An Improved Sibling Model for Forecasting Chum Salmon and Sockeye Salmon Abundance

North American Journal of Fisheries Management 27:634 642, 2007 Ó Copyright by the American Fisheries Society 2007 DOI: 10.1577/M06-094.1 [Management Brief] An Improved Sibling Model for Forecasting Chum Salmon and Sockeye Salmon Abundance STEVEN L. HAESEKER,* 1 BRIGITTE DORNER, RANDALL M. PETERMAN, AND ZHENMING SU 2 School of Resource and Environmental Management, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada Abstract. The sibling model is often one of the best methods for calculating preseason forecasts of adult return abundance (recruits) for populations of Pacific salmon Oncorhynchus spp. This model forecasts abundance of a given age-class for a given year based on the abundance of the previous age-class in the previous year. When sibling relations fit historical data well, the sibling model generally performs better than other forecasting methods, such as stock recruitment models. However, when sibling relations are weak, better forecasts are obtained by other models, such as naïve models that simply use an historical average. We evaluated the performance of a hybrid model that used quantitative criteria for switching between a sibling model and a naïve model when generating forecasts for 21 stocks of chum salmon O. keta and 37 stocks of sockeye salmon O. nerka in the northeastern Pacific Ocean. Compared with the standard sibling model, the hybrid model reduced the root mean square error (RMSE) of forecasts by an average of 27% for chum salmon stocks and 28% for sockeye salmon stocks. Compared with a naïve model, the hybrid model reduced the RMSE of forecasts by an average of 16% for chum salmon stocks and 15% for sockeye salmon stocks. Our results suggest that hybrid models can improve preseason forecasts and management of these two species. Fishery management agencies often generate preseason forecasts of returns of Pacific salmon Oncorhynchus spp. for initial guidance on the fishing regulations that may be needed to achieve spawner abundance (escapement) or harvest-rate goals. The fishing industry has also used these forecasts to regionally allocate harvesting and processing equipment and personnel. However, the predictive performance of preseason forecasts is often poor (Adkison and Peterman 2000; Haeseker et al. 2005), due in part to high interannual variation in survival and recruitment rates (Peterman 1987; Bradford 1995). As a result of forecasting error, * Corresponding author: steve_haeseker@fws.gov 1 Present address: U.S. Fish and Wildlife Service, Columbia River Fishery Program Office, 1211 Southeast Cardinal Court, Suite 100, Vancouver, Washington 98683, USA 2 Present address: Institute for Fisheries Research, 1109 North University Avenue, 212 Museums Annex Building, Ann Arbor, Michigan 48109-1084, USA Received March 24, 2006; accepted September 20, 2006 Published online April 26, 2007 management agencies may have difficulty achieving escapement goals or harvest-rate targets, and the fishing industry may experience economic losses (Bocking and Peterman 1988; Eggers 1993). The standard sibling model (Peterman 1982) has often demonstrated relatively good forecasting ability for sockeye salmon O. nerka and chum salmon O. keta (Bocking and Peterman 1988; Wood et al. 1997). Adults of both species return to their natal streams and rivers across two or more ages (e.g., as 4 5-year-olds), but the proportion at each age is not constant across years, thereby creating a challenge for forecasting total abundances. Furthermore, juvenile sockeye salmon rear in lakes for 1 or 2 years before migrating to the ocean; thus, the number of years spent in freshwater (i.e., before ocean entry) varies. Ages of sockeye salmon and chum salmon are designated as x.y, where x is the number of winters spent in freshwater and y is the number of winters spent in the ocean. The chum salmon stocks evaluated in this analysis mainly consisted of ages 0.2, 0.3, and 0.4 (Table 1), while the sockeye salmon stocks mainly consisted of ages 1.1, 1.2, 1.3, 2.1, 2.2, and 2.3 (Table 2). The sibling model forecasts the abundance of adult recruits of a given age returning in year t from the abundance of the previous adult age-class that returned in year t 1. By using data on such siblings from the same brood year that also spend the same number of winters in freshwater (but one less winter in salt