M. Drew LaMar

April 27, 2016

Introduction to Biostatistics, Spring 2016

- Student evaluations on blackboard (http://evals.wm.edu)
- Reading for Friday: https://qubeshub.org/collections/post/1233
- Projects

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.

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**.

**Example 18.3: Interaction zone**

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.

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

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

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