Data S1
Question 1: Is the boldness of the single boldest member of the group (the keystone) associated with # of attackers that respond to simulated prey? Model Structure: GLMM, normal error distribution, identity link function. Response variable: Average number of attackers across three staged trials. Predictor Variables: keystone boldness, arid/wet site distinction, and their interaction term. Random effects: site ID
Effect Summary
Source / LogWorth / PValue /Keystone Boldness / 9.407 / / 0.00000
Keystone Boldness *Arid or Wet site / 9.385 / / 0.00000
Arid or Wet site / 1.116 / / 0.07654
Summary of Fit
RSquare / 0.465393RSquare Adj / 0.458626
Root Mean Square Error / 1.285881
Mean of Response / 3.08574
Observations (or Sum Wgts) / 241
Parameter Estimates
Term / Estimate / Std Error / DFDen / t Ratio / Prob>|t| /Intercept / 1.7436143 / 0.355166 / 13.32 / 4.91 / 0.0003*
Arid_Wet[Arid] / 0.6201628 / 0.290176 / 5.988 / 2.14 / 0.0765
Keystone Boldness / 0.0038837 / 0.000594 / 231.9 / 6.54 / <.0001*
(Keystone Boldness -341.593)*Arid_Wet[Arid] / 0.0038783 / 0.000594 / 231.9 / 6.53 / <.0001*
Random Effect / Var Ratio / Var Component / Std Error / 95% Lower / 95% Upper / Wald p-Value / Pct of Total /
Site ID / 0.3738864 / 0.6182176 / 0.3893398 / -0.144874 / 1.3813096 / 0.1123 / 27.214
Residual / 1.6534906 / 0.1538581 / 1.3889824 / 2.0019055 / 72.786
Total / 2.2717081 / 0.4166959 / 1.6340297 / 3.3738367 / 100.000
-2 LogLikelihood = 848.54112299
Note: Total is the sum of the positive variance components.
Total including negative estimates = 2.2717081
Fixed Effect Tests
Source / Nparm / DF / DFDen / F Ratio / Prob > F /Arid or Wet site / 1 / 1 / 5.988 / 4.5676 / 0.0765
Keystone Boldness / 1 / 1 / 231.9 / 42.7528 / <.0001*
Keystone Boldness *Arid or Wet site / 1 / 1 / 231.9 / 42.6326 / <.0001*
Q1: Arid sites only
Figure: Regression plots depicting the relationship between the boldness of the single boldest member of the group (x-axis) and the average number of attackers that respond to simulated prey (y-axis) for arid sites in aggregate (top panel) and for each site individually (bottom panel).
Summary of Fit
RSquare / 0.394294RSquare Adj / 0.388981
Root Mean Square Error / 1.509835
Mean of Response / 3.775833
Observations (or Sum Wgts) / 116
Parameter Estimates
Term / Estimate / Std Error / DFDen / t Ratio / Prob>|t| /Intercept / 1.0424515 / 0.56745 / 10.18 / 1.84 / 0.0955
Keystone Boldness / 0.0077515 / 0.001094 / 111.4 / 7.08 / <.0001*
REML Variance Component Estimates
Random Effect / Var Ratio / Var Component / Std Error / 95% Lower / 95% Upper / Wald p-Value / Pct of Total /Site ID / 0.2673344 / 0.6094163 / 0.5638608 / -0.495731 / 1.7145632 / 0.2798 / 21.094
Residual / 2.2796029 / 0.3060172 / 1.7813014 / 3.0219344 / 78.906
Total / 2.8890192 / 0.6362135 / 1.9578994 / 4.6901996 / 100.000
-2 LogLikelihood = 443.17547084
Note: Total is the sum of the positive variance components.
Total including negative estimates = 2.8890192
Fixed Effect Tests
Source / Nparm / DF / DFDen / F Ratio / Prob > F /Keystone Boldness / 1 / 1 / 111.4 / 50.1754 / <.0001*
Q1: Wet sites only
Leverage Plot
Figure: Regression plots depicting the relationship between the boldness of the single boldest member of the group (x-axis) and the average number of attackers that respond to simulated prey (y-axis) for wet sites in aggregate (top panel) and for each site individually (bottom panel).
