M 1 M 2. x 3 f(x)dx. M 3

Transcription

1 APPENDIX B The Delt method... Suppose you hve conducted mrk-recpture study over 4 yers which yields 3 estimtes of pprent nnul survivl (sy, ˆϕ 1, ˆϕ 2, nd ˆϕ 3 ). But, suppose wht you re relly interested in is the estimte of the product of the three survivl vlues (i.e., the probbility of surviving from the beginning of the study to the end of the study)? While it is esy enough to derive n estimte of this product (s [ ˆϕ 1 ˆϕ 2 ˆϕ 3 ]), how do you derive n estimte of the vrince of the product? In other words, how do you derive n estimte of the vrince of trnsformtion of one or more rndom vribles, where in this cse, we trnsform the three rndom vribles (in this cse,ϕ 1,ϕ 2 ndϕ 3 ) by considering their product? One commonly used pproch which is esily implemented, not computer-intensive, nd cn be robustly pplied in mny (but not ll) situtions is the so-clled Delt method (lso known s the method of propgtion of errors). In this ppendix, we briefly introduce some of the underlying bckground theory, nd the ppliction of the Delt method. B.1. Men nd vrince of rndom vribles Our interest here is developing method tht will llow us to estimte the vrince for functions of rndom vribles. Let s strt by considering the forml pproch for deriving these vlues explicitly, bsed on the method of moments. For continuous rndom vribles, consider continuous function f(x) on the intervl[,+ ]. The first four moments of f(x) cn be written s: M 0 M 1 M 2 M f(x)dx, x f(x)dx, x 2 f(x)dx, x 3 f(x)dx. We briefly discuss some compute-intensive pproches in n Addendum to this ppendix. In simple terms, moment is specific quntittive mesure, used in both mechnics nd sttistics, of the shpe of set of points. If the set of points represents probbility density, then the moments relte to mesures of shpe nd loction such s men, vrince, skewness, nd so forth. Cooch & White (2017)

2 B.1. Men nd vrince of rndom vribles B - 2 In the prticulr cse tht the function is probbility density (s for continuous rndom vrible), then M 0 1 (i.e., the re under the pdf must equl 1). For exmple, consider the uniform distribution on the finite intervl [, b]. A uniform distribution (sometimes lso known s rectngulr distribution), is distribution tht hs constnt probbility over the intervl. The probbility density function (pdf) for continuous uniform distribution on the finite intervl[, b] is: Integrting the pdf for p(x) 1/(b ): M 0 M 1 M 2 M 3 0 for x< P(x) 1/(b ) for <x< b 0 for x> b. b b b b b b b b p(x)dx 1 dx 1, b xp(x)dx x + b dx b 2, x 2 p(x)dx x 2 1 b dx 1 ( 2 + b + b 2), 3 x 3 p(x)dx x 3 1 b dx 1 ( b + b 2 + b 3). 4 If you look closely, you should see tht M 1 is the men of the distribution. Wht bout the vrince? How do we interpret/use the other moments? Recll tht the vrince is defined s the verge vlue of the fundmentl quntity [distnce from men] 2. The squring of the distnce is so the vlues to either side of the men don t cncel out. The stndrd devition is simply the squre-root of the vrince. Given some discrete rndom vrible x i, with probbility p i, nd menµ, we define the vrince s: vr (xi µ ) 2 pi. Note we don t hve to divide by the number of vlues of x becuse the sum of the discrete probbility distribution is 1 (i.e., p i 1).

3 B.1. Men nd vrince of rndom vribles B - 3 For continuous probbility distribution, with men µ, we define the vrince s: vr b Given our moment equtions, we cn then write: vr b b b b (x µ) 2 p(x)dx (x µ) 2 p(x)dx. ( x 2 2µx +µ 2) p(x)dx b b x 2 p(x)dx 2µxp(x)dx + µ 2 p(x)dx b b x 2 p(x)dx 2µ xp(x)dx +µ 2 p(x)dx. Now, if we look closely t the lst line, we see tht in fct the terms represent the different moments of the distribution. Thus we cn write: vr b b (x µ) 2 p(x)dx b b x 2 p(x)dx 2µ xp(x)dx +µ 2 p(x)dx M 2 2µ ( M 1 ) +µ 2 ( M 0 ). Since M 1 µ, nd M 0 1 then: vr M 2 2µ ( ) M 1 +µ 2 ( ) M 0 M 2 2µ(µ)+µ 2 (1) M 2 2µ 2 +µ 2 M 2 µ 2 M 2 ( ) 2. M 1 In other words, the vrince for the pdf is simply the second moment (M 2 ) minus the squre of the first moment ((M 1 ) 2 ). Thus, for continuous uniform rndom vrible x on the intervl[, b]: vr M 2 ( M 1 ) 2 ( b)2. 12 It turns out, most of the usul mesures by which we describe rndom distributions (men, vrince,...) re functions of the moments.

4 B.2. Trnsformtions of rndom vribles nd the Delt method B - 4 B.2. Trnsformtions of rndom vribles nd the Delt method OK tht s fine. If the pdf is specified, we cn use the method of moments to formlly derive the men nd vrince of the distribution. But, wht bout functions of rndom vribles hving poorly specified or unspecified distributions? Or, situtions where the pdf is not esily defined? In such cses, we my need other pproches. We ll introduce one such pproch here (the Delt method), by considering the cse of simple liner trnsformtion of rndom norml distribution. Let X 1, X 2,... N(10,σ 2 2). In other words, rndom devites drwn from norml distribution with men of 10, nd vrince of 2. Consider some trnsformtions of these rndom vlues. You might recll from some erlier sttistics or probbility clss tht linerly trnsformed norml rndom vribles re themselves normlly distributed. Consider for exmple, X i N(10, 2) which we then linerly trnsform to Y i, such tht Y i 4X i + 3. Now, recll tht for rel sclr constnts nd b we cn show tht i. E(), E(X + b) E(X)+ b ii. vr() 0, vr(x + b) 2 vr(x). Thus, given X i N(10, 2) nd the liner trnsformtion Y i 4X i + 3, we cn write: Y N ([ 4(10) ], [ (4 2 )(2) ]) N(43, 32). Now, n importnt point to note is tht some trnsformtions of the norml distribution re close to norml (i.e., re liner) nd some re not. Since liner trnsformtions of rndom norml vlues re norml, it seems resonble to conclude tht pproximtely liner trnsformtions (over some rnge) of rndom norml dt should lso be pproximtely norml. OK, to continue. Let X N(µ,σ 2 ), nd let Y g(x), where g is some trnsformtion of X (in the previous exmple, g(x) 4X + 3). It is hopefully reltively intuitive tht the closer g(x) is to liner over the likely rnge of X (i.e., within 3 or so stndrd devitions ofµ), the closer Y g(x) will be to normlly distributed. From clculus, we recll tht if you look t ny differentible function over nrrow enough region, the function ppers pproximtely liner. The pproximting line is the tngent line to the curve, nd its slope is the derivtive of the function. Since most of the mss (i.e., most of the rndom vlues) of X is concentrted roundµ, let s figure out the tngent line tµ, using two different methods. First, we know tht the tngent line psses through (µ, g(µ)), nd tht its slope is g µ (we use the prime nottion, g, to indicte the first derivtive of the function g). Thus, the eqution of the tngent line is Y g X + b for some b. Replcing (X, Y) with the known point (µ, g(µ)), we find g(µ) g (µ)µ + b nd so b g(µ) g (µ)µ. Thus, the eqution of the tngent line is Y g (µ)x + g(µ) g (µ)µ. Now for the big step we cn derive n pproximtion to the sme tngent line by using Tylor series expnsion of g(x) (to first order) round X µ: Y g(x) g(µ)+ g (µ)(x u)+ǫ g (µ)x + g(µ) g (µ)µ+ǫ. OK, t this point you might be sking yourself so wht?. [You might lso be sking yourself wht

5 B.2. Trnsformtions of rndom vribles nd the Delt method B - 5 the heck is Tylor series expnsion?. If so, see the- sidebr -, below.] Well, suppose tht X N(µ,σ 2 ) nd Y g(x), where g (µ)0. Then, whenever the tngent line (derived erlier) is pproximtely correct over the likely rnge of X (i.e., if the trnsformed function is pproximtely liner over the likely rnge of X), then the trnsformtion Y g(x) will hve n pproximte norml distribution. Tht pproximte norml distribution my be found using the usul rules for liner trnsformtions of normls. Thus, to first order: vr(y) vr ( g(x) ) E(Y) g (µ)µ+ g(µ) g (µ)µ g(µ) ( ) 2 g(x) g(µ) ( g (µ)(x µ) ) 2 ( g (µ)) 2 (X µ ) 2 ( g (µ) ) 2 vr(x). In other words, for the expecttion (men), the first-order pproximtion is simply the trnsformed men clculted for the originl distribution. For the first-order pproximtion to the vrince, we tke the derivtive of the trnsformed function with respect to the prmeter, squre it, nd multiply it by the estimted vrince of the untrnsformed prmeter. These first-order pproximtions to the expecttion nd vrince of trnsformed prmeter re usully referred to s the Delt method. Tylor series expnsions? begin sidebr Briefly, the Tylor series is power series expnsion of n infinitely differentible rel (or complex) function defined on n open intervl round some specified point. For exmple, one-dimensionl Tylor series is n expnsion of rel function f(x) bout point x over the intervl ( r, + r), is given s: f(x) f() + f ()(x ) 1! + f (x)(x ) 2 2! +..., where f () is the first derivtive of f with respect to, f (x) is the second derivtive of f with respect to, nd so on. For exmple, suppose the function is f(x) e x. The convenient fct bout this function is tht ll its derivtives re equl to e x s well (i.e., f(x) e x, f (x) e x, f e x,...). In prticulr, f (n) (x) e x so tht f (n) (0) 1. This mens tht the coefficients of the Tylor series re given by: nd so the Tylor series is given by: n f(n) (0) n! 1 n!, 1+ x + x2 2 + x3 6 + x4 xn n! For n interesting review of the history of the Delt method, see Ver Hoef, Jy M. (2012) Who Invented the Delt Method? The Americn Sttisticin, 66, n0 x n n!.

