Cod and climate: effect of the North Atlantic Oscillation on recruitment in the North Atlantic

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1 Ths appendx accompanes the artcle Cod and clmate: effect of the North Atlantc Oscllaton on recrutment n the North Atlantc Lef Chrstan Stge 1, Ger Ottersen 2,3, Keth Brander 3, Kung-Sk Chan 4, Nls Chr. Stenseth 1,5,6, * 1 Centre for Ecologcal and Evolutonary Synthess (CEES), Department of Bology, Unversty of Oslo, PO Box 1066 Blndern, 0316 Oslo, Norway 2 Insttute of Marne Research, PO Box 1870, 5817 Bergen, Norway 3 Internatonal Councl for the Exploraton of the Sea, Palægade 2-4, 1261 Copenhagen K, Denmark 4 Department of Statstcs and Actuaral Scence, Unversty of Iowa, 263 Schaeffer Hall, Iowa Cty, Iowa 52242, USA 5 Department of Coastal Zone Studes, Insttute of Marne Research, Flødevgen Marne Research Staton, 4817 Hs, Norway 6 Present address: Centre for Ecologcal and Evolutonary Synthess (CEES), Department of Bology, Unversty of Oslo, PO Box 1066 Blndern, 0316 Oslo, Norway *Correspondng author. Emal: n.c.stenseth@bo.uo.no Marne Ecology Progress Seres 325: (2006) The appendx ncludes some supplementary text on the robustness of the obtaned results and on the methodology used, plots showng tme seres of spawnng stock bomass (SSB) and recrutment for each stock (Fg. A1), resdual dagnostcs (Fgs. A2-A3), combned effects of SSB and year (Fgs. A4-A5), nteracton effects between SSB and clmate (Fg. A6), and results f SSB effects were set to zero for 7 stocks not dsplayng any R-SSB relatonshp (Fgs. A7-A9), and tables wth results when auto-regressve and year effects are excluded from the model (Table A1) or SSB effects were set to zero for 7 stocks not dsplayng any R-SSB relatonshp (Table A2). SUPPLEMENTARY TEXT Robustness of results to the ncluson of SSB effects for stocks for whch no detectable assocaton between recrutment and SSB exsts It has been argued that usng recrutment per spawnng stock bomass (R/SSB) as response when no sgnfcant relaton between R and SSB exsts may mask possble recrutment-envronment relatonshps (Cardnale & Hjelm 2006). Although we used log e (R/SSB) as response, we do not thnk ths argument apples to our analyses. We thnk so because SSB was ncluded among the

2 predctors, and the model thus allowed modellng near-zero responses to SSB (as s evdent for example from Fg. A5). Nonetheless, to ascertan the robustness of the results, we performed an analyss n whch SSB effects were excluded from the model for the stocks for whch no detectable assocaton between R and SSB exsted. We also decded to omt effects of year for stocks for whch no detectable assocaton between R and year exsted. To perform ths analyss, we modfed the fnal model (correspondng to Eq. 4) as follows: log e (R,j ) = a + k log e (SSB,j ) + b SSB,j + c log e (R,j-1 ) + d (Year) + e NAO j + f(long,lat ) NAO j + g(year j,long,lat ) NAO j + _,j. (A1) For stocks dsplayng an R-SSB relatonshp, the constant k was fxed at 1 and the constant b was estmated from the data, modellng a Rcker-type response. For stocks that dd not dsplay any R- SSB relatonshp, both the constants k and b were set to 0. Smlarly, the functon d was set to zero for stocks that dd not dsplay any effect of year. To select whch stocks should have SSB effects, we frst substtuted the two SSB-terms wth a stock-specfc smooth effect of log e (SSB) (n a model not ncludng NAO or year effects). The smooth effects were selected automatcally by addng a shrnkage term to the roughness penalty. The penalty was estmated by mnmzng the GCV. If the true functon was lnear for a gven stock, the shrnkage term would domnate the penalty. And f the true functon was dentcally zero, a hgh penalty would lkely be selected, nducng hgh shrnkage that mght shrnk the functon to a zero functon (estmated degrees of freedom close to 0; c.f. the help manual of gam.selecton n the mgcv lbrary of R developed by Wood 2001). For seven stocks the estmated degrees of freedom were less than 0.1 (Gulf of Mane, S Grand Bank, Flemsh Cap, S Gulf St. Lawrence, N Gulf St. Lawrence, Baltc E, Faroe). These stocks thus dsplayed no relaton between R and SSB, wherefore we set ther values of k and b n Eq. A5 to zero. We then determned whch year effects to exclude, by addng a shrnkage term to the penalty of the smooth effect of year (d n Eq. A1). All stocks dsplayed effects of year (estmated df 0.5), wherefore the d term was retaned for all stocks. Fnally, we added NAO effects to the model. We found that results regardng the effects of the NAO were qualtatvely smlar whether or not the SSB effects were excluded from the model for the seven stocks that dd not dsplay any R-SSB relatonshp (Table A2, Fgs. A7-A9). We therefore judged the results to be robust to the above-mentoned potental problem.

