Some General Results

Appendix Some General Results

A.1 Multinomial Distribution

Consider the following multinomial distribution coming from n independent multinomial trials with xi the number of counts occurring in the cell with probability pi (i = 1, 2,...,k), namely

n! k Pr[{x }] = pxi i k ! i i=1 xi i=1 n k k k = pxi , x = n, p = 1. (A.1) x , x ,...,x i i i 1 2 k i=1 i=1 i=1

Here Pr[{xi }] stands for

Pr[{xi }] = Pr[X1 = x1, X2 = x2,...,Xk = xk ], the “singular” (and symmetrical version) of the distribution as we need the identifia- k = bility constraint i=1 pi 1. We can express this distribution briefly as the singular ( , ,..., ) ∼ ( , ) distribution x1 x2 xk Multinomial n p . = − k−1 = We obtain the nonsingular distribution by writing pk 1 i=1 pi and xk − k−1 n i=1 xi . In this respect, the notations for the singular and nonsingular distributions are sometimes confused in the literature, especially the definition n n! = . {x } k ! i i=1 xi

The binomial distribution, written Binomial(n, p), has k = 2, and the Bernoulli distribution, written Bernoulli(x, p), is Binomial(1, p) for discrete random x taking the values of 1 and 0.

© Springer Nature Switzerland AG 2019 571 G. A. F. Seber and M. R. Schoﬁeld, Capture-Recapture: Parameter Estimation for Open Animal Populations, Statistics for Biology and Health, https://doi.org/10.1007/978-3-030-18187-1 572 Appendix: Some General Results

A.1.1 Some Properties

By adding appropriate cells together we see that the marginal distribution of any subset of a multinomial distribution is also multinomial with the appropriate pi ’s added together. If we finally end up with just two pooled cells we have the binomial distribution. We now show how a nonsingular multinomial distribution can be represented by the product of conditional binomial distributions. To do this we note first that if x1 and x2 have a multinomial distribution and x1 has a binomial distribution, we find that

Pr[x2, x1] Pr[x2 | x1]= Pr[x1] − − n − x p x2 p n x1 x2 = 1 2 1 − 2 . x2 1 − p1 1 − p1

Now

Pr[x1, x2,...,xk ]=Pr[x1] Pr[x2 | x1] Pr[x3 | x1, x2]···Pr[xk | x1, x2,...,xk−1] and we have shown that for k = 2wehave

Pr[x1, x2]=Pr[x1] Pr[x2 | x1] where both distributions are binomial. We can then show by induction that the fac- torization of Pr[x1, x2,...,xk ] gives a product of conditional binomial distributions. We now consider two useful techniques applied by Robson and Youngs (1971) and referred to as “peeling” and “pooling” by Burnham (1991) in using multinomial distributions.

A.1.2 Peeling Process

The peeling process with multinomial distributions can be described as follows. k = Suppose we wish to peel off the probability distribution of X1.If i=2 xi x and k = − = i=2 pi 1 p1 q1, then Appendix: Some General Results 573

k Pr[X1 = x1, X2 = x2,...,Xk = xk with xi = n] i=1

= Pr[X1 = x1] Pr[X1 = x1,...,Xk = xk | X1 = x1]

= Pr[X1 = x1] Pr[X2 = x2,...,Xk = xk with x = n − x1] k xi n − n − x p = px1 qn x1 1 i . x 1 1 x , x ,...,x q 1 2 3 k i=2 1

A.1.3 Pooling Process

The pooling process begins with two independent singular multinomial distributions with the same number of cells and the same cell probabilities, namely n k k Pr[{x }] = 1 pxi , x = n i x , x ,...,x i i 1 1 2 k i=1 i=1 and n k k Pr[{y }] = 2 pyi , y = n . i y , y ,...,y i i 2 1 2 k i=1 i=1

If we “add” the two distributions together (the convolution) we get

+ n1 n2 k n + n + Pr[{x + y }] = 1 2 pxi yi , (x + y ) = n + n . i i x + y ,...,x + y i i i 1 2 1 1 k k i=1 i=1

This can be proved using moment generating functions as the moment generating function of the sum is the product of the two moment generating functions, or simply observe that we have n1 + n2 multinomial trials with the same set of probabilities {pi }. Now because of the independence we have

Pr[{xi }] Pr[{yi }] k n n + = 1 2 pxi yi x , x ,...,x y , y ,...,y i 1 2 k 1 1 k i=1 k n + n + = 1 2 xi yi pi x1 + y1, x2 + y2,...,xk + yk i=1 + × n1 n2 / n1 n2 , x1, x2,...,xk y1, y2,...,yk x1 + y1, x2 + y2,...,xk + yk 574 Appendix: Some General Results that is, the product of the two probability distributions is the product of the distribution of their sum times a hypergeometric distribution.

A.1.4 Conditional Distribution

Suppose we have a nonsingular distribution

n! − − [ , ]= x1 x2 ( − − )n x1 x2 . Pr x1 x2 p1 p2 1 p1 p2 x1!x2!(n − x1 − x2)!

If y = x1 + x2, then y has probability function n Pr[y]= (p + p )y(1 − p − p )n−y y 1 2 1 2 and Pr[x , y] Pr[x | y]= 1 1 Pr[y] [ , ] = Pr x1 x2 [ ] Pry x1 x2 = y p1 p2 , x1 p1 + p2 p1 + p2 which is a binomial distribution.

A.2 Delta Method

We consider general ideas only without getting too involved with technical details about limits (see also Agresti (2013: Sect. 16.1). Let X be a random variable with mean μ and variance σ2, and let Y = g(X) be a “well-behaved” function of X that has a Taylor expansion

1 g(X) − g(μ) = (X − μ)g(μ) + (X − μ)2g(X ), 2 0

where X0 lies between X and μ, g (μ) is the derivative of g evaluated at X = μ, and g (X0) is the second derivative of g evaluated at X = X0. Then taking expected values, 1 E[g(X)]≈g(μ) + σ2g(μ). (A.2) 2 Appendix: Some General Results 575

Assuming second order terms can be neglected, we have E[Y ]≈g(μ) and

var(Y ) ≈ E[(g(X) − g(μ))2] ≈ E[(X − μ)2][g(μ)]2 = σ2[g(μ)]2. (A.3)

For example, if g(X) = log X then, for large μ,

σ2 var(log X) ≈ . (A.4) μ2

If X = (X1, X2,...,Xk ) is a vector with mean μ, then for suitable g,wehave the ﬁrst order Taylor expansion

k = ( ) − (μ) ≈ ( − ) (μ), Y g X g Xi μi gi (A.5) i=1

(μ) / = μ where gi is ∂g ∂ Xi evaluated at X . Then

[ ]≈ [( ( ) − (μ))2] var Y E ⎡g X g ⎤ k k ≈ ⎣ ( − )( − ) (μ) (μ)⎦ E Xi μi X j μ j gi g j i=1 j=1 k k = [ , ] (μ) (μ). cov Xi X j gi g j (A.6) i=1 j=1

A “quick and dirty” method for a product or ratio of two variables is a follows. If Y = X1/ X2 then taking logs and differentials we get Y = X1 − X2 . Y X1 X2

Squaring and taking expected values gives us X var[X ] var[X ] cov[X , X ] 1 ≈ 2 1 + 2 − 1 2 , var μy 2 2 2 (A.7) X2 μ1 μ2 μ1μ2 where μy ≈ μ1/μ2. For a product X1 X2 we simply replace the minus sign by a plus sign and μy by μ1μ2. Sometimes we wish to derive asymptotic variances and covariances of parameters using Taylor expansions and the delta method. This can be very onerous, but there are a few shortcuts or “rules” that have been brought together by Seber (1967) and Jolly (1965) that we now mention. 576 Appendix: Some General Results

A.2.1 Application to the Multinomial Distribution

Suppose X has the Multinomial distribution given by (A.1) and

X X ···X g(X) = 1 2 r (s ≤ k). Xr+1 Xr+2 ···Xs

Then, using (A.5),

( ) − (μ) r − s − g X g ≈ Xi μi − Xi μi . g(μ) μ μ i=1 i i=r+1 i

Now squaring the above equation, taking expected values, and using μi = npi and 2 σ = npi (1 − pi ),wehave [ ] ( − ) var Xi = npi 1 pi = 1 − 1 2 2 2 μi n pi npi n and [ , ] cov Xi X j =−npi p j =−1 . 2 μi μ j n pi p j n

We now have an expression like (A.6) involving three sets of covariances (except when s = r + 1) so that s s 2 r s − r var[g(X)]≈[g(μ)]2 μ−1 − − + δ − r(s − r) , i n n 2 2 i=1 where δ is 1 when s − r ≥ 2 and zero when s = r + 1. Then (Seber 1982: 8–9), [g(μ)]2 s var[g(X)]≈ p−1 − (s − 2r)2 , (A.8) n i i=1 for all s > r. Another way of describing the above method is, as in (A.7) above, to take loga- rithms and then differentials so that

r s log(X) = log Xi − log Xi , i=1 i=r+1 and r s δXi = δXi − δXi . X X X i i=1 i i=r+1 i Appendix: Some General Results 577

We then square both sides and take expected values. This is the approach used by Cormack (1993a, b), for example, for log-linear models. Two multinomial cases of interest in this monograph are, s = 2, r = 1 and s = 4, r = 2. In the ﬁrst case g(X) = X1/ X2 and 1 1 var[g(X)]≈[g(μ)]2 + . (A.9) μ1 μ2

For example, if Y = log g(X), then from (A.3) and (A.4),

var[g(X)] 1 1 var(Y ) ≈ = + . (A.10) 2 [g(μ)] μ1 μ2

We can estimate var(Y ) by replacing each μi by Xi in (A.10). For the other case of interest, we ﬁnd that X X 4 1 var log 1 2 ≈ . (A.11) X X μ 3 4 i=1 i

A.2.2 Several Multinomial Distributions

In practice the Xi in (A.6) are usually sums of various multinomial variables making any derivation very onerous. However, there are two particular rules that are helpful (see for example Jolly, 1965: 236) that we now mention. Suppose that Xi is the sum of some random variables from a multinomial group of size G, then Xi being a sum of multinomial variables is binomial, Binomial(G, pi ) say, so that

2 var[Xi ]=Gpi (1 − pi ) = E[Xi ]−E[Xi ] /G.

If X1 and X2 are sums from a group of G with X12 in common so that Xi = Xi0 + X12 and pi = pi0 + p12, then

cov[X1, X2]=cov[X10 + X12, X20 + X12]

= cov[X10, X20]+cov[X10, X12]+cov[X12, X20]+var[X12]

=−Gp10 p20 − Gp10 p12 − Gp20 p12 + Gp12(1 − p12)

= Gp12 − Gp1 p2

= E[X12]−E[X1]E[X2]/G. 578 Appendix: Some General Results

A.3 Sufﬁcient and Ancillary Statistics

Given a set of data x with probability function Pr[x | θ] with unknown parameter θ, we say that a statistic T(x), a function of x,issufﬁcient for θ if the conditional probability distribution of x given T, namely Pr[x | T] does not depend on θ.This leads to the famous factorisation theorem, namely T is sufﬁcient for θ if and only if

Pr[x | θ]=g(x)h(T, θ), where g and h are nonnegative functions. Differentiating log Pr[x | θ] with respect to θ we see that maximum likelihood estimator of θ is a function of T. We note that the entire data set x is trivially sufficient for θ and any one-to- one function of a sufficient statistic is also sufficient so that we can have a wide choice of sufficient statistics. What we want is a sufficient statistic with the smallest dimension. This is achieved through a minimal sufficient statistic, if it exists. A sufficient statistic is minimal sufficient if it can be represented as a function of any other sufficient statistic. A basic result is that T is minimal sufficient for θ if and only if Pr[x | θ] Pr[y | θ] is not a function of θ, if and only if T(x) = T(y). While a sufficient statistic contains all the information available about θ,anancil- lary statistic contains no information about θ, so that its distribution does not depend on θ. However, in conjunction with other statistics, such as the maximum likelihood estimator, it can provide valuable information about θ. For example, if T is a maximum likelihood estimator that is not sufficient, we may have an ancillary statistic U such that (T, U) is sufficient. Then U is referred to as an ancillary complement of T. For further comments see Schofield and Barker (2016).

