.8331

So far: mean countsNoKids: expo(-0.092)= 0.91211 Withkids: expo(-0.092 + 0.606) = 1.6720

But also as IRR; comparing the ratio of those with and without kidsIRR = 1.6720/0.91211 = 1.8331

That is Withkids have a 83% higher rate

Rules: a) exponeniating the estimates for the (constant plus the contrasted

category) gives the mean rate for the contrasted category b) ) exponeniating the contrasted category gives IRR in comparison to base

category

19

• As always, the estimated coefficient is the change in response corresponding to a one unit change in the predictor

• Response is underlying logged count

• When Xi is 0 (Nokids); log count| Xi is 0 = β0 • But when Xi is 1(Withkids); log count| Xi is 1 = β0 + β1 X1i

• Subtracting the first equation from the second gives (log count| Xi is 1)-(log count| Xi is 0) = β1

• Exponentiating both sides gives (note the division sign)

(count| Xi is 1)/(count| Xi is 0) = exp(β1)

• Thus, exp(β1) is a rate ratio corresponding to the ratio of the mean number of affairs for a with-child person to the mean number of

affairs without-child person

• Incidence: number of new cases• Rate: because it number of events per time or space• Ratio: because its is ratio of two rates

Why is the exponentiated coefficient a IRR?

20

Modeling number of extra-marital affairs: changing the base category

Previously contrasted category: Withkids: +0.606 Now contrasted category: Nokids : -0.606 Changing base simply produces a change of sign on the loge scale Exponentiating the contrasted category: Before: expo(+0.606) = 1.8331 Now: expo(-0.606) = 0.5455

Doubling the rate on loge scale is 0.693; Halving the rate on loge scale

is -0.693

IRR of 0.111 = IRR of 9-fold increase, difficult to appreciate Advice: choose base category to be have the lowest mean rate;

get positive contrasted estimates;always then comparing a larger value to a base of 1

21

Affairs: modeling a set of categorical predictors

Customised predictions: mean rate, IRR, & graph with 95% CI’s

YrsMarried A: Log estimate

Mean rate(expo A)

B: Differential Log Estimate

IRR(Expo B)

< 4 years -0.297 0.743 0.000 1.000

4-5 years -0.297 + 0.773 1.616 0.773 2.166

6-10 years -0.297 + 0.932 1.888 0.932 2.540

10-15 years -0.297 + 1.041 2.103 1.041 2.832

A model with ‘years married’ included with < 4 years as base

22

Affairs: modeling a continuous predictor Age

To get mean rate as it changes with ageExpo (-0.990 + 0.149*(Age-17) -0.003*(Age-17)2)

To get IRR in comparison for a person aged 33 compared to 17Expo( 0.149*16) –(0.003 *162) (drop the constant!)

Easiest interpreted as graphs!

Age as a 2nd order polynomial centred around 17 years (the youngest person in survey; also lowest rate

23

Affairs: a SET of predictors & models

Term Poisson SE Extra SE NBD SE

FixedCons

-1.33 0.20 -1.33 0.53 -1.56 0.51

Years Married4-5 years

0.89 0.13 0.89 0.34 1.20 0.36

6-10 years 1.00 0.14 1.00 0.38 1.01 0.41

10-15 years 1.31 0.15 1.31 0.40 1.42 0.42

ReligiousSomewhat

-0.00 0.15 -0.00 0.39 0.06 0.37

Slightly 0.95 0.14 0.95 0.37 1.28 0.39

Non 0.87 0.14 0.87 0.37 1.16 0.38

Anti 1.36 0.16 1.36 0.41 1.52 0.47

ChildrenWithKids

-0.06 0.11 -0.06 0.28 0.25 0.29

Age(age-17)^1

0.05 0.02 0.05 0.05 0.01 0.05

(age-17)^2 -0.00 0.00 -0.00 0.00 0.00 0.00

Random Part

Var 1.00 0.00 6.78 0.39 1.00 0.00

NBD var (v) 5.15 0.36

Notice substantial overdispersion

Poisson & extra-Poisson no change in estimates; some change to NBD

Notice larger SE’s when allow for overdispersion; NBD most conservative

In full model, WithKids not significant

24

NBD model for Marital Affairs

IRR of 1 for Under 4 years married, Very religious, No children, aged 17

Previous Age effect is really length of marriageUsed comparable vertical axes, range of 4

25

With 95% confidence intervals NB that they are asymmetric on the unlogged scale

NBD model for number of Extra-Marital Affairs

26

• Likelihood and hence the Deviance are not available for Poisson and NBD models fitted by quasi-likelihood DIC criterion available though MCMC; typically needs larger number of iterations than Normal & Binomial (suggested default is 50k not 5k)