3 Generalzed Cross Valdaton (GCV) The method of cross valdaton (CV) conssts of settng asde a data case, predctng ts response value by the model ftted to the other data cases, computng the squared predctve error, and then repeatng ths procedure for each data case to obtan the average sum of squared predctve error: 1 ( ) 2 CV = ( y yˆ ) (A2) n where ˆ ( ) y s the predcted value of the th data case from the model ftted to all data except the th data case. The calculaton can be speeded up by usng the formula: CV = 1 n ( y yˆ ) 2 /(1 a ) 2 (A3) where a s the th dagonal element of the hat (nfluence) matrx, H: H = X (X'X) -1 X' (A4) where X s the vector of predctor varables. Each of the a elements measures the dstance between a data pont and the centrod of the X-space, and thus the nfluence of the data case. The generalzed cross valdaton s calculated by replacng the nflaton factors a by ther average value, a : GCV = 1 n ( y yˆ ) 2 /(1 a) 2 (A5) Besdes beng computatonally smpler, the GCV has the advantage compared to the CV that t s nvarant wth respect to orthogonal transformatons of the space to whch the response vector and the covarate vectors belong (Wahba 1990). The GCV of a model s thus a proxy for the model s out-of-sample predctve mean squared error. A model wth lower GCV has more explanatory power, and hence s preferred, compared to a model wth hgher GCV. The GCV s analogous to Akake s Informaton

4 Crteron (AIC, Akake 1974) n that t ams at optmsng the trade-off between the numbers of parameters n a model and the goodness-of-ft of the model. In our analyses the GCV crteron was used for two purposes: () fnd the optmal roughness of each smooth term n a generalzed addtve model (Wood 2001, 2004), and () fnd the optmal model structure, e.g. when comparng models wth dfferent predctor varables. Fg. A1. Estmated annual number of recruts (R; whole lnes) and spawnng stock bomass (SSB; stppled lnes) for each cod stock. For R, year refers to the spawnng year. See Table 2 n man text for data sources.

5 Fg. A2. Stock-specfc autocorrelaton functons of resduals from the fnal cod recrutment model (Eq. 3). Stppled lnes: 95% confdence ntervals.

6 Fg. A3. Resduals and ftted values from the fnal cod recrutment model (Eq. 3). The resduals are weghted by an teratvely determned functon of stock and spawnng stock bomass.

7 Fg. A4. The combned effects of year and spawnng stock bomass on recrutment (log e [R/SSB]). R s annual number of recruts and SSB s spawnng stock bomass. Whole, thn lnes: observed tme trends. Bold, broken lnes: the predcted combned effect of year and SSB n a generalzed addtve model (Eq. 2) at NAO (North Atlantc Oscllaton Index) = 0.

8 Fg. A5. The effects of spawnng stock bomass (SSB) and year on annual numbers of recruts (R). Stppled, whole and broken lnes are, respectvely, predctons for the frst, mddle and last year wth data for each stock. The predctons are from a generalzed addtve model (Eq. 2) at NAO (North Atlantc Oscllaton Index) = 0. See Fg. 3 n the man text for the predcted year effects on recrutment when SSB s fxed.