Summary of Fit
RSquare / 0.322064RSquare Adj / 0.316553
Root Mean Square Error / 1.03654
Mean of Response / 2.445333
Observations (or Sum Wgts) / 125
Parameter Estimates
Term / Estimate / Std Error / DFDen / t Ratio / Prob>|t| /Intercept / 2.4513428 / 0.450501 / 4.556 / 5.44 / 0.0038*
Keystone Boldness / -3.539e-6 / 0.000595 / 120.5 / -0.01 / 0.9953
REML Variance Component Estimates
Random Effect / Var Ratio / Var Component / Std Error / 95% Lower / 95% Upper / Wald p-Value / Pct of Total /Site ID / 0.5798506 / 0.6230007 / 0.5369725 / -0.429446 / 1.6754475 / 0.2460 / 36.703
Residual / 1.0744159 / 0.1387054 / 0.8470466 / 1.4079489 / 63.297
Total / 1.6974166 / 0.5534854 / 0.9793695 / 3.6375289 / 100.000
-2 LogLikelihood = 386.49688222
Note: Total is the sum of the positive variance components.
Total including negative estimates = 1.6974166
Fixed Effect Tests
Source / Nparm / DF / DFDen / F Ratio / Prob > F /Keystone Boldness / 1 / 1 / 120.5 / 0.0000 / 0.9953
Question 2A: Is colony foraging aggressiveness (average # attackers) associated with persistence in situ?
Model Structure: GLMM, binomial error distribution, logit link function. Response variable: colony persistence (1/0). Predictor Variables: Average number of attackers across three staged trials, arid/wet site distinction, and their interaction term.
Whole Model Test
Model / -LogLikelihood / L-R ChiSquare / DF / ProbChiSq /Difference / 32.8466439 / 65.6933 / 3 / <.0001*
Full / 134.149956
Reduced / 166.9966
Goodness Of Fit Statistic / ChiSquare / DF / ProbChiSq /
Pearson / 247.8279 / 237 / 0.3014
Deviance / 268.2999 / 237 / 0.0794
AICc /
276.4694
Effect Summary
Source / LogWorth / PValue /Average # Attackers / 8.327 / / 0.00000
Average # Attackers*Arid_Wet / 4.846 / / 0.00001
Arid_Wet / 0.016 / / 0.96451 / ^
Effect Tests
Source / DF / L-R ChiSquare / ProbChiSq /Arid_Wet / 1 / 0.0019802 / 0.9645
Average # Attackers / 1 / 34.30709 / <.0001*
Average # Attackers*Arid_Wet / 1 / 18.836747 / <.0001*
Parameter Estimates
Term / Estimate / Std Error / L-R ChiSquare / ProbChiSq / Lower CL / Upper CL /Intercept / -2.2953226 / 0.4252576 / 39.270925 / <.0001* / -1.508585 / -3.1874977
Arid_Wet[Arid] / -0.0071864 / 0.161516 / 0.0019802 / 0.9645 / -0.308751 / 0.3269206
Average # Attackers / 0.684522 / 0.1337731 / 34.30709 / <.0001* / 0.965555 / 0.437222
(Average # Attackers-3.08574)* Arid_Wet[Arid] / 0.528898 / 0.1337731 / 18.836747 / <.0001* / 0.808943 / 0.279979
Q2A: Arid sites only
Model Structure: GLM, binomial error distribution, logit link function. Response variable: colony persistence (1/0). Predictor Variables: Average number of attackers across three staged trials, Site ID, # attackers * site ID. Site ID was designated as a random effect in the combined model because we were merely interested in contrasting wet vs. dry sites and were disinterested in the particulars of any given site. For the separate wet vs. dry models this designation was changed to a fixed effect because we were specifically interested in testing for differences in the relationship between # attackers and colony survival across sites.