A.4 Conditional Expectations

If x and y are any pair of random variables, then

E[x]=Ey {E[x | y]} (A.12) and var[x]=Ey{var[x | y]} + var y[{E[x | y}], (A.13) where Ey etc. denotes taking the expected value with respect to the distribution of y. We note that if E[x | y] does not depend on y (for example just a constant) then the second term of (A.13) is zero and, by the delta method above, Appendix: Some General Results 579

var[x]=Ey[{var[x | y]}

= Ey[g(y)], say, ≈ g(θ)

={var[x | y]}y=θ, (A.14) where θ = E[y]. If z is independent of x and y, then we have

cov[x, yz]=Ez{cov[x, yz | z]} = E[z]cov[x, y]. (A.15)

A.5 Maximum Likelihood Estimation

Let L(θ) be a likelihood function with unknown vector parameter θ and let θ0 be the true value of θ. The maximum likelihood estimatorθ of θ is usually found by solving the equations ∂ log L(θ/∂θ = 0 with respect to θ. Since in this book L(θ) is usually a product of multinomial distributions, it is true under fairly general conditions that θ is asymptotically distributed with a multivariate Normal distribution having mean θ −1 (θ) = ( ) 0 and variance–covariance matrix B0 , where B bij ,

∂2 log L(θ) bij =−E ∂θi ∂θj and B0 is B(θ0). The matrix B(θ) is referred to as the expected information matrix (or Fisher information matrix), and it can also be expressed in the form

∂ log L(θ) ∂ log L(θ) B(θ) = E · . ∂θ ∂θ

This matrix can be estimated by the so-called “observed information matrix” with 2 ijth element −∂ log L(θ)/∂θi ∂θj evaluated at θ = θ. This matrix without the minus sign is also referred to as the estimated Hessian, or just the Hessian of log L(θ) if θ is used instead of θ. If the maximum likelihood estimator θ is not available explicitly, it can be found iteratively using the so-called method of scoring, namely

θ(k+1) = θ(k) + B−1(θ)(k)g(θ(k), where θ(k) is the estimate of θ at the kth iteration and

∂ log L g(θ) = . ∂θ 580 Appendix: Some General Results

If the estimated Hessian is used instead of B(θ)), we have the Newton or Newton– Raphson method. For a detailed description of this method see Sect. 15.2.1. In this book it is particularly used to estimate standard errors (Sect. 15.2.2), though with some capture– recapture models there can be a problem with multiple maxima (Sect. 15.2.3)as mentioned elsewhere, for example with multistate models. For more general computational methods see Seber and Wild (1989: Chap. 13).

A.6 Proﬁle Likelihood Intervals

Confidence intervals for a parameter θ are generally based on θ, the maximum likelihood estimate of θ. In the past, since θ is asymptotically normal with mean θ and variance estimate σ2(θ), the 95% confidence interval used has been θ ± 1.96σ, sometimes referred to as a Wald confidence interval. However, it is well known that this interval can perform badly for small samples in open (single-state) or closed populations, and in other contexts, due to such things as the bias in θ, a poor variance estimate, and asymmetry in the sampling distribution of θ. Various transformations of θ have been suggested such as the logarithm and, if θ is a probability, the logit transformation η = logit(θ) = log[θ/(1 − θ)] (though it is usually expressed linearly in terms of other parameters). We then back-transform the interval using θ = 1 − (1 + eη)−1. An alternative method of constructing confidence intervals that is generally better than the Wald method is to use the so-called profile likelihood, and we first consider having just a single parameter θ.IfL(θ) is the likelihood function of θ, the likelihood- ratio test for the null hypothesis H0 : θ = θ0 does not reject H0 at the α level of significance if, for large samples,

− [ ( ) − ()] < 2( − ), 2 log L θ0 log L θ χ1 1 α or 1 log L(θ )>log L(θ) − χ2(1 − α)(= ∗), 0 2 1

2( − ) ( − ) where χ1 1 α is the upper 100 1 α th quantile of the chi-square distribution with one degree of freedom. The 100(1 − 2α)% profile confidence interval for θ is then the set of all θ0 not rejected by the test, that is satisfies the above inequality, ∗ namely log L(θ0)> . The bisection method is commonly used to find the end ∗ points of the interval by finding the solutions of log L(θ0) − = 0. For a vector parameter θ = (θ1,...,θp) , we can find the profile interval for a single element, say β = θi , as follows. We first find the maximum likelihood estimate of θ subject to θi = β,sayθ(β). Then our profile interval for θi is the set of all β satisfying log L(θ(β)) − ∗ = 0. Appendix: Some General Results 581

The bisection method is very time consuming as it requires solutions of the equation log L(θ(β)) − ∗ = 0 for each element of θ. However, Gimenez, Choquet et al. (2005) use a better algorithm proposed by Venzonand Moolgavkar (1988)asfollows. The end points are solutions of the nonlinear equations F(θ) = 0, where ⎛ ⎞ log L(θ) − ∗ (θ) = ⎝ (θ) ⎠ , F ∂ log L . ∂θ−i where θ−i is θ without θi . The reader is referred to Gimenez, Choquet et al. (2005) for further details such as finding the best starting value of θ for the iterative process, coping with problems like boundary estimates, and allowing for over-dispersion. They found from their simulation results that both the profile and Wald-type intervals achieve the nominal coverage as long the size of the release cohort exceeded a hundred. Furthermore, when it is less than this there was no gain in using the profile instead of the Wald intervals. An algorithm is available for computing the profile intervals in M-SURGE (Choquet et al., 2003). Clearly this topic needs further investigation.

A.7 Generalized Least-Squares

Suppose we have a random vector y = (y1, y2,...,yn) where the yi have a common mean μ and known variance–covariance matrix V. Then we can express this information as a linear model y = 1nμ + ε, with variance–covariance matrix var[ε]=V. The generalized least-squares estimate −1 μ of μ, obtained by minimizing (y − 1nμ) V (y − 1nμ), is given by

= ( −1 )−1 −1 μ 1nV 1n 1nV y

( −1 )−1 with variance 1nV 1n . This model arises when the yi are estimates (e.g., survival or abundance estimates) and the model for constant μi (= E[yi ]) is accepted as reasonable. Usually V is replaced by its estimate V. If we now assume y is multivariate normal N(μ, V), which is approximately the case if the yi are maximum likelihood estimators, a large sample test for hypothesis H0 : μ = 1nμ is given by

−1 Q = (y − 1nμ) V (y − 1nμ),

2 which is asymptotically distributed as χn−1 when H0 is true. This can be proved by noting that Q is the residual sum of squares for a generalized least-squares ﬁt (with V 582 Appendix: Some General Results

estimated by V) and is the likelihood-ratio test statistic for H0. It can also be proved directly by using a generalized inverse.

A.8 Large Sample Hypothesis Tests

In the past there has been two standard general procedures for testing hypotheses in capture–recapture modeling, the Pearson goodness-of-fit test and the likelihood- ratio test that leads to the concept of deviance. More recently the Score test has also been used, which we first describe generally. Given the likelihood function L(θ),the Score test for testing a null hypothesis H0 about θ is ∂ log L(θ) ∂ log L(θ) B , ∂θ ∂θ θ=θ0 where θ0 is the maximum likelhood estimate of θ when H0 is true, B is −n−1E[∂2 log L(θ)/∂θ∂θ], the expected information matrix, and n is the sample size. It is not recommended to replace B by the observed information matrix −∂2 log L (θ)/∂θ∂θ (cf. Morgan, Palmer, and Ridout, 2007, for a simple illustration of what can go wrong if this is done). There is a general principle involved with exponential families, which includes the multinomial. If θ is the unknown vector parameter and T denotes a vector of sufficient statistics, then the likelihood function can be factorized so that

L(θ; data) = Pr[data | T]×Pr[T; θ], where the second component is used to estimate θ, while the ﬁrst component is used to test for model adequacy, usually involving hypergeometric distributions and contingency tables. We now consider the likelihood-ratio and goodness-of-ﬁt tests with respect to a single multinomial distribution and then for the product of several independent multinomial distributions.

A.8.1 Single Multinomial Distribution

Using the nonsingular version of the multinomial distribution (Appendix A.1), the likelihood function (ignoring constants) is

k ( ) = xi , L p pi i=1 Appendix: Some General Results 583 k−1 where p = (p1, p2,...,pk−1) and pk = 1 − = pi . Suppose we wish to test the i 1 hypothesis H0 that p = p(θ) where θ is a vector of q unknown parameters. If θ is the maximum likelihood estimate of θ, the likelihood-ratio test for H0 is k p (θ)xi = i=1 i , k xi i=1 pi where pi = xi /n. We then ﬁnd that (cf. Seber, 2013: 46)

G2 =−2log k p = 2n p log i i (θ) i=1 pi k [x − np (θ)]2 ≈ i i for large n (A.16) (θ) i=1 npi = X 2, say, the Pearson goodness-of-ﬁt test. We note that

k np G2 = 2 np log i i (θ) i=1 npi k x = 2 x log i . i (θ) i=1 npi

The two test statistics therefore take the memorable forms O (O − E)2 G2 = 2 O log and X 2 = , E E where O stands for “Observed” and E for “Expected.” The two test statistics are asymptotically equivalent and have a large sample χ2 distribution with k − 1 − q degrees of freedom. Here X 2 is, in this instance, also the Score test or the Lagrange Multiplier test. (For some background theory of large sample tests see Seber, 2015: Chaps. 10–12: Chap. 12 refers to the multinomial distribution.) The original model (A.1)isreferredtoasthesaturated model as the number of mathematically independent observations (k − 1) equals the number of unknown mathematically independent pi , and we have a perfect ﬁt of the data to the model. If Ls = log L(p), the maximum value of log likelihood for the saturated model, 2 and L0 is the maximum value when H0 : pi = pi (θ) is true for all i, then G = 2(Ls − L0), and this difference is called the deviance. It has an approximate chi- square distribution when H0 is true. If H01 is a hypothesis nested in H0, then in testing H01 versus H0 we look at the difference in the two deviances and note the 584 Appendix: Some General Results

2 magnitude of any reduction G .AsLs cancels out we end up with the likelihood- ratio test 2(L0 − L01).

A.8.2 Product of Independent Multinomial Distributions

The above theory extends naturally to a product of I independent multinomial distributions with the ith having ki cells and ni trials. If xij is the frequency observed in the (ij)th cell that has probability pij, then to test H0 that pij = pij(θ) we ﬁnd that O becomes xij and E becomes ni pij(θ) and we sum over i and over j (from 1 2 2 to ki ) to obtain G and X .