water), the sibling model takes advantage of the covariation in response to critical environmental influences that these age-groups of fish share during their early life stages (Peterman et al. 1998; Pyper et al. 2002, 2005). Nevertheless, there are examples of naïve time series models outperforming sibling models (Noakes et al. 1990; Wood et al. 1997). In this context, naïve models are those that do not require statistical parameter estimation but rather simply summarize past observations to make forecasts. For example, naïve models could be based on average recruitment over the previous 5 years or the prior year s recruits. Recently, some biologists have adopted hybrid approaches that use both sibling and naïve models for forecasting the age-specific components of salmon returns (Eggers 2003). However, there is little guidance on when to use a sibling model, a naïve model, or a hybrid approach. 634

MANAGEMENT BRIEF 635 TABLE 1. Chum salmon stocks used in a comparison of recruit abundance models (abbreviations used in figures are in parentheses), number of years of data available for estimation (n), number of years forecasted (n f ), ages that could and could not be forecasted using a sibling model, and variance thresholds that achieved the minimum root mean squared error (RMSE min ) for each stock. Chum salmon stock n n f Sibling Ages Non-sibling Thresholds achieving RMSE min Washington Willapa (WIL) 27 16 0.3, 0.4 0.2 7.27 7.30 Grays Harbour (GRH) 24 11 0.3, 0.4 0.2 2.68 2.74 Skagit (SKA) 30 16 0.3, 0.4 0.2 0.00 1.06 Nooksack (NOK) 30 16 0.3, 0.4 0.2 0.44 0.88 Stillaguamish (STI) 30 16 0.3, 0.4 0.2 1.54 1.55 Hood Canal (HC) 30 16 0.3, 0.4 0.2 0.00 1.42 South sound fall (SSF) 30 16 0.3, 0.4 0.2 0.55 0.56 South sound early (SSE) 29 16 0.3, 0.4 0.2 0.00 1.46 South sound winter (SSW) 29 16 0.3, 0.4 0.2 0.49 0.50 British Columbia Fraser total (FTO) 35 24 0.3, 0.4 0.2 0.64 Fraser inner (FIN) 35 24 0.3, 0.4 0.2 1.14 10.00 Area 8 (A8) 21 11 0.3, 0.4 0.2 0.84 Area 6 (A6) 17 7 0.3, 0.4 0.2 0.92 1.01 Alaska Prince William Sound (PWS) 29 18 0.3, 0.4, 0.5 0.2 6.53 6.58 Nushagak (NUS) 21 8 0.3, 0.4, 0.5 0.2 0.73 Togiak (TOG) 19 5 0.3, 0.4, 0.5 0.2 0.00 0.28 Yukon (YUK) 22 10 0.3, 0.4, 0.5 0.2 0.30 Anvik (ANV) 22 10 0.3, 0.4, 0.5 0.2 0.51 0.58 Andreafsky (ADR) 24 12 0.3, 0.4, 0.5 0.2 0.84 10.00 Kwiniuk (KWI) 31 19 0.3, 0.4, 0.5 0.2 0.74 Kotzebue (KOT) 19 8 0.3, 0.4, 0.5 0.2 0.29 10.00 To improve the accuracy of preseason forecasts for chum salmon and sockeye salmon, we retrospectively evaluated the performance of a hybrid model that used the sibling model when sibling relations were strong (small residual variation) and a naïve model when sibling relations were weak (large residual variation). We developed quantitative criteria for determining whether to use a sibling model or a naïve model for generating a forecast in a particular year, stock, and age-stanza. Finally, we compared the performance of this hybrid sibling model with that of both the standard sibling model and a naïve model to determine whether the hybrid model improves upon existing methods for forecasting chum salmon and sockeye salmon returns. Methods Data. We analyzed previously compiled data on age-specific recruitment of 21 chum salmon stocks and 37 sockeye salmon stocks in the northeast Pacific Ocean, distributed from northwestern Alaska to northwestern Washington and including British Columbia (Tables 1, 2). Abundances of recruits included both catch and spawning escapement of both sexes. The time series ranged in duration from 17 to 50 years; the average was 26 years for chum salmon stocks within the period 1962 1998 and 43 years for sockeye salmon stocks within the period 1951 2001. Further details on sources and derivation of data can be found in Peterman et al. (1998), Mueter et al. (2002), and Pyper et al. (2002). Forecasting models. The sibling model (Peterman 1982) assumes a linear relation between raw abundances of two sibling groups in log space, log e ðr d;t Þ¼a þ blog e ðr c;t 1 Þþe t ; ð1þ where R d,t is the abundance of the later-returning siblings of age d in year t, R c,t 1 is the abundance of the earlier-returning siblings of age c in year t 1 (where c ¼ d 1), a and b are estimated parameters, and e t ;N(0, r 2 e ). This form of error term is based on the frequently observed multiplicative lognormal variation of marine survival rates of salmon (Peterman 1981). Because of the occasional presence of zeros in the agespecific return data, a constant of 1 was added to the R d,t and R c,t 1 observations to allow for logarithms. The linear regressions were fit using maximum likelihood estimation, minimizing the error sum of squares. We did our analyses for both chum salmon and sockeye salmon using data series aligned such that the later and earlier returns (R d,t and R c,t 1 ) spent the same number of winters in freshwater but the earlier returns spent one winter less in salt water and thus returned 1 year earlier. For instance, one sibling model for sockeye salmon reflected the relation between