Logistic Fit for Survival Arid Sites
Effect Summary
Source / LogWorth / PValue /Average # Attackers / 17.424 / / 0.00000
Site ID / 1.832 / / 0.01471
Average # Attackers*Site ID / 1.286 / / 0.05177
Whole Model Test
Model / -LogLikelihood / DF / ChiSquare / ProbChiSq /Difference / 38.446849 / 7 / 76.8937 / <.0001*
Full / 41.336423
Reduced / 79.783271
RSquare (U) / 0.4819
AICc / 100.019
BIC / 120.702
Observations (or Sum Wgts) / 116
Parameter Estimates
Term / Estimate / Std Error / ChiSquare / ProbChiSq /Intercept / -6.1798731 / 1.3648482 / 20.50 / <.0001*
Average # Attackers / 2.03512189 / 0.4566824 / 19.86 / <.0001*
Site ID[Boegoeberg] / -1.7362606 / 0.6295414 / 7.61 / 0.0058*
Site ID[Kalkrand] / -0.4402858 / 0.7341707 / 0.36 / 0.5487
Site ID[Rehoboth] / 1.35395271 / 1.0960651 / 1.53 / 0.2167
(Average # Attackers-3.77583)*Site ID[Boegoeberg] / -1.2632103 / 0.5068731 / 6.21 / 0.0127*
(Average # Attackers-3.77583)*Site ID[Kalkrand] / 0.03547491 / 0.6834384 / 0.00 / 0.9586
(Average # Attackers-3.77583)*Site ID[Rehoboth] / 1.07289067 / 1.0993798 / 0.95 / 0.3291
For log odds of 1/0
Effect Likelihood Ratio Tests
Source / Nparm / DF / L-R ChiSquare / ProbChiSq /Average # Attackers / 1 / 1 / 75.4411321 / <.0001*
Site ID / 3 / 3 / 10.5071617 / 0.0147*
Average # Attackers*Site ID / 3 / 3 / 7.73718063 / 0.0518
Q2A: Wet sites only
Model Structure: GLM, binomial error distribution, logit link function. Response variable: colony persistence (1/0). Predictor Variables: Average number of attackers across three staged trials, Site ID, # attackers * site ID. Site ID was designated as a random effect in the combined model because we were merely interested in contrasting wet vs. dry sites and were disinterested in the particulars of any given site. For the separate wet vs. dry models this designation was changed to a fixed effect because we were specifically interested in testing for differences in the relationship between # attackers and colony survival across sites.
Nominal Logistic Fit for Survival Wet Sites
Source / LogWorth / PValue /Average # Attackers*Site ID / 0.381 / / 0.41634
Site ID / 0.212 / / 0.61399 / ^
Average # Attackers / 0.067 / / 0.85681 / ^
Whole Model Test
Model / -LogLikelihood / DF / ChiSquare / ProbChiSq /Difference / 2.424603 / 7 / 4.849207 / 0.6784
Full / 83.059204
Reduced / 85.483807
RSquare (U) / 0.0284
AICc / 183.36
BIC / 204.745
Observations (or Sum Wgts) / 125
Parameter Estimates
Term / Estimate / Std Error / ChiSquare / ProbChiSq /Intercept / -0.5419647 / 0.5202265 / 1.09 / 0.2975
Average # Attackers / 0.03956414 / 0.2191554 / 0.03 / 0.8567
Site ID[Ladysmith] / -0.0976133 / 0.4206221 / 0.05 / 0.8165
Site ID[Outjo] / 0.46738333 / 0.3583988 / 1.70 / 0.1922
Site ID[Rundu] / -0.2957814 / 0.4220266 / 0.49 / 0.4834
(Average # Attackers-2.44533)*Site ID[Ladysmith] / -0.2934648 / 0.436802 / 0.45 / 0.5017
(Average # Attackers-2.44533)*Site ID[Outjo] / 0.21042826 / 0.3904579 / 0.29 / 0.5899
(Average # Attackers-2.44533)*Site ID[Rundu] / 0.40479425 / 0.2873783 / 1.98 / 0.1590
For log odds of 1/0
Effect Likelihood Ratio Tests
Source / Nparm / DF / L-R ChiSquare / ProbChiSq /Average # Attackers / 1 / 1 / 0.03255682 / 0.8568
Site ID / 3 / 3 / 1.80434864 / 0.6140
Average # Attackers*Site ID / 3 / 3 / 2.84378567 / 0.4163
Question 2B: Is colony foraging aggressiveness (average # attackers) associated with persistence in situ?
Model Structure: Logistic regression model for each site. Response variable: colony persistence (1/0). Predictor Variables: Average number of attackers across three staged trials.
