QUESTION 1 You are interested in the relationship (if any) between small mammal species richness (richness) and the connectivity of local habitat patches in which these mammals are found (Patch_conn)....

QUESTION 1

You are interested in the relationship (if any) between small
mammal species richness (richness) and the connectivity of
local habitat patches in which these mammals are found
(Patch_conn). To help test your ideas, four sizes of traps
(Trap_size; 1=small, 2=medium, 3=large, 4= extra-large) were
deployed to trap the small mammals. Below are the type I sum
of squares table and the parameter estimates for the model
relating species richness to trap size and patch connectivity.
Note that the rows in shade would not appear in R output, while the intercept would be evaluated by aliasing off the first rather than last trap size.

Tests of Between-Subjects Effects

Dependent Variable: richness

Type I Sum of
Source Squares df Mean Square

Trap_size 3.724 3 1.241 Patch_conn 2.006 1 2.006 Error 4.037 65 .062

Corrected Total9.76769 a. R Squared = .587 (Adjusted R Squared = .561)

F Sig.

19.988 .000 32.304 .000

Parameter Estimates

Dependent Variable:

richness

Parameter
Intercept
[Trap_size=1]
[Trap_size=2]
[Trap_size=3]
[Trap_size=4]
Patch_conn
a. This parameter is set to zero because it is redundant.

Sig. .000

.000 .000 .002

.

.000

Lower Bound .520

-.809 -.533 -.448

.

.653

Upper Bound .789

-.472 -.196 -.107

.

1.360

B
.655

Std. Error t
.067 9.716

95% Confidence Interval

-.641 -.364 -.278

.084 -7.599 .084 -4.322 .085 -3.250

a0

1.007

..

.177 5.684

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1. 
   

   a) What is the term used to describe general linear models with one continuous and one categorical predictor?

   
   


2. 
   

   b) Write out the formula for the full fitted model

   
   


3. 
   

   c) Describe two null hypotheses being tested and whether they can be rejected.

   
   


4. 
   

   d) You are planning a new study, this time using patch connectivity as a categorical

   
   

   predictor with three levels (high, medium and low). You have a total of 120 traps to allocate to the 4 different size treatments (nominally 1-4) and patch connectivity treatments. Produce a table to show you would distribute your 120 replicates and explain the advantages of your design.

   
   


5. 
   

   e) Species richness is effectively a count of the number of species caught in a given trap. As such, the response may not be normally distributed, especially when low numbers of species are observed. In addition, the number of species caught in neighboring traps may not be independent. If these concerns are valid, how would you address them?

   
   

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QUESTION 2

Using salt on roads and sidewalks is a common winter practice, but high levels of salt in freshwater ponds can be harmful to aquatic organisms. You have tested salt concentration at several ponds in Ottawa, as well as several other variables, and want to know which statistical model is best for predicting salt concentration. Your data has the following structure:

18198 20Highway 84 6NIndustrial 287341 12B.Road 149 3YResidential ........................

1. 
   

   a) Name one method (there are several) that you could use to select which variables are included in your best predictive model, and briefly explain what the method does.

   
   


2. 
   

   b) Why should you not just use the global model with all the variables included in the model instead of searching for the best model?

   
   


3. 
   

   c) What predictor variables are likely to be co-linear and why?

   
   


4. 
   

   d) Common sense suggests that the salt concentration of a pond and distance from a

   
   

   road will not be linear. If distance from the road is an important determinant of salt concentration but the relationship is non-linear, what additional predictor variable not on the list would you consider adding?

   
   


5. 
   

   e) You are comparing the output of two potential linear models, (salt level ~ pond

   
   

   area, and salt level ~ pond area + max depth), and you notice that the model with

   
   

   22
   max depth has a higher r , but a lower adjusted r . Why might this be the case?

   
   

   Would this suggest that “max depth” explains additional variance beyond that explained by “pond area”?

   
   

4

QUESTION 3

Jill tagged a large number of walleye fish with transmitters to let her know when an individual walleye is hanging out in the spawning reef areas. She wants to know if the two sexes can be distinguished on the basis of their spawning behaviour, which might help identify the sexes.

a) Below are her raw withsex(0 or 1) treated as the dependent variable and time of residency (residency_hours2) as a predictor variable. In no more than 5 sentences, describe the model she should fit to help answer her question, and describe what this model does.

After fitting an appropriate model, Jill’s results were as follows:

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b) What are her null and alternative hypotheses? What can you conclude from the above output?
c) Provide the fitted model equation.
d) List one assumption of the type of test that Jill employed.

e) As to be expected in ecology, there are other factors that might influence the amount of time a fish spends on the spawning habitat, such as temperature of the water and size (area) of the reef. What statistical model could she use to explore these relationships?

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QUESTION 4

The following table comes from analyzing an experiment as a randomized block (5 treatments A-E, 4 blocks).

a) Calculate the missing entries: #1, #2 & #3

b) What proportion of the variation is explained by the fitted model?

c) Why was an interaction term (treatment*block) not fitted?

d) Should the block effect be dropped from the model if it is not used for prediction? Explain.

e) What do you conclude from the above study?

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QUESTION 5

On fitting a general linear model with a categorical predictor and a continuous predictor, the following residual plots were obtained.

a) Describe and explain the above residual plots. What is a standardized residual, and what is a Q-Q plot?

b) Are the assumptions of the general linear model reasonable in this case? c) What transformation to the response variable would you suggest and why?

d) You suspect based on underlying biology that the slope of the relationship between the continuous response and continuous predictor will differ for different levels of the categorical predictor. How would you test for this possibility statistically?

e) In this case, the levels of the categorical predictor are entirely arbitrary (they refer to specific locations) and of no real interest (they would likely be different were we to repeat the experiment again). Should the categorical predictor be treated as “fixed” or “random”? Explain your reasoning.