Modelling running speed for carnivors and herbivors
S. Robin
dec 2021
Data
\(n = 49\) species (19 carnivors, 30 herbivors); \(d = 3\) covariates \(x_{ij}\):
- log10(body mass) (log10(kg)),
- metatarsal / femur ratio (mfratio)
response variable: \(y_i =\) running speed (km /h)
## clade runningspeed log10bodymass hindlength mtfratio
## Ursus_maritimus Carnivore 40 2.4232459 84.24 0.214
## Ursus_horribilis Carnivore 48 2.4001925 65.20 0.220
## Ursus_americanus Carnivore 48 1.9703469 66.83 0.200
## Nasua_narica Carnivore 27 0.6434527 22.89 0.265
## Procyon_lotor Carnivore 24 0.8450980 26.15 0.294
## Mephitis_mephitis Carnivore 16 0.3979400 14.65 0.265
##
## Carnivore Herbivore
## 19 30
Gaussian linear model for the running speed
Setting \(x_{i0} \equiv 1\) and \(\beta = [\beta_0 \dots \beta_d]^\intercal\), \[ Y_i \sim \mathcal{N}(x_i^\intercal \beta, \sigma^2). \] Or, denoting by \(i\) the clade (carnivor or herbivor) and numbering species by \(j\) within each clade \[ \mathbb{E}Y_{ij} =: \mu_{ij} = \mu + \alpha_i + \beta_2 x_{ij, 2} + + \beta_3 x_{ij, 3}. \]
##
## Call:
## lm(formula = runningspeed ~ clade + log10bodymass + mtfratio,
## data = carniherb)
##
## Residuals:
## Min 1Q Median 3Q Max
## -41.051 -7.395 1.051 7.656 49.082
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 26.188 9.387 2.790 0.00771 **
## cladeHerbivore -24.275 9.232 -2.629 0.01166 *
## log10bodymass 4.887 4.273 1.144 0.25881
## mtfratio 59.550 12.387 4.808 1.74e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 16.42 on 45 degrees of freedom
## Multiple R-squared: 0.3714, Adjusted R-squared: 0.3295
## F-statistic: 8.864 on 3 and 45 DF, p-value: 9.964e-05Gaussian linear model for the running speed with interaction
Introducing interaction so that the effect of the continuous covariates differs from one clade to another \[ \mu_{ij} = \mu + \alpha_i + \beta_2 x_{ij, 2} + \beta_3 x_{ij, 3} + \gamma_{i2} x_{ij, 2} + \gamma_{i3} x_{ij, 3}. \]
##
## Call:
## lm(formula = runningspeed ~ clade * (log10bodymass + mtfratio),
## data = carniherb)
##
## Residuals:
## Min 1Q Median 3Q Max
## -29.7331 -9.3277 0.4579 6.2951 31.0392
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -46.352 18.010 -2.574 0.013590 *
## cladeHerbivore 80.694 24.846 3.248 0.002259 **
## log10bodymass 18.938 5.382 3.519 0.001039 **
## mtfratio 210.354 38.852 5.414 2.58e-06 ***
## cladeHerbivore:log10bodymass -22.463 7.306 -3.075 0.003653 **
## cladeHerbivore:mtfratio -167.377 40.399 -4.143 0.000157 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.82 on 43 degrees of freedom
## Multiple R-squared: 0.5746, Adjusted R-squared: 0.5251
## F-statistic: 11.62 on 5 and 43 DF, p-value: 3.904e-07Model comparison
## Analysis of Variance Table
##
## Model 1: runningspeed ~ clade + log10bodymass + mtfratio
## Model 2: runningspeed ~ clade * (log10bodymass + mtfratio)
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 45 12127.9
## 2 43 8208.1 2 3919.7 10.267 0.0002264 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Analysis of covariance
With only one continuous covariate
##
## Call:
## lm(formula = runningspeed ~ -1 + clade * (mtfratio), data = carniherb)
##
## Residuals:
## Min 1Q Median 3Q Max
## -29.268 -10.960 0.725 8.487 41.617
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## cladeCarnivore -2.833 14.589 -0.194 0.846905
## cladeHerbivore 23.868 9.819 2.431 0.019116 *
## mtfratio 162.593 40.570 4.008 0.000228 ***
## cladeHerbivore:mtfratio -116.406 42.108 -2.764 0.008236 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.4 on 45 degrees of freedom
## Multiple R-squared: 0.944, Adjusted R-squared: 0.939
## F-statistic: 189.6 on 4 and 45 DF, p-value: < 2.2e-16