9 Fg. A6. Interacton between the effects of spawnng stock bomass and the North Atlantc Oscllaton (NAO) on cod recrutment. Each panel shows the estmated slope of a lnear effect of NAO on log e (R/SSB), where R s annual number of recruts and SSB s spawnng stock bomass. Whole and broken lnes: predctons and 95% bootstrap confdence bands from a model n whch the NAO effect depends on SSB (Eq. 4). Ponts and bars: predctons and 95% bootstrap confdence lmts from a model n whch the NAO effect s fxed (Eq. 2; Fg. 4).

10 Fg. A7. The effects of spawnng stock bomass (SSB) and year on annual numbers of recruts (R). Stppled, whole and broken lnes are, respectvely, predctons for the frst, mddle and last year wth data for each stock. The predctons are from a generalzed addtve model n whch SSB effects were set to zero for 7 stocks not dsplayng any R-SSB relatonshp (marked by astersks). See Supplementary text for detals. The fgure s analogous to Fg. A5.

11 Fg. A8. Spatal pattern of the effect of the North Atlantc Oscllaton (NAO) on cod recrutment. The soclnes represent the slope of a lnear effect of NAO on log e (R), where R s annual number of recruts. SSB s spawnng stock bomass. The plot s based on a model n whch SSB effects were set to zero for 7 stocks not dsplayng any R-SSB relatonshp (see Supplementary text), and s analogous to Fg. 4.

12 Fg. A9. Temporal change n the mpact of the North Atlantc Oscllaton (NAO) on cod recrutment. The lnes show estmated slopes of a lnear effect of NAO on log e (R), where R s annual number of recruts. The effect of NAO s modelled to be lnear for any gven locaton and year, but the slope of the effect may vary spatally and temporally. Bold and broken lnes: estmated slope and 95% bootstrap confdence bands for spatally and temporally varyng NAO effect. Ponts and bars: estmated slope and 95% bootstrap confdence lmts for spatally varyng NAO effect. The plot s based on a model n whch effects of spawnng stock bomass (SSB) were set to zero for 7 stocks not dsplayng any R-SSB relatonshp (Eq. A1), and s analogous to Fg. 5.

13 Table A1. Comparson of orgnal results (Table 3 n man text) wth results obtaned when autoregressve and year effects were not ncluded n the model ( alternatve results ). See Table 3 for explanatons of acronyms and terms. Orgnal results Alternatve results Predctors GCV R 2 GCV R 2 Stock % % Spawnng stock bomass % % Auto-regressve effect % - - Year effect % - - Fxed NAO effect % % Non-lnear NAO effect % % NAO effect _ SSB % % NAO effect _ Year % % Table A2. Comparson of orgnal results (Table 3 n man text) wth results obtaned when SSB effects were set to zero for 7 stocks not dsplayng any R-SSB relatonshp ( alternatve results ). See Supplementary text and Table 3 for detals and explanatons of acronyms and terms. Note that n the orgnal analyses the response varable was log e (R/SSB) whle n the alternatve analyss t was log e (R). The GCV and R 2 values are therefore not drectly comparable across analyses, although the relatve merts of the dfferent sub-models are. Orgnal results Alternatve results Predctors GCV R 2 GCV R 2 Stock % % Spawnng stock bomass % % Auto-regressve effect % % Year effect % % Fxed NAO effect % % Non-lnear NAO effect % % NAO effect _ SSB % % NAO effect _ Year % %

14 LITERATURE CITED Akake H (1974) A new look at statstcal model dentfcaton. IEEE Transactons on Automatc Control AU-19: Cardnale M, Hjelm J (2006) Marne fsh recrutment varablty and clmate ndces. Mar Ecol Prog Ser 309: Wahba G (1990) Splne models for observatonal data. SIAM, Phladelpha Wood SN (2001) mgcv: GAMs and Generalzed Rdge Regresson for R. R News 1/2:20-25 Wood SN (2004) Stable and effcent multple smoothng parameter estmaton for generalzed addtve models. J Am Stat Ass 99: