Maximum likelihood estimation of mark-recapture-recovery models in the presence of continuous covariates

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1 Maximum likelihood estimation of mark-recapture-recovery models in the presence of continuous covariates & Ruth King University of St Andrews

2 1 Outline of the problem 2 MRR data in the absence of covariates 3 MRR data in the presence of covariates 4 An application to Soay sheep MRR data

3 Mark-recapture-recovery (MRR) data: repeated surveyings of a population (e.g. yearly) animals are tagged at initial encounter observations: seen/not seen, survival status (alive, \recent death", \long dead"), potentially covariates Aim: gain understanding of the underlying system! estimate survival rates! detect population trends! etc. here we deal with a statistical problem arising from missing individual-specic time-varying covariate values (e.g., body mass of an animal)

4 Mark-recapture-recovery (MRR) data: repeated surveyings of a population (e.g. yearly) animals are tagged at initial encounter observations: seen/not seen, survival status (alive, \recent death", \long dead"), potentially covariates Aim: gain understanding of the underlying system! estimate survival rates! detect population trends! etc. here we deal with a statistical problem arising from missing individual-specic time-varying covariate values (e.g., body mass of an animal)

5 1 Outline of the problem 2 MRR data in the absence of covariates 3 MRR data in the presence of covariates 4 An application to Soay sheep MRR data

6 MRR data in the absence of covariates Example MRR encounter history: 0: not seen 1: seen alive 2: recovered dead Associated likelihood: L = 1 p 2 2 (1 p 3 ) 3 (1 p 4 ) 4 p 5 (1 5 ) 6 t : prob. of surviving from t to t + 1 p t : prob. of being seen when alive at time t t : prob. of recovery at time t if death in (t 1; t]

7 MRR data in the absence of covariates Example MRR encounter history: 0: not seen 1: seen alive 2: recovered dead Associated likelihood: L = 1 p 2 2 (1 p 3 ) 3 (1 p 4 ) 4 p 5 (1 5 ) 6 t : prob. of surviving from t to t + 1 p t : prob. of being seen when alive at time t t : prob. of recovery at time t if death in (t 1; t]

8 MRR data in the absence of covariates Likelihood for encounter history x 1 ; : : : ; x T if survival process, s 1 ; : : : ; s T, is not fully known: where s t = 8 >< >: L = X X TY 2S c s 2f1;2;3g t=2 1 if alive at time t; f (s t js t 1 )f (x t js t ) ; 2 if dead at time t; but was alive at time t 1; 3 if dead at time t; and was dead already at time t 1; and S c = ft j s t is unknowng.

9 HMM formulation of MRR data in the absence of covariates In hidden Markov-type matrix product form: and Q(x t ) = L = where t = 8 >< >: TY t=2 t 1Q(x t )! 1 3 ; t 1 t A diag(1 p t ; 1 t ; 0) if x t = 0; diag(p t ; 0; 0) if x t = 1; diag(0; t ; 0) if x t = 2:

10 HMM formulation of MRR data in the absence of covariates In hidden Markov-type matrix product form: and Q(x t ) = L = where t = 8 >< >: TY t=2 t 1Q(x t )! 1 3 ; t 1 t A diag(1 p t ; 1 t ; 0) if x t = 0; diag(p t ; 0; 0) if x t = 1; diag(0; t ; 0) if x t = 2:

11 MRR data in the presence of covariates The \interesting" case: what if survival probabilities i are aected by individual-specic continuous covariates (e.g., body mass)? Example MRR encounter history: Animal not seen! covariate value unknown Associated likelihood: L = 1 p 2 2 (1 p 3 ) 3 (1 p 4 ) 4 p 5 (1 5 ) 6 covariate unknown! likelihood can't be computed

12 MRR data in the presence of covariates The \interesting" case: what if survival probabilities i are aected by individual-specic continuous covariates (e.g., body mass)? Example MRR encounter history: Animal not seen! covariate value unknown Associated likelihood: L = 1 p 2 2 (1 p 3 ) 3 (1 p 4 ) 4 p 5 (1 5 ) 6 covariate unknown! likelihood can't be computed

13 MRR data in the presence of covariates The \interesting" case: what if survival probabilities i are aected by individual-specic continuous covariates (e.g., body mass)? Example MRR encounter history: Animal not seen! covariate value unknown Associated likelihood: L = 1 p 2 2 (1 p 3 ) 3 (1 p 4 ) 4 p 5 (1 5 ) 6 covariate unknown! likelihood can't be computed

