Complete Block Design
Prof Randi Garcia
March 21, 2018
Reading Free-Write (5 minutes)
- What are the three ways to create blocks in a design? Give an example of each.
- Explain how blocking converts naisuance variance into a factor of the design?
Announcements
- HW 5 is due on Friday at 5pm on Moodle (HTML file)
- Rachel Shutt's talk tomorrow cancelled due to snow
- Recheduling for the 2nd or 3rd week in April
Agenda
- ANOVA for the The BF[2] design
- Blocking principal
Inside-outside Factors
- Draw the factor diagram vertically and label all inside and outside factors with arrows. Include the universal factors (benchmark and residuals).
Mathematical notation
\[ {y}_{ijk}={\mu}+{\alpha}_{i}+{\beta}_{j}+{\alpha\beta}_{ij}+{e}_{ijk} \]
- 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 BF[2]
\[ {y}_{ijk}={\mu}+{\alpha}_{i}+{\beta}_{j}+{\alpha\beta}_{ij}+{e}_{ijk} \]
| 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 | \( \sum_{i=1}^{a}\sum_{j=1}^{b}\sum_{k=1}^{n}({y}_{ijk}-\bar{y}_{ij.})^{2} \) | \( ab(n-1) \) | \( \frac{{SS}_{E}}{{df}_{E}} \) |
BF[2] in R
Example
Modern zoos try to reproduce natural habitats in their exhibits as much as possible. They try to use appropriate plants, but these plants can be infested with inappropriate insects. Cycads (plants that look vaguely like palms) can be infected with mealybugs, and the zoo wishes to test three treatments: 1) water, 2) horticultural oil, and 3) fungal spores in water. Nine infested cycads are taken to the testing area. Three branches are randomly selected from each tree, and 3 cm by 3 cm patches are marked on each branch. The number of mealybugs on the patch is counted. The three treatments then get randomly assigned to the three branches for each tree. After three days the mealybugs are counted again. The change in number of mealybugs is computed (\( before-after \)).
Example - Basic Factorial Design
Modern zoos try to reproduce natural habitats in their exhibits as much as possible. They try to use appropriate plants, but these plants can be infested with inappropriate insects. Cycads (plants that look vaguely like palms) can be infected with mealybugs, and the zoo wishes to test three treatments: 1) water, 2) horticultural oil, and 3) fungal spores in water. Nine infested cycads are taken to the testing area. The number of mealybugs on each tree is counted. The three treatments then get randomly assigned to the three trees each. After three days the mealybugs are counted again. The change in number of mealybugs is computed (\( before-after \)).
Example
Male albino laboratory rats are used routinely in many kinds of experiments. This experiment was designed to determine the requirements for protein and animo acid threonine in their food. Specifically, the experiment is interested in testing the combinations of eight levels of threonine (.2 through .9% of diet) and five levels of protein (8.68, 12, 15, 18, and 21% of diet). Baby rats were separated into five groups of 40 to form groups of approximately the same weight. The 40 rats in each group were randomly assigned to each of the 40 conditions. Body weight and food consumtption were measured twice weekly, and the average daily weight gain over 21 days was recorded.
Example - Basic Factorial Design [2]
Male albino laboratory rats are used routinely in many kinds of experiments. This experiment was designed to determine the requirements for protein and animo acid threonine in their food. Specifically, the experiment is interested in testing the combinations of eight levels of threonine (.2 through .9% of diet) and five levels of protein (8.68, 12, 15, 18, and 21% of diet). 200 baby rats were randomly assigned to each of the 40 conditions. Body weight and food consumtption were measured twice weekly, and the average daily weight gain over 21 days was recorded.
Example
This experiment is interested in the blood consicentration of a drug after it has been administered. The concentration will start at zero, then go up, and back down as it is metabolized. This curve may differ depending on the form of the drug (a solution, a tablet, or a capsule). We will use three subjects, and each subject will be given the drug three times, once for each method. The area under the time-concentration curve is recorded for each subject after each method of drug delivery.
Example - Basic Factorial Design
This experiment is interested in the blood consicentration of a drug after it has been administered. The concentration will start at zero, then go up, and back down as it is metabolized. This curve may differ depending on the form of the drug (a solution, a tablet, or a capsule). We will use nine subjects, and randomly assign subjects to one of the three delivery methods. The area under the time-concentration curve is recorded for each subject after beging given the drug.
Design Principal: Blocking
- Blocking is using a factor that is not of research interest – Affects the response
- A Block is a level of a blocking factor
- We use blocking to improve precision/power
Three Ways to Block
- Sort units into similar groups
- Albino rats
- Subdivide larger chunks of material into sets of smaller peices
- Mealybug trees
- Reuse subjects or chunks of material in each of sveral times slots
- Drug study
Complete Block Desgin, CB[1]
- Experimental units are separated into blocks of similar units
- Then each member of a block is assigned a random treatment
- Complete means that every block x treatment combination is tested
Inappropriate Insects
Modern zoos try to reproduce natural habitats in their exhibits as much as possible. They try to use appropriate plants, but these plants can be infested with inappropriate insects. Cycads (plants that look vaguely like palms) can be infected with mealybugs, and the zoo wishes to test three treatments: 1) water, 2) horticultural oil, and 3) fungal spores in water. Nine infested cycads are taken to the testing area. Three branches are randomly selected from each tree, and 3 cm by 3 cm patches are marked on each branch. The number of mealybugs on the patch is counted. The three treatments then get randomly assigned to the three branches for each tree. After three days the mealybugs are counted again. The change in number of mealybugs is computed (\( before-after \)).
Inappropriate Insects
| treatment | tree1 | tree2 | tree3 | tree4 | tree5 |
|---|---|---|---|---|---|
| oil | 4 | 29 | 14 | 14 | 7 |
| spores | -4 | 29 | 4 | -2 | 11 |
| water | -9 | 18 | 10 | 9 | -6 |
Draw the factor diagram, labeling inside outside factors.