1

Maarten Cruyff∗, Guus Cruts‡,

Peter G.M. van der Heijden∗ ,

* Utrecht University

‡ Trimbos

ISI 2011

One-Source Capture-Recapture:

Models, applications and software

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Outline

1. One-source data

2. Models and assumptions

3. Software

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One-source CRC data

� Observed data �� � 1,2,3, . . ., � 1, . . . , �

� Individual event count

o drug-related hospital admissions

o visits at rehabilitation center

� Unobserved data �� � 0, � � � 1, . . . , �

o PDU not in hospital

o PDU not in rehabilitation

y

Hospital admissions

Rehabilitation

center

0 ? ?

1 1480 1206

2 155 474

3 41 198

4 11 95

5 10 29

6 3 19

7 2 5

8 0 2

9 0 0

10 1 1

11 2

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Distributional assumption (1)

� Counts follow Poisson distribution:

� � ������

�!

� Poisson parameter �:

o Assigns probabilities to the counts y = 0,1,2,...

�� � 0.5 � � 1

0 .607 .368

1 .303 .368

2 .076 .184

3 .013 .061

4 .002 .015

5 .000 .003

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Distributional assumption (2)

� Model for zero-truncated data

� �|� � 0 �����

1 � ��0�

� Probabilities sum to 1

� Estimation of Poisson parameter �

�� � 0.5 � � 1.0

0 - -

1 .770 .582

2 .192 .291

3 .033 .097

4 .005 .024

5 .000 .005

�� � 0.5 � � 1.0

0 .607 .368

1 .303 .368

2 .076 .184

3 .013 .061

4 .002 .015

5 .000 .003

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Estimation population size

� Given estimate of �

���� �!"#$%#& ��'��()(*+�,-�.�

�'�(*+�,-�.��

�' � � 0 �̂�

�' � � 0 �̂�

� For example

o Suppose �' (*+�,-�. � 1/4

o 1 out 4 individuals observed, so ���� �!"#$%#& � 3

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Example hospital admissions (1)

� Estimation � such that

o fitted frequencies ≈ observed frequencies

� For�̂ � 0.5�' � 2633

� For�̂ � 1.0�' � 993

� Neither model fits very well

o Potential violations of model assumptions

y

Hospital admissions

Fitted

� 6=0.5

Fitted

� 6=1.0

0 - 2633 993

1 1480 1311 993

2 155 329 496

3 41 56 165

4 11 9 40

5 10 0 8

6 3 0 3

7 2 0 0

8 0 0 0

9 0 0 0

10 1 0 0

11 2 0 0

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Assumptions Poisson distribution

� Homogeneity

o Identical Poisson parameter for all � 1, . . , �

o If violated, underestimation population size

� Closed population

o Presence in population during entire observation period

o If violated, overestimation population size

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Models for heterogeneity (1)

� Poisson regression model

o Each individual has its own Poisson parameter

�� � �7897:;<:97=;<=9...

o Insight in composition of population in terms of covariates

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Models for heterogeneity (2)

� Negative binomial (regression) model

o Additional parameter allowing for more variation in counts (longer tail)

o Results in higher population size estimate

o Drawback: rarely estimable

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Models for heterogeneity

� Zelterman (regression) model

o Estimation based on counts 1 and 2 only

o Rationale: use only counts closest to zero

o Population size estimate in between Poisson and negative binomial model

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Model for open population (in progress)

� Recurrent events model

o Analysis of event history

o Requires additional data

� Example illegal immigrants (work in progress)

o Detention times

o Extradition

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Hospital admissions: data

Zelterman

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Parameter estimates

�> � 6695 �> � 10415

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Composition of population

Effect of

covariates

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Rehabilitation data

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Parameter estimates

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Population size estimates

Strong effect dispersion parameter

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Estimated population composition

No strong effects

of the covariates

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Software

� Truncated Poisson/negative binomial models

o R package GAMLSS (not straightforward)

o Simple r-code (next slides)

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Simple R-code (truncated Poisson model)

y n x 1 vector with zero-truncated counts

X n x k matrix with covariates (including constant)

pars k x 1 vector with start values for the regression parameters

loglP <- function(pars){

u <- exp(X%*%pars)

loglike <- log(dpois(y,u))/(1-dpois(0,u))

-sum(loglike)

}

estimates <- optim(pars,loglP)

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Simple R-code (truncated negative binomial model)

y n x 1 vector with zero-truncated counts

X n x k matrix with covariates (including constant)

pars (k+1) x 1 vector with start values for regression parameters and dispersion parameter

loglNB <- function(pars){

u <- exp(X%*%pars[1:k])

a <- exp(pars[k+1])

loglike <- log(dnbinom(y,size=a,mu=u))/(1-dnbinom(0,size=a,mu=u))

-sum(loglike)

}

estimates <- optim(pars,loglNB)

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Software

� Zelterman model

o Simple estimator (no covariates)

�̂ �?�=

�:

where

n1 is observed frequency of 1-count

n2 is observed frequency of 2-count

o Gauss & Stata code for regression in supplement to Bӧhning and Van der Heijden (2009)

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Conclusions

� One-source CRC well suited for PDU estimation

� Potential data sources

o Rehabilitation centers

o Hospital admissions

o Police records (drug-related offences)

� Software not straightforward, but possible

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References

Boehning, D. And P.G.M. van der Heijden (2009). A Covariate Adjustment for Zero-truncated Approaches to Estimating the

Size of Hidden and Elusive Populations. Annals of Applied Statistics, 3, 595-610.

Cruyff, M.J.L.F. and P.G.M. van der Heijden. (2008). Point and interval estimation of the population size using a zero-

truncated negative binomial regression model. Biometrical Journal, 50 (6), 1035-1050.

Van der Heijden, P.G.M., Bustami, R., M. Cruyff, G. Engbersen and H. van Houwelingen (2003b). Point and interval

estimation of the truncated Poisson regression model. Statistical Modelling, 3, 305-322.

Van der Heijden, P.G.M., Cruts, G. and Cruyff, M. (in press) Methods for population size estimation of problem drug users

using a single registration. International Journal of Drug Policy,