In the previous exercise you fitted a pre-conceived model which
included the main effects of the area of the forest patch
(LOGAREA), the grazing intensity (FGRAZE) and
the interaction between these two explanatory variables
(FGRAZE:LOGAREA). This was useful as a training exercise,
and might be a viable approach when analysing these data if the
experiment had been designed to test these effects only. However, if
other potentially important variables are not included in the model this
may lead to biased inferences (interpretation). Additionally, if the
goal of the analysis is to explore what models explain the data in a
parsimonious way (as opposed to formally testing hypotheses), we would
also want to include relevant additional explanatory variables.
Here we revisit the previous loyn data analysis, and ask if a
‘better’ model for these data could be achieved by including additional
explanatory variables and by performing model selection. Because we
would like to test the significance of the interaction between
LOGAREA, and FGRAZE, whilst accounting for the
potential effects of other explanatory variables, we will also include
LOGAREA, FGRAZE and their interaction in the
model as before. Including other interaction terms between other
variables may be reasonable, but we will focus only on the
FGRAZE:LOGAREA interaction as we have relatively little
information in this data set (67 observations). This will hopefully
avoid fitting an overly complex model which will estimate many
parameters for which we have very little data. This is a balance you
will all have to maintain with your own data and analyses (or better
still, perform a power analysis before you even collect your data). No
4-way interaction terms in your models please!
It’s also important to note that we will assume that all the explanatory variables were collected by the researchers because they believed them to be biologically relevant for explaining bird abundance (i.e. data were collected for a reason). Of course, this is probably not your area of expertise but it is nevertheless a good idea to pause and think what might be relevant or not-so relevant and why. This highlights the importance of knowing your study organism / study area and discussing research designs with colleagues and other experts in the field before you collect your data. What you should try to avoid is collecting heaps of data across many variables (just because you can) and then expecting your statistical models to make sense of it for you. As mentioned in the lecture, model selection is a relatively controversial topic and should not be treated as a purely mechanical process. Your expertise needs to be woven into this process otherwise you may end up with a model that is implausible or not very useful (and all models need to be useful!).
1. Import the ‘loyn.txt’ data file into RStudio and assign it to a
variable called loyn. Here we will be using all the
explanatory variables to explain the variation in bird density. If
needed, remind yourself of your data exploration you conducted
previously. Do any of the remaining variables need transforming
(i.e. AREA, DIST, LDIST) or
converting to a factor type variable (i.e. GRAZE)? Add the
transformed variables to the loyn dataframe.
2. Let’s start with a very quick graphical exploration of any
potential relationships between each explanatory variable (collinearity)
and also between our response and explanatory variables (what we’re
interested in). Create a pairs plot using the function
pairs()of your variables of interest. Hint: restrict the
plot to the variables you actually need. An effective way of doing this
is to store the names of the variables of interest in a vector
VOI <- c("Var1", "Var2", ...) and then use the naming
method for subsetting the data set Mydata[, VOI]. If you
feel like it, you can also add the correlations to the lower triangle of
the plot as you did previously (don’t forget to define the function
first).
3. Now, let’s fit our maximal model. Start with a model of
ABUND and include all explanatory variables as main
effects. Also include the interaction LOGAREA:FGRAZE but no
other interaction terms as justified in the preamble above. Don’t forget
to include the transformed versions of the variables where appropriate
(but not the untransformed variables as well otherwise you will have
very strong collinearity between these variables!). Perhaps, call this
model M1.
4. Have a look at the summary table of the model using the
summary() function. You’ll probably find this summary is
quite complicated with lots of parameter estimates (14) and P values
testing lots of hypotheses. Are all the P values less than our cut-off
of 0.05? If not, then this suggests that some form of model selection is
warranted to simplify our model.
5. Let’s perform a first step in model selection using the
drop1() function and use an F test based model
selection approach. This will allow us to decide which explanatory
variables may be suitable for removal from the model. Remember to use
the test = "F" argument to perform F tests when
using drop1(). Which explanatory variable is the best
candidate for removal and why? What hypothesis is being tested when we
do this model selection step?
6. Update and refit your model and remove the least significant
explanatory variable (from above). Repeat single term deletions with
drop1() again using this updated model. You can update the
model by just fitting a new model without the appropriate explanatory
variable and assign it to a new name (M2). Alternatively
you can use the update() function instead. I’ll show you
both ways in the solutions below.
7. Again, update the model to remove the least significant
explanatory variable (from above) and repeat single term deletions with
drop1().
8. Once again, update the model to remove the least significant
explanatory variable (from above) and repeat single term deletions with
drop1().
9. And finally, update the model to remove the least significant
explanatory variable (from above) and repeat single term deletions with
drop1().