636 HAESEKER ET AL. TABLE 2. Sockeye salmon stocks used in a comparison of recruit abundance models (abbreviations used in figures are in parentheses), number of years of data available for estimation (n), number of years forecasted (n f ), ages that could and could not be forecasted using a sibling model, and variance thresholds that achieved the minimum root mean squared error (RMSE min ) for each stock. Sockeye salmon stock n n f Sibling Ages Non-sibling Thresholds achieving RMSE min Washington Lake Washington (LWA) 27 15 1.2, 1.3 1.1 1.42 British Columbia Adams (ADA) 50 37 1.2, 1.3, 2.2 1.1, 2.1 0.88 Birkenhead (BIR) 50 37 1.2, 1.3, 2.2, 2.3 1.1, 2.1 0.79 0.80 Bowron (BOW) 50 37 1.2, 1.3 1.1 0.43 Chilko (CHL) 50 37 1.2, 1.3, 2.2, 2.3 1.1, 2.1 4.14 4.31 Cultus (CUL) 50 37 1.2, 1.3, 2.2 1.1, 2.1 0.54 Gates (GAT) 50 35 1.2, 1.3 1.1 2.57 2.58 Horsefly (HOR) 50 37 1.2, 1.3 1.1 8.31 8.41 Nadina (NAD) 50 37 1.2, 1.3 1.1 2.50 2.51 Pitt (PIT) 50 37 1.2, 1.3 1.1 0.96 10.00 Portage (POR) 50 33 1.2, 1.3 1.1 2.98 Raft (RAF) 50 37 1.2, 1.3 1.1 0.00 0.19 Seymour (SEY) 50 37 1.2, 1.3 1.1 0.00 0.86 Stellako (STO) 50 37 1.2, 1.3 1.1 0.44 Stuart early (STE) 50 37 1.2, 1.3 1.1 1.18 Stuart late (STL) 50 37 1.2, 1.3 1.1 3.32 Weaver (WEA) 50 37 1.2, 1.3 1.1 0.46 0.65 Long Lake (LOL) 21 8 1.3 1.2, 2.3, 3.1 0.47 10.00 Skeena (SKE) 46 26 1.2, 1.3 1.1 0.38 Nass (NAS) 26 15 1.3, 2.3 1.2, 2.2 0.48 0.49 Alaska Copper (COP) 35 23 1.2, 1.3, 1.4, 2.2, 2.3 0.3, 1.1, 2.1 0.69 Cook (COK) 31 15 1.2, 1.3, 1.4, 2.2, 2.3 0.3, 1.1, 2.1 0.58 Ayakulik (AYA) 30 19 1.2, 1.3, 2.2, 2.3 0.3, 1.1, 2.1, 3.2 2.36 2.41 Frazer (FRA) 30 19 1.2, 1.3, 2.2, 2.3 1.1, 2.1 1.86 1.99 Early Upper Station (EUS) 26 15 1.2, 1.3, 2.2, 2.3 0.3, 1.1, 2.1 0.48 0.49 Late Upper Station (LUS) 26 14 1.2, 1.3, 2.2, 2.3 0.3, 1.1, 2.1 0.62 Black (BLA) 48 34 1.2, 1.3, 1.4, 2.2, 2.3 1.1, 2.1 0.88 Chignik (CHI) 48 34 1.2, 1.3, 1.4, 2.2, 2.3 1.1, 2.1 3.39 3.49 Branch (BRA) 45 30 1.2, 1.3, 2.2, 2.3 1.1, 2.1 2.21 2.30 Egegik (EGE) 45 30 1.2, 1.3, 1.4, 2.2, 2.3 0.3, 1.1, 2.1, 3.2 9.24 9.26 Igushik (IGU) 45 30 1.3, 1.4, 2.3 0.3, 1.2, 2.2 2.42 2.57 Kvichak (KVI) 45 30 1.2, 1.3, 2.2, 2.3 1.1, 2.1 4.18 4.27 Naknek (NAK) 45 30 1.2, 1.3, 1.4, 2.2, 2.3 1.1, 2.1 1.05 10.00 Nuyakuk (NUY) 32 17 1.3, 1.4, 2.3 0.3, 1.2, 2.2 2.41 10.00 Togiak (TOG) 45 30 1.3, 2.3 0.3, 1.2, 2.2 0.54 Ugashik (UGA) 45 30 1.2, 1.3, 1.4, 2.2, 2.3 0.3, 1.1, 2.1 2.36 2.38 Wood (WOD) 45 30 1.2, 1.3, 1.4, 2.2, 2.3 0.3, 1.1, 2.1 6.73 6.75 abundances in ages 1.2 and 1.3. The model was parameterized using data from numerous brood years for which both sibling groups had already returned and for which abundances were estimated. Generating forecasts of adult recruits (^R d;t ) based on equation (1) required back-transformation to estimate the forecasted mean number of recruits on the arithmetic scale. Accounting for the well-known bias associated with back-transforming lognormally distributed variables (Hayes et al. 1994), forecasts were generated using the equation ^R d;t ¼ exp ^R d;t þ ^r2 e 2 ; ð2þ where ^R d;t is the forecast of the mean number of age-d recruits in year t, Rˆ d,t is the estimate of R d,t and ^r 2 e is the variance of the residuals estimated from fitting equation (1). The sibling model can be used to make forecasts for older age-classes that have younger siblings, but other methods must be used to forecast abundance of ageclasses for which no data exist on abundance of younger siblings. For example, if a stock is composed of ages 1.1, 1.2, and 2.1, a sibling model can be used to forecast the abundance of age-1.2 returns based on the age-1.1 returns from the previous year, but other methods would have to be used to estimate the abundance of age-1.1 and age-2.1 returns. To forecast the total annual abundance of returns, some method is

