Fighting Avoidance and Dual Utility of Secondary Sexual Characters As Direct Determinants

1S1 Appendix

3 Path analysis tests the patterns of dependence and independence between measured

4variables predicted by the topological structure of direct and indirect causal relationships

5specified by a multivariate causal hypothesis. We constructed a hypothetical causal path

6model assuming that high social rank would be reached by the most competitive (heavy with

7long horns) males. Consequently, our model tested if the interaction between mass and horn

8length was the only direct cause of rank, with age having no direct effect (Fig. 1, solid and

9dashed arrows). We tested this hypothetical model against a “full” model with all possible

10logical causal paths (Fig. 1). We included a free covariance between horn length and body

11mass, because we expected that differences in male quality would act as a common

12unmeasured cause leading to a correlation between horn length and mass.

13 We tested the path models using a d-sep test (Shipley 2000a; Shipley 2003; Shipley

142009) instead of classical structural equations (SEM) models, which are based on fitting

15predicted covariance matrices using maximum likelihood (Shipley 2000b), because two

16properties of our data prevent the use of standard SEM confirmatory path analysis (see also

17Favre et al. 2008 for an example in a similar context). First, our data had a hierarchical

18structure because we observed 18 males for more than one year. Second, horn length, mass

19and rank increased non-linearly with age. Shipley’s d-sep test involves a simultaneous test of

20all patterns of dependence and independence logically implied by the causal graph. It tests

21those independence claims in the BU basis set that, together, imply all others. The BU basis set

22consists of the k pairs of variables in the causal graph (Fig. 1) that do not have an arrow

23between them, conditioned on the direct causes (causal parents) of each. Since these k claims

24are mutually independent, an overall test involves obtaining the k null probabilities (pi) of

1 1 K 25independence claims (Table 1) and combining them as: C  2ln pi which follows a chi- i1

26squared distribution with 2k degrees of freedom if the data are generated by the hypothesized

27causal graph. A non significant p - value (p > 0.05) for the C statistic means that the observed

28and predicted patterns do not statistically differ and that the data support the model. A

29significant p - value would indicate that the model does not provide a good fit to the data.

30 If the basis set predicts that two variables (X,Y) are independent conditional on a set

31of conditioning variables (Z1, Z2), the null probability of this independence claim is obtained

32by fitting a linear mixed model with ibex ID as a random term. We included year as a factor

33in the fixed part of all equations to account for potential year effects. The fixed component of

34this model was Y~Z1+Z2+X, and we calculated the probability that the partial slope of X was

35zero in the statistical population (thus, conditional independence) using a t-test (Table 1). If

36X had potentially nonlinear relationships with Y, a test of conditional independence between

37X and Y was obtained by fitting a mixed model whose fixed component was Y~ Z1+Z2+

38(X+X2) and testing the null hypothesis that the partial slopes associated with both X and X2

39were zero, using an F-ratio. Once we obtained a causal graph that adequately fit the data, we

40calculated the coefficients of the final model (Fig. 1) by fitting linear mixed models in which

41each dependent variable was a function of its direct causes as specified by the causal graph,

42giving the path coefficients for each path, with ID as a random term and year as a fixed term.

44Results

45 The path model including a direct effect of the interaction between body mass and

46horn length on social rank (model 1, C1 = 1.85, df = 8, P = 0.99, Fig. 1) was not the only

47selected final model. Another model, including only the main effects of these two variables

48on rank (model 2, C2 = 3.85, df = 6, P = 0.70, Fig. 1), also fit the data better than the full

49model (CFull = 0.69, df = 2, P = 0.71; likelihood ratio test (LRT) between the full and the two

2 2 50final models: C1 - CFull = 1.16, df = 6, P = 0.98; C2 - CFull = 3.16, df = 4, P = 0.53). Table 1

51presents the independence tests for the basis sets. Once horn length and body mass were

52accounted for, neither model included a direct causal relationship between age and rank (LRT

53of C1 and C2 without and with age: C1 – C1.Age = 0.07, df = 2, P = 0.97; C2 – C2.Age = 0.19, df

54= 2, P = 0.91). Age, however, had a substantial indirect effect on rank through its direct

55effects on both horn length and body mass (Fig. 1). The direct covariance path between horn

56length and body mass, after accounting for age (Fig. 1), confirmed the strong individual

57heterogeneity suggested by the age-independent variation in morphological traits. Therefore,

58both final models suggest that a male’s social rank is directly determined by the size of its

59secondary sexual characters and not by age.

61References

63Favre M, Martin JGA, Festa-Bianchet M (2008) Determinants and life-history consequences

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65Shipley B (2000a) A new inferential test for path models based on directed acyclic graphs.

66 Struct equa model 7:206-218

67Shipley B (2000b) Cause and correlation in biology: A user's guide to path analysis, structural

68 equations, and causal inference. Oxford University Press, Oxford

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74Table 1: Statistics describing the test of conditional independence in the basis set implied by 75the hypothesized path models in Fig. 1.

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

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

78(D) and Rank (E). Each conditional independence (d-sep) claim is presented as X _||_ Y | {Z1,

79Z2} meaning that variables Y and X are independent conditional on the combined set of Z1

80and Z2 (“causal parents”). Therefore, we evaluated conditional independence based on the null

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

4 4 Age

Within: 0.79 (0.53) Between: 0.83 (1.30) Mass Horn

0.47 (0.02) 0.59 (0.02)

Mass x Horn 0.47 (0.09) 0.49 (0.09)

0.91 (0.06)

Rank 89

90 91Fig. 1: Path analyses of the determinants of social rank in male Alpine ibex. Solid arrows

92indicate common significant paths of the two final best-fit models, with either the interaction

93of Mass and Horn (dashed arrows, model 1) or their direct effects (dashed-dotted arrows,

94model 2) on Rank. Values along each path report the standardized path coefficients and

95standard errors. The correlation between Mass and Horn (within and between males) and

96their covariance are reported as the variance within and between males. Dotted arrows

97represent nonsignificant path of the two models tested in the full model. All data were

98collected in Levionaz, Italy, in 2003, 2006 and 2007 (62 observations of rank from 36

99individuals).

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