QK Box 8.10

We will use these data to examine the regression of local species richness against regional species richness just for North America and at a sampling scale of 10% of the region. Although there was some evidence that both local and regional species richness were skewed, we will, like the original authors, analyze untransformed variables. Caley and Schluter (1997) forced their models through the origin, but because that can make interpretation more difficult, we will include an intercept in the models.

This example was used in the first edition; the data file is here

Caley, M. J. & Schluter, D. (1997). The relationship between local and regional diversity. Ecology, 78, 70-80.

Preliminaries

First, load the required packages (car)

Import caley data file (caley.csv)

caley <- read.csv("../data/caley.csv")
head(caley,10)

NA

Fit polynomial model

Create addtional predictor that is regional richness² and fit model with two predictors

caley$rspp10sq <- (caley$rspp10)^2
caley.lm <- lm(lspp10~rspp10 + rspp10sq, data=caley)
tidy(caley.lm, conf.int=TRUE)

Check residuals for quadratic model

plot(caley.lm)

augment(caley.lm)

Check collinearity

vif(lm(lspp10~rspp10 + rspp10sq, data=caley))

  rspp10 rspp10sq 
    15.2     15.2 

Fit simpler model

caley.lm1 <- lm(lspp10~rspp10, data=caley)
tidy(caley.lm1)

Compare fit of two models - test of whether quadratic makes a difference

anova(caley.lm1, caley.lm)

Analysis of Variance Table

Model 1: lspp10 ~ rspp10
Model 2: lspp10 ~ rspp10 + rspp10sq
  Res.Df  RSS Df Sum of Sq    F Pr(>F)  
1      6 1299                           
2      5  377  1       923 12.2  0.017 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Now use centred predictor

caley$rspp10c <- scale(caley$rspp10, center=TRUE, scale=FALSE)
caley$rspp10csq <- (caley$rspp10c)^2
caley.lm3 <- lm(lspp10~rspp10c + rspp10csq, data=caley)
tidy(caley.lm3, conf.int=TRUE)

vif(lm(lspp10~rspp10c + rspp10csq, data=caley))

  rspp10c rspp10csq 
     1.02      1.02