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<br />2000, age-3 in 1999, or age-2 in 1998. However, as the model
<br />is currently formulated, all age-6 fish recaptured in 2002 are
<br />pooled for a single observation. Assuming that the age- and
<br />time-specific captures of marked and umnarked fish LIfe Pois-
<br />son distributed, the log-likelihood, ignoring terms involving
<br />only the data, is computed as:
<br />
<br />InL(Olm,r)= fI[-lilaJ + mil.! In(~iltl,l)]
<br />
<br />a~.t I.d
<br />
<br />j T
<br />+ II[-~1.I +01.1 InV~".I)]'
<br />
<br />,,~.l 1~2
<br />
<br />where ma., is the observed number of age-a, unmarked fish
<br />'captured in year t, In"" is the predicted number of unmarked
<br />fish captured, r f is the observed number of marked fish cap-
<br />tured (i,e" recaptures), fa.! is the predicted number of marked
<br />fish captured. and ~} is the parameter vector to be estimated.
<br />Notice in the second term that the individual log-likelihood
<br />terms are summed over age and time. However, it may be
<br />more informative to stratify the recapture data by tagging
<br />cohort. The proposed log-likelihood is then:
<br />
<br />. ..l T .
<br />InL(elm,r)= II[- /il".1 + ma.lln(/ilaJ)]
<br />
<br />,,=11=1
<br />
<br />.1 T T~l
<br />+ III[-~>,I'( +t;u.( In(!:u.JJ.
<br />
<br />a~I [:::;2 t~'~l
<br />
<br />where c is the tag cohort (i.e.., all tish marked in year t). In
<br />principle, this modified log-likelihood should provide addi-
<br />tional information on time-specific capture probability and
<br />may improve parameter estimation.
<br />
<br />Evaluating Model Fit
<br />
<br />Following Baillargeon and Rivest (2007), standardized
<br />Pearson residuals of ohserved and predicted age composi-
<br />tion for hoth unmarked and marked fish were used to evalu-
<br />ate model fit among the three different ASMR models. The
<br />standardized Pearson residual is the difference between the
<br />observed and predicted values scaled by an estimate of the
<br />standard deviation as:
<br />
<br />r :::::
<br /><:::tJ
<br />
<br />O"J-P",r
<br />
<br />
<br />where Ilt is the number of observations (e.g., the number of
<br />marked tish recaptured each year) and oa,t and pa,f are the
<br />proportions of fish in each ye:rr and age dass observed and
<br />
<br />5
<br />
<br />(1)
<br />
<br />predicted, respectively. The individual Pearson residuals for
<br />each combination of age and time were plotted to look for
<br />consistent bias for individual brood-year cohorts. In addition,
<br />quantile-quantile (Q-Q) plots were used to compare the distli-
<br />bution of the Pearson residuals to a theoretical normal distri-
<br />bution. The intercept of the theoretical curve is approximately
<br />the standard deviation of the distribution of Pearson residuals,
<br />where a small value of the intercept indicates a narrow distri-
<br />bution of the residuals. Deviations from the theoretical curve
<br />indicate a non-normal distribution of the Pearson residuals and
<br />imply that the model error is not well distributed (e.g., tending
<br />to more often either over- or under-predict age proportions)
<br />and possihly inducing bias in parameter estimates.
<br />In addition to examination of model iit using Pearson
<br />residuals, information theory was also used to aid in model
<br />evaluation. The use of this approach is increasingly common
<br />in ecological studies to arbitrate an1(mg competing models
<br />and is primarily concerned with estimating the Kullhack-
<br />Leibler (K-L) distance between the model and the "u'uth" a~
<br />a measure of model support (Burnhmn and Andersen, 20(2).
<br />The Akaike information criterion (AIC; Akaike, 1973) is the
<br />standard estimator for the relative K-L distance and is com-
<br />puted as a function of model likelihood and number of model
<br />pm-ameters. Following review of the ASMR method in 2003
<br />(Kitchell and others, 2(03), it was pointed put that although
<br />ASMR uses a quasi-likelihood structure of estimating equa-
<br />tions and true likelihood. estimates of relative K-L distance
<br />using Ale. though not strictly appropriate, would be useful for
<br />model arbitration (c. SchwLlftz, Simon Fraser University, wIit-
<br />tcn commun., 20(3). Therefore, in addition to the evaluation
<br />based on Pearson residuals, an AlC evaluation was conducted.
<br />
<br />(2)
<br />
<br />Section 2-Estimating the Humpback Chub
<br />Growth Function Using Mark-Recapture Data
<br />
<br />Capture-recapture data have long been used by biolo-
<br />gists in an attempt to characterize growth rates of fish. Thc
<br />basic technique for estimating growth model parameters from
<br />capture-recapture data is to predict the amount of growth in
<br />the elapsed time between capture and recapture. Assuming
<br />the standard van Bertalanffy growth curve (Bertalanffy, J 938)
<br />predictions oflength at time t and at time r+/lr, Fabens (1965)
<br />developed the most basic model where the predicted growth
<br />increment is given as:
<br />
<br />(4)
<br />
<br />(3)
<br />
<br />AI. =:; L(t+At)- L(t) =:; (Lfr(; -L(t)XI-e-k\l),
<br />
<br />where L", and k are the asymptotic length and the rate at
<br />which length approaches Le<,' respectively (Quinn and Deriso,
<br />1999). Parameter estimates are found by minimizing the dif-
<br />ference between predicted and observed growth increments.
<br />Though this technique has been widely applied, numer-
<br />ous authors have pointed out that resulting parameter estimates
<br />will be biased if individual fish exhibit growth variability (e.g.,
<br />
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