6 B.2. Trnsformtions of rndom vribles nd the Delt method B - 6 The primry utility of such power series in simple ppliction is tht differentition nd integrtion of power series cn be performed term by term nd is hence prticulrly (or, t lest reltively) esy. In ddition, the (truncted) series cn be used to compute function vlues pproximtely. Now, let s look t n exmple of the fit of Tylor series to fmilir function, given certin number of terms in the series. For our exmple, we ll expnd the function f(x) e x, t x 0, on the intervl[ 2, + 2], for n 0, n 1, n 2,... (where n is the number of terms in the series). For n 0, the Tylor expnsion is sclr constnt (1): f(x) 1, which we nticipte to be poor pproximtion to the function f(x) e x t ny point. The reltionship between the number of terms, nd the fit of the Tylor series expnsion to the function f(x) x x, is shown clerly in the following figure. The solid blck line in the figure is the function f(x) e x, evluted over the intervl[ 2, 2]. The dshed lines represent different orders (i.e., number of terms) in the expnsion. The red dshed line represents the 0th order expnsion, f(x) 1, the blue dshed line represents the 1st order expnsion, f(x) 1+ x, nd so on. f( x)= e x f( x)» 1+ x+ x f( x)» 1+ x+ x + x f( x)» 1+ x f( x )» 1 We see tht when we dd more terms (i.e., use higher-order series), the fit gets progressively better. Often, for nice, smooth functions (i.e., those nerly liner t the point of interest), we don t need mny terms t ll. For this exmple, the 3rd order expnsion (n 4) yields reltively good pproximtion to the function over much of the intervl [ 2, 2]. Another exmple suppose the function of interest is f(x) (x) 1/3 (i.e., f(x) 3 x). Suppose we re interested in f(x) (x) 1/3 where x 27 (i.e., f(27) 3 27). Now, it is strightforwrd to show tht f(27) But suppose we wnt to know f(25) 3 25, using Tylor series pproximtion? We recll tht to first order: f(x) f() + f ()(x ), where in this cse, 25 nd x 27. The derivtive of f with respect to x for this function f() () 1/3 is: f () 2/ x 2.

7 B.3. Trnsformtions of one vrible B - 7 Thus, using the first-order Tylor series, we write: f(25) f(27)+ f (27)(25 27) 3+( )( 2) Clerly, is very close to the true vlue of f(25) In other words, the first-order Tylor pproximtion works well for this function. As we will see lter, this is not lwys the cse, which hs importnt implictions. end sidebr B.3. Trnsformtions of one vrible OK, enough bckground for now. Let s see some pplictions. Let s check the Delt method out in few cses where we (probbly) know the nswer. Assume we hve n estimte of density ˆD nd its conditionl smpling vrince, vr( ˆD). We wnt to multiply this by some constnt c to mke it comprble with other vlues from the literture. Thus, we wnt ˆD s g(d) c ˆD, nd vr(d s ). The Delt method gives: vr( ˆD s ) (g (D)) 2 σ 2 D ( ) ˆD s 2 vr( ˆD) ˆD c 2 vr( ˆD), which we know to be true for the vrince of rndom vrible multiplied by rel constnt. For exmple, suppose we tke lrge rndom norml smple, with menµ 1.5, nd true smple vrince ofσ We re interesting in pproximting the vrince of this smple multiplied by some constnt, c From the preceding, we expect tht the vrince of the trnsformed smple is pproximted s: vr( ˆD s ) c 2 vr( ˆD) (0.25) The following snippet of R code simultes this sitution the vrince of the (rther lrge) rndom norml smple (smple), multiplied by the constnt c 2.25 (where the trnsformed smple is trns_smple) is , which is quite close to the pproximtion from the Delt method ( ). > smple <- rnorm( ,1.5,0.5) > c < > trns_smple <- c*smple > vr(trns_smple) [1]

8 B.3. Trnsformtions of one vrible B - 8 Another exmple of much the sme thing consider known number of N hrvested fish, with n verge weight (ˆµ w ) nd vrince. If you wnt n estimte of totl biomss (B), then ˆB N ˆµ w nd pplying the Delt method, the vrince of ˆB is pproximted s N 2 vr(ˆµ w ). So, if there re N 100 fish in the ctch, with n verge mss ˆµ w 20 pounds, with n estimted vrince of ˆσ 2 w 1.44, then by the Delt method, the pproximte vrince of the totl biomss ˆB (100 20) 2, 000 is: vr(ˆb) N 2 vr(ˆµ w ) (100) 2 (1.44) 14,400. The following snippet of R code simultes this prticulr exmple the estimted vrince of the (rther lrge) numericlly simulted smple (biomss), is very close to the pproximtion from the Delt method (14, 400). N <- 100 # size of smple of fish mu <- 20; # verge mss of fish v <- 1.44; # vrince of sme... reps < # number of replicte smples desired to generte dist of biomss # set up replicte smples - recll tht biomss is the product of N nd # the verge mss, which is smpled from rnorm with men mu nd vrince v biomss <- replicte(reps,n*rnorm(1,mu,sqrt(v))); # output result from the simulted biomss dt print(vr(biomss)); [1] One finl exmple you hve some prmeterθ,which you trnsform by dividing it by some constnt c. Thus, by the Delt method: vr ( ) ˆθ c ( ) 2 1 vr(ˆθ). c So, using norml probbility density function, forθ µ1.8, ndσ 2 θ c , then vr ( ) ( ) 2 ˆθ 1 (1.5) c , where the constnt The following snippet of R code simultes this sitution the vrince of the (rther lrge) rndom norml smple (smple), divided by the constnt, c (where the trnsformed smple is trns_smple) is , which is gin quite close to the pproximtion from the Delt method ( ).

9 B.3.1. A potentil compliction violtion of ssumptions B - 9 > smple <- rnorm( ,men=1.8,sd=sqrt(1.5)) > c < > smple_trns <- smple/c > vr(smple_trns) [1] B.3.1. A potentil compliction violtion of ssumptions A finl nd conceptully importnt exmple for trnsformtions of single vribles. The importnce lies in the demonstrtion tht the Delt method does not lwys work. Remember, the Delt method ssumes tht the trnsformtion is pproximtely liner over the expected rnge of the prmeter. Suppose one hs MLE for the men nd estimted vrince for some prmeterθwhich is bounded rndom uniform on the intervl[0, 2]. Suppose you wnt to trnsform this prmeter such tht: ψ e θ. [Recll tht this is convenient trnsformtion since the derivtive of e x is e x, mking the clcultions very simple. Also recll for the preceding- sidebr - tht the Tylor series expnsion to first-order my not do prticulr well with this function.] Now, bsed on the Delt method, the vrince for ψ would be estimted s: vr( ˆψ) ( ) 2 ˆψ vr(ˆθ) ˆθ ( e θ) 2 vr(ˆθ). Now, suppose tht ˆθ 1.0, nd vr(ˆθ) Then, from the Delt method: vr( ˆψ) (e θ) 2 ( ) vr ˆθ ( )(0.33) Is this resonble pproximtion? The only wy we cn nswer tht question is if we know wht the true (correct) estimte of the vrince should be. There re two pproches we might use to come up with the true (correct) vrince: (1) nlyticlly, or (2) by numericl simultion. We ll strt with the forml, nlyticl pproch, nd derive the vrince ofψusing the method of moments introduced erlier. To do this, we need to integrte the pdf (uniform, in this cse) over some rnge. Since the vrince of uniform distribution is (b ) 2 /12 (s derived erlier in this ppendix), nd if b nd re symmetric round the men (1.0), then we cn show by lgebr tht given vrince of 0.33, then 0 nd b 2 (check: (b ) 2 /12 (2 0) 2 / ).

10 B.3.1. A potentil compliction violtion of ssumptions B - 10 Given uniform distribution, the pdf is p(θ) 1/(b ). Thus, by the method of moments: M 1 M 2 b b g(x) b dx e b e b, g(x) 2 dx b ( ) 1 2 e2 e 2b. b Thus, by moments, vr(e(ψ) is: vr ( E(ψ) ) M 2 ( M 1 ) 2 ( 1 2 If 0 nd b 2, then the true vrince is given s: ) e2b + e 2 b + ( e b e ) 2 ( b) 2. vr ( E(ψ) ) M 2 (M 1 ) 2 ( ) 1 e2b + e 2 2 b , ( e b e ) 2 ( b) 2 which is not prticulrly close to the vlue estimted by the Delt method ( ). Let s lso consider coming up with n estimte of the true vrince by numericl simultion. The steps re pretty esy: (i) simulte lrge dt set, (ii) trnsform the entire dt set, nd (iii) clculte the vrince of the trnsformed dt set. For our present exmple, here is one wy you might set this up in R: > sim.dt <- runif( ,0,2); > trnsformed.dt <- exp(sim.dt); > vr(trnsformed.dt); [1] which is pretty close to the vlue derived nlyticlly, bove ( ) the slight difference reflects Monte Crlo error (generlly, the bigger the simulted dt set, the smller the error). Ok, so now tht we hve derived the true vrince in couple of different wys, the importnt question is why the discrepncy between the true vrince of the trnsformed distribution ( ), nd the first-order pproximtion to the vrince using the Delt method ( )? As discussed erlier, the Delt method rests on the ssumption the first-order Tylor expnsion round the prmeter vlue is effectively liner over the rnge of vlues likely to be encountered. Since in this exmple we re using uniform pdf, then ll vlues between nd b re eqully likely. Thus, we might nticipte tht s the intervl between nd b gets smller, then the pproximtion to the vrince (which will clerly decrese) will get better nd better (since the smller the intervl, the more likely it is tht the function is pproximtely liner over tht rnge). Forexmple,if 0.5 nd b 1.5 (sme men of1.0),then the true vrince ofθwillbe Thus,by the Delt method, the estimted vrince ofψwill be , while by the method of moments (which is exct), the vrince will be Clerly, the proportionl difference between the two vlues hs declined mrkedly. But, we chieved this improvement by rtificilly reducing the true vrince of the untrnsformed vrible θ. Obviously, we cn t do this in generl prctice.

11 B.3.1. A potentil compliction violtion of ssumptions B - 11 So, wht re the prcticl options? Well, one possible solution is to use higher-order Tylor series pproximtion by including higher-order terms, we cn (nd should) chieve better fit to the function (see the preceding- sidebr -). In other words, our pproximtion to the vrince should be improved by using higher-order Tylor expnsion. The only technicl chllenge (which relly isn t too difficult, with some prctice, potentilly ssisted by some good symbolic mth softwre) is coming up with the higher-order terms. One convenient pproch to deriving those higher-order terms for the present problem is to express the trnsformtion functionψin the form vr[g(x)] vr[g(µ+x µ)], which, fter somefirly tedious bits of lgebr, cn be Tylor expnded s (written sequentilly below, ech row representing the next order of the expnsion): vr[g(µ + X µ)] g (µ) 2 vr(x) + 2g (µ) g (µ) 2 [ g (µ) [ g (µ) g(4) (µ) 4! ( (X ) ) 3 E µ + 2g (µ) g (µ) 3! + 2 g (µ) 2 ] ( (X ) ) 4 E µ ] g (µ) ( (X ) ) 5 E µ 3! Now, while this looks little ugly (ok, mybe more thn little ugly), it ctully isn t too bd the whole expnsion is written in terms of things we know : the derivtives of our trnsformtion (g, g,...), simple sclrs nd sclr fctorils, nd, expecttions of sequentil powers of the devitions of the dt from the men of the distribution (i.e., E ( (X µ) n) ). You lredy know from elementry sttistics tht E ( (X µ) 1) 0, nd E ( (X µ) 2) σ 2 (i.e, the vrince). But wht bout E ( (X µ) 3), or E ( (X µ) 4). The higher-order terms in the Tylor expnsion re often functions of the expecttion of these higher-power devitions of the dt from the men. How do we clculte these expecttions? In fct, it isn t hrd t ll, nd involves little more thn pplying n pproch you ve lredy seen look bck few pges nd hve nother look t how we derived the vrince s function of the first nd second moments of the pdf. Remember, the vrince is simply E ( (X µ) 2) σ 2. Thus, we might nticipte tht the the sme logic used in deriving the estimte E ( (X µ) 2) s function of the moments could be used for E ( (X µ) 3 ), or E ( (X µ) 4), nd so on. The mechnics for E ( (X µ) 3) re lid out in the following- sidebr -. You cn sfely skip this section if you wnt, nd jump hed to the the clcultion of the vrince using higher-order expnsion, but it might be worth t lest skimming through this mteril, if for no other reson tht to demonstrte tht this is quite doble. It is lso pretty nifty demonstrtion tht lot of interesting things cn be nd re developed s function of the moments of the pdf. pproximting E ( (X µ) 3 ) begin sidebr Here, we demonstrte the mechnics for evluting E ( (X µ) 3). While it might look bit dunting, in fct it is reltively strightforwrd, nd is It is convenient demonstrtion of how you cn use the For simplicity, we re dropping (not showing) the terms involving Cov[(x µ) m,(x µ) n ] thus, the expression s written isn t complete expnsion to order n, but it is close enough to demonstrte the point.