MANAGEMENT BRIEF 637 necessary to generate a forecast for all age-classes in a given return year. In practice, stock recruit relations and naïve time series models (e.g., long-term averages) have been used to forecast those age-classes that cannot be forecasted using sibling models (Wood et al. 1997; Eggers 2003). To forecast such age-classes, we used a naïve time series model referred to as R(yr 4): R d;yr ¼ R d;yr 4 þ w yr ; ð3þ where R d,yr is the abundance of age-d fish returning in year yr, R d,yr 4 is the abundance of age-d fish in year yr 4, and w yr ;N(0, r 2 w ). In an initial exploratory analysis that compared retrospective model performance for various naïve models, the R(yr 4) model typically matched or outperformed other naïve models. For sockeye salmon, the root mean square error (RMSE) of the forecasts was on average 29% greater for the R(yr 3) model than the R(yr 4) model and 28% greater for the R(yr 5) model than the R(yr 4) model. For chum salmon, the RMSE of the forecasts was on average 12% greater for the R(yr 3) model than the R(yr 4) model and 23% greater for the R(yr 5) model than the R(yr 4) model. Because this model does not assume lognormal errors, no backtransformation (e.g., equation 2) was required when making forecasts. In our preliminary research that used retrospective analyses (described below), we observed that the sibling model had smaller forecasting errors than the R(yr 4) model when the estimated sibling relations were strong (i.e., characterized by high coefficient of determination [R 2 ] values and low mean square error or residual variance, ^r 2 e ). However, when sibling relations were weak, the R(yr 4) model often performed better than the standard sibling model. These observations provided the rationale for developing the hybrid model that used a sibling model when the sibling relation was strong for a particular stock and age-stanza but used a naïve R(yr 4) model when the sibling relation was weak. Our derivation of the hybrid sibling model involved determining the optimal quantitative criteria for strong sibling relations (i.e., when to choose one model over the other). We used retrospective analysis to evaluate performance of the standard sibling model, the hybrid sibling model, and the naïve R(yr 4) model. In this retrospective analysis, we attempted to simulate the implementation of the modeling approaches as if they had been implemented historically. That is, only the data that would have been available to make a forecast for some past year were used to estimate model parameters and generate the forecast of total adult returns across ages for that year. Each sibling regression based initial parameter estimates on the first 10 years of data. By sequentially adding a year to the data set used for parameter estimation, generating a forecast, and comparing the forecast with the observed value, our retrospective analysis produced a time series of forecasting errors for each model. This method produced out-of-sample forecasts, thereby providing a rigorous assessment of each forecasting model s performance as if it had been used historically. We calculated the variance of the residuals, ^r 2 e (in units of log e [fish] 2 ), for the log e log e relation in equation (1) and assessed model performance based on the RMSE of the forecasts of total adult recruits. One advantage of using RMSE as the performance metric is that errors are expressed in the same units as the response variable (i.e., in numbers of fish). The number of years for which abundances were forecasted varied among stocks, ranging from 5 to 37 years (average ¼ 14 years for chum salmon stocks, 29 years for sockeye salmon stocks; Tables 1, 2). In total, 295 stock-years were forecasted for chum salmon and 1,080 stock-years were forecasted for sockeye salmon. For the hybrid sibling model, we used ^r 2 e estimated for each year and age-stanza during the retrospective analysis to determine whether to use the naïve R(yr 4) model or the sibling model to make the age-specific forecasts for the subsequent year. If the most recent estimated residual variance for an age-stanza was below a threshold, then the sibling model was used to generate the forecast of abundance for that stanza s predicted age-class in the next year. If the residual variance estimate was above that threshold, then the naïve R(yr 4) model was used to generate the forecast. For each stock, we examined thresholds for ^r 2 e