A.8.3 Diagnostic Residuals

In regression, residuals of the form observation minus the fitted value, or scaled versions, have been used effectively to provide a check on the assumptions underlying the model. In multinomial models we focus on scaled versions of the raw residuals xi − npi (θ) or Oi − Ei . For example, the Pearson residuals ei = (xi − npi (θ))/ npi (θ) √ ( − )/ or Oi Ei Ei are called the (scaled) Pearson residuals (though terminology is 2 = 2 not consistent in the literature) and satisfy i ei X , the Pearson goodness-of- fit test. If the data come from Poisson distributions, for which the mean equals the variance, then the ei are approximately distributed as N(0, 1). For multinomial data, the variance is smaller than the mean (compare npq for the binomial distribution with np for a Poisson) so that the ei will be too small in this case. Instead we can divide each ei by its estimated standard deviation to get the√ so-called adjusted (or standardized) residual (Haberman, 1973), namely ri = ei / vii where vii is an estimate of the variance of ei . An expression for ri and a derivation are given by Agresti (2013: Sect. 16.3.2, Eq. (16.18)), which we summarize as follows. Assuming H0 is true, let θT be the true value of θ and let pT = p(θT ).Let ∂p ∂p ∂ p (θ) = ( (θ))−1/2 , = r , A diag p where ∂θ ∂θ ∂θs and diag p is a diagonal matrix with elements pi (i = 1, 2,...,k). We then find that for large n, e is asymptotically multivariate normal Nk (0, VT ), where 1/2 1/2 −1 VT = Ik − p(θ) (p(θ) ) − A(A A) A . θ=θT Appendix: Some General Results 585

Estimating θT by θ we get k k 1 ∂ p ∂ p v = 1 − p (θ) − i i νrs , ii i p (θ) ∂θ ∂θ r=1 s=1 i r s θ where νrs is the (r, s)th element of (AA)−1. Another type of residual is the so-called deviance residual that is deﬁned by 2 = k 2 G i=1 i , where = ( − (θ)) | xi |, i sign xi npi 2xi log npi (θ) where sign(xi − npi (θ)) is +1 if the difference is positive, 0 if the difference is 0, and −1 if the difference is negative. If an xi is zero and all the pi ’s are positive, we 2 ignore the residual for xi in G . In capture–recapture studies we record yω the number of individuals with capture history ω. For example, if s = 4 and ω = (1010), we consider those individuals caught in the ﬁrst and third samples, but not in the second√ and fourth. If E[yω]=μω, then we can obtain the Pearson residual (yω − μω)/ μω and the deviance residual

1/2 sign(yω − μω)[2yω log(yω/μω)] .

Various box plots can be constructed with these residuals, for example all the residuals with frequency of capture i. When ω takes values (1100), (1010), (1001), etc. we have i = 2 in each case. When carrying out an omnibus goodness-of-ﬁt test for a contingency table and obtaining signiﬁcance, we need to look at cell residuals to see where the departures lie. Sharpe (2015) has some helpful comments on the matter and comes up with four suggestions including a residual analysis.

A.8.4 Contingency Table

In this book we frequently come across likelihoods of the form ai ci k b d L = i i , ai + ci i=1 bi + di a product of Hypergeometric distributions. This can then be used to give k 2 × 2 contingency table tests of the form Table A.1, each with an approximate chi-square 586 Appendix: Some General Results

Table A.1 Contingency table ai ci ai + ci for Hypergeometric + distribution bi di bi di ai + bi ci + di ai + bi + ci + di

test with one degree of freedom. If the tests are independent, they can be pooled to give a single test with k degrees of freedom. In the case of ai ci ei k b d f L = i i i , ai + ci + ei i=1 bi + di + fi we simply add another column to the 2 × 2 part of the Table A.1.

A.9 Prior Distributions

In contrast to a frequentist approach to statistics we have a Bayesian approach whereby an unknown parameter is assumed to have a distribution called the prior distribution. As most of the parameters in this book are probabilities, a very useful prior distribution is the beta distribution as it is deﬁned on [0, 1] and can assume a wide variety of shapes. This distribution takes the form

(a + b) f (p) = pa−1(1 − p)b−1, 0 ≤ p ≤ 1, a > 0, b > 0, (a))(b) which we denote by Beta(a, b). To see the variety of shapes we note that a = b = 1 gives the uniform distribution, a = 2, b = 1ora = 1, b = 2 give us triangular distributions, while a = 2, b = 2 produces a parabola. Another useful prior distribution is the gamma distribution, Gamma(α, β), with density function

βα f (τ) = τ α−1e−βτ , α > 0, β > 0, 0 < τ < ∞. (α)

Sometimes we can use an improper prior density, that is one that does not integrate to 1 (or sum to one if discrete). For example we can have

f1(μ) = 1, −∞ < μ < +∞, Appendix: Some General Results 587 which is a ﬂat prior that integrates to inﬁnity, and

f2(σ) = 1/σ, σ > 0, where the prior for log σ is flat. They can generally be used in some situations and do express complete ignorance, but they sometimes can cause problems especially with model selection and hypothesis testing. WinBUGS does not allow their use, but it does use vague priors that have densities with a wide and mostly flat spread. This is achieved by having a large variance. For example, μ ∼ N(0, 106) gives a density function that is essentially flat but integrates to 1, while the inverse prior f2(σ) = 1/σ can be approximated by a gamma distribution Gamma(α, β) where α and β are very small. In practice a gamma prior can be used for 1/σ2 (the so- called precision) or an inverse gamma for σ2. A uniform prior can also be used for σ (Gelman, 2006). There is also a Jeffrey’s prior and a modification that overcomes some of its shortcomings called a reference prior. It should be noted that flat priors are not invariant to transformations. For example, if φi is a survival probability and log(φi /1 − φi ) = bxi , then a uniform prior on b will not give a uniform prior on φi . King, Morgan et al. (2009: Chap. 8) discuss priors for covariate parameters. We can also have hierarchical or two-stage priors where a prior is placed on a prior. For example if the probability p has a prior Beta(a, b) distribution and priors are put on a and b, then the latter are called hyper-priors and a and b hyper-parameters. The use of hyper-priors “dilutes the influence on the posterior of any prior assumptions made and essentially creates random-effects of the model parameters” (King, Morgan et al., 2009: see Chap. 4). The influence of the prior distribution should generally be assessed via a sensitivity analysis, where the effect of changing the priors is considered (e.g., a wider range for a uniform distribution or a larger variance for a normal distribution).

A.10 Bayesian Methods

There are a number of books on Bayesian methods but Barker and Link (2010a), King, Morgan et al. (2009), Morgan (2008: Chap. 7), and McCarthy (2007), focus on ecological applications where much of the following can be found. Some general comments are made in Sect. 15.6. We ﬁrst begin with univariate posterior distributions.

A.10.1 Univariate Priors

Suppose x is a discrete random variable with probability distribution Pr[x; θ], usually referred to as the likelihood function, that depends on an unknown parameter θ. 588 Appendix: Some General Results

With a Bayes’ approach, θ is given a probability distribution Pr[θ] called the prior distribution of θ. The so-called posterior distribution of θ is then given by

Pr[θ, x] Pr[x | θ] Pr[θ] Pr[θ | x]= = Pr[x] Pr[x] ∝ Pr[x | θ] Pr[θ].

We can use the above, for example, to maximize the posterior distribution with respect to θ as we can omit anything that does not involve θ. In general, θ is a continuous variable so that Pr[θ] should be replaced by a density function f (θ) and Pr[θ | x] by f (θ | x). (We use the same f for notational convenience even though they are different functions. In the chapters we have usually used π(θ) to emphasize that we are dealing with priors.) We therefore have the general expression used repeatably in this book posterior ∝ likelihood × prior. (A.17)

In practice we shall sometimes write the likelihood as L(θ) or L(θ; x), when the emphasis is on θ and we omit any product terms not involving θ. We then write

f (θ | x) ∝ L(θ; x) f (θ).

We note that Pr[x] is the marginal distribution of x given by Pr[x]= Pr[x | θ] f (θ)dθ.

The percentiles of a posterior distribution can be used to construct an interval for θ called a highest-posterior-density interval. If .05 and u.95 are the lower and upper 5% percentiles of the posterior distribution, then

Pr[.05 < θ < u.95 | x]=0.9 and θ lies in the interval [.05, u.95], usually called a credible interval, with a probability of 0.90, or loosely 90%. We could also use what we might describe as one-sided 90% intervals such as (0, u.90) and (.10, 1). If x ∼ Bin(n, p) and p ∼ Beta(a, b), it follows from (A.17) that

Pr[p | x]∝px (1 − p)n−x pa−1(1 − p)b−1 = px+a−1(1 − p)n−x+b−1, which is clearly of the form of a beta distribution with parameters x + a and n − x + b. We have thus highlighted another useful property of the beta distribution, Appendix: Some General Results 589 namely a beta distribution combined with a binomial distribution gives another beta distribution. The beta family of distributions is said to be conjugate for the binomial parameter p as the prior and posterior distributions are in the same family, namely the beta family. If one can choose a conjugate prior, the form of the posterior is known and conjugacy completely solves the computational problem. As Barker and Link (2009: 38) comment: “The beauty of this is that the posterior from one study is ready-made to serve as prior in the next.” Also, this property of the beta distribution is particularly useful since a multinomial distribution can be expressed as a product of conditional binomial distributions (cf. Appendix A.1). We note that if the prior for p is the uniform distribution, then the posterior distribution is simply a scaled version of the likelihood function, namely of pa−1(1 − p)b−1. The mode of the posterior distribution is then the maximum likelihood estimate of p. A uniform prior can be useful as it reflects the fact that we have no preference for any particular value of p. Other conjugate priors are the gamma distribution for the mean of Poisson distribution, the beta distribution for the parameter of a geometric distribution, the normal distribution for the mean of a normal distribution, and the inverse gamma for the variance of a normal distribution (King, Morgan et al., 2009: Appendix A). Another interesting family is the family of so-called normal-gamma distributions. To avoid an estimate of p lying outside the [0, 1] interval it is common practice to take a logit transformation of p, namely log[p/(1 − p)], which can now take values in (−∞, +∞). We then find that x above can replaced by y, which we can assume to be normally distributed as N(μ, σ2) so that θ = (μ, τ), where τ = 1/σ2 is the precision. Suppose we choose a bivariate prior for θ, namely f (μ | τ) f (τ), where −1 f (μ | τ) has a normal distribution N(μ0, τ ) and f (τ) has a gamma distribution Gamma(α, β). Then we find that θ has a normal-gamma distribution

f (θ) = f (μ | τ) f (τ), and it is the conjugate prior with respect to the normal distribution having unknown mean and precision. Alternatively we can use σ2 and use the inverse gamma distribution for the prior distribution of σ2. For more general prior distributions we can use the population mean of the posterior distribution to estimate θ. This, along with the percentiles, requires us to fully know the posterior distribution, which means knowing the normalizing constant Pr[x] described above so that the integral of the posterior density is unity. As the normalizing constant may be difficult to find, an alternative approach would be to somehow obtain a sample from the posterior distribution and then use sample statistics such as the sample mean and sample percentiles to estimate the population parameters. We discuss this approach in Sect. A.11. We finally mention the term “full conditional” distribution of a parameter, which applies to the posterior distribution of a parameter conditional on the values of the other parameters. 590 Appendix: Some General Results

A.10.2 Multivariate Priors

The theory in the previous section still holds if we replace x and θ by vectors x and θ. However, the computations become more difﬁcult. If θ = (θ1, θ2,...,θk ) , the joint prior distribution chosen is often the product of the individual priors of the (θ) = k ( ) θi so that f i=1 fi θi . A typical example of this independence is when the likelihood f (x | θ) is from a multivariate normal distribution with μ and variance– covariance matrix Σ. Then we can either assume the independence of μ and Σ so that f (θ) = f (μ, Σ) = f (μ) f (Σ), or else follow the lead of the univariate normal case considered above and assume f (θ) = f (μ | Σ) f (Σ). To make an inference about a single parameter θ1 we not only need the normalizing constant Pr[x] obtained by integrating the joint posterior prior, but we also need the marginal posterior distribution

f (θ1 | x) = ... f (θ1, θ2,...,θk | x)dθ2 ...dθk

obtained by integrating out the other θi ’s. Both processes tend to be analytically intractable, and numerical methods generally need to be used. These are discussed in the next section. For general vectors x = (x1, x2,...,xk ) with elements in [0, 1], a useful prior is the so-called Dirichlet distribution Dir(α), a multivariate beta distribution with density function k k k ( α ) − f (x; α) = i=1 i xαi 1, x = 1, 0 < x < 1, i = 1,...,k. k ( ) i i i i=1 αi i=1 i=1

This is a conjugate prior for a multinomial distribution. If the xi are independent [ , ] = k = k ∼ [ , ] Gamma αi β , x· i=1 xi , and α· i=1 αi , then x· Gamma α· β and the yi = xi /x· have a joint Dir[α1,...,αk ] distribution. We ﬁnally add that an inverse Wishart distribution is a conjugate prior of the covariance matrix of a multivariate normal distribution.