Arid Sites Only — The association is always significant
Kalkrand:
Parameter Estimates
Term / Estimate / Std Error / ChiSquare / ProbChiSq /Intercept / 6.7541062 / 2.3837654 / 8.03 / 0.0046*
ColonyAgg / 2.0705968 / 0.7190677 / 8.29 / 0.0040*
For log odds of 0/1
Effect Likelihood Ratio Tests
Source / Nparm / DF / L-R ChiSquare / ProbChiSq /ColonyAgg / 1 / 1 / 23.5110378 / <.0001*
Rehoboth:
Parameter Estimates
Term / Estimate / Std Error / ChiSquare / ProbChiSq /Intercept / 8.87697676 / 4.1647738 / 4.54 / 0.0331*
ColonyAgg / 3.1080126 / 1.4142681 / 4.83 / 0.0280*
For log odds of 0/1
Effect Likelihood Ratio Tests
Source / Nparm / DF / L-R ChiSquare / ProbChiSq /ColonyAgg / 1 / 1 / 22.3907114 / <.0001*
Upington:
Parameter Estimates
Term / Estimate / Std Error / ChiSquare / ProbChiSq /Intercept / 5.94194716 / 2.1514318 / 7.63 / 0.0057*
ColonyAgg / 2.1899666 / 0.8502974 / 6.63 / 0.0100*
For log odds of 0/1
Effect Likelihood Ratio Tests
Source / Nparm / DF / L-R ChiSquare / ProbChiSq /ColonyAgg / 1 / 1 / 21.3227386 / <.0001*
Boegoeberg:
Parameter Estimates
Term / Estimate / Std Error / ChiSquare / ProbChiSq /Intercept / 3.14646227 / 1.4658218 / 4.61 / 0.0318*
ColonyAgg / 0.7719116 / 0.3110034 / 6.16 / 0.0131*
For log odds of 0/1
Effect Likelihood Ratio Tests
Source / Nparm / DF / L-R ChiSquare / ProbChiSq /ColonyAgg / 1 / 1 / 8.93916964 / 0.0028*
Wet Sites Only — The association is never significant
Ladysmith:
Parameter Estimates
Intercept / -0.0780413 / 1.0234081 / 0.01 / 0.9392
ColonyAgg / 0.25390066 / 0.5343537 / 0.23 / 0.6347
For log odds of 0/1
Effect Likelihood Ratio Tests
Source / Nparm / DF / L-R ChiSquare / ProbChiSq /ColonyAgg / 1 / 1 / 0.22710561 / 0.6337
Outjo:
Parameter Estimates
Term / Estimate / Std Error / ChiSquare / ProbChiSq /Intercept / 0.58914861 / 1.0251724 / 0.33 / 0.5655
ColonyAgg / -0.2499924 / 0.4570083 / 0.30 / 0.5844
For log odds of 0/1
Effect Likelihood Ratio Tests
Source / Nparm / DF / L-R ChiSquare / ProbChiSq /ColonyAgg / 1 / 1 / 0.30263975 / 0.5822
Rundu:
Parameter Estimates
Term / Estimate / Std Error / ChiSquare / ProbChiSq /Intercept / 1.82760376 / 1.0462322 / 3.05 / 0.0807
ColonyAgg / -0.4443586 / 0.2628961 / 2.86 / 0.0910
For log odds of 0/1
Effect Likelihood Ratio Tests
Source / Nparm / DF / L-R ChiSquare / ProbChiSq /ColonyAgg / 1 / 1 / 3.19033264 / 0.0741
Weenen:
Parameter Estimates
Term / Estimate / Std Error / ChiSquare / ProbChiSq /Intercept / -0.1708516 / 1.0664076 / 0.03 / 0.8727
ColonyAgg / 0.28219357 / 0.4527259 / 0.39 / 0.5331
For log odds of 0/1
Effect Likelihood Ratio Tests
Source / Nparm / DF / L-R ChiSquare / ProbChiSq /ColonyAgg / 1 / 1 / 0.39922245 / 0.5275
Q2B: Are the colony aggressiveness vs. persistence relationships different between wet and arid sites?
Model Structure: Simple T-test
Data Matrix:
Site ID / wet_arid / AggVSurvival / AggVEggKalkrand / arid / 2.07 / 0.37
Rehoboth / arid / 3.11 / 0.74
Upington / arid / 2.19 / 0.97
Boegoeberg / arid / 0.77 / 0.94
Ladysmith / wet / 0.25 / -0.51
Outjo / wet / -0.25 / 0.17
Rundu / wet / -0.44 / -0.07
Weenen / wet / 0.28 / 0.48
Colony Aggressiveness vs. Colony Survival