Chapter 6 - Binomial Regression

Moth Coloration and Natural Selection

Lab Setup

  1. Copy the project lab folder located at Home -> STA363_inst_files-> labs. If you check the box next to the folder name, then click the small gear icon you can “copy to” and put a copy of the folder in your newly created folder.
  2. Now, click File-> Open Project and navigate to the project file in the folder you just copied.
  3. You can place your responses in the file qmd file included.

Introduction

An article in the Journal of Animal Ecology by Bishop (1972) investigated whether moths provide evidence of “survival of the fittest” with their camouflage traits. Researchers glued equal numbers of light and dark morph moths in lifelike positions on tree trunks at 7 locations from 0 to 51.2 km from Liverpool. They then recorded the numbers of moths removed after 24 hours, presumably by predators. The hypothesis was that, since tree trunks near Liverpool were blackened by pollution, light morph moths would be more likely to be removed near Liverpool. The data is found in moth.csv, and relevant R code can be found under Moths.Rmd. Variables include:

  • MORPH = light or dark
  • DISTANCE = kilometers from Liverpool
  • PLACED = number of moths of a specific morph glued to trees at that location
  • REMOVED = number of moths of a specific morph removed after 24 hours

Questions

  1. What do you think of the study design? Any suggestions for improvement?

  2. What are logits, and why would we want to plot logits vs. distance (rather than, say, proportion removed vs. distance)?

  3. What can we conclude from the empirical logit plots?

Models breg2 and breg2a

  1. Interpret the 3 coefficient estimates from model “breg2”. Note that breg2 and breg2a provide two alternative ways to express the same model…

  2. What are the implications to using MORPH rather than “dark” in breg2b?

  3. How do the predicted logits from breg2 fit the actual data? Note that the predict() function with type=”link” returns predicted logits, while type=”response” returns predicted probabilities…

Models breg3

  1. Interpret the 4 coefficients estimates from model breg3.

  2. Test the significance of the interaction term in breg3 in two ways. Do both methods agree?

  3. Test the goodness of fit for model breg3. What can we conclude about this model?

  4. Is there evidence of extra-binomial variation (overdispersion) in breg3?

Models breg4 and breg4a

  1. Regardless of your answer to (9), repeat (7) after adjusting for overdispersion (breg4).

  2. Compare confidence intervals for the interaction term in breg3 with and without adjusting for overdispersion, and with and without using profile likelihoods.

Model breg5 and breg5a

  1. What are the implications in breg5 of including DISTANCE as a factor variable? How does this change model interpretations? Does it lead to an improved model?

  2. What happens if we expand the data set to contain one row per moth (968 rows)? Now we can run a logistic regression model. How does the logistic regression model “lreg1” compare to the binomial regression model breg3? What are similarities and differences? Would there be any reason to run a binomial regression rather than a logistic regression in a case like this?