89 



Abstract— We present a method to 

 integrate environmental time series 

 into stock assessment models and to 

 test the significance of correlations 

 between population processes and the 

 environmental time series. Param- 

 eters that relate the environmental 

 time series to population processes 

 are included in the stock assessment 

 model, and likelihood ratio tests are 

 used to determine if the parameters 

 improve the fit to the data significantly. 

 Two approaches are considered to 

 integrate the environmental relation- 

 ship. In the environmental model, 

 the population dynamics process (e.g. 

 recruitment) is proportional to the 

 environmental variable, whereas in 

 the environmental model with pro- 

 cess error it is proportional to the 

 environmental variable, but the model 

 allows an additional temporal varia- 

 tion (process error) constrained by a 

 log-normal distribution. The methods 

 are tested by using simulation analy- 

 sis and compared to the traditional 

 method of correlating model estimates 

 with environmental variables out- 

 side the estimation procedure. In the 

 traditional method, the estimates of 

 recruitment were provided by a model 

 that allowed the recruitment only to 

 have a temporal variation constrained 

 by a log-normal distribution. We illus- 

 trate the methods by applying them to 

 test the statistical significance of the 

 correlation between sea-surface tem- 

 perature (SST) and recruitment to the 

 snapper (Pagrus auratus) stock in the 

 Hauraki Gulf-Bay of Plenty, New Zea- 

 land. Simulation analyses indicated 

 that the integrated approach with 

 additional process error is superior to 

 the traditional method of correlating 

 model estimates with environmental 

 variables outside the estimation pro- 

 cedure. The results suggest that, for 

 the snapper stock, recruitment is posi- 

 tively correlated with SST at the time 

 of spawning. 



A general framework for integrating 

 environmental time series into 

 stock assessment models: model description, 

 simulation testing, and example 



Mark N. Maunder 



George M. Watters 



Inter-Amencan Tropical Tuna Commission 



Scnpps Institution of Oceanography 



8604 La Jolla Shores Drive 



La Jolla, California 92037-1508 



E-mail address (for M N Maunder) mmaunderfgiiattcorg 



Manuscript accepted 20 September 2002. 

 Fish. Bull. 101:89-99 (2003). 



Identifying a clear relationship between 

 an environmental variable and pro- 

 cesses in the dynamics of the population 

 (recruitment, natural mortality, growth) 

 or the fishery (catchability) would al- 

 low improved estimation and prediction 

 of model parameters and derived quan- 

 tities. It is well known that the environ- 

 ment plays a large role in the population 

 dynamics and catchability offish stocks. 

 Many researchers (Green, 1967; Joseph 

 and Miller, 1989; Hinton and Nakano, 

 1996; Lehodey et al., 1997; Shepherd 

 et al., 1984) have identified correlations 

 between population processes and envi- 

 ronmental factors, and others (Hunter, 

 1983; Bertignac et al., 1998; Lehodey 

 et al., 1998) have suggested hypotheses 

 for the underlying causes of these cor- 

 relations. Incorporation of environmen- 

 tal time series into stock assessment 

 models may provide additional informa- 

 tion to help estimate model parameters, 

 particularly when fishing observations 

 (catch, effort, length-frequencies) are 

 missing. For the management of fish 

 stocks, it can be advantageous to be able 

 to predict future catch rates and popula- 

 tion sizes. Because there is often a delay 

 due to the propagation of the recruit- 

 ment signal in the population structure 

 or because statistical and numerical 

 models can provide predictions for some 

 environmental variables (e.g. tempera- 

 ture) (or for both reasons), the relation- 

 ship can be used to predict future catch 

 rates or population sizes. 



Statistical catch-at-age analysis (e.g. 

 Fournier and Archibald, 1982; De- 

 riso et al, 1985; Methot, 1990) is more 

 appropriate than cohort analysis (vir- 



tual population analysis) to include 

 relationships between an environ- 

 mental variable and processes in the 

 dynamics of the population. In cohort 

 analysis, if there are missing data, they 

 are simply extrapolated without any 

 statistical methods, which may cause 

 bias in the parameter estimates. Also, 

 the potential correlation with an envi- 

 ronmental series is calculated outside 

 of the estimation procedure, producing 

 several disadvantages, including the 

 loss of information and the difficulty 

 of propagating uncertainty (Maunder, 

 1998a, 2001a, 2001b). However, in sta- 

 tistical catch-at-age analysis, there are 

 robust statistical methods (maximum 

 likelihood, with all the parameters esti- 

 mated together by obtaining the best fit 

 between predicted and observed data) 

 that allow inclusion of multiple data 

 sets and the integration of the environ- 

 mental series into the stock assessment 

 model. These methods automatically al- 

 low for missing data and provide confi- 

 dence inteivals, and the hypotheses can 

 be easily incorporated and tested. 



The methods used to integrate the 

 environmental series into the stock as- 

 sessment model can be applied to differ- 

 ent processes in the population, but are 

 illustrated here with the case of recruit- 

 ment. Recruitment is the fundamental 

 process in the population dynamic that 

 is responsible for the fluctuations of the 

 stock size. Many studies (e.g. Francis, 

 1993) show that environmental vari- 

 ables affect the recruitment. In statisti- 

 cal catch-at-age analysis, recruitment 

 combines an average value with an an- 

 nual deviate, constrained by using a 



