1. Either create a new R script (perhaps call it ‘linear_model_1’) or
continue with your previous data exploration R script in your RStudio
Project. Again, make sure you include any metadata you feel is
appropriate (title, description of task, date of creation etc) and don’t
forget to comment out your metadata with a # at the
beginning of the line.
2. Import the data file ‘loyn.txt’ you used in the previous exercise
into R and remind yourself of the data exploration of these data you
have already performed (graphical data exploration exercise). The aim of
this exercise is to get familiar with fitting a simple linear model with
a continuous response variable, bird abundance (ABUND) and
a single continuous explanatory variable forest area (AREA)
in R. Ignore the other explanatory variables for now.
3. Create a scatterplot of bird abundance and forest patch area to
remind yourself what this relationship looks like. You may have to
transform the AREA variable to address potential issues
that you identified in your previous data exploration. Try to remember
which is your response variable (y axis) and which is your explanatory
variable (x axis). Now fit an appropriate linear model to describe this
relationship using the lm() function. Remember to use the
data = argument. Assign this linear model to an
appropriately named object (loyn_lm if you imagination
fails you!).
4. Display the ANOVA table by using the anova() function
on your model object. What is the null hypothesis you are testing here?
Do you reject or fail to reject this null hypothesis? Explore the ANOVA
table and make sure you understand the different components. Refer back
to the lectures if you need to remind yourself or ask an instructor to
take you through it.
5. Now display the table of parameter (coefficient) estimates using
the summary() function on your model object. Again, make
sure you understand the different components of this output and be sure
to ask if in doubt. What are the estimates of the intercept and slope?
Write down the word equation of this linear model including your
parameter estimates (hint: think y = a + bx).
6. What is the null hypothesis associated with the intercept? What is the null hypothesis associated with the slope? Do you reject or fail to reject these hypotheses?
7. Looking again at the output from the summary()
function how much variation in bird abundance is explained by your log
transformed AREA variable?
8. Now onto a really important part of the model fitting process.
Let’s check the assumptions of your linear model by creating plots of
the residuals from the model. Remember, you can easily create these
plots by using the plot() function on your model object
(loyn_lm or whatever you called it). Also remember that if
you want to see all plots at once then you should split your plotting
device into 2 rows and 2 columns using par(mfrow = c(2,2))
before you create the plots (Section
4.4).
Can you remember which plot is used to check the assumption of normality of the residuals? What is you assessment of this assumption? Next, check the homogeneity of variance of residuals assumption. Can you see any patterns in the ‘residuals versus fitted’ or the ‘Scale-Location’ plots? Is there more or less equal spread of the residuals? Finally, take a look at the leverage and Cooks distance plot to assess whether you have any residuals with high leverage or any influential residuals. What is your assessment? Write a couple of sentences to summarise your assessment of the modelling assumptions as a comment in your R code.
9. Using your word equation from Question 5, how many birds do you
predict if AREA is 100?
10. Calculate the fitted values from your model using the
predict() function and store these predicted values in an
object called pred_vals. Remember, you will first need to
create a dataframe object containing the values of log transformed
AREA you want to make predictions from. Refer back to the
model interpretation video if you need a quick reminder of how to do
this or ask an instructor to take you through it if you’re in any doubt
(they’d be happy to take you through it).
11. Now, use the plot() function to plot the
relationship between bird abundance (ABUND) and your log
transformed AREA variable. Also add some axes labels to aid
interpretation of the plot. Once you’ve created the plot then add the
fitted values calculated in Question 10 as a line on the plot (you will
need to use the lines() function to do this but only after
you have created the plot).
12. OK, this is an optional question so feel free to skip if you’ve
had enough! (you can find the R code for this question in the exercise
solutions if you want to refer to it at a later date). Let’s recreate
the plot you made in Question 11, but this time we’ll add the 95%
confidence intervals in addition to the fitted values. Remember, you
will need to use the predict() function again but this time
include these.fit = TRUE argument (store these new values
in a new object called pred_vals_se). When you use the
se.fit = TRUE argument with the predict()
function the returned object will have a slightly different structure
compared to when you used it before. Use the str() function
on the pred.vals.se to take a look at the structure. See if
you can figure out how to access the fitted values and the standard
errors. Once you’ve got your head around this you can now use the
lines() function three times to add the fitted values (as
before) and also the upper and lower 95% confidence intervals. Don’t
forget, if you want the 95% confidence intervals then you will need to
multiply your standard error values by the critical value of 1.96. If
you need a reminder then take a look at the video on confidence
intervals on the course MyAberdeen site (in the ‘Introductory statistics
lectures’ folder which you can find in the ‘Course Information’ learning
module).
13. And another optional question (honestly, it’s optional!). This
time plot the relationship between bird abundance (ABUND)
and the original untransformed AREA variable. Now
back-transform your fitted values (remember you got these with the
predict() function) to the original scale and add these to
the plot as a line. Hint 1: you don’t need to reuse the
predict() function, you just need to back-transform your
my.data$LOGAREA values. Hint 2: remember if you used a
log10 transformation (log10()) then you can
back-transform using 10^my.data$LOGAREA and if you used a
natural log transformation then use exp(my.data$LOGAREA) to
back-transform. Comment on the differences between the plot on the
transformed (log) scale and the plot on the back-transformed scale in
your R script.
End of the linear model with single continuous explanatory variable exercise