ranging from 0 to 10 in increments of 0.01 and calculated the RMSE of the resulting forecasts associated with each threshold. The same variance threshold was used for all age-stanzas for a given stock. For example, if (1) the residual variance for the 1.2 versus 1.3 age-stanza in a given year of the retrospective analysis was 0.8 and the residual variance for the 1.3 versus 1.4 age-stanza was 1.7 and (2) we were evaluating a residual variance threshold of 1.5, then the sibling model would be used to forecast the abundance of age-1.3 fish and the naïve R(yr 4) model would be used to forecast the age-1.4 fish abundance for that particular stock and year. We defined the optimum threshold for each stock as that which minimized the RMSE of forecasts for that stock. In contrast, we used two criteria to determine which threshold for residual variance was optimal across all stocks within a species for sockeye salmon and chum salmon. We developed these criteria in an effort to derive a forecasting method that could be applicable

638 HAESEKER ET AL. FIGURE 1. The sum of relative root mean square errors (RMSEs; solid line) and the number of stocks (count) within 10% of their minimum RMSEs (dashed line) across candidates for the variance threshold in a hybrid sibling model of chum salmon recruit abundance. The vertical dashed line denotes the resulting optimum variance threshold, ^r 2 e. generally to sockeye salmon and chum salmon and that would not be limited to only those stocks considered in this analysis. First, for each stock, we calculated the relative RMSE as follows: Relative RMSE i ¼ RMSE i =RMSE min ; ð4þ where RMSE i is the RMSE of the hybrid sibling model using threshold value i and RMSE min is the minimum RMSE across all threshold values explored for that stock. To determine the appropriate residual variance threshold across stocks within each species, we summed the relative RMSE values across all stocks. For a given species, the best threshold was defined as the one that minimized the sum of the relative RMSE values across stocks within a species. Because the profiles of this relative RMSE sum across thresholds did not always have well-defined minima (Figures 1, 2), we developed a second criterion that counted the number of stocks for which the RMSE i was within 10% of the RMSE min. The threshold that maximized this count was judged to be best for this criterion. Results For chum salmon stocks, the sum of relative RMSEs was minimized at variance thresholds of 1.08 and 1.09, but it was nearly as low for a wide range of candidate thresholds from 0.5 to 7.5 (Figure 1). The count of stocks within 10% of their stock-specific RMSE min was maximized with candidate thresholds of 1.08 1.16 (Figure 1). Therefore, we judged 1.08 and 1.09 to be appropriate thresholds for use in the hybrid sibling model across chum salmon stocks because both objective functions were met with these values. We also determined the stock-specific optimum thresholds (i.e., those candidate thresholds that, when applied to all age-stanzas, achieved RMSE min ; Table 1). In some stocks, a wide range of thresholds gave almost the same RMSE min. For sockeye salmon stocks, the minimum sum of relative RMSEs was better defined; candidate variance thresholds of 0.7 3.0 resulted in the lowest values (Figure 2). While the minimum was achieved with a candidate threshold of 2.41, values from 2.40 to 2.54 were within 1% of the minimum. The count of stocks within 10% of their stock-specific RMSE min was maximized with candidate thresholds of 2.31 2.34, 2.38 2.39, and 2.53 2.54 (Figure 2). Because the threshold of 2.53 maximized the count of stocks within 10% of RMSE min and also achieved a near-minimal sum of relative RMSEs, we judged it to be an appropriate threshold for use in the hybrid sibling model across sockeye salmon stocks. However, stockspecific optimum thresholds for sockeye salmon varied considerably among stocks (Table 2). Using general thresholds of 1.09 for chum salmon and 2.53 for sockeye salmon, we found that switching between the sibling model and the naïve R(yr 4) model during the time series occurred in 16% of the age-stanzas for sockeye salmon and 18% of the agestanzas for chum salmon. Thus, for the majority of age-

