Andrew R. Jacobson, Antonello Provenzale, Achaz von Hardenberg, Bruno Bassano, and Marco Festa-Bianchet. 2004. Climate forcing and density dependence in a mountain ungulate pPopulation. Ecology 85:1598–1610.

Since standard tests for the significance of a serial correlation assume that the variables are normally distributed, we established a correlation test using techniques entirely analogous to those discussed in Methods: Tests for density dependence of the main text. A general dependence y_i=f(v_i) of population change on a given climate variable v may be expanded in a Taylor series. The first two terms of this expansion yield the linear dependence y_i = a + c v_i, with a and c parameters to be determined. We add a stochastic term to form the model of interest,

 

yi = a + cvi + i,

(C.1)

in which _i are independent samples of a normally distributed random variable with mean zero and unit variance, and the parameter weights the stochastic contribution. The maximum-likelihood values of the parameters a, c, and are determined using standard least-squares regression. The question we pose is whether the optimal value of the climate dependence parameter c differs significantly from zero. We use a randomization technique similar to that proposed by Pollard et al. (1987) to test this hypothesis. If there is no dependence of y on v, then a random reordering of the v values should have a correlation with y which is equally as likely as the actual one. From a large number of different shufflings of the N – 1 values of v, we compute a distribution of c_null values representing the null hypothesis of no correlation.

The actual maximum-likelihood value of c may be compared with the distribution of c_null to empirically determine confidence limits. Because the reordering should destroy any correlation, we expect the distribution of c_null values to have a mean of zero. In Table C1, we report the two-sided error probability P_s, which is the proportion of c_null values with absolute value less than that of c.

Since linear correlation analysis is sensitive to outliers (e.g., Huber 1981), it is interesting to ask how much the maximum-likelihood value of c would change if only a portion of the data were used. The jackknife procedure establishes a distribution c_j of possible c values from a large number of randomly chosen subsets of the data. We report a Student's t value to express the amount by which the maximum-likelihood value c_ml varies from the mean of the jackknife distribution, <c_j>:

 

tj = (< cj> - cml) / j,

(C.2)

where _j is the standard deviation of the jackknife distribution. It is common to drop only one observation to form the jackknife subsets (e.g., Sokal and Rohlf 1995). This test of parameter robustness is made more conservative by dropping considerably more observations (the d-delete jackknife; see Miller 1974; Efron and Tibshirani 1993). In this analysis, we dropped half of the data points. There is no evidence from this procedure that the parameter values are strongly influenced by outliers.

Table C1. Significance of correlations of relative changes in total fall ibex counts with climate variables from stations at Teleccio and Serrù based on a distribution of 100,000 random reorderings of the climate data.

Climate Variable	Teleccio		Serrù
	Ps	tj	Ps	tj
Average winter snow depth	0.0185	0.0349	0.0031	0.0204
Average winter maximum temperature	0.3054	-0.0358	0.1330	0.0759
Average winter minimum temperature	0.3982	0.0304	0.5769	0.194
Average summer maximum temperature	0.1414	-0.0122	0.4222	0.0504
Average summer minimum temperature	0.5511	0.0161	0.9617	0.0977
Total spring precipitation	0.0167	0.195	0.0609	0.164
Total winter precipitation	0.0092	0.0107	0.0126	0.116
Total summer precipitation	0.0196	-0.0633	0.028	0.128
Days of snow depth above level 1	0.0106	-0.0714	0.0041	-0.0876
Days of snow depth above level 2	0.0517	0.0401	0.0143	0.139

   Notes: Snow-depth level 1 and level 2 are defined in Methods: Data of the main text. The tabulated two-sided error probability P_s is underlined for results significant at the = 0.05 level. For all significant results, the correlations are negative. t_j is the Student's t value for the deviation of the optimal estimate from the jackknife estimate of that value. t_j values which approach +/- 1 indicate that the parameter estimate is significantly affected by outliers.

Literature cited

Efron, B., and R. J. Tibshirani. 1993. An introduction to the bootstrap. Chapman and Hall, New York, New York, USA.

Huber, P. J. 1981. Robust statistics. John Wiley and Sons, New York, New York, USA.

Miller, R. G. 1974. The jackknife--A review. Biometrika 61:1–15.

Pollard, E., K. H. Lakhani, and P. Rothery. 1987. The detection of density-dependence from a series of annual censuses. Ecology 68:2046–2055.

Sokal, R. R., and F. J. Rohlf. 1995. Biometry. Third edition. Freeman, New York, New York, USA.