A.11 Markov Chain Monte Carlo Sampling

In the one-dimensional situation a key problem is to ﬁnd the posterior distribution Pr[θ | x] so that we can sample from it. One way this is done is to set up a Markov chain which, in the long run, produces the required distribution. A kth order Markov chain is a sequence of random variables y1, y2,...such that

Pr[yt | yt−1, yt−2,...,y1]=Pr[yt | yt−1, yt−2,...,yt−k ], Appendix: Some General Results 591 that is the probability distribution of the next variable depends only on the previous k variables (called a kth order chain). For a first-order chain the distribution of yt depends only on yt−1. The following technical conditions are assumed, namely the chain is irreducible (ergodic) so that all the states can communicate with each other, is positive recurrent (each state i can return to state i in a finite time, which follows from irreducibility if the chain is finite), and is aperiodic. These conditions are discussed further in Sect. 15.7.2. Under these conditions, the Markov chain settles down as t increases on a distribution called the stationary distribution that does not depend on the starting value y1. (In practice periodic behavior is rarely an issue but non-recurrence can be a problem.) For large t,sayt > t0, a set of values of yt will approximate observations from the stationary distribution, though successive values suffer from autocorrelation (that we would endeavor to minimize) and they only approximate a random sample because of the Markov property. However, for a first order Markov chain we could take every second value to reduce the dependence, or for a kth order chain take every (k + 1)th observation. It is important to run a long chain and to allow a “burn in” before using the observations. The trick is to ensure that the stationary distribution turns out to be the required posterior distribution so that observations from the chain approximate a random sample from the posterior distribution. This simulation process for examining probability distributions is called Markov chain Monte Carlo or MCMC (cf. Link, Cam et al., 2002;Morgan(2008: Chap. 7); King, Morgan et al., 2009, Chap. 5; King, 2011). The process can be carried out using software such as WinBUGS and a method called the Metropolis–Hastings (MH) algorithm with the special case of Gibbs sampling discussed below, can be used. The beauty of the MH algorithm is that we can draw samples from a probability density function f (y) provided we can compute the value of a function that is proportional to f (y), which means we avoid the need for a normalizing constant. At each iteration the algorithm picks a candidate for the next sample value based on the current sample value. Then, with a certain probability, the candidate is either accepted (in which case the candidate value is used in the next iteration) or rejected. In the latter case, the candidate value is discarded and the current value is reused in the next iteration. We now provide some details (Barker and Link, 2010b). Let f (y) be the target distribution, that is the distribution we wish to sample from, and let g(y | z) be the “proposal” distribution (to be specified) to provide candidate generating distributions that describe probabilities for candidate values y,giventhe current value z. Then, fixing a value of y0, we generate yt for t = 1, 2,...according to the following rules: Step 1: Generate a candidate value, y by sampling from g(y | yt−1). Step 2: Calculate ( ) ( | ) = f y g yt−1 y . r f (yt−1)g(y | yt−1)

Step 3: Generate a value of u from the uniform distribution on [0, 1]. Step 4:Ifu < r,setyt = y , otherwise set yt = yt−1. 592 Appendix: Some General Results

We note that the target distribution only occurs in r in the numerator and denominator, and, if f (y) is a posterior density function, the pesky normalizing constant cancels out of r. Clearly the performance of the above algorithm will depend on the choice of the value of y0 and the function g. Use of the method for the CJS model is described by Link and Barker (2008). For a helpful introduction to the above topics see Brooks (1998). We note in passing that a related method called data cloning has appeared in the literature (Lele, Dennis, and Lutscher, 2007). Knape and de Valpine (2012) and (Finke et al. 2017) describe an adaption called Markov chain Monte Carlo with particle ﬁlters (PFMCMC) that has been used for statespace and integrated population models. Some background theory for the MH algorithm is given in Sect. 15.8, and examples are given in Sects. 5.6.2 and 9.2.3. A method that also makes use of auxiliary variables called Hamiltonian Monte Carlo is described in Sect. 15.11.

A.11.1 Gibbs Sampling

A special case of the previous section is Gibbs sampling that is designed for multivariate posterior distributions and is therefore particularly appropriate for using Bayes’ methods in capture–recapture (see also Barker and Schoﬁeld, 2007). If X is the data set and θ = (θ1, θ2,...,θk ) , the goal is to sample from the posterior distribution f (θ | X). A detailed discussion of this method is given in Sect. 15.9, and a brief summary of the Gibbs algorithm is given below. An application is given in Sect. 13.7.2. Let θ(−i) denote the k − 1 dimension vector made up of all the components of θ but omitting θi .Thefull conditional distribution for θi is

f (θi | θ(−i), X), abbreviated to f (θi |·), which is the distribution of the ith component of θ, having fixed values of all the other components, and having been informed by the data. Like the posterior distribution it is proportional to Pr[X | θ] f (θ), the difference being that the normalizing constant is now f (θ(−i), X) rather than Pr[X]. It turns out that full conditionals f (θi | θ(−i), X) are often easily identified by inspection of Pr[X | θ] f (θ), when the marginal posterior distribution f (θi | X) of the joint posterior density f (θ | X) is not readily found. For example, there are situations for a pair (μ, τ), namely the mean and precision, where the prior and the full conditional distribution for the pair both belong to the same normal-gamma family. Although the prior and the posterior distributions do not belong to this same family, the use of conjugate families can be very useful for Gibbs sampling, which is described below. Full conditional distributions can exist even when improper priors are used The Gibbs sampling algorithm proceeds as follows for drawing samples from the (θ | ) θ(0) = ( (0), (0),..., (0)) joint posterior distribution f X .Wefirstfixavalue θ1 θ2 θk . Then for t = 1, 2,..., generate θ(t) according to the following rules: Appendix: Some General Results 593

(t) ( | θ(t−1), ) Step 1: Sample θ1 from the full conditional f θ1 (−1) X . (t) ( | θ(t−1), ) Step 2: Sample θ2 from the full conditional f θ2 (−2) X . ... (t) ( | θ(t−1)) Step k: Sample θk from the full conditional f θk (−k) . + θ(t) = ( (t), (t),..., (t)) Step k 1: Set θ1 θ2 θk . It has been suggested that we can sequentially update θ(t) after each step in the proceeding algorithm and then use the partially updated θ(t) in sampling full (t) conditionals. For example, θ3 could be sampled from the full conditional distribution

( | (t), t), (t−1),..., (t−1), ) f θ3 θ1 θ2 θ4 θk X rather than ( | (t−1), (t−1),..., (t−1), ). f θ3 θ1 θ2 θk X

Unfortunately this method is now not recommended, as the Markov chain may not converge to the posterior distribution. In the case of random-effects models, Browne (2004) and Browne, Steele et al. (2009) have shown that two reparameterizing methods, hierarchical centering mentioned above and parameter expansion (Liu, Rubin and Wu, 1998) can improve the efﬁciency of the Gibbs sampling algorithm described above. The methods can be combined to create a more efﬁcient MCMC estimation algorithm in which posterior correlations are reduced with a consequent increase in the effective sample size. A survey of Monte Carlo methods is given by de Valpine (2004, 2008).

A.11.2 Test for Stationarity

We have seen above that under general conditions the Markov chain converges to a stationary distribution that is the posterior density function. After a settling in (“burn in”) period when initial values are ignored, members of the chain then behave like a sample from the posterior distribution that can be used for inference purposes. What we need are criteria for determining the number of iterations for an adequate burn in time, the number of additional iterations, and the value of k, where every kth iteration is chosen. Unfortunately it seems that all we can do at present is to run some simulations and check to see whether the chain has reached an adequate level of convergence. One method for doing this is to use the Brooks-Gelman-Rubin diagnostic (Gelman and Rubin, 1992b; Gelman, Carlin et al., 2014),whichwenow describe. Suppose we have the posterior density f (θ | x) for a parameter θ. Then using different starting values that are over-dispersed with respect to the posterior distribution, we run m Markov processes {yit}, (i = 1, 2,...,m; t = 1, 2,...)that are values of θ, each for 2n iterations, and we use just the last n, labeled as t = 1, 2,...,n.We 594 Appendix: Some General Results run multiple sequences, as a single sequence can be misleading (Gelman and Rubin 1992a). The method used is essentially the same as a one-way analysis of variance where we compare two estimates of variance of the stationary distribution. If

n 2 1 2 s = (y − y ·) , i n − 1 it i t=1 then their sample mean is 1 m W = s2, n m i i=1 and we ﬁnd that, before convergence, Wn underestimates the posterior variance var[θ | x].LetBn be the between-chain sum of squares, where

m Bn 1 2 = (y · − y··) n m − 1 i i=1 is the variance estimate for the m sequence means. The weighted average of Wn and Bn, n − 1 1 v [θ | x]= W + B , n n n n n overestimates var[θ | x], but becomes unbiased when the chains have reached their stationary distribution. A diagnostic test is based on the assumption that the target posterior distribution 2 is normal with mean μ and variance σ .Ifμ = y··, the sample mean of the mn observations, and using an estimate√σ of the scaling factor, we assume that can we ( − )/ v √use the t-distribution for yt μ n with df degrees of freedom and scale factor vn, where B v = v [θ | x]+ n . n n mn The degrees of freedom df is estimated by the method of moments, giving us

2v2 df = n . var[vn]

An expression for var[vn] is given by Gelman and Rubin (1992b: 461), namely n − 1 2 1 m + 1 2 2 var[v ]= var[s2]+ B2 n n m i mn m − 1 n ( + )( − ) m 1 n 1 n 2 2 2 +2 {cov[s , y ]−2y··cov[s , y ·]}, mn2 m i i· i i Appendix: Some General Results 595 where the estimated variances and covariances are obtained from the m sample 2 values of yi· and si . Since the posterior distribution of the transformed yt is now approximated by this t-distribution, we can use it to obtain a credible interval for yt . A measure of the scale factor for this distribution that indicates how it might be 1/2 reduced if n →∞is the so-called shrink factor Rn , where v ( + ) − + ( + ) n df 3 n 1 Bn m 1 df 3 Rn = · = + . Wn (df + 1) n Wn mn (df + 1)