MANAGEMENT BRIEF 639 FIGURE 2. The sum of relative root mean square errors (RMSEs; solid line) and the number of stocks (count) within 10% of their minimum RMSEs (dashed line) across candidates for the variance threshold in a hybrid sibling model of sockeye salmon recruit abundance. The vertical dashed line denotes the resulting optimum variance threshold, ^r 2 e. stanzas (84% for sockeye salmon, 82% for chum salmon), the model chosen at the onset of the time series was used throughout the time series. Therefore, the thresholds generally determine at the onset whether a sibling model or a naïve model should be used for a particular age-stanza, although switching between models during the time series based on changes in the residual variance does occur. We compared the RMSEs for four cases: the standard sibling model, the hybrid sibling model using the general across-stock thresholds as determined above, the hybrid sibling model using the stockspecific optimum thresholds (Tables 1, 2), and the naïve R(yr 4) model. The general hybrid sibling model with a threshold of 1.09 for chum salmon and 2.53 for sockeye salmon performed the same as or better than (i.e., had a lower RMSE) the standard sibling model in 20 of 21 chum salmon stocks (Figure 3) and 33 of 37 sockeye salmon stocks (Figures 4, 5). On average, the general hybrid sibling model reduced the RMSE by 27% for chum salmon stocks and 28% for sockeye salmon stocks relative to the standard sibling model. The naïve R(yr 4) model had an average RMSE reduction of 10% compared with the standard sibling model for chum salmon stocks and a reduction of 9% compared with the standard sibling model for sockeye salmon stocks. The naïve R(yr 4) model performed as well as or better than the standard sibling model in 13 of 21 chum salmon stocks and 19 of 37 sockeye salmon stocks. Compared with the naïve R(yr 4) model, the general hybrid sibling model reduced the RMSE by 16% on average for chum salmon and by 15% for sockeye salmon; the general hybrid model performed as well as or better than the naïve R(yr 4) model in 18 of 21 chum salmon stocks and 30 of 37 sockeye salmon stocks. Finally, use of the hybrid model with stock-specific optimal thresholds (rather than the across-stock optimal values) improved the hybrid model RMSE by a further 5% on average for chum salmon and 8% for sockeye salmon. Discussion We have proposed quantitative criteria for an algorithm to decide for each year whether to use a sibling model rather than a naïve model to generate preseason forecasts when data to estimate sibling relations are available. Our findings suggest that there are cases when sibling relations are uninformative for generating forecasts of a given age-class due to the model s large previous residual variance; in those situations, it may be prudent to use a naïve time series model to forecast that age-class. In several chum salmon and sockeye salmon stocks, much of this unexplained variance was due to estimates of zero returns for particular age-classes. However, our findings also suggest that there are cases when the sibling relations are highly informative and should therefore be used. Using our criteria for thresholds in residual variance of the general sibling model, the RMSE of forecasts from the hybrid sibling model was

640 HAESEKER ET AL. FIGURE 3. Root mean square error (RMSE) values for the naïve model, general hybrid sibling model, and stock-specific hybrid sibling model of chum salmon recruit abundance; each RMSE value is divided by the RMSE for the standard sibling model. The horizontal line at 1.0 represents the RMSE for the standard sibling model; therefore, models with values below that line made better forecasts. less than or equal to that of the standard sibling model for 53 of the 58 stocks analyzed. The RMSE of the naïve R(yr 4) model was lower than that of the general hybrid sibling model in only 10 of the 58 stocks analyzed. These results suggest that there is little chance of degrading forecasts by adopting the general hybrid sibling model instead of the standard sibling model or the naïve R(yr 4) model. FIGURE 4. Root mean square error (RMSE) values for the naïve model, general hybrid sibling model, and stock-specific hybrid sibling model of sockeye salmon recruit abundance in stocks from Lake Washington and British Columbia (see Table 2 for stock code descriptions); each RMSE value is divided by the RMSE for the standard sibling model. The horizontal line at 1.0 represents the RMSE for the standard sibling model; therefore, models with values below that line made better forecasts.

