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Confidence Intervals for Capture. Recapture Data With Matching Stephen Sharp, National Records of Scotland Confidence Intervals for Capture. Recapture Data With Matching Stephen Sharp, National Records of Scotland

The Problem (i) • You have undertaken a (presumably imperfect) enumeration of a given The Problem (i) • You have undertaken a (presumably imperfect) enumeration of a given population. • You then undertaken a second (also presumably imperfect) coverage survey. • You have matched the two so that you know how many people were in both surveys (N 12); in the first survey only (N 1); and in the second survey only (N 2). • You require to estimate the number of people in neither survey (N 0).

Summary S 1 out in S 2 in N 12 N 2 out N Summary S 1 out in S 2 in N 12 N 2 out N 1 N 0

The Problem (ii) • The classical estimate of N 0 is the product of The Problem (ii) • The classical estimate of N 0 is the product of N 1 and N 2 divided by N 12. • However this assumes that absence from the first survey does not change the probability of absence from the second. • For humans, this is very unlikely.

A Bayesian approach • As we do not know N 0, we require its A Bayesian approach • As we do not know N 0, we require its probability distribution conditional on N 12, N 1 and N 2 which we do know. • We get this from Bayes’ theorem. • p(N 0 | N 12 N 1 N 2) = constant x p(N 12 N 1 N 2 | N 0) x p(N 0). • Posterior is proportional to likelihood x prior. • We need a likelihood and a prior.

The likelihood function (i) • The distribution of N 12, N 1 and N The likelihood function (i) • The distribution of N 12, N 1 and N 2 conditional on N 0 is multinomial with probability parameters p 12, p 1, p 2 and p 0. • The four probabilities must sum to one so we need three constraints to specify uniquely three parameters. • We assume that p 12, p 1 and p 2 stand in the same proportions as N 12, N 1 and N 2. • This gives us two constraints.

The likelihood function (ii) • Instead of imposing a third constraint however we let The likelihood function (ii) • Instead of imposing a third constraint however we let the posterior distribution of N 0 depend on the dichotomous correlation ϕ, which measures stochastic dependency. • We can now specify the likelihood for a given value of ϕ and watch the effect of changing it.

The prior distribution • What did we know about the likely size of the The prior distribution • What did we know about the likely size of the population before we took the two surveys? • This knowledge is reflected in the prior distribution. • A safe bet would be an uninformative prior (perhaps a normal or uniform distribution with a very big variance). • If you are confident though you might be better to use an informative prior (i. e. a smaller prior variance). • This reduces the variance of the posterior distribution (though be careful to check that the prior is consistent with the likelihood).

Some examples S 1 out in S 2 in 388 56 out 75 k Some examples S 1 out in S 2 in 388 56 out 75 k

Example 1 – Using a Poisson prior with l = 550 Complete independence (f Example 1 – Using a Poisson prior with l = 550 Complete independence (f = 0. 00) Low dependence (f = 0. 10) Low dependence (f = 0. 20) Low dependence (f = 0. 30) Medium dependence (f = 0. 40) Medium dependence (f = 0. 50) Medium dependence (f = 0. 60) High dependence (f = 0. 70) High dependence (f = 0. 84) Cut off points 2. 5% 50% 97. 5% 4 11 18 12 22 36 54 81 123 193 336 444 21 33 49 71 101 146 222 374 487 31 46 64 89 122 171 253 413 532

Example 2 – Using a totally uninformative prior Complete independence (f = 0. 00) Example 2 – Using a totally uninformative prior Complete independence (f = 0. 00) Low dependence (f = 0. 10) Low dependence (f = 0. 20) Low dependence (f = 0. 30) Medium dependence (f = 0. 40) Medium dependence (f = 0. 50) Medium dependence (f = 0. 60) High dependence (f = 0. 70) High dependence (f = 0. 84) Cut off points 2. 5% 50% 97. 5% 4 10 17 11 22 37 59 94 158 306 994 3, 721 20 33 51 76 117 189 353 1, 106 4, 084 30 45 66 96 141 221 402 1, 224 4, 467

Further work (i) • So we can model the point estimate and confidence intervals Further work (i) • So we can model the point estimate and confidence intervals as a function of the dichotomous correlation f. • But what is the value of f? • This will vary from one subgroup to another within the population. • It will depend on the diversity within the subgroup of the propensity to take part in public surveys like the Census and the coverage survey.

Further work (ii) • Attempts to model this have suggested that typical values for Further work (ii) • Attempts to model this have suggested that typical values for f vary between 0. 25 and 0. 40. • This suggests that for an uninformative prior, the population point estimate might be 560 against 520 with the independence assumption; an underestimate of about 7%. • The confidence intervals are ± 14 or 15 as opposed to ± 6 or 7; about twice as wide.

Conclusion • The assumption of independence introduces error into both the point estimate and Conclusion • The assumption of independence introduces error into both the point estimate and the confidence intervals when population size is estimated from capture-recapture data. • The CI error is in the “wrong” direction (i. e. not on the side of caution). • Departure from independence arises because those members of the population unlikely to be included in one sample are less likely to be included in the other. • Assessing the extent of dependence is difficult but its effects make it important to try.

Confidence Intervals for Capture-Recapture Data With Matching Stephen Sharp National Records of Scotland Ladywell Confidence Intervals for Capture-Recapture Data With Matching Stephen Sharp National Records of Scotland Ladywell House Ladywell Road Edinburgh EH 12 7 TF 0131 314 4649 Stephen. [email protected] gsi. gov. uk