Then lim Rn = 1. n←∞ If Rn is much greater than 1 we should increase n. The correction (df + 3)/(df + 1) by Brooks and Gelman (1998) will be minor as df tends to be large at convergence. The program BUGS will construct an ergodic chain that can have rapid convergence if the model is properly specified. It also has an algorithm for tuning chain construction to try and minimize autocorrelation. The computations can be carried out using for example the gelman.diag program in the R package. A review of 13 methods for detecting convergence was given by Cowles and Carlin (1996)who concluded that several methods should be used at the same time. Brooks and Roberts (1998) also provided a review, but focusing on theoretical aspects of the methods. Giakoumatos, Vrontos et al. (1999) provided a diagnostic based on a subsample of the chain. A sequential plot of the yt , called a trace plot, will give some idea as to how yt is settling down as t increases. Warning signs of non-stationarity are the chain wandering around, with changes in variability, sudden changes in mean behavior, or high autocorrelation that does not seem to decay. Here an autocorrelation plot of the yt and of other measures can be helpful. We note that Wn and vn should both settle down as n increases. Brooks and Gelman (1998: 438–439) have several graphical methods based on splitting up the sequences into batches. For a given sequence, the 1/2 R program gelman.plot calculates the shrink factor Rn for the first 50 observations, for the first 50 + k observations, for the first 50 + 2k observations, and in general for the first 50 + rk observations, where k is called the “bin width.” The shrink factor is then plotted against r. Such a plot avoids the potential problem of the shrink factor being close to 1 by chance, leading to a misdiagnosis by gelman.diag; it will show whether the shrink factor has really converged or is still fluctuating. Once convergence has been approximately reached, the mn observations can be used to obtain a 100(1 − α)% credible interval using the α/2 and 1 − α/2 quantiles of the observations. We can carry out the above procedure for each parameter θ in the model. The above method has been extended by Brooks and Guidici (2000) using a two- way analysis of variance method that incorporates two processes, the one mentioned above and the Bayesian modeling choice that involves the reversible jump algorithm. In addition to Bm and Wm there are three other sums of squares allowing for model variation. 596 Appendix: Some General Results

A.11.3 Reversible Jump Algorithm

There are times when we want to apply MCMC to situations where we do not know the dimension of the parameter set, or we wish to jump from one dimension to another to compare different models. In the presence of model uncertainty, the posterior distribution can be extended to be deﬁned jointly over both parameter and model space, so that exploring the model and parameter space can be carried out simultaneously. This can be done using the so-called reversible jump MCMC (RJMCMC) method that extends the scope of the Metropolis–Hastings method (Green, 1995). The main criterion is that the process is reversible so that if we jump from one parameter space to another we want to be able to jump back. The idea is to add a bit to the smaller space essentially increasing its dimensions so that there is a one-to-one relationship between the two spaces. For a detailed discussion of the topic see King, Morgan et al. (2009: Chap. 7), and Barker and Link (2010a) involving Gibbs sampling. (Oedekoven et al. 2016) described using hierarchical centering (Gelfand, Sahu, and Carlin, 1995) for RJMCMC in the presence of random effects, where the zero-mean of the random- effect component, typically assumed to be normally distributed, is replaced with a model consisting of an intercept and one or more ﬁxed effect covariates. This leads to a reduction in autocorrelations within the MCMC algorithm. For a detailed description of the reversible jump method, and transdimensional methods in general, see Sect. 15.12.

A.11.4 Model Averaging

In the past, using classical frequentist methods, the aim was to obtain the best ﬁtting model according to some criterion. This is problematic when several different models ﬁt almost equally well, so that model averaging has become popular (cf., Kabailaa, Welsh, Abeysekera, 2016; Fletcher, 2018). With a Bayesian approach, especially involving Markov chain Monte Carlo (MCMC), one can use prior probabilities for the different models, obtain posterior probabilities, and then use some kind of model averaging. Following Brooks, Catchpole, and Morgan (2000a: Appendix A2), let M j denote the jth model ( j = 1,...,J) with likelihood function L j θ; x),prior distribution π j (θ) for the parameter θ, and prior probability Pr[M j ]=p j . (Strictly speaking, the probability is conditional on the model set.) If x is the data set, then the posterior probability is

Pr[x | M ]p Pr[M | x]= j j . j Pr[x] where

Pr[x | M j ]= L j (θ; x)π j (θ)dθ = c j (x). Appendix: Some General Results 597

If we simulate n values θ1, θ2,...,θn from the prior for θ, we can estimate c j (x) by, for example, 1 n c = L (θ ; x). j n j k k=1

We can then estimate the relative posterior model probabilities

Pr[x | M ]p p c (x) Pr[M | x]= j j = j j j J [ | ] J ( ) k=1 Pr x Mk pk k=1 pk ck x to compare the models. Typically p j = 1/J, and Link and Barker (2009)givethe approximation exp[−BIC /2]p Pr[M | x]≈ j j , j J [− / ] k=1 exp BICk 2 pk where BICk is deﬁned in Sect. 14.4.2. If we want to obtain a model average we begin with π(θ | M j , x), the posterior distribution for θ under model M j and then obtain the average posterior distribution

J π(θ | x) = Pr[M j | x]π(θ | M j , x). j=1

We can then obtain estimates of the model-averaged posterior means and standard deviations of the parameters using

Further comments about using model-averaged estimates are given in Sect. A.11.4. The above ideas are discussed further by Link and Barker (2009). In the case of prediction, if Q is a quantity about which we wish to make a prediction based on the observation x for model M j , then we would use Pr[Q | x, M j ]. However, if we were interested in in combining predictions over the entire model set we would use

J Pr[Q | x]= Pr[Q | x, M j ] Pr[M j | x], j=1 598 Appendix: Some General Results where the posterior model probabilities Pr[M j | x] serve as weights. If we were interested in comparing two models M j and Mk we might consider the ratio of their posterior probabilities

Pr[M j | x] Pr[x | M j ] Pr[M j ] Pr[M j ] = = BFjk , Pr[Mk | x] Pr[x | Mk ] Pr[Mk ] Pr[Mk ] where BFjk is called the Bayes factor for comparing models j and k.Itisthe multiplicative factor by which prior model odds are converted to posterior model odds. Barker and Link (2013) presented a simple and intuitive “palette” version of reversible jump Markov chain Monte Carlo (RJMCMC) that allows models to be ﬁtted one at a time using an ordinary MCMC method and effectively Gibbs sampling. The models are then post-processed to compute Bayes factors or model weights for comparing the models.

A.11.5 Bayesian Diagnostics

Gelman, Meng, and Stern (1996) and Gelman and Meng (1996) described a general diagnostic based on the idea of discrepancies using a p-value approach to goodness of ﬁt. Here Gibbs sampling is used to take a posterior sample θi , θ2,..., θn of parameter estimates. For each θi we simulate a set of recovery data xi from the assumed model matched to the original data set x by using the same release cohort sizes. We then let D(x, θ) be some measure of discrepancy and consider plotting Di = D(xi ; θi ) for i = 1, 2,...,n versus Di = D(x, θi ). The Bayesian p-value for the model is then the fraction of times that Di > Di . As noted by Brooks, Catchpole et al. (2000b), if the model is adequate this fraction should be around 0.5. They used the Freeman–Tukey statistic (Freeman and Tukey, 1950) ( , θ) = ( 1/2 − 1/2)2, D x x j e j j where e j is the expected frequency of x j . In capture–recapture models many cells may contain few observations, and the Freeman–Tukey measure avoids the need to pool small cells to avoid overweighting. Also the square root serves to stabilize the variance in the models considered.

A.12 Bootstrap and Monte Carlo Estimates

In this book we frequently have a sample of data x1, x2,...,xn that we use to calculate an estimate θ of an unknown parameter θ. Often the computation is complex, and the variance and distribution of θ are unknown. One way around the problem is to use Appendix: Some General Results 599 so-called “bootstrapping”, a type of resampling method introduced by Efron (1979). Here we take a random sample with replacement of size n from the n values of xi so that some of the values may be repeated, and calculate an estimate θ∗ from this new sample that has an underlying probability function of

Pr[X = xi ]=1/n, i = 1, 2,...,n.

∗ = , ,..., We then repeat the process k times and obtain k estimates θi (i 1 2 k). This gives k observations from the distribution of θ∗ representing a so-called Monte Carlo approximation to the distribution of θ∗. We can then calculate the sample mean θ and sample standard deviation sθ of these k values to obtain a confidence interval or hypothesis tests for θ. It transpires that for large k, this Monte Carlo approximation for the distribution of θ∗ − θ generally approximates well the distribution of θ − θ. ∗ We can also use the values of (θ − θ)/sθ and their percentiles to obtain a confidence interval for θ (Efron and Tibshirani, 1993: 322–325). Chernick (2008:12) believes that “many simulation studies indicate that the bootstrap can safely be applied to a large number of problems even where strong theoretical justification does not yet exist.” Sometimes a bootstrap solution is better than nothing, even if only approximate! For a helpful review of simulation and bootstrapping methods see Morgan (2008: Chap. 6).

A.13 Instantaneous Mortality

If we have a general mortality process that is Poisson with parameter μ, then we have that the probability that the animal dies in a time interval (t, t + δt) is μδt + o(δt), where o(δt) is some function of δt such that o(δt)/δt → 0asδt → 0. It can be −μt shown that the probability of surviving in the interval (0, t) is φt = e .IfT is the time that an individual dies (its lifetime), then

F(t) = Pr[T ≤ t] = 1 − Pr[T > t] = 1 − Pr[animal survives until time t] = 1 − exp(−μt), and T has probability density function

f (t) = F (t) = μe−μt ,(y ≥ 0).

Therefore the mean life expectancy is

∞ −μt EL = E[T ]= μte dy = 1/μ =−1/ log φ1. 0 600 Appendix: Some General Results

A.14 Hazard Functions

We begin with a survival function φ(t), the probability of survival to time t.Inthe previous section the lifetime T of an individual has a probability density function f (t) ( ) = t ( ) and a distribution function F t 0 f s ds. Survival distributions are sometimes deﬁned in terms of the hazard function (force of mortality) or “hazard rate” H(t), where f (t) H(t) = . 1 − F(t)

For example, if f (t) has the exponential density function μe−μt then H(t) = μ for all values of t.Cox(1972) introduced a log-linear model that expressed the hazard as a function of intrinsic and extrinsic baseline covariates, namely the regression model

( | ) Hh t X = β, log xh H0(t) where Hh(t | X) is the hazard experienced by the hth individual, and H0(t) is the hazard experienced by the“standard” animal. As relative risk is of importance we do not need to know H0(t). Here the regressors could consist of continuous variables such as xi = age, and dummy variables such as sex with x2 = 1 for females and 0 for males, and three dummy variables x3, x4, and x5 for seasons, where x3 = 1for spring and 0 otherwise, x4 = 1 for summer and 0 otherwise, x5 = 1 for autumn and 0 otherwise, and x3 = x4 = x5 = 0 for winter. Hazard rates are tricky with animal populations because of such things as seasonal variation caused by changes in the environment and in behavior of the animals, in predator populations, in hunting pressure, and in the behavior during breeding or nesting seasons. For this reason, Tsai, Brownie et al. (1999) used a smoothing nonparametric technique for estimating the hazard function. For further details about the related topic of risk ratios and their estimation see Riggs and Pollock (2001).

A.15 Hidden Markov Models

A comparatively recent innovation has been the use of so-called hidden Markov models (HMM’s) in capture–recapture and other areas of statistics, where it is more generally referred to as latent variable models (Sect. 15.3). It assumes that individuals in a population move independently of each other over a ﬁnite set of A states, 1, 2 ...,A, through s sampling occasions. In addition to the states we have, with the s, samples an observed random encounter or event history for each individual that will depend on its state. For example, an individual seen for the last time in a particular sample will have an unknown future state as we won’t know if it is alive later. The individual’s state then becomes unknown and can therefore be described as “hidden”; hence the name HMM. We now look at the Markov part of the name. Appendix: Some General Results 601

An HMM consists of two parts. First, there a random variable xt (t = 1, 2,...) representing the state of an individual at time t, and this is a ﬁrst order Markov chain as Pr[xt+1 | xt ,...,x1]=Pr[xt+1 | xt ], t = 1, 2,....

