1. 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 sciences
Number of children; Number of marriages
Number of arrests; Number of traffic accidents
Number 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. Theoretical Poisson distribution I
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 London
Key research question: falling at random or under a guidance system
If random independent events should be distributed spatially as a Poisson distribution
Divide south London into 576 equally sized small areas (0.24km2)
Count the number of bombs in each area and compare to a Poisson
Mean rate =  = [229(0) + 211(1) + 93(2) + 35(3) + 7(4) +1(5)]/576
= hits per unit square 1 2 3 4 5+
Observed
229
211 93 35 7
Poisson
227.5
211.3
98.15
30.39
7.06
1.54
Very close fit; concluded random

4. Theoretical Poisson distribution II
Probability mass function(PMF) for 3 different mean occurrences
When mean =1; very positively skewed
As mean occurrence increases (more common event), distribution approaches Gaussian;
So use Poisson for ‘rareish’ events; mean below 10
Fundamental property of the Poisson: mean = variance
Simulated 10,000 observations according to Poisson
Mean Variance Skewness
Variance is not a freely estimated parameter as in Gaussian

5. Death by Horse-kick I: the data
Bortkewicz L von.(1898) The Law of Small Numbers, Leipzig
No 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
Frequency
Mean Variance Number of obs
Interpretation: mean rate of deaths per cohort year (ie rare)
Mean equals variance, therefore a Poisson distribution

6. Death by Horse-kick II: as a Poisson
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
With a mean (and therefore a variance) of 0.61
Deaths
Frequency
Theory
Formula for Poisson PMF
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?

7. Horse-kick III: as a single level Poisson model
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,
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)

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
is the mean rate of occurrence on the log scale
Exponentiate 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. Overdispersion I: Types and consequences
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
apparent: mis-specified fixed part, not separated out distributions with different means
true: genuine stochastic property of more inherent variability
in practice model fixed part as well as possible, and allow for overdisperion
Who?
Mean
Var
Comment
Men&
Women
0.55
0.76
Overdisp
Men
1.00
Poisson
0.10

10. Overdispersion II: the unconstrained Poisson
Deaths by horse-kick
estimate an over- dispersed Poisson
allow the level-1 variance to be estimated
Not significantly different from 1;
No evidence that this is not a Poisson distribution

11. Overdispersion III: the Negative binomial
Instead of fitting an overdispersed Poisson, could fit a NBD model
Handles long-tailed distributions;
An explicit model in which variance is greater than the mean
Can even have an over-dispersed NBD
Same log-link but NBD has 2 parameters for the level-1 variance; that is quadratic level-1 variance, v is the overdispersion parameter

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. Linking the Binomial and the Poisson: I
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 =
0.5
0.5*0.5 =
0.99
0.99/0.01 =
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
Least variation when denominator is large (more reliable), and as underlying probability approaches 0 or 1

14. Linking the Binomial and the Poisson: II
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]

15. Taking stock: 4 distributions for counts
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
Mean variance relations for 4 different distributions that could be used for counts

16. Modeling number of extra-marital affairs Single- level Poisson with Single categorical Predictor
Extract of raw data (601 individuals from Fair 1978)
Affairs
Cons
Children
Religious
YearsMarried
Age 1
NoKids
Slightly
10-15 years
37.0
Somewhat
< 4 years
27.0 3
WithKids
Anti
6-10 years
32.0
Very
57.0
Fair, R C(1978) A theory of extramarital affairs, Journal of Political Economy, 86(1), 45-61
Single categorical predictor Children with NoKids, as the base
Understanding customized predictions…………….

17. Single- level Poisson with Single categorical Predictor
Understanding customized predictions
Log scale:
NoKids: WithKids = 0.514
First 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)=
Withkids: expo( ) =
Those with children have a higher average rate of affairs (but have they been married longer?)

18. Modeling number of extra marital affairs:
the incidence rate ratio (IRR)
So far: mean counts
NoKids: expo(-0.092)=
Withkids: expo( ) =
But also as IRR; comparing the ratio of those with and without kids
IRR = / =
That is Withkids have a 83% higher rate
BUT can get this directly from the model by exponentiating the estimate for the contrasted category
expo(0.606) =
Rules:
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. Why is the exponentiated coefficient a IRR?
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

20. Modeling number of extra-marital affairs: changing the base category
Previously contrasted category: Withkids:
Now contrasted category: Nokids :
Changing base simply produces a change of sign on the loge scale
Exponentiating the contrasted category:
Before: expo(+0.606) =
Now: expo(-0.606) =
Doubling the rate on loge scale is 0.693; Halving the rate on loge scale is
IRR of = 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
A model with ‘years married’ included with < 4 years as base
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
1.616
0.773
2.166
6-10 years
1.888
0.932
2.540
10-15 years
2.103
1.041
2.832

22. Affairs: modeling a continuous predictor Age
Age as a 2nd order polynomial centred around 17 years (the youngest person in survey; also lowest rate
To get mean rate as it changes with age
Expo ( *(Age-17) *(Age-17)2)
To get IRR in comparison for a person aged 33 compared to 17
Expo( 0.149*16) –(0.003 *162) (drop the constant!)
Easiest interpreted as graphs!

