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Paruelo & Lauenroth (1996) analyzed the geographic distribution and the effects of climate variables on the relative abundance of a number of plant functional types (PFT's) including shrubs, forbs, succulents (e.g. cacti), C3 grasses and C4 grasses. They used data from 73 sites across temperate central North America (see pareulo.syd) and calculated the relative abundance of C3 grasses at each site as a response variable
Open the paruelo data file. HINT.
We obviously cannot easily incorporate all 6 predictors into the one model, because of the collinearity problem. Paruelo and Lauenroth (1996) separated the predictors into two groups for their analyses. One group included LAT and LONG and the other included MAP, MAT, JJAMAP and DJFMAP. We will focus on the relationship between the square root relative abundance of C3 plants and latitude and longitude. This relationship will investigate the geographic pattern in abundance of C3 plants.
Loyn (1987) modeled the abundance of forest birds with six predictor variables (patch area, distance to nearest patch, distance to nearest larger patch, grazing intensity, altitude and years since the patch had been isolated).
Open the loyn data file. HINT.
Since none of the predictor variables are highly correlated to one another, we can include all in the linear model fitting.
In this question, we are not so interested in hypothesis testing. That is, we are not trying to determine if there is a significant effect of one or more predictor variables on the abundance of forest birds. Rather, we wish to either (or both);
Question 3 - Hierachical partitioning An alternative model selection procedure is called hierarchical partitioning. Hierarchical partitioning essentially determines the contributions of each predictor variable as both an individual predictor as well as a joint predictor in explaining the variation in the response variable. We will use hierarchical partitioning for model selection on the Loyn (1987) data set.
Rademaker and Cerqueira (2006), compiled data from the literature on the reproductive traits of opossoms (Didelphis) so as to investigate latitudinal trends in reproductive output. In particular, they were interested in whether there were any patterns in mean litter size across a longitudinal gradient from 44oN to 34oS. Analyses of data compiled from second hand sources are called metaanalyses and are very usefull at revealing overal trends across a range of studies.
Open the rademaker data file.
The main variables of interest in this data set are MLS (mean litter size) and LATITUDE. The other variables were included so as to enable you to see how meta data might be collected and collated from a number of other sources.
The relationship between two continuous variables can be analyzed by simple linear regression, as was seen in question 1. Before performing the analysis we need to check the assumptions. To evaluate the assumptions of linearity, normality and homogeneity of variance, construct a scatterplot of MLS against LATITUDE including a lowess smoother and boxplots on the axes. (HINT)
To get an appreciation of what a residual plot would look like when there is some evidence that the linearity assumption has been violated, perform the simple linear regression (by fitting a linear model) purely for the purpose of examining the regression diagnostics (particularly the residual plot)
For this sort of trend that is clearly non-linear (yet the boxplots suggest normal data), transformations are of no use. Therefore, rather than attempt to model the data on a simple linear relationship (straight line), it is better to attempt to model the data on a curvilinear linear relationship (curved line). Note it is important to make the distinction between line (relationship) linearity and model linearity
Peake and Quinn (1993) investigated the relationship between the size of mussel clumps (m2) and the number of other invertebrate species supported.
Open the peake data file. HINT.