6 ERROR MODELS AND ESTIMATION

Transcription

1 6 ERROR MODELS AD ESTIMATIO Model specfcaton usuall ncludes a descrpton of the behavour of the mean n the dstrbuton and a random effect or error. Ths error s the dfference between the el epected value and the erved value, and can be classfed as: Structural error. The estmaton procedure uses structural relatons between the populatons at dfferent tmes. These relatons wll onl be appromatel correct. For nstance, CPUE s unlel to be precsel proportonal to abundance, although ths el maes estmaton easer. Measurement error. Ths error s not onl caused b the samplng gear, (e.g. orgnatng from varaton n weather condtons durng surves), but also b patch dstrbutons of the populaton. Process error. Random effects ma also affect underlng dnamcs els. For nstance, random weather and oceanographc changes ma ncrease or decrease natural mortalt and recrutment n an unpredctable wa. The dfference between process and measurement error s that process error ntroduces a real change n the sstem, whereas measurement errors ntroduce no underlng change and therefore do not affect future ervatons. 6. LIKELIHOOD In fttng els to data, the els descrbng the stoc and ervatons cease to become descrptons of the process n ther own rght, but nstead become descrptons of how parameters n probablt els change. For eample, n man applcatons a el s used to descrbe the parameter n the normal dstrbuton, whch also happens to be the mean. These probablt els descrbe how lel the erved data are, gven the parameters. The lelhood concept smpl turns ths on ts head. Lelhood s the probablt that a set of parameters s correct gven the data. Ths maes no substantve dfference to the probablt dstrbuton, but conceptuall underpns most crtera for fttng els to data. For eample, mamum lelhood defnes a set of parameters when the lelhood functon reaches ts mamum pont, and a Baesan estmator uses lelhood, along wth pror probabltes and a cost functon, to defne a set of parameters where the epected cost s mnmsed. Although usng lelhood (.e. a full probablt el) s theoretcall better, leastsquares s often used n the analss of fsheres data. Ths can be ustfed b the followng ponts: least-squares s mamum lelhood where the probablt dstrbuton s the ormal. true lelhoods can rarel be specfed wth an certant. Least-squares wth some approprate transformaton s probabl as good as an alternatve wthout 34

2 more nformaton on error structure. Therefore, a more comple procedure ma not be ustfed. mamum lelhood for man parametrc lelhoods (e.g. Posson etc.) can be reformulated n terms of least-squares. least-squares numercal procedures are much smpler and less lel to brea down than more general approaches, partcularl where there are man parameters to estmate. The fact that least-squares are relatvel eas to use, adaptable, obectve and ma often be close to the mamum lelhood soluton has resulted n ts wde use. evertheless, there s ncreasng nterest n alternatve approaches that are theoretcall more appealng and for whch robust numercal methods are beng developed. However, even f other more complcated methods are used to ft els, n most cases least-squares can stll form the startng pont of the analss, so the methods dscussed here are alwas lel to be of use. 6. LEAST-SQUARES ESTIMATIO Least-squares estmaton s based on the same prncple as curve fttng. Gven a set of ervatons, defne a el and then establsh the set of the parameters that gves the best ft of ths el to the erved data. The goodness of ft s usuall the sum of squared dfferences between erved and calculated dependent varables (.e. squared resduals). The sum of squares (SSQ) s also sometmes called the Eucldean norm. The best ft s the set of parameters that mnmses the squared dfference between the ervatons and the el s epected values. There s no guarantee that the best ft el s correct, and the el ma be entrel nadequate for reflectng the dnamcs. Analses of a data set therefore should alwas nclude comparson between ervatons and the ftted el for nspecton. Ths s often done graphcall b plottng the resduals aganst the ndependent varables or b plottng on the same graph both the ervatons and the ftted curve. The resduals should show a random scatter and should not ehbt an remanng pattern. Mamum lelhood estmaton (e.g. Lehman 983) ncludes the same elements as a least-squares estmaton, onl the goodness of ft measure wll often dffer based on the eplct assumpton of the form of the error el. The elements n least-squares estmaton are: The el: Model( ) (47) where s the vector of parameters, and s the dfference (error) between the ervatons and value calculated from the el. In practce ths means all uncertant s treated as ervaton error whether created b measurement nose, el ms-specfcaton or otherwse. The mean of the erved quantt can be defned as: 35