MANAGEMENT BRIEF 641 FIGURE 5. Root mean square error (RMSE) values for the naïve model, general hybrid sibling model, and stock-specific hybrid sibling model of sockeye salmon recruit abundance in Alaska (see Table 2 for stock code descriptions); each RMSE value is divided by the RMSE for the standard sibling model. The horizontal line at 1.0 represents the RMSE for the standard sibling model; therefore, models with values below that line made better forecasts. Although additional reduction in RMSE can be achieved by using stock-specific optimal thresholds within the hybrid sibling model for the stocks listed, the average improvement in RMSE over the speciesspecific hybrid sibling model is only 5 8% on average. This result supports the view that the optimal acrossstock variance thresholds determined here (1.09 for chum salmon; 2.53 for sockeye salmon) could be used to generate forecasts of abundance for stocks not included in this analysis and that those species-specific thresholds represent a reasonable trade-off between generality across stocks in the northeastern Pacific Ocean and accuracy for individual stocks. However, if it is feasible, stock-specific values should be derived by analysts for their stocks of interest. We also investigated using optimized age-stanza-specific variance thresholds but found little improvement in precision of forecasts over the across-stanza variance thresholds. Variability and uncertainty in escapement, reproduction, freshwater survival, and marine survival all contribute to the difficulty of making accurate preseason forecasts of salmon returns (Fried and Yuen 1987). Forecasts based on stock recruitment models are typically fitted from data on escapement and adult returns, which include all of these sources of error. The advantage of sibling models (Alaska Department of Fish and Game 1981; Peterman 1982; Bocking and Peterman 1988) is that many of these sources of error are avoided by using age-specific return data in successive years; the relative abundances of different adult age-classes have already been largely determined by the demographic factors listed above during periods when siblings that mature at different ages share common environmental conditions. The only exception is marine survival during the last year of life. Despite these intuitive reasons why standard sibling models should perform well, they do not always outperform stock recruitment-based forecasting models or naïve models, perhaps due to inaccuracies in the age-specific return data, inherently weak sibling relations, or temporally changing parameters of component relations (Noakes et al. 1990; Wood et al. 1997; Holt and Peterman 2004). In such cases, naïve time series models (e.g., moving averages or lagged returns) can outperform sibling models as well as other forecasting methods (Noakes et al. 1990; Haeseker et al. 2005; S.L.H., unpublished data). Because the hybrid sibling model examined here includes both types of models, it can perform better across stocks and agestanzas than either model alone. While the hybrid sibling model presented shows some promise for improving the accuracy of preseason forecasts for chum salmon and sockeye salmon, considerable uncertainty remains. Recognizing this

642 HAESEKER ET AL. uncertainty, a precautionary approach to managing chum and sockeye salmon stocks is warranted (FAO 1995). Implementing a precautionary approach would involve explicit accounting of the uncertain components of a fishery system, including the large uncertainties in preseason forecasts of abundance, when evaluating fishing regulation options. Appropriate adjustments to early-season fishing plans can be determined through consideration of such uncertainties by means of risk assessments or decision analyses (e.g., Robb and Peterman 1998). Acknowledgments We thank the numerous biologists and technicians in various management agencies for gathering and processing the lengthy time series data used here and providing them to us. This research was supported by the Natural Sciences and Engineering Research Council of Canada. References Adkison, M. D., and R. M. Peterman. 2000. Predictability of Bristol Bay, Alaska, sockeye salmon returns one to four years in the future. 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