14 Maximum likelihood? Idea: model covariate process and integrate over possible values Bonner, Morgan and King (2010, Biometrics): \except when few values are missing, the large number of integrals [...] will make it impossible to perform maximum likelihood estimation"

15 Most popular existing estimation methods 0 Trinomial approach! conditioning on only observed covariate values! closed-form likelihood, but throwing away information Bayesian imputation approach! assume model for covariate process, for example AR(1)! impute missing covariate values within MCMC! model selection dicult! prior distributions need to be specied

16 ML approach Assuming some (Markovian) model for the covariate process, e.g., y t = y t 1 + ( y t 1 ) + t ; the likelihood for an encounter history x 1 ; : : : ; x T Z L = : : : Z X TY t=2 X 2S c s 2f1;2;3g f (y 1 ) I f12w c g f (s t js t 1 ; y t 1 )f (x t js t )f (y t jy t 1 )dy W c ; where W c = ft j y t is unobserved; t 2 S; s t 6= 2; 3g. is ) multiple integral, cannot be evaluated...

17 ML approach Assuming some (Markovian) model for the covariate process, e.g., y t = y t 1 + ( y t 1 ) + t ; the likelihood for an encounter history x 1 ; : : : ; x T Z L = : : : Z X TY t=2 X 2S c s 2f1;2;3g f (y 1 ) I f12w c g f (s t js t 1 ; y t 1 )f (x t js t )f (y t jy t 1 )dy W c ; where W c = ft j y t is unobserved; t 2 S; s t 6= 2; 3g. is ) multiple integral, cannot be evaluated...

18 ML approach idea: discretize covariate space, thereby reducing R to P split \essential range" into m intervals jth interval: B j = [b j 1 ; b j ), with midpoint b j L X mx X X 2W c j =1 2S c s 2f1;2;3g TY t=2 f (y 1 2 B j1 ) I f12w c g TY t=2 Q (m) (x t ) f (s t js t 1 ; y t 1 ) I f(t 1)2Wg f (s t js t 1 ; b j ) I f(t 1)2W c g f (x t js t ) f (y t jy t 1 ) I ft2w;(t 1)2Wg f (y t jb j t 1 )I ft2w;(t 1)2W c g f (y t 2 B jt jy t 1 ) I ft2w c ;(t 1)2Wg f (y t 2 B jt jb j t 1 )I ft2w c ;(t 1)2W c g!

19 ML approach idea: discretize covariate space, thereby reducing R to P split \essential range" into m intervals jth interval: B j = [b j 1 ; b j ), with midpoint b j L > m jw c j summands!!! / (m) T Y TY t=g +1 t=g +1 (m) t 1 Q(m) (x t ) 1 A 1m+2 f (s t js t 1 ; y t 1 ) I f(t 1)2Wg f (s t js t 1 ; b j ) I f(t 1)2W c g f (x t js t ) f (y t jy t 1 ) I ft2w;(t 1)2Wg f (y t jb j t 1 )I ft2w;(t 1)2W c g f (y t 2 B jt jy t 1 ) I ft2w c ;(t 1)2Wg f (y t 2 B jt jb j t 1 )I ft2w c ;(t 1)2W c g

20 ML approach idea: discretize covariate space, thereby reducing R to P split \essential range" into m intervals jth interval: B j = [b j 1 ; b j ), with midpoint b j 0 Y L > m jw c j summands!!! T (m) TY t=g +1 t=g +1 (m) t 1 Q(m) (x t ) 1 A 1m+2 f (s t js t 1 ; y t 1 ) I f(t 1)2Wg f (s t js t 1 ; b j ) I f(t 1)2W c g f (x t js t ) f (y t jy t 1 ) I ft2w;(t 1)2Wg f (y t jb j t 1 )I ft2w;(t 1)2W c g f (y t 2 B jt jy t 1 ) I ft2w c ;(t 1)2Wg f (y t 2 B jt jb j t 1 )I ft2w c ;(t 1)2W c g

21 ML approach idea: discretize covariate space, thereby reducing R to P split \essential range" into m intervals jth interval: B j = [b j 1 ; b j ), with midpoint b j L (m) T Y TY t=g +1 t=g +1 (m) t 1 Q(m) (x t ) 1 A 1m+2, f (s t js t 1 ; y t 1 ) I f(t 1)2Wg f (s t js t 1 ; b j ) I f(t 1)2W c g f (x t js t ) f (y t jy t 1 ) I ft2w;(t 1)2Wg f (y t jb j t 1 )I ft2w;(t 1)2W c g f (y t 2 B jt jy t 1 ) I ft2w c ;(t 1)2Wg f (y t 2 B jt jb j t 1 )I ft2w c ;(t 1)2W c g