12 B.3.1. A potentil compliction violtion of ssumptions B - 12 lgebr of moments to derive some interesting nd useful things. E ( (X µ) 3) Since M 1 µ, nd M 0 1, then b (x µ) 3 p(x)dx b ( x 3 + 3µ 2 x 3µx 2 µ 3) p(x)dx b b b b x 3 p(x)dx + 3µ 2 xp(x)dx 3µx 2 p(x)dx µ 3 p(x)dx b b b b x 3 p(x)dx + 3µ 2 xp(x)dx 3µ x 2 p(x)dx µ 3 p(x)dx M 3 + 3µ 2 (M 1 ) 3µ(M 2 ) µ 3 (M 0 ). E ( (X µ) 3) b (x µ) 3 p(x)dx M 3 + 3µ 2 (M 1 ) 3µ(M 2 ) µ 3 (M 0 ) M 3 + 3M 3 1 3M 1 M 2 M3 1. At this point, ll tht remins is substituting in the expressions for the moments corresponding to the prticulr pdf (in this cse, U(, b), s derived few pges bck), nd you hve your function for the expecttion E ( (X µ) 3 ). We ll leve it to you to confirm the lgebr the nswer is E ( (X µ) 3) 2 0. ( ) 3 ( 1 2 b ) ( 1 2 b b + 1 ) 3 b b b b3 Yes, lot of work nd some lgebr, for wht seems like n entirely nti-climtic result: E ( (X µ) 3) for the pdf U(, b) is 0. But you re hppier knowing how it s done (no, relly). We use the sme procedure for E ( (X µ) 4), nd so on. In fct, if you go through the exercise of clculting E ( (X µ) n) for n 4, 5,..., you ll find tht they generlly lternte between 0 (e.g., E ( (X µ) 3) 0 for U(, b)), nd non-zero (e.g., E ( (X µ) 4) 0.2, for U(0, 2)). This cn be quite helpful in simplifying the Tylor expnsion. end sidebr How well does higher-order pproximtion do? Let s strt by hving nother look t the Tylor expnsion we presented few pges bck we ll focus on the expnsion out to order 5: vr[g(µ + X µ)] g (µ) 2 vr(x) + 2g (µ) g (µ) ( (X ) ) 3 E µ 2

13 B.3.1. A potentil compliction violtion of ssumptions B [ g (µ) 2 [ 4 2g (µ) g(4) (µ) 4! ] + 2g (µ) g (µ) ( (X ) ) 4 E µ 3! + 2 g (µ) 2 ] g (µ) ( (X ) ) 5 E µ. 3! If you hd look t the preceding- sidebr -, you d hve seen tht some of the expecttion terms (nd products of sme) equl 0, nd thus cn be dropped from the expnsion. So, our order 5 Tylor series expnsion cn be written s: vr[g(µ + X µ)] g (µ) 2 vr(x) + 2g (µ) g (µ) ( (X ) ) 3 E µ 2 [ ] g (µ) g (µ) g (µ) 3! ( (X ) ) 4 E µ [ ] + 2g (µ) g(4) (µ) 4! + 2 g (µ) g (µ) ( (X ) ) 5 2 3! E µ [ ] g (µ) 2 g (µ) 2 vr(x)+ + 2g (µ) g (µ) ( (X ) ) 4 E µ. 4 3! So, how much better does this higher-order pproximtion do? If we run through the mth, nd for U(, b) where 0, b 2, such thtµ 1,σ , E (( X µ ) 4) 0.2, we end up with vr[g(µ + X µ)] g (µ) 2 vr(x)+ [ g (µ) 2 ( [( e 1) 2 e 1 ) 2 (0.3 3)+ + 2 ( ] e 1) e 1 (0.20) 4 3! , 4 ] + 2g (µ) g (µ) ( (X ) ) 4 E µ 3! which is much closer to the true vlue of (the fct tht the estimted vlue is slightly lrger thn the true vlue is somewht odd, nd possibly reflects not including the Cov[(x µ) m,(x µ) n ] terms in the Tylor expnsion). Regrdless, it is much better pproximtion thn the first-order vlue of OK, the preceding is rgubly somewht rtificil exmple. Now we ll consider more relistic sitution where the first-order pproximtion my be insufficient to our needs. Delt method pplied to the expecttion of the trnsformed dt Consider the following sitution. Suppose you re interested in simulting some dt on the logit scle, where vrition round the men if norml (so, you re going to simulte logit-norml dt). Suppose the men of some prmeter on the rel probbility scle isθ 0.3. Trnsformed to the

14 B.3.1. A potentil compliction violtion of ssumptions B - 14 logit scle, the men of the smple you re going to simulte would be log(θ/(1 θ)) So, you wnt to simulte some norml dt, with some specified vrince, on the logit scle, centered on µ lo git Here, using R, we generte vector (which we ve clled smp, below) of 50,000 logit-norml devites, with µ lo git , nd stndrd devition of ofσ lo git 0.75 (corresponding to vrince of σ 2 lo git ). We ll set the rndom number seed t1234 so you cn try this yourself, if inclined: > set.seed(1234); > smp <- rnorm(50000, ,0.75); If we check the men nd vrince of our rndom smple, we find they re quite close to the true prmeters used in simulting the dt (perhps not surprising given the size of the simulted smple). > men(smp) [1] > vr(smp) [1] If we plot histogrm of the simulted dt, we see symmetricl distribution centered round the true menµ lo git (verticl red line): Histogrm of smp Frequency Wht is the vrince of the bck-trnsformed estimte of the men, on the rel probbility scle? We know from wht we ve covered so fr tht if we try to clculte the vrince of the bck-trnsform of these dt from the logit scle rel probbility scle, by simply tking the bck trnsform of the estimted vrince ˆσ 2 lo git , we ll get the incorrect nswer. If we do tht, we would get e e

15 B.3.1. A potentil compliction violtion of ssumptions B - 15 How cn we confirm our developing intuition tht this vlue is incorrect? Well, if we simply bcktrnsform the entire rndom smple, nd then clculte the vrince of this bck trnsformed smple (which we cll bck) directly, > expit=function(x) exp(x)/(1+exp(x)); > bck <- expit(smp) > vr(bck) [1] we get vlue which, s we might hve expected, isn t remotely close to the vlue of we obtined by simply bck-trnsforming the vrince estimte. Of course, we know by now we should hve used the Delt method here. First, we recll tht the bck-trnsform f from the logit to the rel scle is: Then, we pply the Delt method s: vr( f) f eθ 1+ e θ. ( ) 2 f vr(ˆθ) ˆθ e ˆθ (1+ ) vr(ˆθ) e ˆθ 2 2 e (1+ e ) , 2 which is very close to the vlue we derived (bove) by clculting the vrince of the entire bcktrnsformed smple ( ). However, the min point we we wnt to cover here is pplying the Delt method to other moments specificlly, the men. Recll tht the men from our logit-norml smple ws Cn we simply bck-trnsform this men from the logit rel probbility scle? In other words, e e Now, compre this vlue to the men of the entire bck-trnsformed smple: > men(bck) [1]

16 B.3.1. A potentil compliction violtion of ssumptions B - 16 You might think tht the two vlues ( , ) re close enough for government work (lthough the difference is roughly 6%), but since we don t work for the government, let s pply the Delt method to generte correct pproximtion to the bck-trnsformed men. First, recll tht the trnsformtion function f (from logit rel) is f eθ 1+ e θ. Next, remember tht the Delt method s we ve been pplying generlly (nd in the preceding for the vrince) it is bsed on the first-order Tylor series pproximtion. Wht is the first-order Tylor series expnsion for f, ifθ µ? In fct, it is simply: e µ 1+ e µ + O( (θ µ) 2). where there term O ( (θ µ) 2) is the symptotic bound of the growth of the error. But, more to the point, the first-order pproximtion is bsiclly our bck-trnsformtion, with some (possibly lot) of error dded. In fct, we might expect this error term to be incresingly importnt if the ssumptions under which the first-order pproximtion pplies re strongly violted. In prticulr, if the trnsformtion function is highly non-liner over the rnge of vlues being exmined. Do we hve such sitution in the present exmple? Compre the histogrms of our simulted dt, on the logit (smp) nd bck-trnsformed rel scles (bck), respectively: Histogrm of smp Histogrm of bck Frequency Frequency smp bck Note tht the men of the bck-trnsformed distribution (verticl blue line) is somewht to the right of the mss of the distribution, which is firly symmetricl. This suggests tht the bck-trnsformtion might be sufficiently non-liner tht we need to use higher-order Tylor series pproximtion.

17 B.3.1. A potentil compliction violtion of ssumptions B - 17 If you do the mth (which isn t tht difficult), the second-order pproximtion is given s e µ µ ( 1+ e µ) + e ( (1 e µ) (θ µ)+ O (θ µ) 2). 2 Now, while the preceding might look bit complicted, the key here is remembering tht we re deling with expecttions. Wht is the expecttion of (θ µ)? In this sitution, θ is rndom vrible where ech estimted men from set of replicted dt sets on the logit scle represents θ, nd µ is the overll prmetric men. We know from even the most bsic sttistics clss tht the expecttion of the difference of rndom vrible X i from the men of the set of rndom vribles, X, is 0 (i.e., E(X i X) 0). By the sme logic, then, the expecttion of E(θ µ) 0. And, nything multiplied by 0 is 0, so, fter dropping the error term, our second-order pproximtion reduces to e µ e µ (1+ e µ) (1 e + µ) (θ µ) 2 e µ (1+ e µ), which brings us right bck to our stndrd first-order pproximtion. Wht bout third-order pproximtion? After bit more mth, we end up with e µ µ e µ( ) e µ 1 ( 1+ e µ) + e 1 ( (1 e µ) (θ µ) 2 2 (1 e µ) (θ 3 µ)2 + O (θ µ) 3). Agin, the expecttion for E(θ µ) 0. So, tht terms drops out: e µ e µ (1+ e µ) (1 + e µ) (θ µ) e µ( ) e µ 1 ( 1 e µ) 3 (θ µ)2 Wht bout for the term E(θ µ) 2? Look closely vrite minus men, squred. Look fmilir? It should it s the vrince! So, E(θ µ) 2 ˆσ 2. Thus, fter dropping the error term, our third-order pproximtion to the men is given s e µ ( 1+ e µ) 1 2 e µ( ) e µ 1 (1 e µ) 3 σ2. So, given our estimte of ˆµ nd ˆσ on the logit-scle, our third-order Delt method pproximtion for the expecttion (men) on the bck-trnsformed rel probbility scle, using this third-order pproximtion is e ( 1+ e ) 1 2 e ( ) e (1 e ) 3 ( ) ,

18 B.4. Trnsformtions of two or more vribles B - 18 which is quite bit closer to the empiricl estimte of the men derived from the entire bck-trnsformed smple ( ) thn ws our first ttempt using the first-order pproximtion ( ). So, we see tht the clssicl Delt method, which is bsed on first-order Tylor series expnsion of the trnsformed function, my not do prticulrly well if the function is highly non-liner over the rnge of vlues being exmined. Of course, it would be fir to note tht the preceding exmple mde the ssumption tht the distribution ws rndom uniform over the intervl. For most of our work with MARK, the intervl is likely to hve ner-symmetric mss round the estimte, typicllyβ. As such, most of dt, nd thus the trnsformed dt, will ctully fll closer to the prmeter vlue in question (the men in this exmple) thn we ve demonstrted here. So much so, tht the discrepncy between the first order Delt pproximtion to the vrince nd the true vlue of the vrince will likely be significntly smller thn shown here, even for strongly non-liner trnsformtion. We leve it to you s n exercise to prove this for yourself. But, this point notwithstnding, it is importnt to be wre of the ssumptions underlying the Delt method. If your trnsformtion is non-liner, nd there is considerble vrition in your dt, the firstorder pproximtion my not be prticulrly good. B.4. Trnsformtions of two or more vribles We re often interested in trnsformtions involving more thn one vrible. Fortuntely, there re lso multivrite generliztions of the Delt method. Suppose you ve estimted p different rndom vribles X 1, X 2,...,X p. In mtrix nottion, these vribles would constitute (p 1) rndom vector: which hs men vector: nd the (p p) vrince-covrince mtrix is: X 1 X 2 X,. X p EX 1 µ 1 EX 2 µ 2 µ...,. EX p µ p vr(x 1 ) cov(x 1, X 2 )... cov(x 1, X p ) cov ( ) cov(x 2, X 1 ) vr(x 2 )... cov(x 2, X p ) X 1, X cov(x p, X 1 ) cov(x p, X 2 )... vr(x p ) Note tht if the vribles re independent, then the off-digonl elements (i.e., the covrince terms) re ll zero.

19 B.4. Trnsformtions of two or more vribles B - 19 Thus, for (k p) mtrix of constnts A ij, the expecttion of rndom vector Y AX is given s: with vrince-covrince mtrix: EY 1 EY 2 Aµ,. EY p cov(y) AΣA. Now, using the sme logic we first considered for developing the Delt method for single vrible, for ech x i nerµ i, we cn write: g 1 (x) g 2 (x) y. g p (x) g 1 (µ) g 2 (µ) + D(x µ),. g p (µ) where D is the mtrix of prtil derivtives of g i with respect to x j, evluted t (x µ). As with the single-vrible Delt method, if the vrinces of the X i re smll (so tht with high probbility Y is ner µ, such tht the liner pproximtion is usully vlid), then to first-order we cn write: EY 1 g 1 (µ) EY 2 g 2 (µ).. EY p g p (µ) vr(ŷ) DΣD. In other words, to pproximte the vrince of some multi-vrible function Y, we (i) tke the vector of prtil derivtives of the function with respect to ech prmeter in turn (generlly known s the Jcobin), D, (ii) right-multiply this vector by the vrince-covrince mtrix, Σ, nd (iii) right-multiply the resulting product by the trnspose of the originl vector of prtil derivtives, D. Note: interprettion of the vrince estimted using the Delt method is dependent on the source of the vrince-covrince mtrix, Σ, used in the clcultions. If Σ is constructed using stndrd ML estimtes of the vrinces nd covrinces, then the resulting Delt method estimte for vrince is n estimte of the totl vrince, which is the sum of smpling + biologicl process vrince. In contrst, if Σ is bsed on estimted process vrinces nd covrinces only, then the Delt method estimte for

20 B.4. Trnsformtions of two or more vribles B - 20 vrince is n estimte of the process vrince. Decomposition of totl vrince into smpling nd process components is covered in detil in Appendix D. lterntive lgebrs for Delt method begin sidebr There re lterntive formultions of this expression which my be more convenient to implement in some instnces. When the vriblesθ 1,θ 2...θ k (in the function, Y) re independent, then vr(ŷ) vr ( f(θ 1,θ 2,...θ k ) ) k i1 vr ( θ i ) ( f θ i where f/ θ 1 is the prtil derivtive of Y with respect toθ i. When the vriblesθ 1,θ 2...θ k (in the function, Y) re not independent, then the covrince structure mong the vribles must be ccounted for: vr(ŷ) vr ( f(θ 1,θ 2,...θ k ) ) k i1 vr ( θ i ) ( f θ i ) 2 k + 2 i<j end sidebr ) 2, cov ( ) ( )( ) f f θ i,θ j θ i θ j Exmple (1) vrince of product of survivl probbilities Let s consider the ppliction of the Delt method in estimting smpling vrinces of firly common function the product of severl prmeter estimtes. From the preceding, we see tht: vr(ŷ) DΣD [ ] [ ] (Ŷ) (Ŷ) Σ, (ˆθ) ( ˆθ) where Y is some liner or nonliner function of the k prmeter estimtes ˆθ 1, ˆθ 2,... ˆθ k. The first term, D, on the RHS of the vrince expression is row vector contining prtil derivtives of Y with respect to ech of these k prmeters (ˆθ 1, ˆθ 2,... ˆθ k ). The right-most term of the RHS of the vrince expression, D, is simply trnspose of this row vector (i.e., column vector). The middle-term, Σ is simply the estimted vrince-covrince mtrix for the prmeters. To demonstrte the steps in the clcultion, we ll use estimtes from model {ϕ t p. } fit to the mle Europen dipper dt set. Suppose we re interested in the probbility of surviving from the strt of the first intervl to the end of the third intervl. The estimte of this probbility is esy enough: Ŷ ( ˆϕ 1 ˆϕ 2 ˆϕ 3 ) ( )

21 B.4. Trnsformtions of two or more vribles B - 21 So, the estimted probbility of mle Dipper surviving over the first three intervls is 14% (gin, ssuming tht our time-dependent survivl model is vlid model). To derive the estimte of the vrince of the product, we will lso need the vrince-covrince mtrix for the survivl estimtes. You cn generte the mtrix esily in MARK by selecting Output Specific Model Output Vrince-Covrince Mtrices Rel Estimtes. The vrince-covrince mtrix for the mle Dipper dt, generted from model{ϕ t p. }, s output to the defult editor (e.g., Windows Notepd), is shown below: Here, the vrince-covrince vlues re below the digonl, wheres the stndrdized correltion vlues re bove the digonl. The vrinces re given long the digonl. However, it is very importnt to note tht the V-C mtrix tht MARK outputs to the editor is rounded to 5 significnt digits. For the ctul clcultions, we need to use the full precision vlues. To get those, you need to either (i) output the V-C mtrix into dbse file (which you could then open with dbse, or Excel), or (ii) copy the V-C mtrix into the Windows clipbord, nd then pste it into some other ppliction. Filure to use the full precision V-C mtrix will lmost lwys led to significnt rounding errors. The full precision V-C mtrix for the 3 Dipper survivl estimtes is shown below. ĉov(ŷ) vr( ˆϕ 1 ) ĉov( ˆϕ 1, ˆϕ 2 ) ĉov( ˆϕ 1, ˆϕ 3 ) ĉov( ˆϕ 2, ˆϕ 1 ) vr( ˆϕ 2 ) ĉov( ˆϕ 2, ˆϕ 3 ) ĉov( ˆϕ 3, ˆϕ 1 ) ĉov( ˆϕ 3, ˆϕ 2 ) vr( ˆϕ 3 ) The vrince-covrince estimtes MARK genertes will occsionlly depend somewht on which optimiztion method you use (i.e., defult, or simulted nneling), nd on the strting vlues used to initilize the optimiztion. The differences in the reported vlues re often very smll (i.e., pprent only severl deciml plces out from zero), but you should be wre of them. For ll of the exmples presented in this Appendix, we hve used the defult optimiztion routines, nd defult strting vlues.

22 B.4. Trnsformtions of two or more vribles B For this exmple, the trnsformtion we re pplying to our 3 survivl estimtes (which we ll cll Y) is the product of the estimtes (i.e., Ŷ ˆϕ 1 ˆϕ 2 ˆϕ 3 ). Thus, our vrince estimte is given s vr(ŷ) [( ) ( ) ( )] (Ŷ) (Ŷ) (Ŷ) ˆϕ 1 ˆϕ 2 ˆϕ 3 ( ) ( Ŷ) ˆϕ 1 ( ) (Ŷ) ˆϕ. 2 ( ) (Ŷ) ˆϕ 3 Ech of the prtil derivtives for Ŷ is esy enough to derive for this exmple. Since Ŷ ˆϕ 1 ˆϕ 2 ˆϕ 3, then Ŷ/ ˆϕ 1 ˆϕ 2 ˆϕ 3. And so on. So, vr(ŷ) [( ) ( ) ( )] (Ŷ) (Ŷ) (Ŷ) ˆϕ 1 ˆϕ 2 ˆϕ 3 ( ) (Ŷ) ˆϕ 1 ( ) (Ŷ) ˆϕ 2 ( ) (Ŷ) ˆϕ 3 ( ) [ ( ) ( ) ( ) ] vr( ˆϕ 1 ) ĉov( ˆϕ 1, ˆϕ 2 ) ĉov( ˆϕ 1, ˆϕ 3 ) ˆϕ2 ˆϕ 3 ( ) ˆϕ2 ˆϕ 3 ˆϕ1 ˆϕ 3 ˆϕ1 ˆϕ 2 ĉov( ˆϕ 1, ˆϕ 1 ) vr( ˆϕ 2 ) ĉov( ˆϕ 2, ˆϕ 3 ) ˆϕ1 ˆϕ 3. ĉov( ˆϕ 3, ˆϕ 1 ) ĉov( ˆϕ 3, ˆϕ 2 ) vr( ˆϕ 3 ) ( ) ˆϕ1 ˆϕ 2 Clerly, the estimtor is getting more nd more impressive s we progress. The resulting expression (written in piecewise fshion to mke it esier to see the bsic pttern) is shown below: vr(ŷ) ˆϕ 2 2 ˆϕ2 3 [ vr( ˆϕ 1 )] + 2 ˆϕ 2 ˆϕ 2 3 ˆϕ 1 [ĉov( ˆϕ 1, ˆϕ 2 )] + 2 ˆϕ 2 2 ˆϕ 3 ˆϕ 1 [ĉov( ˆϕ 1, ˆϕ 3 )] + ˆϕ 2 1 ˆϕ2 3 [ vr( ˆϕ 2 )] + 2 ˆϕ 2 1 ˆϕ 3 ˆϕ 2 [ĉov( ˆϕ 2, ˆϕ 3 )] + ˆϕ 2 1 ˆϕ2 2 [ vr( ˆϕ 3 )].