3 Model( estmated ) Model( ) E (48) Therefore, the mean error becomes: E{} = 0. Where the epected error s not zero, ths s often referred to as bas. For the goodness of ft measure, the sum-of-squares, s used: Model( ) L (49) The parameters n the el are chosen so that ths sum-of-squares s at ts mnmum. Ths goodness of ft measure s often etended to account for the ervatons havng dfferent varances. In ths case, the goodness-of-ft measure becomes: L Model( ) where data subset contans all ervatons wth the same varance. A least-squares estmator wll produce mamum lelhood estmates and confdence ntervals f =(0, ), that s normall dstrbuted wth mean 0 and varance. It wll even produce mamum lelhood estmates f the varance s constant over the range of eplanator varables or the proportonal change n varance s nown n a weghted least-squares scheme. However, f the true error dstrbuton s not smmetrcal, the varance changes n an unnown manner, or there are process or structural errors (as there almost certanl alwas are), the estmates wll not be mamum lelhood. It has been found wth fsheres data that least-squares b tself provdes a poor ft. For ths reason, t s a common practce to use transformatons to appromate alternatve dstrbutons. The transformatons form part of the ln el and often are used to represent alternatve error dstrbutons besdes the normal. Ellott (983) suggests the followng transformatons for stablsng the varance: (50) Error dstrbutons Observaton Transformaton Log-normal : contnuous ln () Posson : dscrete / Bnomal h: frequenc arcsn(h) General frequenc dstrbuton h: frequenc ln( h/(-h) ) Talor epanson : dscrete 36

4 The most commonl assumed error s the log-normal, whch s dealt wth b tang logarthms of the data and then assumng that errors are ormall dstrbuted. The use of the lognormal mght be theoretcall ustfed n some nstances. For eample, consder a cohort beng subect to random survval rates between egg release and recrutment: M t R t R0 e (5) If the mortalt s made up of a large sum of small random effects (M ), the fnal total mortalt, b the Central Lmt Theorem, wll be normall dstrbuted even f the ndvdual random components are not. Hence, ths wll result n a lognormal dstrbuton. As well as the practcal ervaton that els ftted to log transformed data ft the data better, the log-normal has several other advantages: Tang logs often maes errors smmetrcal around the mean. The ormal dstrbuton does not dscrmnate aganst negatve values, so for eample, t allows for negatve populatons of fsh, whch are clearl mpossble. In effect, ths produces a bas towards larger ervatons n the analss. The log-normal assumes negatve values are mpossble and corrects ths bas. As a result, note that the eponent of the log-normal parameter, ep(), s not the same as the arthmetc mean, but lower as s the mean of the log values. The arthmetc mean wll depend on both the log-mean () and the varance. The log-normal does not assume a constant error varance, but assumes the varance ncreases wth the arthmetc mean. Agan, ths has generall been erved n fsheres data, where ncreasng catches and effort tends to produce greater varablt. The log-normal corrects estmates for ths effect. evertheless, the man argument for the log-normal remans pragmatc. It represents the erved error dstrbuton better than would the normal, producng better estmates. However, ou should alwas chec whch transformaton f an s approprate for our data b eamnng el resduals. The fact that a procedure s wdel used s not a ustfcaton for ts use n an partcular case. The el for whch we want to estmate the parameters, provdes the mean n the unnown probablt dstrbuton. Hence, for eample, CPUE mght be elled as: ln CPUE = ln q + ln P + (5) where = measurement error. Based on the above el we epect that the mean value, the logarthmc mean CPUE over man statons randoml spread out n the surve area, wll be a lnear functon of the populaton P and that ths mean value can be measured wthout bas. The estmaton equaton becomes: 37

