Jaume Forcada, P. N. Trathan, K. Reid, and E. J. Murphy. 2005. The effects of global climate variability in pup production of Antarctic fur seals. Ecology 86:2408–2417.

Appendix B. A description of the Antarctic fur seal pup production mixed-effects model and responses to SST anomalies.

Every pupping season (each lasting slightly over two months), the daily number of pupping females was described by a roughly symmetric distribution curve with an optimum, corresponding to a peak or mean birth date. Given the irregular shape and long tails of this distribution (Fig. B1), we modeled the cumulative number of pups born P(t) with a simple logistic model. The model, which smoothed out the daily variation in P(t), was

,

(B.1)

where t was day since November 1^st;  was the horizontal asymptote of the accumulative births and, as , it estimated total annual pup production;  was the value at which the response was , the inflection point of the curve, and corresponded to the expected mean birth date; and  was a scale parameter that measured the distance between the inflection point and approximately . This parameter provided a measure of variability in birth date.

To assess inter-annual differences in the three parameters, we initially considered random effects in all of them, i.e.,

,

(B.2)

for years i=1,…,M, and newborn pup counts in day j of year i, with j=1,…n_i. Each  parameter was defined as , where  were fixed effects or long-term means, and b_i were random effects or deviations of the  from the long-term mean, with .  was the variance-covariance matrix of random effects, which in our analysis was a general symmetric positive-definite matrix (Pinheiro and Bates 2000).

In our models, random effects were inter-annual deviates of the long-term mean caused by environmental variability and other stochastic effects, but were assumed independent of intra-annual variability in daily newborn pup counts. Thus, they were assumed to be independent between years and independent of the within-year error terms . Possible environmental autocorrelation structures were considered from the results of the cross-correlation analysis described in Appendix A. For instance, linear and quadratic SST anomaly effects with no lag on annual pup production were modeled as

,

(B.3)

where the b_1i was the seasonal random deviate. A simplified stepwise model-building approach allowed us to test for different combinations of mixed, random, and environmental effects.

The  terms were assumed to be independent for different i and j and to be homoscedastic. However, daily records were evaluated as accumulated numbers and we expected serial autocorrelation in  and heteroscedasticity. To model heteroscedasticity, we considered different within-year variances, , with S+1 parameters to represent S years. To account for autocorrelation, , we considered structures based on ARMA models (autoregressive moving-average models; e.g., Chatfield 2004), where

,

with p+q correlation parameters in , corresponding to p autoregressive parameters  and q moving average parameters .  The number of  parameters was the same as of “noise” terms a_t. Under this general structure we tested two simpler nested options, autoregressive errors of lag p, AR(p), and moving averages of order q, MA(q).

The most adequate structure was investigated with autocorrelation plots of the model fit residuals.

Models were fitted using maximum-likelihood methods and the appropriate modeling options were selected by examining residual variability and autocorrelation plots of the residuals.  The best models were chosen from comparison of deviances and the Akaike information criterion (AIC) of competing models. The comparison of AIC is described in Table B1.

Our best model included random effects in all the  parameters, with SST anomaly quadratic effects in  and and linear in . This model accounted for heteroscedasticity with a different variance in  by year and a correlation structure best modeled by a moving average of order 6. The inference from this model consisted of estimates of fixed and random effects (Table 1, main text) and within-year residual errors. The mean within-year standard error ( ) was estimated as 7.48 (95% CI: 6.78 - 8.26), and inter-annual deviates in variance ( ) ranged from 0.85 (95% CI: 0.70 - 1.02) to 1.45 (95% CI: 1.21 - 1.75). The approximate 95% confidence intervals of the parameters were obtained from the distributions of their maximum likelihood estimates, conditional on their significance as measured by t tests (Pinheiro and Bates 2000).

TABLE B1. Comparison and selection of nonlinear mixed-effects models of the accumulative number of Antarctic fur seal pups born at a study site in Bird Island, South Georgia.  are fixed effects, b are random effects, sst is the mean summer value of the SST anomaly.  is the model likelihood, df are degrees of freedom, AIC is the Akaike information criterion. Heteroscedasticity ( ) and serial correlation ( ) structures are described in the text.

	Parameter					Model fit
						df		AIC
1						4	-8214.65	16437.30
2						5	-8108.30	16226.61
3						5	-7632.39	15274.77
4						6	-7035.66	14083.32
5						5	-6970.71	13951.42
6						6	-6938.99	13889.98
7						6	-5299.88	10611.76
8						7	-5114.88	10243.75
9						8	-5110.93	10237.86
10						11	-5102.20	10226.41
11						10	-5104.32	10228.63
12						9	-5104.56	10227.12
13						10	-5102.76	10225.52
14						12	-5097.73	10219.46
15						11	-5097.98	10217.96
16						13	-5095.60	10217.21
17						12	-5095.76	10215.51
18						31	-5026.30	10114.60
19					AR(1)	13	-4340.89	8707.77
20					MA(6)	18	-4320.84	8677.67
21					AR(1)	32	-4305.12	8674.24
22					MA(6)	37	-4291.95	8656.43

 

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FIG. B1. Inter-annual variability in observed daily number of Antarctic fur seal pups born at the study site in Bird Island, South Georgia.

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FIG. B2. Adequacy of the best fitted model. Continuous lines are predictions of the population mean modeled with SST anomaly as a covariate, and the dashed lines are predicted annual deviates based on modeled random-effects. The “+” signs correspond to observed counts. Click here for larger sized image of figure for viewing.

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LITERATURE CITED

Chatfield, C. 2004. The analysis of time series: an introduction. Sixth edition. Chapman and Hall/CRC, London, UK.

Pinheiro, J. C., and D. M. Bates. 2000. Mixed-effects models in S and S-Plus. Springer-Verlag, New-York , New York, USA.

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