23 B.4. Trnsformtions of two or more vribles B - 23 After substituting in our estimtes forϕ i nd the vrinces nd covrinces, our estimte for the vrince of the product Ŷ ( ˆϕ 1 ˆϕ 2 ˆϕ 3 ) is (pproximtely) vr(y) Exmple (2) vrince of estimte of reporting rte In some cses nimls re tgged or bnded to estimte reporting rte the proportion of tgged nimls reported (sy, to conservtion mngement gency), given tht they were killed nd retrieved by hunter or ngler (see chpter 8 for more detils). Thus, N c nimls re tgged with norml (control) tgs nd, of these, R c re recovered the first yer following relese. The recovery rte of these control nimls is merely R c /N c nd we denote this s f c. Another group of nimls, of smple size N r, re tgged with specil rewrd tgs; these tgs indicte tht some mount of money (sy, $50) will be given to people reporting these specil tgs. It is ssumed tht$50 is sufficientto ensure thtllsuchtgs willbe reported,thus these serve s bsis forcomprison nd the estimtion of reporting rte. The recovery probbility for the rewrd tgged nimls is merely R r /N r, where R r is the number of recoveries of rewrd-tgged nimls the first yer following relese. We denote this recovery probbility s f r. The estimtor of the reporting rte is rtio of the recovery rtes nd we denote this sλ. Thus: ˆλ fˆ. f cˆ r Now, note tht both recovery probbilities re binomils. Thus: vr ( ) fc ˆ (1 ˆ ) f c fc ˆ nd vr ( ) fr ˆ (1 ˆ ) f r fr ˆ. N c N r In this cse, the smples re independent, thus cov ( f c, f r ) nd the smpling vrince-covrince mtrix is digonl: [ vr ( ) ] fc ˆ 0 0 vr ( ) fr ˆ Next, we need the derivtives ofλwith respect to f c nd f r :. ˆλ f ˆ 1ˆ, nd c fr ˆλ fˆ c f ˆ r ˆ f 2 r. Thus, vr(ˆλ) [ 1, fˆ r ˆ f c fˆ 2 r ] [ vr ( ) fc ˆ 0 0 vr ( ) fr ˆ ] 1 ˆ f r fˆ. c ˆ f 2 r

24 B.4. Trnsformtions of two or more vribles B - 24 Exmple (3) vrince of bck-trnsformed estimtes - simple In Chpter 6, we demonstrted how we cn bck-trnsform from the estimte ofβon the logit scle to n estimte of some prmeterθ (e.g.,ϕ or p) on the probbility scle (which is bounded[0, 1]). But, we re clerly lso interested in n estimte of the vrince (precision) of our estimte, on both scles. Your first thought might be to simply bck-trnsform from the link function (in our exmple, the logit link), to the probbility scle, just s we did bove. But, s discussed in chpter 6, this does not work. For exmple, consider the mle Dipper dt. Using the logit link, we fit the time-invrint model {ϕ. p. } to the dt. Let s consider only the estimte for ˆϕ. The estimte for ˆβ forϕis Thus, our estimte of ˆϕ on the probbility scle (which is wht MARK reports) is: ˆϕ e e But, wht bout the vrince? Well, if we look t theβestimtes, MARK reports tht the stndrd error for the estimte of β corresponding to survivl is If we simply bck-trnsform this from the logit scle to the probbility scle, we get: ŜE e e However, MARK reports the estimted stndrd error for ϕ s , which isn t even remotely close to our bck-trnsformed vlue of Wht hs hppened? Well, hopefully you now relize tht you re trnsforming the estimte from one scle (logit) to nother (probbility). And, since you re working with trnsformtion, you need to use the Delt method to estimte the vrince of the bck-trnsformed prmeter. Since ˆϕ e ˆβ 1+ e ˆβ, then vr( ˆϕ) ( ) 2 ˆϕ vr(ˆβ) ( ˆβ e ˆβ ( e ˆβ ) 2 )2 1+ e ˆβ 1+ ( e ˆβ ) vr(ˆβ) 2 ( e ˆβ 2 ( 1+ e ˆβ 2) ) vr(ˆβ). It is gin worth noting tht if ˆϕ e ˆβ 1+ e ˆβ,

25 B.4. Trnsformtions of two or more vribles B - 25 then it cn be esily shown tht ˆϕ(1 ˆϕ) e ˆβ ( 1+ e ˆβ ) 2, which is the derivtive of ϕ with respect to β. So, we could rewrite our expression for the vrince of ˆϕ conveniently s 2 vr( ˆϕ) e ˆβ ( (1+ ) vr(ˆβ) ˆϕ ( 1 ˆϕ )) 2 e ˆβ 2 vr(ˆβ). From MARK, the estimte of the SE for ˆβ ws Thus, the estimte of vr(β) is( ) Given the estimte of ˆβ of , we substitute into the preceding expression, which yields vr( ˆϕ) ( e ˆβ ( 1+ e ˆβ ) 2) 2 vr(ˆβ) ( ) So, the estimted SE for ˆϕ is , which is wht is reported by MARK. SE nd 95% CI begin sidebr The stndrd pproch to clculting 95% confidence limits for some prmeter θ is θ ±(1.96 SE). Is this how MARK clcultes the 95% CI on the rel probbility scle? Well, tke the exmple we just considered the estimted SE for ˆϕ ws So, you might ssume tht the 95% CI on the rel probbility scle would be ± ( ) - [ , ]. However, this is not wht is reported by MARK - [ , ], which is quite close, but not exctly the sme. Why the difference? The difference is becuse MARK first clculted the 95% CI on the logit scle, before bck-trnsforming to the rel probbility scle. So, for our estimte of ˆϕ, the 95% CI on the logit scle for ˆβ is[ , ], which, when bck-trnsformed to the rel probbility scle is[ , ], which is wht is reported by MARK. In this cse, the very smll difference between the two CI s is becuse the prmeter estimte ws quite close to 0.5. In such cses, not only will the 95% CI be nerly the sme (for estimtes of 0.5, it will be identicl), but they will lso be symmetricl. However, becuse the logit trnsform is not liner, the reconstituted 95% CI will not be symmetricl round the prmeter estimte, especilly for prmeters estimted ner the [0, 1] boundries. For exmple,consider the estimte for ˆp On the logit scle,the 95% CI for theβcorresponding to p (ŜE ) is [ , ]. The bck-trnsformed CI is [ , ], which is wht is reported by MARK. This CI is clerly not symmetric round ˆp The degree of symmetry is function of how close the estimted prmeter is to either the 0 or 1 boundry.

26 B.4. Trnsformtions of two or more vribles B - 26 Further, the estimted vrince for ˆp: vr(ˆp) [ˆp(1 ˆp) ] 2 vr(ˆβ) [ ( ) ] , yields n estimted SE of on the norml probbility scle (which is wht is reported by MARK). Estimting the 95% CI on the probbility scle simply s ±( ) yields[ , ], which is clerly quite bit different, nd more symmetricl, thn wht is reported by MARK (from bove, [ , ]). MARK uses the bck-trnsformed CI to ensure tht the reported CI is bounded [0, 1]. As the estimted prmeter pproches either the 0 or 1 boundry, the degree of symmetry in the bck-trnsformed 95% CI tht MARK reports will increse. end sidebr Exmple (4) vrince of bck-trnsformed estimtes - hrder In Chpter 6 we considered the nlysis of vrition in the survivl of the Europen Dipper,s function of whether or not there ws flood in the smpling re. Here, we consider just the mle Dipper dt (the encounter dt re contined in ed_mles.inp). Recll tht flood occurred during over the second nd third intervls. For convenience, we ll ssume tht encounter probbility is constnt over time, nd tht survivl is liner function of flood. Using logit link function, where flood yers were coded in the design mtrix using 1, nd non-flood yers were coded using 0, the estimted liner model for survivl on the logit scle ws: logit( ˆϕ) (flood) So, in flood yer: logit( ˆϕ flood ) (flood) (1) Bck-trnsforming onto the rel probbility scle yields the precise vlue reported by MARK: ˆϕ flood e e Now, wht bout the estimted vrince forϕ flood? First, wht is our trnsformtion function (Y)? Simple it is the bck-trnsform of the liner eqution on the logit scle. Given tht: logit( ˆϕ) β 1 +β 2 (flood) (flood),

27 B.4. Trnsformtions of two or more vribles B - 27 then the bck-trnsform function Y is Ŷ e (flood) 1+ e (flood). Second, since our trnsformtion clerly involves multiple prmeters (β 1,β 2 ), the estimte of the vrince is given to first-order by vr(ŷ) DΣD [ ] (Ŷ) (ˆθ) [ (Ŷ) Given our liner (trnsformtion) eqution, then the vector of prtil derivtives is (we ve trnsposed it to mke it esily fit on the pge): [( ) ( )] (Ŷ) (Ŷ) (ˆθ) ] ˆβ 1 ˆβ 2 e β 1 +β 2 (flood) 1+ e β 1 +β 2 (flood) flood e β 1 +β 2 (flood) 1+ e β 1 +β 2 (flood) (e β 1 +β 2 (flood)) 2 (1+ e β 1 +β 2 (flood)) 2 flood (e β 1 +β 2 (flood)) 2 (1+ e β 1 +β 2 (flood)) 2 While this is firly ugly looking, the structure is quite strightforwrd the only difference between the 2 elements of the vector is tht the numertor of both terms (on either side of the minus sign) re multiplied by 1, nd flood, respectively. Where do these sclr multipliers come from? They re simply the prtil derivtives of the liner model (we ll cll it Y) on the logit scle: Y logit ( ˆϕ ) β 1 +β 2 (flood), with respect to ech of the prmeters (β i ) in turn. In other words, Y/ β 1 1, nd Y/ β 2 flood. Substituting in our estimtes for ˆβ nd ˆβ ,nd settingflood=1 (to indicte flood yer ) yields: [( ) ( )] (Ŷ) (Ŷ) [ ] ˆβ 1 ˆβ 2 From the MARK output (fter exporting to dbse file nd not to the Notepd in order to get full precision), the full V-C mtrix for the prmetersβ 1 ndβ 2 is: ĉov (ˆβ 1, ˆβ 2 ) [ ]