5 a ln Catcha ln Catcha a ln Cpue ln Cpue... MIel parameters a a (53) where subscrpt = abundance nde, a = age and = ear of the ervaton. Most often there are several tunng data seres avalable. A basc feature of the ADAPTIVE framewor (Chapter 8) s to sum these ndvdual contrbutons as n Equaton 53. It s the researcher s responsblt to buld the estmaton equatons relevant for each ndvdual stoc assessment. 6.. Weghts When there s more than one tunng tme seres avalable the data of the dfferent seres are usuall not obtaned wth the same measurement varance. In ths case, t s preferable to ntroduce a weghtng of the data seres, b specfng a weght parameter,, for each data seres. In theor, these weghts should be the nverse of the varance of the measurements. Estmates of the true varances are often avalable from abundance surve data, but the are more dffcult to estmate for data from the commercal fsheres. However, t s unnecessar to obtan the absolute weghts, onl relatve weghtng wth respect to some prmar data seres. In least-squares theor, the varances can be estmated from the sum-of-squares as: X X ( n p) where n s the number of ervatons and p s the number of parameters n the el (e.g. Lehmann 983). It s not possble to estmate weghts wthn the estmaton procedure, onl once the el s ftted. Ths s llustrated b a smple sstem wth two tunng data seres CPUE and CPUE : mo d ln CPUE ln CPUE ln CPUE ln CPUE MI (54) (55) Ths sum-of-squares clearl has a mnmum for = 0 ( 0), as ths elmnates the second contrbuton to the sum of squares. Therefore, weghts need to be treated as eternal varables estmated through some other means. Etended Survvor Analss (Darb and Flatman 994) ncludes an nternal weghtng procedure, treatng each age group and each data seres separatel. These weghts are ntroduced n a double teraton nherent n the method. Detals of ths procedure are dscussed n Secton 8.3. All data tpes above can be ntegrated n the combned estmaton least-squares epresson: 38

6 ln Catcha ln Catcha aage ear CPUE ln CPUEa Abundance ndces aage ear bomass ln I ln I bomass ndces ear effort ndces effort nde ear ln CPUE a ln E ln E MIel parameters (56) where are the weghts appled to data seres. 6.3 FIDIG THE LEAST-SQUARES SOLUTIO Fndng the least-squares soluton s the common problem of fndng the mnmum for a functon. Model ) MI ( (57) Ths problem s converted nto an equvalent problem of solvng a set of smultaneous equatons. In an functon a mnmum occurs where the partal dfferentals of the parameters are equal to zero, so: f MI wth respect to the set of 's s equvalent to f ( ) 0 for,,... In least-squares, the functon f s the functon (the sum-of-squares), and the parameters are the parameters of the el. An numercal routne could be used to fnd a soluton, and good robust routnes est. Wh not ust use the canned, blacbo routnes avalable n man software pacages? The smple answer s no reason n man cases, and as long as the researcher checs such routnes have successfull found the mnmum, the are recommended. However, n some cases the are not adequate, partcularl when the number of parameters s ver large. Faster, more relable and more accurate methods ma be developed for a problem b consderng the numercal soluton ourself. Wth large numbers of parameters, the -dmensonal parameter space can become ver comple. Routnes wrtten for general functons can mae no assumptons about those functons. The therefore tend to crawl around the parameter space ver slowl to avod oversteppng the mnmum. Ths ma stll not avod mssng the mnmum and can tae nordnate amounts of tme. Routnes to fnd the least-squares mnmum tae advantage of attrbutes of the functon, ncreasng the chance of success and the speed at whch the mnmum s found. 39

7 Canned blac bo routnes are also wdel avalable for fndng least-squares, so wh s the detal of methodolog gven here? The reason s largel the same. The researcher can tae advantage of ther nowledge of the functon (.e. the stoc assessment el) to ncrease the chance of success and speed of the method. An canned routne would treat the stoc assessment el as a sngle functon. However, a researcher wll often see how the functon could be broen down nto smpler components, each amenable to smpler analss. As wll be seen, ths approach s used n XSA, where a smple lnear regresson to estmate parameters of the el lnng the populaton to the CPUE nde, so these parameters can be solved b a sngle calculaton. The parameters belongng to the more comple nonlnear el stll need to be found through teraton, but the number of parameters has been greatl reduced Lnear Models On the face of t, lnear els would be of lttle use n stoc assessment as most realstc populaton els are non-lnear. However, there are often lnear components, whch can be estmated separatel. The advantages of dealng wth lnear parts separatel s purel pragmatc. Lnear parameters can be found b calculaton rather than teratve numercal procedures, whch speeds up el fttng. Where the el s lnear, the least-squares equatons are lnear as well and can be solved drectl through calculaton. The soluton s obtaned b solvng the M smultaneous lnear equatons, where M s the number of parameters or ndependent varables. The soluton of lnear smultaneous equatons s subect to standard lnear algebra technques. Assumng equal varances, we wsh to fnd the soluton to a set of M equatons: L 0 where L M (58) and there are M parameters and data ponts. The set of dfferental equatons can be found easl for a lnear el: L M L 0 (59) 40

