One-Source Capture-Recapture: Models, applications . M. Cru · One-Source Capture-Recapture:...

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  • 1

    Maarten Cruyff, Guus Cruts,

    Peter G.M. van der Heijden ,

    * Utrecht University

    Trimbos

    ISI 2011

    One-Source Capture-Recapture:

    Models, applications and software

  • 2

    Outline

    1. One-source data

    2. Models and assumptions

    3. Software

  • 3

    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

  • 4

    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

  • 5

    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

  • 6

    Estimation population size

    Given estimate of

    !"#$%#& '()(*+,-.

    '(*+,-.

    ' 0

    ' 0

    For example

    o Suppose ' (*+,-. 1/4

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

  • 7

    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

  • 8

    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

  • 9

    Models for heterogeneity (1)

    Poisson regression model

    o Each individual has its own Poisson parameter

    7897:;

  • 10

    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

  • 11

    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

  • 12

    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

  • 13

    Hospital admissions: data

    Zelterman

  • 14

    Parameter estimates

    > 6695 > 10415

  • 15

    Composition of population

    Effect of

    covariates

  • 16

    Rehabilitation data

  • 17

    Parameter estimates

  • 18

    Population size estimates

    Strong effect dispersion parameter

  • 19

    Estimated population composition

    No strong effects

    of the covariates

  • 20

    Software

    Truncated Poisson/negative binomial models

    o R package GAMLSS (not straightforward)

    o Simple r-code (next slides)

  • 21

    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

  • 22

    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

  • 23

    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 Bhning and Van der Heijden (2009)

  • 24

    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

  • 25

    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,