23. Affairs: a SET of predictors & models
Term
Poisson SE
Extra
NBD
Fixed
Cons
-1.33
0.20
0.53
-1.56
0.51
Years Married
4-5 years
0.89
0.13
0.34
1.20
0.36
6-10 years
1.00
0.14
0.38
1.01
0.41
10-15 years
1.31
0.15
0.40
1.42
0.42
Religious
Somewhat
-0.00
0.39
0.06
0.37
Slightly
0.95
1.28
Non
0.87
1.16
Anti
1.36
0.16
1.52
0.47
Children
WithKids
-0.06
0.11
0.28
0.25
0.29
Age
(age-17)^1
0.05
0.02
0.01
(age-17)^2
0.00
Random Part
Var
6.78
NBD var (v)
5.15
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 marriage
Used comparable vertical axes, range of 4

25. NBD model for number of Extra-Marital Affairs
With 95% confidence intervals
NB that they are asymmetric on the unlogged scale

26. Affairs: Evaluating a sequence of models using DIC
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
+2.176 Pd 1 5 8 9 11
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. Titanic survivor data: Taking account of exposure
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!
Here 2 possible measures of exposure
a) the number of potential cases; could use a binomial
b) the expected number if everyone had the same exposure (i indexes cell)
Death rate = Total Deaths/ Total exposed =817/1316 =
Expi = Casesi * Survival rate
Latter often used to treat the exposure as a nuisance parameter & allows calculation of Standardised Rate’s
SRi = Obsi/Expi * 100
Survive
Case
Age
Gender
Class
Exp SR 14
168
Adult
Men
Mid
63.7
22.0 75
462
Low
175.2
42.8 13 48
Child
18.2
71.4 57
175
High
66.4
85.9 31
Women
11.8
119.1 76
165
62.6
121.5 80 93
35.3
226.9
140
144
54.6
256.4
4.93
263.7 1
0.38 11
4.17 5
1.89
Previous examples
Horsekick: Exposure removed by design: 200 cohort years
Affairs: included length of marriage; theoretically interesting

28. Modeling SR’s: the use of the OFFSET
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

29. Surviving on the Titanic as a log-linear model
Ages
Gender
Class
Log
Estimate
Modeled SR
SR=
Obs/Exp
Child
Women
Low
0.17
1.19
Mid
0.97
2.64
High
Men
-0.34
0.71
Adult
0.19
1.21
0.82
2.27
0.94
2.56
-0.85
0.43
-1.52
0.22
-0.15
0.86
Include the offset
As 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

30. Titanic survival: parsimonious model
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

31. Titanic survival: parsimonious model
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
Low
0.42
0.43
Adults
Women
1.21
Child
High
1.71
2.64
2.26
2.27
2.56
2.57
2.62
2.68 *
1.19
0.71
0.86

32. 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. Estimation of multilevel Poisson and NBD in MLwiN I
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

34. Estimation of multilevel Poisson and NBD in MLwiN II
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

35. VPC for Poisson models Can either use
Simulation method to derive VPC (modify the binomial procedure)
Use exact method: (Henrik Stryhn)
VPC for two level random intercepts model (available for other models)
Clearly VPC depends on  and

36. Modeling Counts in MLwiN: HIV in India
Aim: investigate the State geography of HIV in terms of risk Data: nationally representative sample of 100k individuals in 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 applied Predictors of Age, Sex and Education and Urbanity Two-level multilevel Poisson, extra-Poisson & NBD

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
exactly the same procedure as for binary models
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
Yes
Low
Male
Subra 2
High
Nina 3
Fem
Min 4 No
Myles 5
Sarah 12 50
Kat 13
Colin 14
Andy 15

41. Same data as a multilevel structure: a set of tables for each district
GENDER
QUALIF MALE FEMALE Postcode UnErate
LOW 5 out of 6 3 out of G1A %
HIGH 2 out of 7 7 out of 9
LOW 5 out of 9 7 out of 11 G1B %
HIGH 8 out of 8 7 out of 9
LOW 3 out of G99Z %
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 M L K J I H G F E D C B A
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. Scottish Lip Cancer Spatial multiple-membership model
Response: observed counts of male lip cancer for the 56 regions of Scotland ( )
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 effects
1 area random effects; (ie unstructured; non-spatial variation);
2 multiple membership set of random effects for the neighbours of each region
What is not there
Why do we need techniques? ?
Why do we need to condition on variables ?

45. MCMC estimation: 50,000 draws
Poisson model
Fixed effects:
Offset and
Well-supported + relation
Well-supported Residual neighbourhood effect
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
where ni is the number of neighbours for area i and the weights are typically all 1
What is not there
Why do we need techniques? ?
Why do we need to condition on variables ?
MLwiN: limited capabilities for CAR model; ie at one level only (unlike Bugs)

47. MCMC estimation: CAR model, 50,000 draws
Poisson model
Fixed effects:
Offset and
Well-supported + relation
Well-supported Residual neighbourhood effect

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 ( ) 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)
What is not there
Why do we need techniques? ?
Why do we need to condition on variables ?

50. MCMC estimation: repeated measures model, 50,000 draws
General trend
N’hood variance
Variance function for between county time trend
Default priors

51. Respiratory cancer trends in Ohio: raw and modelled
Red: County 41 in 1988; SMR = 77/49 = 1.57
Blue: County 80 in 1988: SMR= 6/19 = 0.31

52. General References on Modeling Counts
Agresti, 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 Press
Hilbe, 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 models
Browne, 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)
What is not there
Why do we need techniques? ?
Why do we need to condition on variables ?