BIOL2001 Chap.12 Multiple Regression, Interactions and Model Selection
Multiple Regression, Interactions and Model Selection
Groups, several covariates and interactions
This chapter extends the linear model in three directions. Binary and categorical covariates enter as dummy variables, with each coefficient a difference from a reference category, and an analysis of variance asks whether a factor matters as a whole. Several covariates enter one model, each coefficient a partial effect with the others held fixed.
Interaction terms, products of covariates, let the effect of one depend on another; a computed figure shows tadpole growth rising more steeply with temperature in temporary than in permanent ponds.
Choosing a model
A worked example reads interaction coefficients by writing one equation per group, and a comparison table covers categorical by continuous, categorical by categorical and continuous by continuous interactions.
Because R-squared rises with every covariate, the chapter introduces adjusted R-squared, AIC and BIC, and works an AIC comparison through weights and an evidence ratio, ending with the rules for keeping factors and main effects together.
What this chapter covers
- 01
Binary covariates coded 0 and 1
- 02
Dummy variables and the reference category
- 03
ANOVA for a factor as a whole
- 04
Partial effects in multiple regression
- 05
Interaction terms and separate slopes
- 06
Adjusted R-squared
- 07
AIC, BIC and evidence ratios
- 08
Rules for keeping terms together
Worked example · free
Two slopes from one interaction model
- 1Permanent ponds set temporary = 0: growth = 2.0 + 0.30 × temperature, slope 0.30.
- 1Temporary ponds set temporary = 1: growth = 2.5 + 0.45 × temperature, slope 0.30 + 0.15 = 0.45.
- 1At 20 °C, permanent ponds give 2.0 + 6.0 = 8.0 mg per day.
- 1At 20 °C, temporary ponds give 2.5 + 9.0 = 11.5 mg per day.
Key terms
- Dummy Variable
- A covariate coded 1 for one category and 0 otherwise.
- Reference Category
- The category without a dummy, against which every dummy coefficient is compared.
- Partial Effect
- The change in the response for a one-unit change in one covariate with the others held fixed.
- Adjusted R-Squared
- R-squared penalised for the number of parameters, so it can fall when a useless covariate is added.
- Akaike Information Criterion
- A model selection measure combining fit and the number of parameters, where smaller values are better.
- Evidence Ratio
- The ratio of AIC weights showing how much more support one model has than another.
- Analysis of Variance
- A test of whether the mean response is the same across all categories of a factor.
Multiple Regression, Interactions and Model Selection FAQ
How many dummy variables does a factor with k categories need?
It needs k minus 1 dummy variables. The category left without a dummy becomes the reference, and each dummy coefficient is the difference in the mean response between its category and that reference. R builds the dummies itself when the covariate is a factor.
Why can R-squared not be used to choose between models?
R-squared increases whenever a covariate is added, even one that is pure noise, so it always favours the largest model. Adjusted R-squared, AIC and BIC add a penalty for each parameter, so a covariate improves them only if it adds more fit than it costs.
What does an interaction coefficient mean?
It is the amount by which the slope of one covariate changes for each unit of the other, or between a category and the reference. For a dummy interaction, the reference group's slope is the main effect and the other group's slope is the main effect plus the interaction.
Should I keep main effects when an interaction is in the model?
Yes. The lecture's rule is to keep both main effects whenever their product is included, to include or drop all categories of a factor together, and to keep lower polynomial terms such as x and x squared when x cubed is used.
Assessment move
For every model output, write the fitted equation first, then the equation for each category by substituting the dummies. Practise one categorical model, one with two covariates and one with an interaction, interpreting every coefficient with its units and what is held fixed. Compute AIC weights for a set of models by hand once to see how the evidence ratio behaves.
Use the hotel occupation models from the Week 8 workshop as a template: interpret the model with temperature alone, then with city added, then with their interaction, and note how each coefficient's meaning shifts.
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