28 B.4. Trnsformtions of two or more vribles B - 28 So, vr(ŷ) [ ] [ ] [ ] The estimted SE for the vrince for the reconstituted vlue of survivl for n individul during flood yer is , which is wht is reported by MARK (to within rounding error). Once gin...se nd 95% CI begin sidebr As noted in the precceding exmple, the stndrd pproch to clculting 95% confidence limits for some prmeterθ isθ±(1.96 SE). However, to gurntee tht the clculted 95% CI is[0, 1] bounded for prmeters tht re [0, 1] bounded (like ϕ), MARK first clcultes the 95% CI on the logit scle, before bck-trnsforming to the rel probbility scle. However, becuse the logit trnsform is not liner, the reconstituted 95% CI will not be symmetricl round the prmeter estimte, especilly for prmeters estimted ner the [0, 1] boundries. For the present exmple, the estimted vlue of survivl for n individul during flood yer ( ˆϕ flood ), MARK reports 95% CI of [ , ]. But, where do the vlues [ , ] come from? Clerly, they re not bsed on ± 1.96(SE). Given ŜE , this would yield 95% CI of [ , ], which is close, but not exctly wht MARK reports. In order to derive the 95% CI, we first need to clculte the vrince (nd SE) of the estimte on the logit scle. In the preceding exmple, this ws very strightforwrd, since the model we considered hd single β term for the prmeter of interest. Mening, we could simply use the estimted SE for β to derive the 95% CI on the logit scle, which we then bck-trnsformed onto the rel probbility scle. For the present exmple, however, the prmeter is estimted from function (trnsformtion) involving more thn one β term. In this exmple, the liner eqution, which for consistency with the preceding we will denote s Y, ws written s: Ŷ logit ( ˆϕ ) β 1 +β 2 (flood) Thus, the estimted vrince of logit( ˆϕ flood ) is pproximted s vr ( Ŷ ) DΣD [ (Ŷ) (ˆβ 1 ) ] (Ŷ) (ˆβ 2 ) [ (Ŷ) (ˆβ 1 ) ] (Ŷ) (ˆβ 2 ) Since [ (Ŷ) (ˆβ 1 ) ] (Ŷ) [1 flood] [1 1], (ˆβ 2 ) nd the VC mtrix for ˆβ 1 nd ˆβ 2 is ĉov(ˆβ 1, ˆβ 2 ) [ ] ,

29 B.4. Trnsformtions of two or more vribles B - 29 then vr(ŷ) DΣD [ 1 1 ] [ ] [ ] 1 1 So, the ŜE on the logit scle! is Thus, the 95% CI on the estimte on the logit scle, logit( ˆϕ flood ) ± 1.96( ) [ , ]. All tht is left is to bck-trnsform the limits on the CI to the rel probbility scle: [ ] e e [ , ] 1+e , [ , ] 1+ e which is wht is reported by MARK (to within rounding error). end sidebr Exmple (5) vrince of bck-trnsformed estimtes - hrder still In Chpter 11, we considered nlysis of the effect of vrious functions of mss,m, nd mss-squred, m2, on the survivl of hypotheticl species of bird (the simulted dt re in file indcov1.inp). The liner function relting survivl to m nd m2, on the logit scle, is: logit( ˆϕ) ( ) ( m s m 2) s Note tht for the two mss terms in the eqution, there is smll subscript s, reflecting the fct tht these re stndrdized msses. Recll tht we stndrdized the covrites by subtrcting the men of the covrite, nd dividing by the stndrd devition (the use of stndrdized or non-stndrdized covrites is discussed t length in Chpter 11). Thus, for ech individul in the smple, the estimted survivl probbility (on the logit scle) for tht individul, given its mss, is given by: ( ) ( m m m 2 m 2 ) logit( ˆϕ) SD m SD 2 m In this expression,mrefers tomss ndm 2 refers tomss2. The output from MARK (preceding pge) ctully gives you the men nd stndrd devitions for both covrites: for mss, men = , nd SD = , while for mss2, the men = 12, , nd the SD = 5, The vlue column shows the stndrdized vlues for mss nd mss2 (0.803 nd 0.752) for the first individul in the dt file. Now let s consider worked exmple of the clcultion of the vrince of estimted survivl. Suppose the mss of the bird ws 110 units, so thtm=110,m2 =(110) 2 12,100. Thus: logit( ˆϕ) ( ) ( ) ( ) (12,100 12, ) ,

30 B.4. Trnsformtions of two or more vribles B - 30 So, if logit( ˆϕ) 0.374, then the reconstituted estimte ofϕ, trnsformed bck from the logit scle is: e e Thus, for n individul weighing 110 units, the expected nnul survivl probbility is pproximtely (which is wht MARK reports if you use the User specify covrite option). Wht bout the vrince (nd corresponding SE) for this estimte? First, wht is our trnsformtion function (Y)? For the present exmple, it is the bck-trnsform of the liner eqution on the logit scle. Given tht: then the bck-trnsform Y is: logit( ˆϕ) β 1 +β 2 (m s )+β 3 (m 2 s ) (m s ) (m 2 s ), Ŷ e (m s ) (m 2 s ) 1+ e (m s ) (m2 s ) As in the preceding exmple,since ourtrnsformtion clerly involves multiple prmeters (β 1,β 2,β 3 ), the estimte of the vrince is given by: vr(ŷ) DΣD [ ] ( Ŷ) (ˆθ) [ ( Ŷ) Given our liner (trnsformtion) eqution (from bove) then the vector of prtil derivtives is is: ( ) (Ŷ) ˆβ 0 ( ) (Ŷ) ˆβ 1 ( ) (Ŷ) ˆβ 2 e β 1 +β 2 (m)+β 3 (m2) 1+ e β 1 +β 2 (m)+β 3 (m2) m e β 1 +β 2 (m)+β 3 (m2) 1+ e β 1 +β 2 (m)+β 3 (m2) m2 e β 1 +β 2 (m)+β 3 (m2) 1+ e β 1 +β 2 (m)+β 3 (m2) (ˆθ) ] [e β 1 +β 2 (m)+β 3 (m2)] 2 [1+ e β 1 +β 2 (m)+β 3 (m2)] 2 m [e β 1 +β 2 (m)+β 3 (m2)] 2 [1+ e β 1 +β 2 (m)+β 3 (m2)] 2 m2 [e β 1 +β 2 (m)+β 3 (m2)] 2 [1+ e β 1 +β 2 (m)+β 3 (m2)] 2 Although this looks complicted, the structure is ctully quite strightforwrd the only difference between the 3 elements of the vector is tht the numertor of both terms (on either side of the minus sign) re multiplied by 1,m, ndm2, respectively, which re simply the prtil derivtives of the liner model (we ll cll it Y) on the logit scle: Ŷ logit( ˆϕ) β 1 +β 2 (m s )+β 3 (m 2 s ), with respect to ech of the prmeters (β i ) in turn. In other words, Y/ β 1 1, Y/ β 2 m, nd Y/ β 3 m2.

31 B.4. Trnsformtions of two or more vribles B - 31 So, now tht we hve our vectors of prtil derivtives of the trnsformtion function with respect to ech of the prmeters, we cn simplify things considerbly by substituting in the stndrdized vlues formndm2, nd the estimted prmeter vlues (ˆβ 1, ˆβ 2, nd ˆβ 3 ). For mss of 110 g, the stndrdized vlues formndm2 re: ( ) m s ( ) m2 s The estimtes for ˆβ i we red directly from MARK: ˆβ , ˆβ ,nd ˆβ Substituting in these estimtes for ˆβ i nd the stndrdizedm ndm2 vlues into our vector of prtil derivtives (which we ve trnsposed in the following to sve spce) yields: [( ) ( ) ( )] (Ŷ) (Ŷ) (Ŷ) ˆβ 1 ˆβ 2 ˆβ From the MARK output (fter exporting to dbse file nd not to the editor in order to get full precision), the full V-C mtrix for the β prmeters is So, vr(ŷ) [ ] So, the estimted SE for vr for the reconstituted vlue of survivl for n individul weighing 110 g is , which is exctly wht is reported by MARK. It is importnt to remember tht the estimted vrince will vry depending on the mss you use - the estimte of the vrince for 110 g individul ( ) will differ from the estimted vrince for (sy) 120 g individul. For 120 g individul, the stndrdized vlues ofmndm2 re nd , respectively. Bsed on these vlues, then: [( ) ( ) ( )] (Ŷ) (Ŷ) (Ŷ) ˆβ 1 ˆβ 2 ˆβ

32 B.4. Trnsformtions of two or more vribles B - 32 Given the vrince covrince-mtrix for this model (shown bove), then vr(ŷ) DΣD Thus, the estimted SE for the reconstituted vlue of survivl for n individul weighing 120 g is , which gin is exctly wht is reported by MARK. Note tht this vlue for the SE for 120 g individul ( ) differs from the SE estimted for 110 g individul ( ), lbeit not by much (the smll difference here is becuse this is very lrge simulted dt set bsed on deterministic model see Chpter 11 for detils). Since ech weight would hve its own estimted survivl, nd ssocited estimted vrince nd SE, to generte curve showing the reconstituted vlues nd their SE, you d need to itertively clculte DΣD over rnge of weights. We ll leve it to you to figure out how to hndle the progrmming if you wnt to do this on your own. For the less mbitious, MARK hs the cpcity to do much of this for you you cn output the 95% CI dt over rnge of individul covrite vlues to spredsheet (see section 11.5 in Chpter 11). Exmple (6) - estimting vrince + covrince in trnsformtions Here, we consider ppliction of the Delt method to joint estimtion of the vrince of prmeter, nd the covrince of tht prmeter with nother, where one of the two prmeters is liner trnsformtion of the other. This is somewht complicted, but quite useful exmple, since it illustrtes how you cn use the Delt method to estimte not only the vrince of individul prmeters, but the covrince structure mong prmeters s well. There re mny instnces where the mgnitude of the covrince is of prticulr interest. Here, we consider such sitution, in terms of different prmeteriztions for nlysis of ded recovery dt. Ded recovery models re covered in detil in Chpter 8 here, we briefly review two different prmeteriztions (the Seber nd Brownie prmeteriztions), nd the context of our interest in the covrince between two different prmeters. The encounter process for the Seber prmeteriztion (1973: 254) is illustrted in the following: Mrked individuls re ssumed to survive from relese i to i + 1 with probbility S i. Individuls my die during the intervl, either due to hrvest or to nturl mortlity. The probbility tht ded mrked individuls re reported during ech period i between releses, nd (most generlly) where the deth is not necessrily relted to hrvest, is r i. In other words, r i is the joint probbility of (i) the mrked individul dying from either hrvest or nturl cuses, nd (ii) being recovered nd reported (i.e., encountered ). Brownie et l. (1985) (herefter, simply Brownie ) developed different prmeteriztion for ded recovery dt, where the sources of mortlity (hrvest, versus nturl or non-hrvest) re modeled seprtely.