The probabilities (ab) γt = Pr[xt+1 = b | xt = a]

(ab) are called transition probabilities, and the A × A matrix Γt = (γt ) is called a (probability) transition matrix with the property that the elements of each row sums to unity as an individual either stays in the same state or goes to another state. Second, we also have a random variable yt representing an observation from an encounter and it has the property that

Pr[yt | yt−1,...,y1, xt ,...,x1]=Pr[yt | xt ].

We see than that xt is an unobserved parameter process and yt is a “state dependent” process such that the distribution of yt depends only on the current state xt and not on previous states or observations. Vectors and matrices can be conveniently introduced into the algebra and we now (a) = [ = ] δ = ( (1), (2),..., (A)) illustrate this. If δt Pr xt a and t δt δt δt , then we ﬁnd that δ = δ Γ . t t−1 t−1

Let

(a) (1) (2) (A) pt (y) = Pr[yt = y | xt = a] and Pt (y) = diag(pt (y), pt (y),...,pt (y)).

Then

A Pr[yt = y]= Pr[xt = a] Pr[yt = y | xt = a] a=1 A (a) (a) = δt pt (y) a=1 = δ ( ) t Pt y 1A = δ Γ ( ) . t−1 t−1Pt y 1A

If y = (y1, y2,...,ys ) , then having a time-speciﬁc transition matrix and using sim- ilar algebra to Zucchini and MacDonald (2009: 37–38) we ﬁnd that

[ = ]=δ ( )Γ ( ) Γ ( ) ···Γ ( ) . Pr y z 1 P1 z1 1 P2 z2 2 P3 z3 s−1 Ps zs 1A 602 Appendix: Some General Results

A good example of such a process is given by Pradel (2005) and Kendall, White et al. (2012)forr = 3 states. Algorithms for applying hidden Markov models are given in Sect. 15.5, and some examples are given in Sects. 6.10, 7.3.1, and 13.13.

A.16 Em Algorithm

Dempster, Laird, and Rubin (1977) gave a general method of iteratively computing maximum likelihood estimates where there is incomplete data due, for example to missing observations (see also McLachlan and Krishnan (1997, 2008). The method gets its name because each iteration of the algorithm consists of an expectation step followed by a maximization step. Let x denote the vector of “complete” data, and let x∗ denote the data actually available. It is assumed that sampling is from an exponential family of which the multinomial distribution is a member. Let θ be the vector of unknown parameters and let t(x) be a sufﬁcient statistic for θ based on the complete data. Since x is only partly known, the EM procedure amounts to simultaneously estimating t and θ.Ifθ(k) is the current value of θ after k cycles of the algorithm, then the next cycle has two steps, the E-step that estimates t(x) by

t(k) = E[t | x∗, θ(k)], and the M-step, which determines θ(k+1) as the solution of

E[t | θ]=t(k).

Eventually, under reasonable conditions, θ(k) converges to the maximum likelihood estimate based on x∗. Clearly the notation for applications of the above theory will be complex involving both superscripts and subscripts. For an example see Sect. 5.8, and for details and extensions see Sect. 15.4.

A.17 Estimating Equations

A method of estimation that uses so-called estimating equations can be readily explained as follows (Seber and Wild, 1989:47). We endeavor to choose a suitable vector function g(y, θ) such that E[g(y, θ)]=0 and then choose an estimatorθ such that g(y,θ) = 0. For example, we could choose the likelihood function L(θ; y) and have ∂L(θ; y) g(y, θ) = . ∂θ

Then E[g(y, θ)]=0 and, under fairly general conditions, θ is the maximum likelihood estimate of θ. More speciﬁcally, if suitable conditions are imposed on the Appendix: Some General Results 603 distribution of y and the function g, then θ will be a consistent estimator and close to the true value of θ for sufﬁciently large sample size n. Then, using a Taylor expansion,

0 = g(y,θ) ≈ G(y, θ) + g(θ − θ), where G = ∂g/∂θ evaluated at θ. If the rows of G are linearly independent, then √ √ n(θ − θ) ≈ n(GG)−1Gg(y, θ) = Hg(y, θ), say. Given that plim H exists and is nonsingular, and g(y, θ) (suitably scaled) tends to normality as n →∞, then θ will be approximately normal.

A.18 Smoothing Splines

In the past a polynomial function f (x) of order q + 1 (degree q), where q is to be determined, was used to fit a smooth curve defined on [a, b] to m data points (xi , yi ). Unfortunately as q increases the design matrix X soon becomes ill-conditioned, that is its columns tend to become linearly dependent, especially when q > 6. Orthogonal polynomials were introduced to try and alleviate this problem as well as to simplify computations (e.g., Seber and Lee, 2003: 165–172). More recently a spline of degree q has been used for curve fitting, where a spline is a piecewise polynomial defined on [a, b].Herea and b are sometimes called boundary knots, and the values of x where the polynomial pieces join up are called (internal) knots, which have to be determined with regard to their number and placing. (It should be noted that in the literature some writers focus on order, others on degree; we choose the latter to avoid confusion with order of differences, as mentioned below.) A helpful approach to fitting a spline is to use a set of basis splines that span the space of qth degree piecewise polynomials in much the same way that the vectors (1, 0) and (0, 1) span the space of two dimensional vectors. There is more than one way of obtaining basis splines as is the case with using, for example, (1, 1) and (1, −1) as basis vectors for two dimensional vectors (cf. Schumacher, 1993; Ruppert, Wand, and Carroll, 2003, for some background information on splines). One method, where each term in the model below is a basis spline, uses the form

J 2 q q f (x) = β0 + β1x + β2x + ...βq x + γ j (x − ξ j )+, j=1 604 Appendix: Some General Results where the ξ j satisfying a < ξ1 <...<ξJ < b represent J knots (frequently chosen q to be equidistant), and (x − ξ j )+ is a truncated polynomial of degree q defined to q be equal to 0 for x < ξ j and (x − ξ j ) for x ≥ ξ j . When a and b are finite, ξ0 = a and ξJ+1 = b are sometimes defined to be part of the set of knots. The form of the polynomial is allowed to change at each knot, but only in a constrained manner that ensures that the entire curve over [a, b] will be smooth. The curve will therefore be continuous at each knot and will have q − 1 continuous derivatives over the whole range, with the qth derivative being continuous except for jumps of size γ j at each knot. The flexibility of the spline comes from these jumps. A popular choice is cubic splines (q = 3), with two continuous derivatives. With q fixed, more flexibility can be introduced by either increasing the number of knots or allowing for larger jumps. Two of the problems with this basis is that the design matrix X can be ill-conditioned when two knots are close together, and the elements of X can be very large so that matrix operations become unstable (Bonner, Thomson, and Schwarz, 2009). Another method uses the so-called B-spline basis giving us a B-spline of degree q with q + 1 polynomial pieces each of degree q, namely

J+q ( ) = q ( ), f x α j B j x j=1

q ( ) where B j x is the jth basis function of degree q. In order to fit the above model it transpires that we have to create q − 1 artificial knots below a and q − 1 above b, which including a and b means that we have J + 2q knots altogether; however only J + q basis splines are needed in f (x) above. Once the knots are given, it is straightforward to compute the B-splines recursively for any desired degree of polynomial (Seber and Lee, 2003: 173–176). This particular basis has a number of advantages, namely (Bonner, Thomson, and Schwarz, 2009): (1) the basis functions are positive and sum to 1 at any single point so that the elements of the design matrix will always be between 0 and 1, and (2) all the basis functions are local in that each is positive only over a subset of [a, b]. We add the following properties of the basis B-splines (Eilers and Marx, 1996): (1) They are positive on a domain spanned by q + 2 knots; everywhere else they are zero; (2) except at the boundaries, they overlap with 2q polynomial pieces of their neighbors; (3) at a given x, q + 1 B-splines are nonzero; (4) each interval between a pair of knots is covered by q + 1 B-splines of degree q. The choice of knots has received much attention as too many knots leads to over-fitting the data and too few to under-fitting. Fitting cubics splines, with the imposition of two continuous derivatives at each knot, can lead to f being “wiggly” or having a rough fit between consecutive points. To overcome this we can impose a penalty function that measures the degree of roughness by utilizing various measures such as the second derivative f .Tofita spline we would then find f by minimizing m b 2 2 S = [yi − f (xi )] + λ [ f (t)] dt, i=1 a Appendix: Some General Results 605 called the penalized residual sum of squares, for different values of λ. The first term essentially measures closeness of fit, while the second term penalizes curvature with the smoothing parameter providing a trade-off between the two criteria. As λ varies between 0 and ∞, f can vary from too many fluctuations to few. Therefore using n q ( ) B-splines B j x , we would choose λ and minimize the expression ⎧ ⎫ 2 m ⎨ n ⎬ b n = − ( ) + { ( )}2 S ⎩yi α j B j xi ⎭ λ α j B j t dt i=1 j=1 a j=1 with respect to the α j . As an alternative to using f in the second term, we can use a simple difference penalty on the coefficients themselves of adjacent B-splines. For example, given J and λ, we now minimize ⎧ ⎫ 2 m ⎨ n ⎬ n = − q ( ) + (k )2, S ⎩yi α j B j xi ⎭ λ α j i=1 j=1 j=k+1 where the kth difference penalty is a good discrete approximation to the integrated square of the kth derivative. Usually k is 2 or 3 (Eilers and Marx, 2010). This approach reduces the dimensionality of the problem to n, the number of B-splines, instead of m, the number of observations. Other penalties can be used, as seen in this book. Compu- tational details are given by Eilers and Marx (1996), who use the Akaike information criterion (AIC) and cross-validation to determine λ. They combine basis B-splines with different “penalties” to get what are called P-splines, and point out that they have no boundary effects, are a straightforward extension of generalized linear regression models, conserve means and variances of the data, and have polynomial curve fits as limits (see their paper for details).

A.19 Computer Software

In Chap. 15 the emphasis is on methods, as software may come and go, or change. Although a variety of specialized software has been written for the different models, and some has fallen by the wayside, some have stayed and grown. Since there is no point in “reinventing the wheel” we shall brieﬂy discuss some of the available packages. One that is now widely used is MARK (White and Burnham, 1999) for both open and closed models and unequal time intervals. The manual can be downloaded from http://www.phidot.org/software/mark/docs/book/, and can handle various combina- tions of recapture, resighting, radio-tracking and dead recovery data, estimate various parameters and their relationships (Franklin, 2001), provide nest survival models 606 Appendix: Some General Results

(Rotella, Chap. 17 of the manual), and determine numerically the number of parameters that are estimable in the model. It has a number of link functions such as the logit function for transforming probabilities, and can handle grouped data and covariates. A number of data summaries are provided such as, for example, estimates, standard errors, goodness-of-ﬁt tests, deviance residuals from the model (including graphics and point and click capability to view the encounter history responsible for a particular residual), likelihood-ratio test and analysis of deviance (ANODEV) between models, adjustments for over-dispersion, AIC with AICc values and related quan- tities for different models, model averaging, bootstrapping, age and cohort models, multistate models (cf., White, Kendall, and Barker, 2006), various JS and superpopulation models (with POPAN incorporated into MARK), robust models, nest and known fate models, occupancy models, and simulation. Information, examples, and very helpful descriptions of the methods used can be obtained from the large online book by the editors Cooch and White (2014). They state that:

While MARK is clearly not a replacement (in some ways) for a more general purpose approach like SURVIV (or, more recently, R, MATLAB or WinBUGS), it is in many respects a replacement for much (if not all) of the ‘canned’ software previously in general use.

