One-Source Capture-Recapture: Models, applications and software

Transcription

1 One-Source Capture-Recapture: Models, applications and software Maarten Cruyff, Guus Cruts, Peter G.M. van der Heijden, * Utrecht University Trimbos ISI

2 Outline 1. One-source data 2. Models and assumptions 3. Software 2

3 One-source CRC data Observed data 1,2,3,..., 1,..., Individual event count o drug-related hospital admissions o visitsat rehabilitationcenter Unobserved data 0, 1,..., o PDU not in hospital o PDU not in rehabilitation y Hospital admissions Rehabilitation center 0??

4 Distributional assumption (1) Counts follow Poisson distribution:! Poisson parameter : o Assigns probabilities to the counts y = 0,1,2,

5 Distributional assumption (2) Model for zero-truncated data Probabilities sum to Estimation of Poisson parameter 5

6 Estimation population size Given estimateof 0 0 For example o Suppose 1/4 o 1 out 4 individualsobserved, so 3 6

7 Example hospital admissions(1) Estimation suchthat o fitted frequencies observed frequencies y Hospital admissions Fitted =0.5 Fitted = For For Neither model fits very well o Potential violations of model assumptions

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 8

9 Models for heterogeneity (1) Poisson regression model o Each individual has its own Poisson parameter... o Insight in composition of population in terms of covariates 9

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 10

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 11

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 12

13 Hospital admissions: data Zelterman 13

14 Parameter estimates

15 Composition of population Effect of covariates 15

16 Rehabilitationdata 16

17 Parameter estimates 17

18 Population size estimates Strong effect dispersion parameter 18

19 Estimated population composition No strong effects of the covariates 19

20 Software Truncated Poisson/negative binomial models o R package GAMLSS (not straightforward) o Simple r-code (next slides) 20

21 Simple R-code (truncated Poisson model) y X pars n x 1 vector with zero-truncated counts n x k matrix with covariates(including constant) 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) 21

22 Simple R-code (truncated negative binomial model) y X pars n x 1 vector with zero-truncated counts n x k matrix with covariates(including constant) (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) 22

23 Software Zelterman model o Simple estimator(no covariates) where n 1 is observedfrequencyof 1-count n 2 is observed frequency of 2-count o Gauss & Stata code for regression in supplement to BӧhningandVan der Heijden (2009) 23

24 Conclusions One-source CRC well suited for PDU estimation Potential data sources o Rehabilitationcenters o Hospital admissions o Police records (drug-related offences) Software not straightforward, but possible 24

25 References Boehning, D. AndP.G.M. van der Heijden (2009). A CovariateAdjustmentforZero-truncatedApproaches to Estimating the Size of Hidden and Elusive Populations. Annals of Applied Statistics, 3, Cruyff, M.J.L.F. and P.G.M. van der Heijden. (2008). Point and interval estimation of the population size using a zerotruncated negative binomial regression model. Biometrical Journal, 50 (6), Van der Heijden, P.G.M., Bustami, R., M. Cruyff, G. EngbersenandH. van Houwelingen (2003b). Point and interval estimation of the truncated Poisson regression model. Statistical Modelling, 3, 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, 25