Terms Null + Relig + YrsMar +Kids +Age2

Cons 0.375 -0.139 -1.131 -1.135 -1.352

Slightly 0.818 1.031 1.040 0.963

Somewhat -0.017 0.050 0.056 0.005

Anti 1.086 1.406 1.412 1.367

Not 0.641 0.917 0.925 0.881

4-5 years 0.918 0.943 0.882

6-10 years 1.077 1.100 0.994

10-15 years 1.265 1.290 1.303

WithKids -0.029 -0.051

(age-17)^1 0.048

(age-17)^2 -0.001

DIC: 3421.5 3296.8 3065.8 3068.0 3052.5

∆DIC -124.653 -231.074 +2.176 -15.440

Pd 1 5 8 9 11

Affairs: Evaluating a sequence of models using DIC

Currently MCMC not available in MLwiN for over-dispersed Poisson nor NBD models; so have to use Wald tests in Intervals and tests window

27

So far, response is observed count, now we want to model a count given exposure: EG only 1 high-class female child survived but only 1 exposed!

Survive Case Age Gender Class Exp SR

14 168 Adult Men Mid 63.7 22.0

75 462 Adult Men Low 175.2 42.8

13 48 Child Men Low 18.2 71.4

57 175 Adult Men High 66.4 85.9

14 31 Child Women Low 11.8 119.1

76 165 Adult Women Low 62.6 121.5

80 93 Adult Women Mid 35.3 226.9

140 144 Adult Women High 54.6 256.4

13 13 Child Women Mid 4.93 263.7

1 1 Child Women High 0.38 263.7

11 11 Child Men Mid 4.17 263.7

5 5 Child Men High 1.89 263.7

Latter often used to treat the exposure as a nuisance parameter & allows calculation of Standardised Rate’s SRi = Obsi/Expi * 100

Titanic survivor data: Taking account of exposure

Here 2 possible measures of exposurea) the number of potential cases; could use a binomialb) the expected number if everyone had the same exposure (i indexes cell) Death rate = Total Deaths/ Total exposed =817/1316 = 0.379179 Expi = Casesi * Survival rate

Previous examplesHorsekick: Exposure removed by design: 200 cohort yearsAffairs: included length of marriage; theoretically interesting

28

......1100 ii xx

......1100 ii xx

...110 xxe ioi

Model: SRi = (Obsi/Expi) = F(Agei, Genderi, Classi)

Where i is a cell, groups with same characteristics

Aim: are observed survivors greater or less than expected, and how these ‘differences’ are related to a set of predictor variables?

As a non-linear model: E(SRi)= E(Obsi/Expi) =

As a linear model (division of raw data is subtraction of a log)Loge (Obsi) - Loge(Expi)) =

As a model with an offset, moving Loge(Expi) to the right-hand side, and

constraining coefficient to be 1; ie Exp becomes predictor variable

Loge (Obsi) = 1.0* Loge(Expi) +

NB MLwiN automatically loge transforms the observed response; you have to create the loge of the expected and declare it as an offset

Sir John Nelder

Modeling SR’s: the use of the OFFSET

29

Include the offsetAs a saturated model; ie Age *Gender*Class, (2*2*3), 12 terms for 12 cells

Make predictions on the loge scale (must

include constant); exponentiate all terms to get departures from the expected rate, that is modeled SR’s

Ages Gender Class LogEstimate

Modeled SR

SR=Obs/Exp

Child Women Low 0.17 1.19 1.19

Child Women Mid 0.97 2.64 2.64

Child Women High 0.97 2.64 2.64

Child Men Low -0.34 0.71 0.71

Child Men Mid 0.97 2.64 2.64

Child Men High 0.97 2.64 2.64

Adult Women Low 0.19 1.21 1.21

Adult Women Mid 0.82 2.27 2.27

Adult Women High 0.94 2.56 2.56

Adult Men Low -0.85 0.43 0.43

Adult Men Mid -1.52 0.22 0.22

Adult Men High -0.15 0.86 0.86

Surviving on the Titanic as a log-linear model

30

Remove insignificant terms starting with 3-way interactions for High*women*children

Customized predictions: Very low rates of survival for Low and Middle class adult men; large gender gap for adults, but not for children

Titanic survival: parsimonious model

31

Modeled SR’s and descriptive SR’s Ordered by worse survival Estimated SR’s only shown if 95% CI’s do not include 1.0