BUGS (Bayesian inference Using Gibbs Sampling) is a package for Markov chain Monte Carlo sampling and is available in two forms WinBUGS (Lunn, Thomas et al., 2000; Gimenez, Bonner et al., 2009;Kéry,2010), which can be recalled from R (R2WinBUGS), and OpenBUGS (embedded in BRugs, which will become the standard version, both of which are readily available online (https://www. mrc-bsu.cam.ac.uk/software/bugs/). Another program, relating to BUGS,isJAGS (cf. https://sourceforge.net/projects/mcmc-jags) by Martyn Plummer (cf. Plummer, 2003; Depaoli, Clifton, and Cobb, 2016), for a review). There is also R2jags for running JAGS from R (Su and Yajima, 2011). When it comes to goodness-of-fit tests for various models, these have been provided in the past by such programs as RELEASE (which can be called up in MARK), SURVIV, JOLLY, and JOLLYAGE. A program U-CARE for Unified Capture- Recapture) that provides several enhancements to RELEASE by Choquet, Lebre- ton et al. (2009) arrived on the scene, and recently multistate models have been been added to it. Multistate models were also considered in MS-SURVIV or more recently in M-SURGE (Choquet et al., 2004), which also includes GEMACO, dedicated to the automatic generation of design matrices for multistate models (Choquet, 2008). There is also E-SURGE (Choquet, Rouan, and Pradel, 2009), which can be used to fit multievent models. The package R is a comparatively recent addition and has been linked with MARK through RMARK (Laake and Rexstad, 2008) and, as mentioned above, with BUGS. It is becoming increasingly adapted to capture–recapture methodology (and ecology and statistics in general), as in R-capture (Baillargeon and Rivest, 2007), at http://artax.karlin.mff.cuni.cz/r-help/library/Rcapture/html/00Index.html, marked (see Laake, Johnson, and Conn, 2013), unmarked (Laake, Johnson, and Conn, 2013) and BaSTA (Colchero, Jones et al. 2012). Some comparisons of R Appendix: Some General Results 607 with BUGS and another package AD Model Builder (ADMB; (Fournier, Skaug et al., 2012; available at http://admb-project.org) are made by Bolker et al. (2013). The application of R to fisheries is given by Ogle (2016), who includes such topics as age comparisons, age-length keys, size structure, weight-length, abundance from capture–recapture, and the estimation of various basic parameters. As movement models have begun to invade capture–recapture as in spatial models (cf. Sect. 11.7) we mention an R movement package called moveHMM by Michelot, Langrock, and Patterson (2017). This section will continue to grow! References

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A Bayesian imputation, 138 Acoustic tags, 17, 181, 290 Bayesian Information Criterion (BIC), 506 Activity center, 381 Bayesian method, 587 Additive hypothesis, 61, 462 penalized cubic spline, 154 Age cohorts, 58 Bayesian model, 7, 202, 502 catch data and, 58 AIC and, 502 Age data autoregressive survival, 194 age- and time-specific survival, 76 averaging, 76 age-at-harvest data, 71 covariates and, 139, 144 multistate combined data and, 449 dead recoveries and, 76 Age-specific survival dead recoveries with heterogeneity, 86 exploited population, 71 diagnostics for, 598 unexploited population, 74 environmental catastrophe and, 183 Akaike Information Criterion (AIC), 500 for Arnason–Schwarz statespace, 432 improved version, 503 for CJS model, 113 problems with, 502 general method, 540 Analytical-numerical method, 510 integrated population model, 417 Ancillary statistic, 578 live and dead recoveries with covariates, Apparent mortality, 41, 334 276 Apparent survival, 114, 253 missing data and, 179 Arnason–Schwarz model mulitsite recapture and, 434 modifications of, 428 multistate, 460 Autocorrelation, 591 parameter association and, 309 plot of, 595 parameter redundancy and, 514 Autocorrelation plot, 76 pros and cons, 7 Automatic differentiation, 512 p-value, 76, 598 Availability parameter, 370 random effects, 92, 327 Average instantaneous survival rate, 192 statespace heterogeneity, 250 survival and, 87 Bayesian model averaging, 506 B Beta prior, 438 Barker’s model Beta prior distribution, 586 for combined data and random emigra- Bias adjustment, 4, 46, 121 tion, 292 CJS model, 121, 130 Batch marking, 413 dead recoveries model, 46, 82, 83 Bayesian covariate methods, 144 Bootstrap, 598 Bayesian diagnostics, 598 Box plot, 585 © Springer Nature Switzerland AG 2019 655 G. A. F. Seber and M. R. Schofield, Capture-Recapture: Parameter Estimation for Open Animal Populations, Statistics for Biology and Health, https://doi.org/10.1007/978-3-030-18187-1 656 Index

Box plot of residuals, 48 six sub-models, 339 Breeding return times, 206 spatial capture–recapture models, 379 Brooks-Gelman diagnostic, 593 sub-models for, 348 Brooks-Gelman-Rubin diagnostic, 593 tag loss, 31 B-splines, 153, 187 telemetry, resighting and, 380 Burnham’s combined model, 257 trace-contrasts models, 22 Burn in time, 591 transience and dispersal, 383 two-sample Bayesian method, 7 two-sample case, 4 C Closed populations Capture–recapture models, 111 review of, 10 with resighting data, 198 CLOSTEST, 378 Capture–recapture–recovery data, 258 Combined data model, 143 CAPWIRE model, 20 Barker’s model, 292 Catch-age data, 66 live and dead recoveries, 367 Catch-and-release fishery, 395 live recaptures and dead recoveries, 257 Catch data, 58 robust design, 367, 370 Catch–effort data, 402 statespace model for, 443 CJA model Combined models, 257 transience and, 127 Combining recaptures and resightings CJS model, 101, 102 with dead recoveries, 300 bias adjustments, 121 with losses, 292 hidden Markov version, 207 Commercial fisheries method of mixtures, 232 boats with observers, 55 missing data and, 179 components in, 55 random effects, 327 incomplete mixing of tagged, 57 statespace format, 418 instantaneous tag-return rates, 55 trap dependence and, 126 multiple components in, 55 unknown or misclassified group, 170 planted tags, 56 Closed population tag returns, 55 aerial censusing, 97 Compensatory hypothesis, 61 Bayesian covariate methods, 145 Complete data likelihood, 19, 146, 149, 156, Bayesian imputation, 19 179, 223, 437, 515 Bayesian methods, 309 imputed data, 143 covariates measured with error, 138 information matrix, 532 data augmentation and, 18, 328 Conditional expectation, 578 definition of, 2 Conjugate prior, 147, 589, 590 derived population parameter, 329 Contingency table, 31, 73 epidemiology and, 2, 10 CJS model and, 106 estimating equations, 184 CJS model and extensions, 127 estimating tag loss, 23 from hypergeometric distributions, 586 genetic tags, 17 low power and, 125 heterogeneity and, 139 range of tests, 127 log-linear model and, 332 small cell numbers, 124 method of mixtures, 89 structural zero, 332 misidentification and, 15 Contingency table test, 299 model requirements, 345 for Type I losses, 212 mother of all models, 515 list for, 265 next-of-kin, 21 minimal sufficient statistics and, 249 pedigree reconstruction and, 18 test 2.CL(i), 128 radio tags and, 17 test 2.CT(i), 128 review of, 2, 111 test 3.SR(i), 128 sampling one at a time, 20 Type II losses, 213 Index 657

Continuous fishery, 55, 66 Dead recovery model age-dependent model, 400 statespace format, 418 grouped recovery times, 393 Delayed recoveries, 272 incomplete mixing of tagged, 57 Delta method, 574 individual recovery times, 386 multinomial distribution and, 576 instantaneous rates defined, 385 Density dependence, 324 Cormack–Jolly–Seber model, 111 Depensatory hypothesis, 61 Covariates, 137, 317, 323 Derivative matrix abundance estimates and, 141 analytic method for, 510 Bayesian live and dead recoveries, 276 Designing a capture–recapture experiment, Bayesian methods, 144 492 continuous, 142 Deterministic imputation, 138 examples using, 139 Deviance live and dead recoveries, 272 definition of, 583 measurement errors, 138 Deviance Information Criterion (DIC), 507 migration and, 140 Deviance residual, 585 missing, 273 Diagnostic residuals, 584 missing values and imputation, 142 Dirichlet distribution, 590 some unobserved, 145 Dispersal strategies for handling, 138 multievent model and, 483 types of, 137 Double tags Credible interval, 7, 8, 113, 149, 487, 588, problems with, 34 595 Crippling losses, 41 Crosbie–Manly model, 112, 160 E Cubic splines, 206 Edge effect with small mammals, 230 Cyclic fixing, 166 Effective trapping area, 230 EM algorithm, 155 CJS model and, 155 extensions of, 536 D Monte Carlo extension, 536 Data augmentation, 149, 151, 328, 350, 382 resighting and, 413 super-population model and, 350 summary, 602 Dead recoveries model, 41 tag loss and, 35 age and time-specific survival, 76 tag-loss model and, 217 age-at-harvest data, 71 theory, 528 age-dependent survival and reporting, 76 Emigration age-specific survival, 71 first-order Markovian, 141 Bayesian time and age factors, 92 Environmental catastrophe, 183 constant survival, 69 Escapement estimation, 167 constant survival and recovery, 69 Estimating equations, 184, 602 delay between recaptures and releases, Estimation of abundance 85 statespace model, 440 exploited population, 41, 45 Expected information matrix, 526, 579 fixed-recapture times, 79 Experimental design for capture–recapture, goodness-of-fit test, 47 492 incomplete mixing of tagged, 57 Extrinsic catchability, 230 method of mixtures, 90 Extrinsic parameter-redundancy, 510 some unidentified ages, 78 Extrinsic redundancy, 509 statespace for, 461 tag loss and, 50 test for constant survival, 48 F time-dependent model, 41 Finite-space Markov chain, 543 unexploited population, 47 irreducibility and aperiodicity of, 543 658 Index

stationary distribution of, 544 Hessian matrix, 579 Fish length, 10, 95 standard errors using, 526 proxy for age, 71 Hessian method for parameter redundancy, tag loss and, 30 509 Frailty model, 36 Heterogeneity, 228 Freeman–Tukey statistic, 598 residuals and, 338 Full conditional distribution, 94, 313, 320, Heterogeneity of reporting tags, 480 325, 326, 589, 592 Hidden Markov model, 207 Gibbs sampling and, 549 backward algorithm, 538 Full conditional posterior distribution, 76 disease dynamics and, 471 forward algorithm, 538 general theory, 600 G hidden process model and, 405 Gamma prior distribution, 586 live and dead recoveries, 277 Gauss–Hermite quadrature, 251 multievent and, 228, 466 Gelman–Rubin diagnostics, 149 multivariate, 476 Generalized least-squares partially, 279 definition, 581 robust design and, 471 test for a constant parameter, 581 statespace CJS model and, 466 Genetic tags, 17 stopover and, 484 capture heterogeneity and, 231 tag-loss and, 214 estimating genotypic error, 19 three properties of, 537 next-of-kin data, 21 Hidden process model, 414, 418 some problems with, 18 Highest-posterior-density interval, 588 Gibbs sampling, 292, 314, 320, 325, 328, Home-range center, 381 549, 591, 598 Hybrid Monte Carlo, 556 Hammersley–Clifford theorem, 549 Hypergeometric distribution, 3 irreducibility and, 550 Hyper-prior, 587 positivity condition need to hold, 551 special case of Metropolis–Hastings algorithm, 553 I Gibbs sampling algorithm, 437, 592 Identifiability constraints GLIM parameters, 332 two-way analysis of variance, 331 Gompertz model, 271 Immature-emigration process, 464 Goodness-of-fit test Incomplete mixing, 400–402 calibrated simulation, 414 Individual recovery times, 33 CJS model, 49, 106 multiple releases, 395 dead recoveries model, 47 nonparametric method, 387 general comments, 497 single release, 386 JMV model, 431 Instantaneous mortality JS model and, 122 probability density function and, 599 random emigration and , 299 Instantaneous mortality rate, 55, 385 recaptures and recoveries model, 265 average, 192 Gross recruitment, 167 Bayes heterogeneity and, 86 Growth ratio, 119, 174, 231, 356 definition of, 385 resighting and robust design, 378 live and dead recoveries with age, 291 Instantaneous natural fishing rate separating mortality rates, 287 H Integrated population model, 413 Hamiltonian Monte Carlo, 556 shortcomings of, 417 Hammersley–Clifford theorem, 549 Inter-birth emigration process, 464 Hazard function, 459, 600 Intrinsic catchability, 230 Hazard rate, 600 Intrinsic redundancy, 508 Heisey-Fuller estimate, 16 Inverse Wishart distribution, 590 Index 659

