# Basic Factorial Design (cont.)

Prof Randi Garcia
February 26, 2018

1. How can you tell if one factor is inside another factor? Give an example—drawing a diagram if it helps.
2. If you have time, write any questions you have from the reading/ideas that are unclear.

### Announcements

• Project proposal due today
• HW 4, due on Wednesday
• Exam 1 passed out on Wed
• Due one week from Wed

### Agenda

• Practice recognizing designs
• Analysis of variance (cont.)
• Inside and outside factors
• ANOVA for the BF[1] Design
• Complete source table
• Checking assumptions in R

### Design Questions

1. How many structural factors are there? What might you call them?
2. How many levels does each factor have? What are they?
3. How are the factors related? (i.e., crossing, blocking)
4. What is the response variable? What's it's level of measurement?
5. What are the experimental units?
6. What's the name of this design.

### Barley Sprouting

Barley seeds are divided into 30 lots of 100 seeds each. The 30 lots are then divided at random into ten different groups of three lots each, with each group receiving a different treatment combination. The amount of water given per day is varied (4 ml or 8 ml). Also, the number of seeds sprouted is measured at different lengths of time randomly assigned to the groups. Thus, the researchers can understand the effect of the age of the seeds (1 week, 3 weeks, 6 weeks, 9 weeks, and 12 weeks), as well as water, on growth.

### 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 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.

The Canada goose is a magnificent bird, but it can be a nuisance in urban areas in large numbers. One method of population control is to addle eggs in nests, but this method can hard adult females. Would removal of the eggs at the usual hatch date prevent harm? It is suspected that females nesting together at different sites are similar to each other. We randomly select 5 different sites, and we then randomly assign 5 nests per site to the addle with no removal condition, and 5 nests per site to the addle plus removal condition. The females at the nests are banded such that survival age can be measured later.

### 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.

### ANOVA Source Table for BF[1]

• Using algebraic Notation

${y}_{ij}=\mu+{\alpha}_{i}+{e}_{ij}$

${H}_{0}:{\alpha}_{1}={\alpha}_{2}=...={\alpha}_{a}$

Source SS df MS F
Treatment $$n\sum_{i=1}^{a}(\bar{y}_{i.}-\bar{y}_{..})^{2}$$ $$a-1$$ $$\frac{{SS}_{Treatments}}{{df}_{Treatments}}$$ $$\frac{{MS}_{Treatments}}{{MS}_{E}}$$
Error $$\sum_{i=1}^{a}\sum_{j=1}^{n}({y}_{ij}-\bar{y}_{i.})^{2}$$ $$N-a$$ $$\frac{{SS}_{E}}{{df}_{E}}$$

### Treatment effects and F-ratios for all designs

• We cannot ALWAYS use the same formula for the treatment effects. It depends on inside and outside factors.
• We do not ALWAYS divide the $${MS}_{factor}$$ by $${MS}_{E}$$. To test some effects in some designs we will need a different denominator.

### Inside-outside Factors

• One factor is inside another if each group of the first (inside) fits completely inside some group of the second (outside) factor.
• Estimated effect for a factor = Average for the factor - sum of estimated effects for all outside factors.
• The sum of estimated effects for all outside factors is called the partial fit.
• Thus, the general rule is:

$Effect = Average - Partial Fit$

### Inside-outside Factors

student animal cute scary
2 cat 5 1
5 cat 5 5
1 dog 5 1
3 dog 4 2
• Draw the factor diagram as a hierarchy of inside and outside factors.
• Use the inside-outside rule to calculate effects for this example we we did for the leafhoppers.

### 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.

### Start HW 4

• Ch4: B1-3, C3, D1, RE CH 3: 3-4 (data in fig 3.21), 11-13, 17-19