8 4 So, now we have M equatons n terms of the and data varables and the parameters {}, each equaton equal to zero. These are rearranged sutable for a matr format: M M M M M M M M (60) All the terms n Equaton 59 now appear n Equaton 60, but arranged as matrces. The soluton for s conceptuall smple. We multpl both sdes of Equaton 60 b the nverted matr appearng on the left-hand sde, solatng the {} vector. Rentroducng the varances for completeness, the soluton can be wrtten: M (6) The s the varance assocated wth each data pont, and often s assumed equal among data ponts. Ths matr equaton allows the least-squares estmate to be obtaned n one teraton. The advantage of lnear els should now be apparent, and occurs because dfferentaton elmnates the parameter from the equaton, enablng the lneart of the equatons to be mantaned. Ths allows an eas soluton. Wth non-lnear els, the set of Equatons 59 wll not be lnear, and therefore no smple soluton ests. In the smplest case wth onl one parameter, Equaton 6 becomes: (6) If the are constant among data ponts ths equaton becomes the sum of the product of the and varables dvded b the sum of squares of the varable. Ths result s often ver useful n estmatng parameters n els lnng erved varables to underlng populaton dnamcs varables. For eample, consder the case where we have generated a populaton tme seres from a el, and we wsh to relate t to a CPUE nde whch requres estmatng a sngle parameter q, as: t t qp CPUE (63)

9 Assumng least-squares, constant varance and tme ndependent errors, q can be calculated as: T CPUEt Pt t q T Pt t (64) Ths avods the need to estmate q as part of the mnmsaton process. Instead q s calculated each ccle, and the numercal routne concentrates on solvng the nonlnear parameters assocated wth the populaton el. A smlar smple procedure can be undertaen wth two parameters. In ths case, we tr to ft the logarthm of the CPUE to the log populaton sze, assumng a non-lnear relatonshp: v CPUEt qpt ln CPUEt ln q v ln Pt (65) In ths case, the parameter v s the slope, but we also have an ntercept parameter (ln q). In the lnear framewor, constant parameters (non-covarates are often called factors) are estmated usng dumm varables. Dumm varables tae on values of or 0 dependng on whether the parameter apples to an partcular ervaton or not. In ths case, the constant apples to all ervatons, so the frst varable s alwas : t t t where t ln CPUEt t, t ln Pt ln q, v (66) Solvng for and usng the general Equaton 6 gves: SS SS SS SS SS SS SS SS (67) where 4

10 T S t T t T S t t T T S S t t t t t T T S t t t t t T S t t t (68) otce that the subscrpts refer to the elements of the matrces and vectors n Equaton 60. S s the element n the frst row and frst column of the matr on the left-hand sde, and S s the element n the frst row of the vector on the rght-hand sde. For two parameters, the matr nverson s ver smple and t s possble to wrte out the result n a smple equaton as above. Ths smplct rapdl dsappears wth larger matrces. Inverson s closel related to calculatng determnants, whch s a sum nvolvng all row-column combnatons of elements. For large matrces these calculatons are not trval and ma tae a consderable amount of tme, although the method remans faster (or at least more eact) than ts non-lnear cousn. A second problem wth nvertng the matr, s t ma be sngular. Ths ma occur through alasng, or hgh correlatons pushng the nverson calculatons beond the computer s precson. An alternatve soluton to removng the offendng parts of the el s to use matr transformaton technques, notabl Sngular Value Decomposton (SVD). These technques do not produce dfferent results, merel srt around sngulart problems (Press et al. 989). However, a detaled descrpton falls beond the scope of ths manual. Ths technque, of estmatng lnear parameters separate from the teratve ft, s wdel used, n XSA for eample. The benefts should not be underestmated. Searches for the mnmum of non-lnear functons s not trval where there are large numbers of parameters and an method that reduces ths number should be used on-lnear Models A usual approach to fndng the mnmum s the ewton teraton scheme: new where H old L H L( old ) (69) 43