J Markov process, 440, 445 Jeffrey’s prior, 587 Markovian migration, 355 JMV model, 468 Markovian temporary emigration test for, 431 robust design and, 364, 367 Jolly–Balser test, 124 Maximum likelihood estimation, 579 Jolly–Seber model, 111 Mean life expectancy, 599 JS model, 114 Measures for model comparison, 498 bias adjustments, 130 Memory model, 427 births only, 131 for captures only, 431 death only, 130 Method of mixtures, 231, 356, 431 method of mixtures, 233 CJS model and, 232 special cases, 129 dead recoveries model, 90 variances and covariances, 119 group stopover, 201 JS model and, 233 Pradel’s reverse time model and, 178 K robust design, 374 Kalman filter, 413, 417 transients and, 253, 475 Kaplan–Meier estimate, 16, 283 Method of scoring, 40, 579 Kullback-Leibler information loss, 500 Metropolis–Hastings proposal density and, 547 Metropolis–Hastings algorithm, 147, 149, L 314, 321, 325, 326, 545, 591 Large sample hypothesis tests, 582 irreducible and aperiodic, 546 Latent variables, 527 proposal density, 546 Latent variables model, 179, 328 Migration live and dead recoveries, 291 CJS model and, 258 Leapfrog method, 558 covariates and, 140 Leslie matrix, 413 models for, 300 Life cycle, 463 multistate single-recovery model and, Life span, 102, 193 441 Lincoln index, 1 Linear mixed model, 380 temporary, 352 Live and dead recoveries types of, 323 age estimation and, 291 Misclassified group membership, 169 Local identifiability, 512 Missing data, 179, 434, 435 Locally identifiable, 508 Mixed-effects model, 376 trap effects and, 484 Mixture models, 89, 431 Log-linear model, 331 Model averaging, 596 closed population, 332 Model JMV, 431 open population and, 334 Mortality switch, 443 sub-models, 339 Mother of all models, 515 Multievent model, 431, 466 dispersal components, 483 M Multinomial distribution Markov chain, 146, 590 conditional distribution, 574 transition kernel and, 146 definition, 571 Markov chain Monte Carlo, 542 peeling and pooling, 571 process correlation and, 61 product of conditional binomials, 572 Markov chain Monte Carlo sampling properties of, 572 summary, 590 singular, 571 Markov chain Monte Carlo simulation, 76, Multiple-component fisheries, 55 113 age cohorts and, 58 life cycles and, 463 catch data and, 58 statespace models, 407 Multiple maxima, 527 660 Index

Multi-sample single-recapture census, 79 robust design and combined data, 368 Multisite model separate estimation of, 262 age and combined data, 449 Petersen estimate, 118, 121, 131, 136, 346 Multistate model Planted tags, 56, 402 combined data and, 445 Plants and capture–recapture, 10 partial state observation, 457 Pooling process, 573 time and age model, 455 Posterior distribution, 7, 588 Multivariate hidden Markov model, 476 two-sample closed population, 7 Multivariate prior distribution, 590 Pradel’s reverse model, 173 Precision, 587 Prior distribution, 147, 486, 586 N autoregressive survival, 195 Natural tags, 15 beta, 88 photo-identification, 15 beta family, 9 trace-contrast models, 21 CJS model and, 113 Natural variation, 211 closed population, 349 Near singularity, 515 covariates and, 317 Newton algorithm, 524 inverse Wishart, 312 Newton–Raphson method, 580 live and dead recoveries, 277 Next-of-kin data, 21 missing data model, 183 Nonpreferential prior, 507 normal-inverse-Wishart, 313 Nonstandard chi-square test, 91, 233 random effects and, 327 Normal-Gamma distributions, 589 spline and, 153 Numerical optimization, 524 statespace model and, 436 stopover model and, 203 weak identifiability and, 514 O Profile likelihood, 509 Observed information matrix, 527, 579, 582 Profile likelihood interval, 580 Occupancy model, 10, 179, 329, 347, 606 Proposal distribution, 147 Open populations P-splines, 152, 186, 605 previous developments, 3 Pulse fishery, 55, 65, 402 Over-dispersion, 504 description of, 254 heterogeneity and, 376 Q random effects and, 92 QAIC, 504 Quasi-Newton algorithms, 526

P Parameter-expanded Data Augmentation R (PX-DA), 329, 485 Radio tags, 16, 57, 283, 461 Parameter redundancy, 508 continuous data, 290 automatic differentiation, 512 dead recoveries and, 57 combining the methods, 513 experimental design for, 496 reduced-form exhaustive summary, 513 instantaneous natural and fishing rates, Parameter sequences, 189 287 Partially observable states, 469 radio failure, 225 Particle filters, 592 statespace model, 443 Pearson goodness-of-fit test, 582, 583 Random effects, 61, 92, 140, 250, 277, 593 Peeling and pooling methods Bayesian model, 327 example of, 131 CJS model and, 113, 327 Peeling process, 572 CJS two-state model, 432 Penalized residual sum of squares, 605 dead recoveries with time and age, 92 Penalized spline, 154 experimental design and, 493 Permanent emigration, 306, 324 hidden process model and, 414 Index 661

hyper-priors and, 587 stopover and, 484 life cycles and, 463 surrogate for age, 271 modeling and, 507 tag effect on survival, 103 parameter sequences and, 189 telemetry and robust design, 380 reversible jump algorithm and, 596 time specific, 97 spatial capture–recapture, 381 with recapture data, 198 spatial model and, 380 Reversetimeanalysis,177 statespace model and, 408, 413 Reversible jump algorithm, 321, 328, 350, survival and density dependence, 191 383, 460, 507, 595 survival sequence, 190 brief introduction to, 596 Random emigration, 324, 355 theory for, 562 CJS model and, 263 Reversible jump method, 154 combined data and, 297 Reward tags, 52 goodness-of-fit test, 299 component fisheries and, 55 short-term marking effect and, 301 Ricker’s two-release method, 67 Random migration, 252 strata and, 441 secondary samples and, 357 Robson-Pollock heterogeneity model, 235 Random temporary emigration, 305 Robust design, 345 robust design and, 367 advantages over the JS model, 345 Rao–Blackwell method, 564 hidden Markov model and, 471 Rcapture, 338 live and dead recoveries and, 367 Recapture models recaptures, resightings, and dead recov- CJS model, 112 eries, 370 MV2, 428 spatial capture–recapture, 379 Recruitment state uncertainty and, 465 components of, 346 transients and, 378 Recruitment analysis, 174 Redundant parameters, 508 Bayesian models and, 514 S Regression models, 400 Saturated model, 583 Regression splines Scan samples, 484 tag loss and, 34 Schwarz–Arnason model, 161, 484 Relocations, 57 groups and, 169 Residence time distribution, 203 misclassified groups, 169 Residuals, 337 unknown groups, 169 “broken stick” plot, 338 Score test, 76, 126, 272, 409, 499 problems with, 338 also the Lagrange multiplier test, 583 Resighting data CJS model, 129 Arnason–Schwarz statespace model, 432 CJS model and transients, 126 state uncertainty, 465 definition of, 582 Resighting models, 97 for homogeneity, 128 closed population using, 97 simplest model and, 498 combined with live captures and dead test for memory, 469 recoveries, 300 SEM algorithm, 533 combined with live recaptures and dead Semi-complete data likelihood, 350 recoveries, 367 Semi-Markov model, 458 dead recoveries and multievent, 520 Seniority parameter, 165, 174, 231 losses on, 292 extended to groups, 235 radio tags and, 306 Separating mortality components recaptures combined with, 292 statespace model using recoveries, 461 residence time and, 203 Sequential importance sampling, 407 robust design and, 350, 375 Shadow effect, 19 statespace and, 413 Short-term mortality test, 125 662 Index

Shrink factor, 595 Bayesian model, 33 Simplex algorithms, 526 general comments and, 35 Singular multinomial distribution heterogeneity of reporting and, 480 dead recoveries model, 40 Hidden Markov model and, 214 pooling process, 573 indistinguishable independent tags, 28 Site fidelity, 259 individual recovery times, 33 Slice sampling, 556 individual tag return times, 399 Smoothing splines, 603 instantaneous rate, 31 Solicited tags, 52 JS model and, 217 Spatial capture–recapture models, 379 permanent tag, 27, 34, 50 Splines, 151, 206 tag attachment skills, 36 B-splines, 187 test for identical tags, 26 cubic, 187 test for independence, 27 definition, 152 two different tags, 23 penalized, 152, 186, 189 two-state model and, 408 regression, 34 Tag loss models smoothing, 603 double and single tags, 31 Standard errors Tag–reporting probabilities, 51 computation of, 531 Tag returns Oakes’ method, 532 commercial fisheries, 55 Statespace model Takeuchi Information Criterion (TIC), 503 state uncertainty and, 465 Target distribution, 552, 591 State uncertainty Temporary emigration robust design for, 466 life cycle and, 464 statespace model for, 465 Markovian, 300 Stationarity test for Markov chain, 593 Stationary distribution, 591 test for, 124 Stationary population, 184 unobservable state, 347 Stopover parameters, 195, 483 Temporary immigration, 352 breeding return times, 206 Test for a constant parameter, 581 residence time, 203 Test for constant survival, 69 reviews, 196 Time series of estimates, 193 super-population and, 196 autoregressive survival, 194 Sufficient statistic, 578, 582 Trace-contrast models, 21 hypothesis testing and, 582 Trace plot, 76, 149, 595 Super-population Transdimensional algorithm, 328, 350 avoids dimension change, 329 Transdimensional sampling, 561 spatial capture–recapture and, 381 Transdimensional Simulated Annealing Super-population approach, 159 (TDSA), 432 Super-population model, 233, 484 Transients, 126, 252 Survival sequence of parameters, 190 effects of, 178 Symbolic rank, 510 log-linear model and, 340 models for, 378 permanent emigration and, 306 T residual plot and, 338 Tag attachment skills, 36 robust design for, 378 Tagging methods, 13 stopover and, 195 acoustic tags, 17 test for, 123, 431 genetic tags, 17 trap avoiders, 230 natural tags, 15 Travel times, 181 Pit tags, 14 Trinomial model, 138 radio tags, 16 Two-stage prior, 587 Tag loss methods, 22, 214 Type I losses, 212 Index 663

U W Unidentifiable species or sex, 428 Wald test, 126 Unified capture–recapture framework, 317 WBWA test for long-term memory, 468 Weak identifiably, 514 Universal variable method, 563 Weibull distribution, 33, 271 Unknown group membership, 169 Weibull model, 30 U-shaped residual plot, 338 Wiener process, 145