22 ML approach idea: discretize covariate space, thereby reducing R to P split \essential range" into m intervals jth interval: B j = [b j 1 ; b j ), with midpoint b j L (m) T Y TY t=g +1 (m) t 1 Q(m) (x t ) 1 A 1m+2 - with (m) t and Q (m) (x f (s t js t 1 ; y t ) suitably t 1 ) I 1)2Wg dened f(t f (s t js t 1 ; b j ) I f(t 1)2W c g f (x t js t ) - corresponds to an ecient HMM-type recursive scheme t=g +1 (idea: augment \alive" survival state by dividing it into m distinct states, associated with the dierent intervals the covariate value may lie in)

23 ML approach idea: discretize covariate space, thereby reducing R to P split \essential range" into m intervals jth interval: B j = [b j 1 ; b j ), with midpoint b j L (m) T Y TY t=g +1 (m) t 1 Q(m) (x t ) 1 A 1m+2, - with (m) t and Q (m) (x f (s t js t 1 ; y t ) suitably t 1 ) I 1)2Wg dened f(t f (s t js t 1 ; b j ) I f(t 1)2W c g f (x t js t ) - corresponds to an ecient HMM-type recursive scheme t=g +1 (idea: augment \alive" survival state by dividing it into m distinct states, associated with the dierent intervals the covariate value may lie in)

24 Trinomial vs. full ML approach { a simulation experiment Survival probability: logit( t ) = y t Table: Sample mean estimates (ME), sample mean widths (CW) of the estimated 95% condence intervals and coverage probabilities (CC) of the condence intervals, in four simulation scenarios. inte. (0 = 3) slope (1 = 0:2) Sc Meth. -ME CW CC ME CW CC p = 0:95 Tri = 0:95 full ML p = 0:9 Tri = 0:3 full ML p = 0:3 Tri = 0:9 full ML p = 0:3 Tri = 0:3 full ML

25 1 Outline of the problem 2 MRR data in the absence of covariates 3 MRR data in the presence of covariates 4 An application to Soay sheep MRR data

Application to Soay sheep MRR data capture histories for 1344 female Soay sheep, 1985{2009 four dierent age groups: lambs (< 1), yearlings (1{2), adults (2{7) and seniors (> 7)

26 Application to Soay sheep MRR data capture histories for 1344 female Soay sheep, 1985{2009 four dierent age groups: lambs (< 1), yearlings (1{2), adults (2{7) and seniors (> 7) Survival probability: covariate process model: logit( t ) = at ;0 + at ;1weight t weight t = weight t 1 + at at weight t 1 + at t year-dependent recapture and recovery probabilities

27 Application to Soay sheep MRR data { estimated survival probability lambs yearlings surv. prob surv. prob weight weight adults seniors surv. prob surv. prob weight weight Figure: Solid lines: ML estimates. Dashed lines: 95% pointwise condence intervals, obtained using the delta method.

28 Application to Soay sheep MRR data { tted covariate process age (in yrs.) weight (in kg) Figure: Empirical 5% and 95% quantiles (big grey dots) and empirical medians (big black dots) of body masses, and model-derived 5% and 95% quantiles (dashed grey lines) and medians (solid black lines) of body mass distributions at those ages.

29 Remarks, future work & references HMM-based discretization strategy applicable to essentially any state-space model multiple covariates computationally challenging! apply more sophisticated numerical integration strategies other state processes can be considered: e.g., additional states (Arnason-Schwarz model), semi-markov components Bonner, R., Morgan, B.J.T., King, R., Continuous covariates in mark-recapture-recovery analysis: a comparison of methods. Biometrics, 66, 1256{1265. Langrock, R., King, R., Maximum likelihood estimation of mark-recapture-recovery models in the presence of continuous covariates. Annals of Applied Statistics, in press. Langrock, R., Some applications of nonlinear and non-gaussian state-space modelling by means of hidden Markov models. Journal of Applied Statistics, 38, 2955{2970.

arxiv: v3 [stat.me] 29 Nov 2013

arxiv: v3 [stat.me] 29 Nov 2013 The Annals of Applied Statistics 2013, Vol. 7, No. 3, 1709 1732 DOI: 10.1214/13-AOAS644 c Institute of Mathematical Statistics, 2013 arxiv:1211.0882v3 [stat.me] 29 Nov 2013 MAXIMUM LIKELIHOOD ESTIMATION