11 {H} - s the nverse matr of second partal dervatves of the sum of squares wth respect to parameter pars, often called the Hessan matr. As n the lnear case, the am s to fnd the pont where the smultaneous partal dervatve equatons are zero. Ths approach to mnmsaton wors on the prncple that the step length movng parameters towards the zero pont should be rato of the dfferental to the slope of the dfferental (.e. the second dervatve) at the current poston, whch wll produce a the correct step length where the el s lnear. On each teraton, {} new s generated and becomes the {} old for the net ccle, so eventuall {} new converges to {} old and the teratons can stop. The startng pont for {} s mportant, but reasonable estmates are often avalable n VPA applcatons (e.g. F=0.5 ear - ). It s onl for a few problems when the second order dervatves are actuall evaluated analtcall. Instead computer-orented methods are based on numercal appromatons to the frst and the second order dervatves, whch are based on calculatons of the functon at small departures (h), e.g. L( ) L( h ) L( ) h (70) defnes the frst dervatve and L L( h, h ) L( h, ) L(, h ) L(, ) H hh (7) defnes the second dervatve and could therefore be used to calculate the Hessan matr, {H}. More sophstcated methods are usuall used as these calculatons ma not be accurate (see Abramowch and Stegun 966). An actual applcaton wll ver often use standard mplementatons (see Press et al. 989), whch wor wth all but the most ll-behaved functons. However, we can tae partcular advantage of what s nown about the least-squares functon to mprove both the speed and chance of success n fndng the mnmum. The mnmsaton problem s frst converted nto the normal equatons. Because the sum of squares s at a mnmum pont, we now that: Model( ) Model( ) 0 for,,..., p L (7) Lewse, eplct dfferentaton to produce the Hessan terms gves: L Model Model Model Model (73) 44

12 The frst term n Equaton 73 contanng the second partal dervatve s generall gnored n estmatng the Hessan matr for two reasons. Frstl, the second dervatves are often small compared to the frst dervatves (the are zero n lnear els for eample), so ther ncluson ma not mprove the effcenc of the fttng. Secondl, n practce the frst term wll sum to a small value when Model() estmates are close to the epected value of the (.e. the mean). Therefore, the procedure ma be most effcent when the ntal estmates are reasonabl close to the best-ft estmates. The fttng process now becomes: new old Model Model Model (74) In some cases, the Hessan n ths form ma be eas to obtan analtcall. For eample, notce that where the el s lnear, the Hessan matr s the same as that n Equaton 60. Usng the true dfferentals should mprove the effcenc of the ft. In other cases, Equaton 74 wll not help as the frst dervatve s ust too complcated to derve and smple numercal methods (e.g. Equaton 70) are nstead used to generate both the Hessan matr and vector of frst dervatves. The scheme proposed n Press et al. (989) s the Levenberg-Marquardt method, whch uses ether the Hessan matr or a smple step routne where the Hessan s a poor appromaton to the shape of the functon. Although ths approach should be used n man cases, t ma well stll be worthwhle eplorng the smultaneous partal dfferental equatons and Hessan matr. Whle t ma not be worthwhle pursung the analtcal approach, some analss ma help n understandng the behavour of the el and potental ptfalls n attemptng to fnd the least-squares soluton numercall. 6.4 ESTIMABLE PARAMETERS Whle t s possble to formulate a least-squares functon for an el t does not follow that all parameters can be estmated. Ths can be nherent n the el formulaton or t can be because of a lac of suffcent nformaton. An eample of a el that cannot be full dentfed s where parameters multpl or add together n a wa that cannot be separated b the data collected, such as Model()=, where onl the product of the two parameters can be estmated. In fsher bolog, an eample s the populaton equaton: e M a F a a, a (75) The equaton contans such a problem n parameters F a and M a unless data can be brought to bear to estmate these parameters separatel. It s ths basc problem that eplans the mnmum data requrement for an analtcal assessment. To separate the two, the catch n numbers b age and b ear combned wth ervatons on ether the fshng mortalt or the stoc n numbers are requred. Usuall M a s ust fed as an eternal parameter. 45

