Question

The article “Estimating Population Abundance in Plant Species With Dormant Life-Stages: Fire and the Endangered Plant Grevillea caleye R. Br.” (T. Auld and J. Scott, Ecological Management and Restoration, 2004:125-129) presents estimates of population sizes of a certain rare shrub in areas burnt by fire. The following table presents population counts and areas (in m^2) for several patched containing the plant:

Area	Population	Area	Population
3739	3015	2521	707
5277	1847	213	113
400	17	11958	1392
345	142	1200	157
392	40	12000	711
7000	2878	10880	74
2259	223	841	1720
81	15	1500	300
33	18	228	31
1254	229	228	17
1320	351	10	4
1000	92

1. Create a scatter plot for the two variables, does there seem to be a linear relationship? (Do this scatter plot by hand)
2. Compute the least-square line. (show your work for the sum of squares, slope and intercept)
3. The scatter plot done in part a) is based on the model form y(hat)=a+bx. Create a scatter plot of y versus lnx. What do you notice? (Note: you can find ln y and x using excel, but show your values. Do this scatter plot by hand)
4. In part c), the data is considered to be “transformed”. Of this “transformed” data, compute the least-square line for predicting lny from lnx. (show all your work)
5. In part c), the data is considered to be “transformed”. Of this “transformed” data, calculate the value of Pearson’s correlation coefficient and interpret your value. Also, calculate the value of Pearson’s correlation coefficient of the original data, interpret your result. (show all your work by hand)
6. Calculate the proportion of observed variation in “y” for the original data. Explain your result. (show your work for SSE)
7. Calculate the residuals of this original data using Microsoft Excel (Print off your residual values in excel). Create a residual plot by hand. Comment on your plot.

Answer 1

a) Create a scatter plot for the two variables, does there seem to be a linear relationship? (Do this scatter plot by hand)

data=read.csv('data.csv')
#install.packages("ggplot2")
library(ggplot2)
ggplot(data, aes(x=data$Area , y=data$Population)) +
geom_point()

b) Compute the least-square line. (show your work for the sum of squares, slope and intercept)

ggplot(data, aes(x=data$Area , y=data$Population)) +
geom_point()+
geom_smooth(method=lm, se=FALSE)

c)

The scatter plot done in part a) is based on the model form y(hat)=a+bx. Create a scatter plot of y versus lnx. What do you notice? (Note: you can find ln y and x using excel, but show your values. Do this scatter plot by hand)

ggplot(data, aes(x=log(data$Area) , y=data$Population)) +
geom_point()+
geom_smooth(method=lm, se=FALSE)

We observe that this model is a better fit for the data than the earlier one.

d) In part c), the data is considered to be “transformed”. Of this “transformed” data, compute the least-square line for predicting lny from lnx. (show all your work)

Solution:

model<-lm(formula= log(Population)~log(Area),data = data)
summary(model)
model