90 



Fishery Bulletin 101(1) 



distributional assumption (e.g. Maunder and Starr, 2001). 

 This constraint allows the estimation when there is no in- 

 formation (i.e. missing data). Traditional methods that re- 

 late recruitment to environmental factors use correlation 

 analysis of an environmental time series with estimates of 

 recruitment from a stock assessment model. For example, 

 cohort analysis is first used to generate a time series of re- 

 cruitment. Then the time series of recruitment is regressed 

 against sea-surface temperature (SST). This two-step pro- 

 cedure has a number of disadvantages (Maunder, 1998a, 

 2001a, 2001b), including the loss of information and the 

 difficulty of propagating uncertainty. 



We introduce a method suggested by Maunder (1998a; 

 see Maunder and Starr, 2001) that incorporates environ- 

 mental time series into stock assessment models and tests 

 the significance of the correlation between the population 

 processes and the environmental time series. We test the 

 model with simulated data and compare the results to 

 correlating model estimates with environmental vari- 

 ables outside the estimation procedure. We illustrate this 

 method with an application that investigates the correla- 

 tion between SST and recruitment within the context of 

 a statistical catch-at-age analyses used to assess snapper 

 (Pagrus auratus) in the Hauraki Gulf-Bay of Plenty, New 

 Zealand (Maunder and Starr, 2001). 



Materials and methods 



Integrating environmental indices into 

 stock assessment models 



Parameters that relate the environmental time series to 

 population processes were included in the statistical catch- 

 at-age stock assessment model. We added additional struc- 

 ture to the stock assessment model for each parameter of 

 the stock assessment model (X) that was hypothesized to 

 1) have temporal variation, 2) be correlated with an envi- 

 ronmental time series, and 3) have sufficient information 

 in the data to be estimated for multiple time periods. This 

 structure included a mean value for the stock assessment 

 model parameter (^i), temporal deviations in the stock 

 assessment model parameter (f,), a parameter that relates 

 the environmental series to the stock assessment model 

 parameter (/3), and a scaling parameter (a) that ensures 

 that ji is the mean value for the stock assessment model 

 parameter over the time period used in the model. 



X, = // exp (a + lili + e,). 



(1) 



where t = time, and 



/,= the value of the environmental time series at 

 time t. 



The parameter a ensures that ^ is equal to the mean over 

 the whole time period (Gilbert'; see Maunder and Starr, 



2001). Therefore, a removes the log normal bias and bias 

 caused by an unnormalized environmental time series 

 and is defined as 



a = In 



(2) 



where n is the number of time periods. 



The additional structure requires that a set of param- 

 eters (f,) that are constrained by a distributional assump- 

 tion and two free parameters (|j, /3) be estimated. The 

 distributional assumption (referred to as a "prior" in the 

 following description and represented by the negative 

 logarithm of the prior probability, see Eq. 3) is a prior on 

 the degree of temporal variation in the stock assessment 

 model parameter The default assumption is a normal 

 distribution (assuming that the stock assessment model 

 parameter is lognormally distributed) with mean zero and 

 given standard deviation. Information about this distribu- 

 tion can be obtained from estimates for similar species 

 (e.g. Myers et al., 1995). The prior 



-In Prior (f |cr) = V 



(£^ 



2ct- 



(3) 



' Gilbert, D. J. 1999. Personal cdiiirmin. National Institute 

 of Water and Atmospheric Research Limited. P.O. Box 14-901, 

 Wellington, New Zealand. 



keeps the temporal deviations close to zero if there is no 

 information in the data to the contrary. It is important 

 to note that the prior is also needed to avoid making /3 a 

 redundant parameter 



The parameters ^ and j5 and the set of parameters £, are 

 estimated simultaneously with the other parameters of 

 the stock assessment model, and the negative logarithm of 

 the prior is added to the negative log-likelihood function of 

 the stock assessment model. The parameter estimates are 

 really the mode of the posterior distribution, but we treat 

 them in a likelihood context. The influence of the environ- 

 mental time series can be removed from the analysis by 

 fixing /3 at zero. Therefore, likelihood ratio tests can be used 

 to determine if the /3 parameter significantly improves the 

 fit to the data. If the addition of /3 reduces the total negative 

 log likelihood by more than about 1.92 units (/-, ^^^y qc^ ), then 

 the additional parameter significantly improves the fit to 

 the data at the 0.05 level, and there is a statistically sig- 

 nificant correlation between the population process and the 

 environmental time series. Similar tests can be performed 

 to test the significance of the set of temporal deviation 

 parameters, f,, by taking into account the number of ad- 

 ditional parameters. Hilborn and Mangel ( 1997 ) provided a 

 simple description of the likelihood ratio test. (The Akaike 

 information criterion or the Bayes information criterion 

 could also be used. ) Therefore, by fixing, or not, /J or f at zero 

 we can define three types of statistical models; 



1 Traditional model 



/J is (ixi'd at zero, the parameter set f, is estimated, and 

 a significant relationship is determined by testing if 

 the correlation coefficient between f, and the environ- 

 mental time series is significantly different from zero. 