33 B.4. Trnsformtions of two or more vribles B - 33 The encounter process for the Brownie prmeteriztion is illustrted in the following: Following Brownie,S i is the probbility thtthe individulsurvives the intervlfrom relese occsion i to i + 1 (note tht the definition for the probbility of survivl is logiclly identicl between the Seber nd Brownie prmeteriztions). The probbility tht the individul dies from either source of mortlity is simply 1 S. However, in contrst to the Seber prmeteriztion, Brownie specified prmeter f, to represent the probbility tht n individul dies specificlly due to hrvest during intervl i, nd is reported ( encountered ). Thus, the probbility tht the individuls dies from nturl cuses is(1 S f). Under the Seber prmeteriztion, the probbility of the encounter history 11 is given s r(1 S). Under the Brownie prmeteriztion, the expected probbility of this event is simply f. Since the encounter history is the sme, we cn set the different prmeteriztions for the expected probbility of the event equl to ech other, generting the following expressions relting the two prmeteriztions: f i r i ( 1 Si ) r i f i ( 1 Si ) Clerly, the prmeter r i is reduced prmeter, nd cn be expressed s function of two other prmeters normlly found in the Brownie prmeteriztion. An obvious prcticl question is, why choose one prmeteriztion over the other, nd does it mtter? This issue is discussed more fully in Chpter 8, but for now, we focus on the left-hnd expression: f i r i ( 1 Si ) So, given estimtes of ˆr i nd Ŝ i from Seber nlysis, we could use this lgebric reltionship (i.e., trnsformtion) to generte estimtes of f ˆ. Nturlly, we wish to be ble to estimte vr( f). ˆ However, in ddition, we re potentilly interested in estimting the covrince ĉov( f,ŝ). ˆ Recll from bove tht the prmeter f reltes in prt to the probbility of being hrvested. We might nturlly be interested in the reltionship between hrvest mortlity f, nd overll nnul survivl, S. For exmple, if hrvest nd nturl mortlity re strictly dditive, then we might expect negtive covrince between survivl nd hrvest (i.e., s the probbility of mortlity due to hrvest increses, nnul survivl will decrese). Whether or not the covrince is negtive hs importnt implictions for hrvest mngement (see full discussion in the Willims, Nichols & Conroy 2001 book). We ll begin by considering estimtion of the vrince for f ˆ only, using the Delt method. Let the trnsformtion g be f (1 S)r. Given Ŝ, ˆr, vr(ŝ) nd vr(ˆr), then the Jcobin for g is [ g S ] [ g ˆr r ] 1 Ŝ,

34 B.4. Trnsformtions of two or more vribles B - 34 nd thus vr( f) ˆ [ ˆr ] [ ] ˆr 1 Ŝ, 1 Ŝ where is the vrince-covrince mtrix for S nd r: [ ] vr(ŝ) ĉov(ŝ, ˆr). ĉov(ŝ, ˆr) vr( ˆr) So, yields vr( f) ˆ [ ˆr ] [ ] ˆr 1 Ŝ 1 Ŝ vr ( ˆ f) ˆr2 vr ( Ŝ) 2ˆr ĉov(ŝ, ˆr ) + 2ˆr ĉov ( Ŝ, ˆr ) Ŝ + vr (ˆr ) 2 vr (ˆr ) S + vr (ˆr ) S 2, which, with little re-rrnging, yields vr ( ˆ f) ˆr2 vr ( Ŝ ) 2 [ (1 Ŝ)ˆr] ĉov ( Ŝ, ˆr ) + ( 1 Ŝ) 2 vr (ˆr ), If you substitute in r f/(1 S) into the preceding expression, we end up with vr ( ( ) 2 f) ˆ fˆ vr ( Ŝ ) 2 f ˆ ĉov ( Ŝ, ˆr ) + ( 1 Ŝ ) 2 ) vr (ˆr. 1 Ŝ Now, wht if insted of vr( ˆ f) only, we re lso interested in estimting the covrince of (sy) f nd S? Such covrince might be of interest since f is function of S, nd there my be interest in the degree to which S vries s function of f (see bove). Thus, we wnt to pply the Delt method to function (the covrince) of two prmeters, f nd S. The key step here is recognizing tht there re in fct two different functions (or, trnsformtions) involved, which we ll cll g 1 nd g 2 : g 1 : S S nd g 2 : (1 S)r f You might be puzzled by g 1 : S S. In fct, this represents null trnsformtion direct, non-trnsformtive 1:1 mpping between S under the Seber prmeteriztion, nd survivl under the Brownie prmeteriztion (since the probbility of surviving is, logiclly, the sme under the two prmeteriztions). This is nlogous to generting the estimte for Ŝ i under one prmteriztion by multiplying the sme estimte under the other prmeteriztion by the sclr constnt 1. Thus, with two trnsformtions, we generte Jcobin mtrix of prtil derivtives of ech trnsformtions with respect to S nd r, respectively: g 1 Ŝ g 2 Ŝ g 1 Ŝ ˆr g Ŝ 2 f ˆ ˆr Ŝ Ŝ ˆr f ˆ ˆr

35 B.5. Delt method nd model verging B r 1 Ŝ 1 0 fˆ. 1 Ŝ (1 Ŝ) Given the vrince-covrince mtrix for Ŝ nd ˆr vr(ŝ) ĉov(ŝ, ˆr) ĉov(ŝ, ˆr), vr( ˆr) we evlute smpling vrince-covrince mtrix for Ŝ nd ˆ f s the mtrix product 1 0 fˆ 1 Ŝ (1 Ŝ) fˆ 1 (1 Ŝ), 0 1 Ŝ which (fter bit of lgebr) yields fˆ vr(ŝ) 1 Ŝ vr( Ŝ ) + ( 1 Ŝ ) ĉov ( Ŝ, ˆr ) fˆ 1 Ŝ vr( Ŝ ) + ( 1 Ŝ ) ĉov ( Ŝ, ˆr ) ˆr 2 vr ( Ŝ ) 2 [ (1 Ŝ)ˆr ] ĉov ( Ŝ, ˆr ) + ( 1 Ŝ ). 2 ) vr (ˆr Here, mtrix elements [1,1] nd [2,2] re the expressions for the pproximte vrince of S nd f, respectively (note tht the expression in element [2,2], for vr( f), ˆ is identicl to the expression we derived on the preceding pge). Elements [1,2] nd [2,1] (which re the sme) re the expressions for the pproximte covrince of f nd S. As noted erlier, interprettion of the estimted vrince nd covrince is dependent on the source of the vrince-covrince mtrix, Σ, used in the clcultions. If Σ is constructed using vrinces nd covrinces from the usul ML prmeter estimtes, then the resulting estimte for vrince is n estimte of the totl vrince (i.e., smpling + process, where process vrition represents the underlying biologicl vrition). In contrst, if Σ is bsed on estimted process (vrinces nd covrinces only, then the estimte for vrince is n estimte of the process vrince. Decomposition of totl vrince into smpling nd process components is covered in detil in Appendix D. B.5. Delt method nd model verging In the preceding exmples, we focused on the ppliction of the Delt method to trnsformtions of prmeter estimtes from single model. However, s introduced in Chpter 4 nd emphsized throughout the reminder of this book we re often interested in ccounting for model selection

36 B.5. Delt method nd model verging B - 36 uncertinty by using model-verged vlues. There re no mjor complictions for ppliction of the Delt method to model-verged prmeter vlues you simply need to mke sure you use modelverged vlues for ech element of the clcultions. We ll demonstrte this using nlysis of the mle dipper dt (ed_mle.inp). Suppose tht we fit 2 cndidte models to these dt: {ϕ. p t } nd {ϕ f lood p t }. In other words, model where survivl is constnt over time, nd model where survivl is constrined to be function of binry flood vrible (see section 6.4 of Chpter 6). Here re the results of fitting these 2 models to the dt: As expected (bsed on the nlysis of these dt presented in Chpter 6), we see tht there is some evidence of model selection uncertinty the model where survivl is constnt over time hs roughly 2-3 times the weight s does the flood model : The model verged vlues for ech intervl re shown below: estimte SE Now, suppose we wnt to derive the best estimte of the probbility of survivl over (sy) the first 3 intervls. Clerly, ll we need to do is tke the product of the 3 model-verged vlues corresponding to the first 3 intervls: ( ) In other words, our best estimte of the probbility tht mle dipper would survive from the strt of the time series to the end of the third intervl is Wht bout the stndrd error of this product? Here, we use the Delt method. Recll tht: vr(ŷ) DΣD [ ] ( Ŷ) ( ˆθ) [ ] ( Ŷ), ( ˆθ) where Y is some liner or nonliner function of the prmeter estimtes ˆθ 1, ˆθ 2,... For this exmple, Y is the product of the survivl estimtes. So, the first thing we need to do is to generte the estimted vrince-covrince mtrix for the model verged survivl estimtes. This is esy enough to do in the Model Averging Prmeter Selection window, you simply need to Export Vrince-Covrince Mtrix to dbse file - you do this by checking the pproprite check box (lower-left, s shown t the top of the next pge).

37 B.6. Summry B - 37 The rounded vlues which would be output to the Notepd re shown below. (Remember, however, tht for the ctul clcultions, you wnt to use the full precision vrince-covrince mtrix from the exported dbse file.) Remember, however, tht for the ctul clcultions you need the full precision vrince-covrince mtrix from the exported dbse file. All tht remins is to substitute our model-verged estimtes for(i) ˆϕ nd (ii) the vrince-covrince mtrix (bove), into vr(ŷ) DΣD. Thus, vr(ŷ) [ ] (Ŷ) ( ˆθ) [ (Ŷ) ( ˆθ) ] [ ( ) ( ) ( ) ] vr( ˆϕ 1 ) ĉov( ˆϕ 1, ˆϕ 2 ) ĉov( ˆϕ 1, ˆϕ 3 ) ( ) ˆϕ 2 ˆϕ 3 ˆϕ 2 ˆϕ 3 ˆϕ 1 ˆϕ 3 ˆϕ 1 ˆϕ 2 ĉov( ˆϕ 1, ˆϕ 2 ) vr( ˆϕ 2 ) ĉov( ˆϕ 2, ˆϕ ( ) 3 ) ˆϕ 1 ˆϕ 3 ĉov( ˆϕ 3, ˆϕ 1 ) ĉov( ˆϕ 3, ˆϕ 2 ) vr( ˆϕ 3 ) ( ) ˆϕ 1 ˆϕ 2 [ ] B.6. Summry In this ppendix, we ve briefly introduced convenient, generlly strightforwrd method for deriving n estimte of the smpling vrince for trnsformtions of one or more vribles. Such trnsformtions re quite commonly encountered when using MARK, nd hving method to derive estimtes of the smpling vrinces is convenient. The most strightforwrd method bsed on first-order Tylor series