13 It s not onl the el structure that maes certan parameters nestmable. The data structure can also have features that prevent the estmaton of all parameters. Ths s the collneart problem, ndcated b hgh parameter correlaton estmates. In etreme cases, parameters ma be alased whch mples the data are nadequate to provde separate parameter estmates. A smple eample where such correlaton occurs s n estmatng fshng power based on vessel characterstcs. Most characterstcs are dctated b vessel sze. So the sze of net, vessel speed, hold sze, number of crew, sophstcaton of gear all relate bac to the sze of vessel. In essence, because we do not have ervatons on catch rates of large vessels wth small engnes or small vessels wth large engnes, t s not possble to separate the effects of engne sze and vessel length. What appears to be a large amount of data, all the dfferent characterstcs of the fleet, bols down to ver lttle real nformaton to separate vessels. Methods such as prncple components analss should be used to reduce a large number of correlated varables nto a few representatve uncorrelated components for ths tpe of analss. A more worrng eample for stoc assessment s the possble relatonshp between stoc sze and catchablt. Vessels aggregate n areas where catchablt s hghest. Fsh aggregate to mprove spawnng success and mnmse ther natural mortalt. There are several cases where t s suspected that as the populaton decreases, fsh denst on the fshng grounds remans constant, so effectvel catchablt s ncreasng as the populaton falls. Whle correlatons n lnear els are relatvel straghtforward, t s much more complcated n non-lnear els such as those used n fsh stoc assessment. It s not clear how termnal cohort szes mght be correlated wth catchablt estmates for CPUE ndces before dong a full analss. Statstcal epermental desgn ensures that such collneart does not occur n epermental data. However n fsheres or oceanographc surves, the researcher does not have the same degree of control over the sstem under nvestgaton and such data, because of the oceanographc or bologcal lns occurrng n nature, often show some degree of correlaton between the ndependent varables. 6.5 ROBUST REGRESSIO An alternatve approach to least-squares s to appl robust regresson (e.g. see Chen and Palohemo 995). The least-squares ft s based on mnmsng the squared sum of resduals and ths sum can be strongl dependent on a few outlers (cf. the eample above). Robust regresson ests n dfferent forms, but s based on ether replacng the sum of squares of the resduals b some other measure of goodness of ft, (e.g. the medan) or gnorng a certan percentage of the largest resduals n the fttng procedure (trmmed LSQ). Usng the medan, the least-squares problem s reformulated to fndng the best curve where 50% of the ervatons have postve and 50% negatve resduals. Obvousl the magntude of the resduals s of no mportance and therefore outlers have less nfluence on the fnal result than when normal least-squares s appled. The approach can be formulated based on fttng a el as: 46

14 Medan Model,,,... (76) 6.6 CATCH In man applcatons catch errors are ether gnored (e.g. n most ADAPT and XSA methods) or the errors are assumed to be log-normal (e.g. n the ICA or n the CAGEA methods). The reason for gnorng these errors n the catch data s that the stochastc error n the catch data s often nsgnfcant compared to the nose n the surve data. Ths s probabl correct n man fsh stoc assessments, but onl for the more abundant age groups. The number of old fsh caught, f consttutng onl a few percent of the total catch, s unlel to be precsel estmated. Methot (990) suggested as part of hs Snthetc Model that the error structure of the catch data be decomposed nto two contrbutons: Estmate of the overall catch n weght Estmate of the age composton The frst contrbuton can be assumed to have lognormal errors. For the second contrbuton, Methot (990) suggests that a multnomal dstrbuton s more approprate. The estmaton of the catch n numbers, C, s often obtaned through a fsheres statstcs programme that provdes total landngs b speces and b tme perod supplemented b a bologcal samplng programme that taes a length sample (n l ) and an age-length e (m la ) (ALK). The estmaton of the catch-at-age for the populaton el s: nl mla C a C (77) n l ml where the dot subscrpt ndcates summaton over that subscrpt (Lew and Lassen 997). However, where the erved and epected catch s ncluded as part of the sum-ofsquares, we can use the age composton ervatons drectl. In the smple case, the age dstrbuton s a random sample of the catches wth erved frequences n numbers of fsh, h a. The catch composton n the el s gven b: a where Ca C Ca a C a a (78) Therefore a s essentall ndependent of the total catch. In ths case, the catch term contrbuton: 47

15 a ln ln C C a a (79) wth the multnomal lelhood becomes: lnc lnc h ln... MIel parameters a a (80) a, Fortunatel, because onl the age sample, but not the catches, appear wth the a parameters, the can be estmated ndependentl of the catch data and populaton el b fndng the mamum of the multnomal lelhood functon. Combnng length samplng wth ALK gves a smlar result, but the formulae are more complcated (Lew and Lassen 997). The varance can be found as: Var( Ca ) Var( C ) Var( ha ) Ca C ha (8) for the smple stuaton when the age sample s a random sample of the catch. If t can be assumed that the varance contrbuton from the total landngs can be neglected compared to the error due to ageng, and f the ageng error can be appromated b a multnomal dstrbuton, then ths can be smplfed to: Ca Ca Ca Ca C Var C ( a ) n C C C n C a C (8) where n s the number of fsh n the sample. 48