Two-Way Factorial Designs II
2024-03-28
Announcements
- Mini-Project 1 due Friday Nov 1 11:59p on Moodle
- Office hours
- Friday 3:00p - 4:00p
- Where to get MP1 help
- Office hours
- Post questions to #mini-project1 channel on Slack!
- Direct message me on Slack.
Agenda
- Finish Assessing conditions in block designs
- Two-Way Factorial Design
Assessing conditions in block designs
New Example Studies
Paper Helicopters
Paper helicopters can be cut from one half of an 8.5 by 11 sheet of paper. We can conduct an experiment by dropping helicopters from a fixed height and clocking the time it takes to drop. We can vary wing length: 4.25 in, 4.0 in, 3.75 in, and 3.5 in, as well as body width: 3.25 in, 3.75 in, 4.0 in, and 4.25 in. We’ll make 32 planes and randomly assign them to the 16 combinations.
Parks and Recreation
Deputy director of the Pawnee Parks and Rec department, Leslie Knope, needs to know how resistant different vegetative types are to trampling so that the number of visitors can be controlled in sensitive areas. Twenty lanes of a park are established, each .5 m wide and 1.5 m long. These twenty lanes are randomly assigned to five treatments: 0, 25, 75, 200, or 500 walking passes. Each pass consists of a 70-kg individual wearing boots, walking in a natural gait. One year after trampling, the average height of the vegetation along the lanes are measured.
Design Name Disambiguation
One-Way Design
- If the factor is a true experimental factor
- One-Way Randomized Design
Block design
- If every block gets every treatment
- Complete Block Design
- If the factor of interest is experimental
- Randomized Complete Block Design
Design Name Disambiguation
Two-Way Factorial Design
- If at least one factor of interest is experimental
- Randomized Two-Way Factorial
- There is no blocking in this design
Piglets
It seems natural to think that adding the right vitamins to a pig’s diet might produce fatter pigs faster. You’ve decided to study the effects of B12 in two doses (0mg and 5mg). But pigs have bacteria living in their intestines that might prevent the uptake of vitamins, so you decided to give antibiotics to the pigs in one of two doses (0mg or 40 mg). You design your experiment in such a way that 3 piglets are randomly assigned to each of the 4 treatment conditions. You measure their weight every day, and take each pig’s average daily weight gain as your final number recorded.
Two-Way Factorial Design Factor Diagram
- Draw the factor diagram
Research Questions in a Two-Way Factorial Design
- Does treatment A have an effect on the response variable?
- Is there a main effect of factor A?
- Does treatment B have an effect on the response variable?
- Is there a main effect of factor B?
- Does being in a specific combination of treatments have an effect over and above the additive effects of treatment A and B alone?
- Is there an interaction between factor A and factor B?
Two-Way Factorial Design Formulas
\[{y}_{ijk}={\mu}+{\alpha}_{i}+{\beta}_{j}+{\alpha\beta}_{ij}+{e}_{ijk}\]
- Where \(i\), from 1 to \(a\), is the level of the first factor,
- \(j\), from 1 to \(b\), is the level of the second factor,
- and \(k\), from 1 to \(n\), is the observation in each cell.
Sum of Squares (SS)
\[{SS}_{A} = \sum_{i=1}^{a}bn(\bar{y}_{i..}-\bar{y}_{…})^{2}\]
\[{SS}_{B} = \sum_{j=1}^{b}an(\bar{y}_{.j.}-\bar{y}_{…})^{2}\]
\[{SS}_{AB} = n\sum_{i=1}^{a}\sum_{j=1}^{b}(\bar{y}_{ij.}-\bar{y}_{i..}-\bar{y}_{.j.}+\bar{y}_{…})^{2}\]
\[{SS}_{E} = \sum_{i=1}^{a}\sum_{j=1}^{b}\sum_{k=1}^{n}({y}_{ijk}-\bar{y}_{ij.})^{2}\]
Degrees of Freedom (df)
\[{df}_{A}=a-1\]
\[{df}_{B}=b-1\]
\[{df}_{AB}=(a-1)(b-1)\]
\[{df}_{E}=ab(n-1)\]
Mean Squares (MS)
\[{MS}_{A}=\frac{{SS}_{A}}{{df}_{A}}\]
\[{MS}_{B}=\frac{{SS}_{B}}{{df}_{B}}\]
\[{MS}_{AB}=\frac{{SS}_{AB}}{{df}_{AB}}\]
\[{MS}_{E}=\frac{{SS}_{E}}{{df}_{E}}\]
F-ratios and the F-distribution
The ultimate statistics we want to calculate is Variability in treatment effects/Variability in residuals. The F-ratio.
\[F = \frac{{MS}_{A}}{{MS}_{E}}\]
\[F = \frac{{MS}_{B}}{{MS}_{E}}\]
\[F = \frac{{MS}_{AB}}{{MS}_{E}}\]
ANOVA Source Table for Two-Way Factorial
\[{y}_{ijk}={\mu}+{\alpha}_{i}+{\beta}_{j}+{\alpha\beta}_{ij}+{e}_{ijk}\]
| Source | SS | df | MS | F |
|---|---|---|---|---|
| Treatment A | \(\sum_{i=1}^{a}bn(\bar{y}_{i..}-\bar{y}_{…})^{2}\) | \(a-1\) | \(\frac{{SS}_{A}}{{df}_{A}}\) | \(\frac{{MS}_{A}}{{MS}_{E}}\) |
| Treatment B | \(\sum_{j=1}^{b}an(\bar{y}_{.j.}-\bar{y}_{…})^{2}\) | \(b-1\) | \(\frac{{SS}_{B}}{{df}_{B}}\) | \(\frac{{MS}_{B}}{{MS}_{E}}\) |
| Interaction AB | \(n\sum_{i=1}^{a}\sum_{j=1}^{b}(\bar{y}_{ij.}-\bar{y}_{i..}-\bar{y}_{.j.}+\bar{y}_{…})^{2}\) | \((a-1)(b-1)\) | \(\frac{{SS}_{AB}}{{df}_{AB}}\) | \(\frac{{MS}_{AB}}{{MS}_{E}}\) |
| Error | \(\sum_{i=1}^{a}\sum_{j=1}^{b}\sum_{k=1}^{n}({y}_{ijk}-\bar{y}_{ij.})^{2}\) | \(ab(n-1)\) | \(\frac{{SS}_{E}}{{df}_{E}}\) |
Interaction Graphs
For each of the following interaction graphs, answer the following questions with YES or NO.
- Is there a main effect of B12?
- Is there a main effect of antibiotics?
- Is there an interaction between B12 and antibiotics?
Practice 1
Practice 1
- YES
- YES
- NO
Practice 2
Practice 2
- NO
- YES
- NO
Two-Way Factorial in R
Teaching Methods
A professor wanted to compare three different teaching methods to determine how students
would perceive the course: 1) instructionist, 2) inquiry-based, and 3) team-based. She randomly assigned the same class (same topic different students) from 6 different semesters to treatments. At the end of the semester students were asked to rate the course on a 5-point scale, and the average class rating was calculated.
Anxiety and Memory
A psychologist wants to study the effect of anxiety on 4 different types of memory. Twelve participants are assigned to one of two anxiety conditions: 1) low anxiety group is told that they will be awarded $5 for participation and $10 if they remember sufficiently accurately, and 2) high anxiety group is told they will be awarded $5 for participation and $100 if they remember sufficiently accurately. All subjects perform four memory trials in random order, testing 4 different types of memory. The number of errors on each trial is recorded.