<p> 1S1 Appendix</p><p>2</p><p>3 Path analysis tests the patterns of dependence and independence between measured </p><p>4variables predicted by the topological structure of direct and indirect causal relationships </p><p>5specified by a multivariate causal hypothesis. We constructed a hypothetical causal path </p><p>6model assuming that high social rank would be reached by the most competitive (heavy with </p><p>7long horns) males. Consequently, our model tested if the interaction between mass and horn </p><p>8length was the only direct cause of rank, with age having no direct effect (Fig. 1, solid and </p><p>9dashed arrows). We tested this hypothetical model against a “full” model with all possible </p><p>10logical causal paths (Fig. 1). We included a free covariance between horn length and body </p><p>11mass, because we expected that differences in male quality would act as a common </p><p>12unmeasured cause leading to a correlation between horn length and mass.</p><p>13 We tested the path models using a d-sep test (Shipley 2000a; Shipley 2003; Shipley </p><p>142009) instead of classical structural equations (SEM) models, which are based on fitting </p><p>15predicted covariance matrices using maximum likelihood (Shipley 2000b), because two </p><p>16properties of our data prevent the use of standard SEM confirmatory path analysis (see also </p><p>17Favre et al. 2008 for an example in a similar context). First, our data had a hierarchical </p><p>18structure because we observed 18 males for more than one year. Second, horn length, mass </p><p>19and rank increased non-linearly with age. Shipley’s d-sep test involves a simultaneous test of </p><p>20all patterns of dependence and independence logically implied by the causal graph. It tests </p><p>21those independence claims in the BU basis set that, together, imply all others. The BU basis set</p><p>22consists of the k pairs of variables in the causal graph (Fig. 1) that do not have an arrow </p><p>23between them, conditioned on the direct causes (causal parents) of each. Since these k claims </p><p>24are mutually independent, an overall test involves obtaining the k null probabilities (pi) of </p><p>1 1 K 25independence claims (Table 1) and combining them as: C  2ln pi which follows a chi- i1</p><p>26squared distribution with 2k degrees of freedom if the data are generated by the hypothesized </p><p>27causal graph. A non significant p - value (p > 0.05) for the C statistic means that the observed</p><p>28and predicted patterns do not statistically differ and that the data support the model. A </p><p>29significant p - value would indicate that the model does not provide a good fit to the data. </p><p>30 If the basis set predicts that two variables (X,Y) are independent conditional on a set </p><p>31of conditioning variables (Z1, Z2), the null probability of this independence claim is obtained </p><p>32by fitting a linear mixed model with ibex ID as a random term. We included year as a factor </p><p>33in the fixed part of all equations to account for potential year effects. The fixed component of</p><p>34this model was Y~Z1+Z2+X, and we calculated the probability that the partial slope of X was </p><p>35zero in the statistical population (thus, conditional independence) using a t-test (Table 1). If </p><p>36X had potentially nonlinear relationships with Y, a test of conditional independence between </p><p>37X and Y was obtained by fitting a mixed model whose fixed component was Y~ Z1+Z2+</p><p>38(X+X2) and testing the null hypothesis that the partial slopes associated with both X and X2 </p><p>39were zero, using an F-ratio. Once we obtained a causal graph that adequately fit the data, we </p><p>40calculated the coefficients of the final model (Fig. 1) by fitting linear mixed models in which </p><p>41each dependent variable was a function of its direct causes as specified by the causal graph, </p><p>42giving the path coefficients for each path, with ID as a random term and year as a fixed term.