Kope and Botsford: Recruitment of Oncorhynchus tshawytscha in central California 



259 



the Pacific have been related to precipitation in Califor- 

 nia, which could in turn influence flow rates. Markham 

 and McLain (1977) found a positive correlation between 

 California rainfall and sea surface temperature off the 

 coast of Washington. However, rainfall in California 

 is only weakly related statistically to equatorial condi- 

 tions associated with ENSO events (Douglas and 

 Englehart 1981, Cayan and Peterson 1989, Redmond 

 1988). 



Environmental influences on fish populations are 

 often explored by computing correlations between en- 

 vironmental and population time-series; however, these 

 are computed and the significance of resulting correla- 

 tions are evaluated in a variety of ways. In attempting 

 to detect environmental influences by computing corre- 

 lations, the probability of obtaining a significant result 

 is often artificially inflated by two problems: (1) intra- 

 series correlation and (2) multiple tests (cf. Walters and 

 Collie 1988). The latter arises from the fact that many 

 environmental series can be tested for correlation with 

 the population series; hence the probability of falsely 

 identifying a significant relationship is greater than the 

 probability level of each test. Methods exist for deal- 

 ing with both problems, but we focus on the former 

 here [see Hollowed et al. (1987) and Drinkwater and 

 Myers (1987), for examples of dealing with the latter]. 

 The former, intraseries correlation, is a lack of statis- 

 tical independence of observations, which implies that 

 the true number of degrees of freedom that should be 

 used in determining the significance of a computed cor- 

 relation is less than the number of points in the series. 

 There are two steps that can be taken to deal with this 

 problem: (1) removing as much of the intraseries cor- 

 relation as possible, and (2) accounting for the remain- 

 ing correlation in the computation of confidence limits. 



Removal of intraseries correlation can be accom- 

 plished by arbitrarily "pre-whitening" the series (i.e., 

 filtering the series so that correlation between adja- 

 cent points is removed) (Box and Jenkins 1976, Fogarty 

 1988), but is more directly justified when there is a 

 physical basis for doing so and the new series repre- 

 sents a meaningful quantity. In the case of an abun- 

 dance or catch series, one can compute a recruitment 

 time-series from the original series using deconvolu- 

 tion (Kope and Botsford 1988). Deconvolution is a pro- 

 cedure based on the observation that catch or abun- 

 dance is a sum over age-classes which is essentially a 

 weighted sum over past recruitment (i.e., a moving 

 average), in which the weighting factors are the 

 (assumed constant) proliabilities of survival to each age. 

 If the abundance or catch series is established using 

 a size limit, rather than an age limit, the numbers at 

 each age that are larger than the size limit must also 

 be accounted for in the weighting factors. Deconvolu- 

 tion inverts this summing process to give recruitment 



in terms of catches or abundances. Deconvolution pro- 

 vides two advantages over using the catch or abun- 

 dance series without deconvolution: (1) it removes the 

 intraseries correlation due to summing ages in abun- 

 dance or catch data, and hence provides more conser- 

 vative estimates of the significance of correlations 

 between recruitment and the environment; and (2) it 

 yields an estimate of recruitment from the abundance 

 or catch data, hence it can also increase the magnitude 

 of estimated correlation coefficients. However, decon- 

 volution cannot yield an exact value of recruitment 

 when noise is present in the abundance or catch data. 

 The error present in the recruitment estimates depends 

 on the error present in the catch or abundance series 

 and on the stability of the deconvolution (Kope and 

 Botsford 1988). Thus, there is a trade-off between the 

 two advantages of removing the effects of multiple age- 

 classes and the disadvantage of amplifying errors that 

 depends on the stability of the deconvolution and the 

 magnitude of measurement errors in abundance. Con- 

 sequently, deconvolution may not always be useful. 



Intraseries correlation is accounted for in the com- 

 putation of confidence limits by estimating the variance 

 of the computed correlation coefficient in a way that 

 accounts for the intraseries correlation. This often in- 

 volves using one of the many approximations to an ex- 

 pression originally due to Bartlett (1946) (Box and 

 Jenkins 1976, Chelton 1984, Botsford 1986). This ap- 

 proach is functionally equivalent to adjusting the num- 

 ber of degrees of freedom using a variation of the 

 expression originally due to Bayley and Hammersley 

 (1946) (Sutcliffe et al. 1976, Drinkwater and Myers 

 1987). We use another method for dealing with this 

 problem that leads to an actual probability of detection 

 that is closer to the specified probability than other 

 methods (Kope and Botsford 1988, Botsford and Wain- 

 wright unpubl.). 



In spite of the fact that greater intraseries correla- 

 tion artificially inflates the probability of detecting a 

 significant relationship, some researchers have added 

 intraseries correlation to the time-series before com- 

 puting correlations. For example, in the freshwater 

 Chinook salmon studies cited above, Dettman et al. 

 (1986) and Reisenbichler (1986) used 2-year moving 

 averages of both spawning escapement and environ- 

 mental variables in an effort to reduce the effects of 

 age structure in the spawning escapement data. Hol- 

 lowed et al. (1987) used five-point moving averages in 

 their analysis of recruitment patterns in the northeast 

 Pacific. Because the effects of intraseries correlation 

 were not accounted for in either of these studies, the 

 use of moving averages would decrease the probability 

 of detecting real correlations and increase the probabil- 

 ity of spurious correlation (Kope and Botsford 1988). 

 Others have used moving averages before computing 



