Multiple explanatory variables (cont'd)

M. Drew LaMar
April 27, 2016

https://xkcd.com/605/

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Factorial designs using GLMs

Linear model Other name Example study design
\( Y = \mu + A + B + A*B \) Two-way, fixed-
effects ANOVA
Factorial experiment
\( Y = \mu + A + b + A*b \) Two-way, mixed-
effects ANOVA
Factorial experiment
\( Y = \mu + X + A \) Analysis of covariance (ANCOVA) Observational study

The explanatory variables are called factors, as they represent treatments of direct interest.

\( \mathrm{A} \) and \( \mathrm{B} \) are called main effects; they represent effects of each factor alone, when averaged over the categories of the other factor.

Factorial designs using GLMs

Linear model Other name Example study design
\( Y = \mu + A + B + A*B \) Two-way, fixed-
effects ANOVA
Factorial experiment
\( Y = \mu + A + b + A*b \) Two-way, mixed-
effects ANOVA
Factorial experiment
\( Y = \mu + X + A \) Analysis of covariance (ANCOVA) Observational study

The explanatory variables are called factors, as they represent treatments of direct interest.

\( \mathrm{A*B} \) is the interaction term.

Factorial designs using GLMs

Factorial designs using GLMs

Example 18.3: Interaction zone

Factorial designs using GLMs

Harley (2003) investigated how herbivores affect the abundance of plants living in the intertidal habitat of coastal Washington using field transplants of a red alga, Mazzaella parksii. The experiment also examined whether the effect of herbivores on the algae depended on where in the intertidal zone the plants were growing. Thirty-two study plots were established just above the low-tide mark, and another 32 plots were set at mid-height between the low- and high-tide marks. Using copper fencing, herbivores were excluded from a randomly chosen half of the plots at each height.

Factorial designs using GLMs

\[ \begin{align} \mathrm{ALGAE} = & \mathrm{CONSTANT} + \mathrm{HERBIVORY} + \mathrm{HEIGHT} +\\ & \mathrm{HERBIVORY*HEIGHT} \end{align} \]

Factorial designs using GLMs

\[ \begin{align} \mathrm{ALGAE} = & \mathrm{CONSTANT} + \mathrm{HERBIVORY} + \mathrm{HEIGHT} +\\ & \mathrm{HERBIVORY*HEIGHT} \end{align} \]

We need to examine the improvement in fit of the model to the data with and without each term (i.e. main effects and interaction effect).

Question: Does herbivory have an effect on mean algal cover?

Null Model (Type I): \[ \mathrm{ALGAE} = \mathrm{CONSTANT} \]

Alt Model: \[ \mathrm{ALGAE} = \mathrm{CONSTANT} + \mathrm{HERBIVORY} \]

Factorial designs using GLMs