10. If all goes well, your final model should be
lm(ABUND ~ LOGAREA + FGRAZE + LOGAREA:FGRAZE) which you
encountered in the previous exercise. Also, you may have noticed that
the output from the drop1() function does not include the
main effects of LOGAREA or FRGRAZE. Can you
think why this might be the case?
11. Now that you have your final model, you should go through your model validation and model interpretation as usual. As we have already completed this in the previous exercise I’ll leave it up to you to decide whether you include it here (you should be able to just copy and paste the code). Please make sure you understand the biological interpretation of each of the parameter estimates and the interpretation of the hypotheses you are testing.
OPTIONAL questions if you have time / energy / inclination!
A1. If we weren’t aiming to directly test the effect of the
LOGAREA:FGRAZE interaction statistically (i.e. test this
specific hypothesis), we could also have used AIC to perform model
selection. This time when we remove each term we are looking for the
model with the lowest AIC (remember that lower AIC values are better).
If you like, you can repeat the model selection you did above, starting
with the same most-complex model (M1) as before, but this time use the
drop1() function and perform model selection using AIC
instead (omitting the test = "F" argument), each time
removing the term that gives a model with a lower AIC.
A2. However, the “superpower” of AIC is the ability to simultaneously compare multiple competing models, something we are not taking advantage of when we perform a stepwise model selection process. So, another approach to selecting our best model is to decide which set of models we are going to compare before we fit any of them, then fit them all, extract the AIC values, and see which model(s) have the lowest.
Fit 5 different models with the following terms, giving each model appropriate names to distinguish from the other models we have fitted so far: 1) “LOGLDIST + YR.ISOL + ALT + LOGAREA + FGRAZE + LOGAREA:FGRAZE”, 2) “LOGLDIST + YR.ISOL + ALT + LOGAREA + FGRAZE + LOGLDIST:YR.ISOL + LOGAREA:FGRAZE”, 3) “YR.ISOL + LOGAREA + FGRAZE”, 4) “LOGAREA + FGRAZE + LOGAREA:FGRAZE”, 5) “LOGAREA + FGRAZE”. If you like, you can add further models using combinations of terms you think might give the best model. Which model do you think will have the lowest AIC?
A3. Now we have fitted our set of models, we can calculate the AIC of
each using the AIC() function. Which has the lowest? Which
has the highest? Are there any within 2 AIC of the lowest AIC? Are there
any models that differ in a single term and have an AIC difference of
about 2? What does this tell us about the additional term?
If all goes well, the best model should be
lm(ABUND ~ LOGAREA + FGRAZE + LOGAREA:FGRAZE). This is the
same model you ended up with when using the F test based model
selection in a stepwise manner. This might not always be the case and
generally speaking AIC based model selection approaches tend to favour
more complicated minimum adequate models compared to F test
based approaches.
We don’t need to re-validate or re-interpret the model, since we have already done this previously.
I guess the next question is how to present your results from the model selection process (using either F tests or AIC) in your paper and/or thesis chapter. One approach which I quite like is to construct a table which includes a description of all of our models and associated summary statistics. Let’s do this here for the AIC based model selection, but the same principles apply when using F tests (although you will be presenting F statistics and P values rather than AIC values).
Although you can use the output from the drop1() (and do
a bit more wrangling) let’s make it a little simpler by fitting all of
our models and then use the AIC() function to calculate the
AIC values for each model rather than drop1(). Note that
the values differ slightly between the two approaches; using either is
fine but best not to mix AIC values from the drop1() and AIC()
functions.
Or if you prefer a prettier version to include directly in your paper
/ thesis. You will need to install the knitr package before
you can do this. See the ‘Installing R Markdown’ in Appendix A in the
Introduction to R book for more details.
| Model | AIC | deltaAIC |
|---|---|---|
| LOGAREA + FGRAZE + LOGAREA:FGRAZE | 413.71 | 0.00 |
| LOGLDIST + YR.ISOL + ALT + LOGAREA + FGRAZE + LOGAREA:FGRAZE | 418.54 | 4.83 |
| LOGLDIST + YR.ISOL + ALT + LOGAREA + FGRAZE + LOGLDIST:YR.ISOL + LOGAREA:FGRAZE | 420.34 | 6.63 |
| LOGAREA + FGRAZE | 422.61 | 8.90 |
| YR.ISOL + LOGAREA + FGRAZE | 424.60 | 10.89 |
Which model selection approach do you prefer? Can you think of pros and cons of either approach?
Whichever model selection approach you chose, what is key to remember is to perform a fully manual and thought-through model selection process, informed by the understanding of theory in the research area, and of the research questions, with rigorous model validation throughout.
End of the model selection exercise