</p><p>43</p><p>44Results</p><p>45 The path model including a direct effect of the interaction between body mass and </p><p>46horn length on social rank (model 1, C1 = 1.85, df = 8, P = 0.99, Fig. 1) was not the only </p><p>47selected final model. Another model, including only the main effects of these two variables </p><p>48on rank (model 2, C2 = 3.85, df = 6, P = 0.70, Fig. 1), also fit the data better than the full </p><p>49model (CFull = 0.69, df = 2, P = 0.71; likelihood ratio test (LRT) between the full and the two </p><p>2 2 50final models: C1 - CFull = 1.16, df = 6, P = 0.98; C2 - CFull = 3.16, df = 4, P = 0.53). Table 1 </p><p>51presents the independence tests for the basis sets. Once horn length and body mass were </p><p>52accounted for, neither model included a direct causal relationship between age and rank (LRT </p><p>53of C1 and C2 without and with age: C1 – C1.Age = 0.07, df = 2, P = 0.97; C2 – C2.Age = 0.19, df </p><p>54= 2, P = 0.91). Age, however, had a substantial indirect effect on rank through its direct </p><p>55effects on both horn length and body mass (Fig. 1). The direct covariance path between horn </p><p>56length and body mass, after accounting for age (Fig. 1), confirmed the strong individual </p><p>57heterogeneity suggested by the age-independent variation in morphological traits. Therefore, </p><p>58both final models suggest that a male’s social rank is directly determined by the size of its </p><p>59secondary sexual characters and not by age. </p><p>60</p><p>61References</p><p>62</p><p>63Favre M, Martin JGA, Festa-Bianchet M (2008) Determinants and life-history consequences </p><p>64 of social dominance in bighorn ewes. Anim Behav 76:1373-1380</p><p>65Shipley B (2000a) A new inferential test for path models based on directed acyclic graphs. </p><p>66 Struct equa model 7:206-218</p><p>67Shipley B (2000b) Cause and correlation in biology: A user's guide to path analysis, structural</p><p>68 equations, and causal inference. Oxford University Press, Oxford</p><p>69Shipley B (2003) Testing recursive path models with correlated errors using d-separation. </p><p>70 Struct Equa Model 10:214-221</p><p>71Shipley B (2009) Confirmatory path analysis in a generalized multilevel context. Ecology </p><p>72 90:363-368</p><p>73</p><p>74Table 1: Statistics describing the test of conditional independence in the basis set implied by 75the hypothesized path models in Fig. 1.</p><p>3 3 76 Conditional independence Partial SE Statistics Null claims slopes Probability Quadratic paths F-value (df) A _||_ D | {B,C} (model 1,2) D~(B+B2)+(C+C2)+(A+A2) A -0.029 0.073 0.351 (2,18) 0.709 A2 0.038 0.065 A _||_ E | {D} (model 1) E~D+(A+A2) A 0.082 0.530 0.037 (2,21) 0.963 A2 -0.134 0.474 A _||_ E | {B,C} (model 2) E~D+(A+A2) A -0.213 0.582 0.099 (2,20) 0.906 A2 0.196 0.502</p><p>Linear paths T- value (df) B _||_ E | {A,D} (model 1) E~(A+A2)+D+B B 0.066 0.203 0.323 (20) 0.750 C _||_ E | {A,D} (model 1) E~(A+A2)+D+C C -0.075 0.254 -0.296 (20) 0.771 D_||_E | {B,C} (model 2) E~B+C+D D 0.802 0.642 1.249 (21) 0.226 77Standardized variables: Age (A), Body mass (B), Horn length (C), Body mass x Horn length </p><p>78(D) and Rank (E). Each conditional independence (d-sep) claim is presented as X _||_ Y | {Z1,</p><p>79Z2} meaning that variables Y and X are independent conditional on the combined set of Z1 </p><p>80and Z2 (“causal parents”). Therefore, we evaluated conditional independence based on the null</p><p>81probability of the relationship between Y and X according to the equation: Y~ Z1 + Z2 + X 82(see methods). We fitted male identity as random terms and year (factor) as a fixed term in all 83d-sep claims. We calculated a single null probability for the quadratic paths using F-statistics 84for the ratio of the mean sum of square of the combined quadratic terms (e.g. A+A2) and the 85residual sum of squares of the random effects. 86 87</p><p>88</p><p>4 4 Age</p><p>Within: 0.79 (0.53) Between: 0.83 (1.30) Mass Horn</p><p>0.47 (0.02) 0.59 (0.02)</p><p>Mass x Horn 0.47 (0.09) 0.49 (0.09)</p><p>0.91 (0.06)</p><p>Rank 89</p><p>90 91Fig. 1: Path analyses of the determinants of social rank in male Alpine ibex. Solid arrows </p><p>92indicate common significant paths of the two final best-fit models, with either the interaction </p><p>93of Mass and Horn (dashed arrows, model 1) or their direct effects (dashed-dotted arrows, </p><p>94model 2) on Rank. Values along each path report the standardized path coefficients and </p><p>95standard errors. The correlation between Mass and Horn (within and between males) and </p><p>96their covariance are reported as the variance within and between males. Dotted arrows </p><p>97represent nonsignificant path of the two models tested in the full model. All data were </p><p>98collected in Levionaz, Italy, in 2003, 2006 and 2007 (62 observations of rank from 36 </p><p>99individuals).</p><p>100</p><p>101 102 103 104</p><p>5 5</p>