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Here is an example from Fowler, Cohen and Parvis (1998). An agriculturalist was interested in the effects of fertilizer load on the yield of grass. Grass seed was sown uniformly over an area and different quantities of commercial fertilizer were applied to each of ten 1 m2 randomly located plots. Two months later the grass from each plot was harvested, dried and weighed. The data are in the file fertilizer.csv.
Open the fertilizer data file.
If there is no evidence that the assumptions of simple linear regression have been violated, fit the linear model YIELD = intercept + (SLOPE * FERTILIZER). At this stage ignore any output.
Christensen et al. (1996) studied the relationships between coarse woody debris (CWD) and, shoreline vegetation and lake development in a sample of 16 lakes. They defined CWD as debris greater than 5cm in diameter and recorded, for a number of plots on each lake, the basal area (m2.km-1) of CWD in the nearshore water, and the density (no.km-1) of riparian trees along the shore. The data are in the file christ.csv and the relevant variables are the response variable, CWDBASAL (coarse woody debris basal area, m2.km-1), and the predictor variable, RIPDENS (riparian tree density, trees.km-1).
Open the christ data file.
If there is no evidence that the assumptions of simple linear regression have been violated, fit the linear model CWDBASAL = (SLOPE * RIPDENS) + intercept HINT. At this stage ignore any output.
Here is a modified example from Quinn and Keough (2002). Peake & Quinn (1993) investigated the relationship between the number of individuals of invertebrates living in amongst clumps of mussels on a rocky intertidal shore and the area of those mussel clumps.
Open the peakquinn data file.
The relationship between two continuous variables can be analyzed by simple linear regression. As with question 2, note that the levels of the predictor variable were measured, not fixed, and thus parameter estimates should be based on model II RMA regression. Note however, that the hypothesis test for slope is uneffected by whether the predictor variable is fixed or measured.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 INDIV against AREA (INDIV on y-axis, AREA on x-axis) including a lowess smoother and boxplots on the axes.
To get an appreciation of what a residual plot would look like when there is some evidence that the assumption of homogeneity of variance 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)
Transform both variables to logs (base 10), replot the scatterplot using the transformed data, refit the linear model (again using transformed data) and examine the residual plot.HINT
Nagy, Girard & Brown (1999) investigated the allometric scaling relationships for mammals (79 species), reptiles (55 species) and birds (95 species). The observed relationships between body size and metabolic rates of organisms have attracted a great deal of discussion amongst scientists from a wide range of disciplines recently. Whilst some have sort to explore explanations for the apparently 'universal' patterns, Nagy et al. (1999) were interested in determining whether scaling relationships differed between taxonomic, dietary and habitat groupings.
Open the nagy data file.
For this example, we will explore the relationships between field metabolic rate (FMR) and body mass (Mass) in grams for the entire data set and then separately for each of the three classes (mammals, reptiles and aves).
Unlike the previous examples in which both predictor and response variables could be considered 'random' (measured not set), parameter estimates should be based on model II RMA regression. However, unlike previous examples, in this example, the primary focus is not hypothesis testing about whether there is a relationship or not. Nor is prediction a consideration. Instead, the researchers are interested in establishing (and comparing) the allometric scaling factors (slopes) of the metabolic rate - body mass relationships. Hence in this case, model II regression is indeed appropriate.
A feeding ecologist wished to investigate the energetic and nutritional consequences of lactation on captive Tasmanian pademelons (Thylogale billardierii - a small wallaby). The researcher was primarily interested in compensatory alterations in food intake and chewing parameters (such as chew rate). Such measures are extremely difficult to obtain accurately and require intense investigation, thereby restricting the sample sizes. Prior to commensing the investigation, the researcher wisely decided to perform a quick power analysis so as to gauge the estimated sample size necessary to detect a linear trend in chewing rate (chews. min-1) with increasing joey mass (g). Previous research (Rose et al, 2005) into changes in milk composition and growth and joey growth in the species had demonstrated a 4-fold increase in the energy content of milk throughout lactation.