38 Addendum: computtionlly intensive pproches B - 38 expnsion is known generlly s the Delt method. However, first-order Tylor series pproximtion my not lwys be pproprite, especilly if the trnsformtion is highly non-liner, nd if there is significnt vrition in the dt. In such cse, you my hve to resort to higher-order pproximtions, or numericlly intensive bootstrpping pproches. Addendum: computtionlly intensive pproches At the strt of this ppendix, we motivted the Delt method s n pproch for deriving n estimte of the expecttion or vrince of function of one or more prmeters specificlly, n pproch tht ws not compute-intensive. While this pproch hs certin elegnce, ppliction to complex functions cn be cumbersome. Further, trnsformtions tht re strongly nonliner ner the mss of the dt my necessitte using higher-order Tylor series expnsion, which gin cn be complex for prticulr function. In such cses, it is useful to t lest be wre of lterntive, compute-intensive pproches. Here, we briefly introduce two different pproches, pplied to the estimtion of the vrince of the product of survivl estimtes, using the dipper exmple presented in section B.4. Agin, we ll use estimtes from model {ϕ t p. } fit to the mle Europen dipper dt set, nd gin, we ll suppose we re interested in the probbility of surviving from the strt of the first intervl to the end of the third intervl. As noted in section B.4, the estimte of this probbility is esy enough: Ŷ ( ˆϕ 1 ˆϕ 2 ˆϕ 3 ) ( ) So, the estimted probbility of mle Dipper surviving over the first three intervls is 14% (gin, ssuming tht our time-dependent survivl model is vlid model). i. using the Delt method... To derive the estimte of the vrince of the product using the Delt method, we require the vrincecovrince mtrix for the survivl estimtes: ĉov(ŷ) vr( ˆϕ 1 ) ĉov( ˆϕ 1, ˆϕ 2 ) ĉov( ˆϕ 1, ˆϕ 3 ) ĉov( ˆϕ 2, ˆϕ 1 ) vr( ˆϕ 2 ) ĉov( ˆϕ 2, ˆϕ 3 ) ĉov( ˆϕ 3, ˆϕ 1 ) ĉov( ˆϕ 3, ˆϕ 2 ) vr( ˆϕ 3 ) For this exmple, the trnsformtion we re pplying to our 3 survivl estimtes (which we ll cll Y) is the product of the estimtes (i.e., Ŷ ˆϕ 1 ˆϕ 2 ˆϕ 3 ).

39 Addendum: computtionlly intensive pproches B - 39 Thus, our vrince estimte is given s [( ) ( ) ( )] vr(ŷ) ( Ŷ) ( Ŷ) ( Ŷ) ˆϕ 1 ˆϕ 2 ˆϕ 3 ( ) (Ŷ) ˆϕ 1 ( ) ( Ŷ) ˆϕ. 2 ( ) ( Ŷ) ˆϕ 3 Ech of the prtil derivtives for Ŷ is esy enough to derive for this exmple. Since Ŷ ˆϕ 1 ˆϕ 2 ˆϕ 3, then Ŷ/ ˆϕ 1 ˆϕ 2 ˆϕ 3. And so on. Expnding the preceding results in: vr(ŷ) ˆϕ2 2 ˆϕ2 3 [ vr( ˆϕ 1 )]+ ˆϕ2 1 ˆϕ2 3 [ vr( ˆϕ 2 )]+ ˆϕ2 1 ˆϕ2 2 [ vr( ˆϕ 3 )] + 2 ˆϕ 2 ˆϕ 2 3 ˆϕ 1 [ĉov( ˆϕ 1, ˆϕ 2 )]+2 ˆϕ2 2 ˆϕ 3 ˆϕ 1 [ĉov( ˆϕ 1, ˆϕ 3 )]+2 ˆϕ2 1 ˆϕ 3 ˆϕ 2 [ĉov( ˆϕ 2, ˆϕ 3 )] After substituting in our estimtes forϕ i nd the vrinces nd covrinces, our estimte for the vrince of the product Ŷ ( ˆϕ 1 ˆϕ 2 ˆϕ 3 ) is (to first-order) vr(y) Now, we consider couple of compute-intensive pproches. ii. simultion from multivrite norml distribution... The bsic ide behind the pproch we illustrte here is strightforwrd: we (i) simulte dt s rndom drws from multivrite norml distribution with known mens nd vrince-covrince, (ii) generte the product of these rndom drws, nd (iii) derive numericl estimtes of the men (expecttion) nd vrince of these products. However, we need to be bit creful here. If we simulte the rndom norml drws on the rel probbility scle, then we run the risk of simulting rndom vlues which re not plusible, becuse they fll outside the[0, 1] intervl (e.g., you could simulte survivl probbility> 1, or< 0, neither of which re possible). To circumvent this problem, we simulte rndom norml vribles on the logit scle (i.e., logit-norml devites) using theβestimtes nd the vrince-covrince mtrix (both estimted on the logit scle), bck-trnsform the rndom devites from the logit rel probbility scle, nd then generte the product on the rel probbility scle. For the mle Dipper dt, the β estimtes using n identity design mtrix (such tht ech bet corresponds to the survivl estimte for tht intervl see Chpter 6 for specifics) re: ˆβ , ˆβ , ˆβ The vrince-covrince mtrix for theβestimtes (which cn be output from MARK) is: ĉov(ŷ) vr(ˆβ 1 ) ĉov(ˆβ 1, ˆβ 2 ) ĉov(ˆβ 1, ˆβ 3 ) ĉov(ˆβ 2, ˆβ 1 ) vr(ˆβ 2 ) ĉov(ˆβ 2, ˆβ 3 ) ĉov(ˆβ 3, ˆβ 1 ) ĉov(ˆβ 3, ˆβ 2 ) vr(ˆβ 3 ) An lternte pproch would be to simulte correlted rndom vlues drwn from bet distribution which is constrined on the intervl [0, 1], with shpe prmeters α nd β determined by estimted prmeters nd vrinces of those estimtes. Computtionlly, this cn be done by first generting stndrd norml vrites with the required covrince structure, nd then trnsforming them to bet vrites with the required men nd stndrd devition. See dos Sntos Dis et l

40 Addendum: computtionlly intensive pproches B The following R script uses the mvtnorm pckge to simulte the multivrite norml dt: # include librry to simulte correlted MV norm librry("mvtnorm") Now, we set up the prmeter vlues needed to specify the simultion: # number of smples to tke from MV norm iter < ; # dipper prmeter vlues nd vr-covr -- on the logit scle -- to use in simultion bet1 < ; bet2 < ; bet < ; vc <- mtrix(c( , , , , , , , , ),3,3,byrow=T); # generte rnnor smples conditionl bets nd VC mtric logit_smples = rmvnorm(iter,men=c(bet1,bet2,bet),sigm=vc,method="svd") Now, we do visul check to confirm our simulted vrince-covrince mtrix is close to the estimted mtrix (bove) which it is: # check to confirm simulted VC is correct... ct("simulted VC mtrix") print(round(cov(logit_smples),10)) simulted VC mtrix [,1] [,2] [,3] [1,] [2,] [3,] Then, we simply bck-trnsform our smples from the logit rel probbility scle, nd proceed from there. # convert logit smples to dt frme logit_smples <- s.dt.frme(logit_smples) # bck-trnsform from logit scle rel_smples <- exp(logit_smples)/(1+exp(logit_smples)); # generte the product of bck-trnsformed devites rel_smples$prod = rel_smples[,1]*rel_smples[,2]*rel_smples[,3];

41 Addendum: computtionlly intensive pproches B - 41 # summry stts ct("expecttion of product =", men(rel_smples$prod)) ct("vrince of product =", vr(rel_smples$prod)) Running this script results in the following estimtes, which re quite close to the expected product ( ), nd vrince of the product derived using the Delt method ( ): expecttion of product = vrince of product = iii. using MCMC... Another pproch mkes use of the Mrkov Chin Monte Crlo (MCMC) cpbilities in MARK. Here, we provide only brief description of the ide, nd mechnics for more complete discussion of the MCMC cpbilities in MARK, see Appendix E. The bsic ide is s follows. We ll fit model{ϕ t p. } to the mle Dipper dt, nd use MCMC to derive estimtes of the survivl nd encounter prmeters, bsed on estimted movements (men, medin, or mode), nd ssocited vrinces, from the posterior distribution for ech of the prmeters. The posterior distribution for ech prmeter is generted by Mrkov smpling over the joint probbility distribution for ll prmeters, given the dt. If we were using specilized MCMC ppliction, like JAGS, or BUGS, we could simply crete derivedprmeters function ofotherstructurlprmeters in the model(sy,prodϕ 1 ϕ 2 ϕ 3 ),nd then nlyze the posterior smples for this derived prmeter (this bility to explicitly code functions of prmeters is one of the rel conveniences of using MCMC in Byesin frmework). The MCMC cpbilities in MARK do not llow the explicit construction of user-specified derived prmeter. However, we cn ccomplish much the sme thing, lbeit in slightly more brute-force wy, but simply (i) tking the individul smple chins from the MCMC simultions for ech of the 3 prmeters (ϕ 1 ϕ 3 ), (ii) tking their product, nd (iii) evluting this product s the posterior distribution for the product (which it is). In fct, this is equivlent to wht JAGS or BUGS does, except tht insted of clculting the product of the survivl prmeters t ech step of the smpler, we simply do it post hoc fter the smplers re finished. OK, let s see how this is done. First, we fit model {ϕ t p. } to the mle Dipper dt. We ll use logit link (for resons discussed in Appendix E). Before submitting the model for numericl estimtion, we first check the MCMC Estimtion box: Once you click the OK to Run button, MARK will respond with window where you specify the

42 Addendum: computtionlly intensive pproches B - 42 MCMC prmeters tht will specify spects of the numericl estimtion (see Appendix E for complete discussion of these prmeters). Wht is generlly importnt is tht we wnt sufficient number of smples (t ll stges) to ensure tht the smplers hve converged on the sttionry joint distribution. For this exmple we ve used 7,000 tuning smples, 3,000 burn in smples, nd 100,000 smples from the posterior distribution. We ve lso specified only single chin, with no convergence dignostics. Once finished, MARK will output the results to the editor. If you scroll down to ner the bottom of the output listing, you ll see vrious mcro vlues tht cn be used for post-processing of the chins for ech prmeter. These mcro vlues re copied into SAS or R progrms tht re provided in the MARK helpfile. We ll demonstrte the mechnics using R. For the mle Dipper dt, nd model {ϕ t p. }, the R mcro vlues re: ncovs <- 7; # Number of bet estimtes nmens <- 0; # Number of men estimtes ndesigns <- 0; # Number of design mtrix estimtes nsigms <- 0; # Number of sigm estimtes nrhos <- 0; # Number of rho estimtes nlogit <- 7; # Number of rel estimtes filenme <- "C:\\USERS\\USER\\DESKTOP\\MCMC.BIN"; # pth MCMC.BIN file So, ll we do is copy this into the pproprite section t the top of the R script provided in the MARK helpfile. The script is firly lengthy, so we won t reproduce it in full here. Insted we ll focus on the dditionl steps you ll need to execute in order to derive n estimte of the vrince for the product of the first 3 survivl estimtes. First, copy the mcro vribles (bove) into the R script, nd execute it s is. This will crete n MCMC object, clled mcmcdt, tht is comptible with one of severl R pckges (e.g.,cod). This object contins ech of the individul Mrkov chins, for ech prmeter.