Age Gender Class Est SR SR

Adult Men Mid 0.22 0.22

Adult Men Low 0.42 0.43

Adults Women Low 1.21 1.21

Child Men High 1.71 2.64

Adults Women Mid 2.26 2.27

Child Women High 2.56 2.64

Adult Women High 2.57 2.56

Child Women Mid 2.62 2.64

Child Men Mid 2.68 2.64

Child Women Low * 1.19

Child Men Low * 0.71

Adults Men High * 0.86

Titanic survival: parsimonious model

32

1)var(

),0(~

....)log(

)(~

)(~

0

5.0^

0

200

01100

00

i

iji

uj

jijijij

ijijijij

ijij

e

z

Nu

uxx

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Poissony

20u

Two-level multilevel Poisson

One new term, the level 2 differential, on the loge scale, is assumed to come from Normal distribution with a variance of

Can also fit Poisson multilevel with offset and NBD multilevel in MLwiN

33

• Same options as for binary and binomial

• Quasi-likelihood and therefore MQL or PQL fitted using IGLS/RIGLS; fast, but no deviance (have to use Wald tests); may be troubled by small number of higher-level units; simulations have shown that MQL tends to overestimate the higher-level variance parameters

• MCMC estimates; good quality and can use DIC to compare Poisson models; but currently MCMC is not possible for extra-Poisson nor for NBD

• MCMC in MLwiN often produces highly correlated chains (in part due to the fact that the parameters of the model are highly correlated; variance =mean) Therefore requires substantial number of simulations; typically much larger than for Normal or for Binomial

Estimation of multilevel Poisson and NBD in MLwiN I

34

• Possibility to output to WinBUGS and use the univariate AR sampler and Gamerman (1997) method which tends to have less correlated chains, but WinBUGS is considerably slower generally [Gamerman, D. (1997) Sampling from the posterior distribution in generalized linear mixed models.

Statistics and Computing 7, 57-68]

• Advice: start with IGLS PQL; switch to MCMC, be prepared to make 500,000 simulations (suggest use 1 in 10 thinning to store the chains); use Effective sample size to assess required length of change, eg need ESS of at least 500 for key parameters of interest; compare results and contemplate using PQL and over-dispersed Poisson

• Freely available software MIXPREG for multilevel Poisson counts including offsets; uses full information maximum likelihood estimated using quadrature http://tigger.uic.edu/~hedeker/mixpcm.PDF

Estimation of multilevel Poisson and NBD in MLwiN II

35

Lev2var: 1.0

Lev2var: 1.5Lev2var: 2.0

543210

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

VP

C

of Level-2 variance on Loge scaleVPC versus mean rate for 3 different values

Mean Rate

e uii xxLower

]2/...)[( 201100

eeuiiuii xxxx

Upper

]...)(2[]2...)(2[ 2

011002

01100

)/( LowerUpperUpperVPC

20u

Can either use Simulation method to derive VPC (modify the binomial procedure) Use exact method: http://people.upei.ca/hstryhn/iccpoisson.ppt (Henrik

Stryhn) VPC for two level random intercepts model (available for other models)

Clearly VPC depends on and

VPC for Poisson models

Aim: investigate the State geography of HIV in terms of risk

Data: nationally representative sample of 100k individuals in 2005-2006

Response: HIV sero-status from blood samples

Structure: 1720 cells within 28 States; cells are a group of people who share common characteristics [Age-Groups(4), Education(4), Sex(2), Urbanity(2) and State (28)]

Rarity: only 467 sero-positives were found

Model: Log count of number of seropositives in a cell related to an offset of Log expected count if national rates appliedPredictors of Age, Sex and Education and UrbanityTwo-level multilevel Poisson, extra-Poisson & NBD

Modeling Counts in MLwiN: HIV in India

37

HIV in India: Standardized Morbidity Rates

Higher educated females have the lowest risk, across the age-groups

38

HIV in India: some results

Risks for different States relative to living in urban and rural areas nationally.

39

Modeling proportions as a binomial in MLwiN

;...1

log 0110 juxe

),0(~ 2

00 uj Nu

• exactly the same procedure as for binary models

1,)ˆ1(ˆ

, 2

eii

iiiiiii n

zzey

• except that observed y is a proportion (not just 1 and 0, the denominator (n) is variable (not just 1) and extra-dispersion at level 1 is allowed (not just exact binomial)

Reading:

Subramanian S V, Duncan C, Jones K (2001) Multilevel perspectives on modeling census data Environment and Planning A 33(3) 399 – 417

40

Data: teenage employment In Glasgow districts• “Ungrouped” data that is individual data • Model binary outcome of employed or not and two individual predictors

Name Person District Employed Qualif Sex

Craig 1 1 Yes Low Male

Subra 2 1 Yes High Male

Nina 3 1 Yes Low Fem

Min 4 1 No Low Fem

Myles 5 1 No High Male

Sarah 12 50 Yes High Fem

Kat 13 50 No Low Fem

Colin 14 50 Yes Low Male

Andy 15 50 No High Male

41

Same data as a multilevel structure: a set of tables for each district

GENDERQUALIF MALE FEMALE Postcode UnErateLOW 5 out of 6 3 out of 12 G1A 15%HIGH 2 out of 7 7 out of 9

LOW 5 out of 9 7 out of 11 G1B 12%HIGH 8 out of 8 7 out of 9

LOW 3 out of 3 - G99Z 3%

HIGH 2 out of 3 out of 5

• Level 1 cell in table• Level 2: Postcode sector• Margins: define the two categorical predictors • Internal cells: the response of 5 out of 6 are

employed

42

Teenage unemployment: some results from a binomial, two-level logit model

43

Spatial Models as a combination of strict hierarchy and multiple membership: counts are commonly

used

Multiple membership defined by common boundary; weights as function of inverse distance between centroids of areas

MLK

J IHG

FED

C

BA Person in A

Affected by A(SH) and B,C,D (MM)

Person in H

Affected by H(SH) and E,I,K,G (MM)

44

• Response: observed counts of male lip cancer for the 56 regions of Scotland (1975-1980)

• Predictor: % of workforce working in outdoor occupations (Agric;For;

Fish) Expected count based on population size

• Structure areas and their neighbours defined as having a common border (up to 11); equal weights for each

neighbouring region that sum to 1

Rate of lip cancer in each region is affected by both the region itself and its nearest neighbours after taking account of outdoor activity

• Model Log of the response related to fixed predictor, with an offset, Poisson distribution for counts;

NB Two sets of random effects1 area random effects; (ie unstructured; non-spatial variation);2 multiple membership set of random effects for the neighbours of

each region

Scottish Lip Cancer Spatial multiple-membership model

45

MCMC estimation: 50,000 draws

Fixed effects:Offset and

Well-supported + relation

Well-supported Residual

neighbourhood effect

Poisson model

NB: Poisson highly correlated chains

46

Scottish Lip Cancer: CAR model

CAR: CAR one set of random effects, which have an expected value of the average of the surrounding random effects; weights divided by the number of neighbours

)(

,

2

/ where

)/,(~

ineighjijjii

iuii

nuwu

nuNu

where ni is the number of neighbours for area i and the weights are typically all 1

MLwiN: limited capabilities for CAR model; ie at one level only (unlike Bugs)

47

MCMC estimation: CAR model, 50,000 draws

Fixed effects:Offset and

Well-supported + relation

Well-supported Residual

neighbourhood effect

Poisson model

48

NB Scales: shrinkage

49

Ohio cancer: repeated measures (space and time!)

• Response: counts of respiratory cancer deaths in Ohio counties• Aim: Are there hotspot counties with distinctive trends? (small

numbers so ‘borrow strength’ from neighbours)

• Structure: annual repeated measures (1979-1988) for counties Classification 3: n’hoods as MM (3-8 n’hoods)Classification 2: counties (88)Classification 1: occasion (88*10)

• Predictor: Expected deaths; Time

• Model Log of the response related to fixed predictor, with an offset, Poisson distribution for counts (C1);Two sets of random effects

1 area random effects allowed to vary over time; trend for each county from the Ohio distribution (c2) 2 multiple membership set of random effects for the

neighbours of each region (C3)

50

MCMC estimation: repeated measures model, 50,000 draws

General trend

Variance function for

between county time

trend

N’hood variance

Default priors

51

Respiratory cancer trends in Ohio: raw and modelled

Red: County 41 in 1988; SMR = 77/49 = 1.57Blue: County 80 in 1988: SMR= 6/19 = 0.31

52

General References on Modeling CountsAgresti, A. (2001) Categorical Data Analysis (2nd ed). New York: Wiley.Cameron, A.C. and P.K. Trivedi (1998). Regression analysis of count data, Cambridge University PressHilbe, J.M. (2007). Negative Binomial Regression, Cambridge University Press. McCullagh, P and Nelder, J (1989). Generalized Linear Models, Second Edition. Chapman & Hall/CRC. 

On spatial modelsBrowne, W J (2003) MCMC Estimation in MLwiN; Chapter 16 Spatial models Lawson, A.B., Browne W.J., and Vidal Rodeiro, C.L. (2003) Disease Mapping using WinBUGS and MLwiN Wiley. London (Chapter 8: GWR)