Split Plot/Repeated Measures Design II

1. There are two types of error in a split plot design. Explain the kids on boats analogy from page 277. Which type of error is represented by what movement in the analogy?

1. There are two movements, the boats moving on the water, and the children moving on the ships. The boat movement is the block variance, and the children's movement is the residual variance.

• Quiz1 grades posted; most MP1 grades posted
• MP2 pre-approval due Friday
  • Changes to mp2 requirements!
• HW6 is due tomorrow morning
  • Sign up for office hours
• HW7 due on Thurs at 11:55p
  • Come with questions to tonight's session
  • RE CH 7: A1-A3, A6, B5-B6, C1-C3, C6-C9
  • C2 example on p. 240

• MP1 feedback
• Latin Square design
• Split plot designs
  • Crossing versus nesting

• Directional hypotheses please.
• Dealing with floor effects and ceiling effects.
• Issue with confusing the N assumption and S assumption.
• Interpret your results in context please.
• If something goes wrong with data collection or the design, give a full description/explanation of what went wrong and why.
• Please proof read. Utilize the Jacobson center!

This experiment is interested in the blood concentration 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.

Treatments:

• Solution is treatment A
• Tablet is treatment B
• Capsule is treatment C

	period 1	period 2	period 3
subject 1	A 1799	C 2075	B 1396
subject 2	C 1846	B 1156	A 868
subject 3	B 2147	A 1777	C 2291

Factor diagram for the Latin Square

The actual data structure for analysis is “long” format

subject	treatment	period	group	c_curve
1	solution	1	A	1799
1	capsule	2	C	1846
1	tablet	3	B	2147
2	capsule	1	C	2075
2	tablet	2	B	1156
2	solution	3	A	1777
3	tablet	1	B	1396
3	solution	2	A	868
3	capsule	3	C	2291

\[ {y}_{ijk}={\mu}+{\alpha}_{i}+{\beta}_{j}+{\tau}_{k}+{e}_{ijk} \]

• \( {\mu} \) is the benchmark
• \( {\alpha}_{i} \) is the row effect
• \( {\beta}_{j} \) is the column effect
• \( {\tau}_{k} \) is the treatment effect
• There are p rows, columns, and treatments

Source	SS	df	MS	F
rows	\( \sum_{i=1}^{p}p(\bar{y}_{i..}-\bar{y}_{...})^{2} \)	\( p-1 \)	\( \frac{{SS}_{A}}{{df}_{A}} \)	\( \frac{{MS}_{A}}{{MS}_{E}} \)
columns	\( \sum_{j=1}^{p}p(\bar{y}_{.j.}-\bar{y}_{...})^{2} \)	\( p-1 \)	\( \frac{{SS}_{B}}{{df}_{B}} \)	\( \frac{{MS}_{B}}{{MS}_{E}} \)
treatment	\( \sum_{k=1}^{p}p(\bar{y}_{..k}-\bar{y}_{...})^{2} \)	\( p-1 \)	\( \frac{{SS}_{T}}{{df}_{T}} \)	\( \frac{{MS}_{T}}{{MS}_{E}} \)
Error	\( \sum_{i=1}^{p}\sum_{j=1}^{p}({y}_{ijk}-\bar{y}_{i..}-\bar{y}_{.j.}-\bar{y}_{..k}+2\bar{y}_{..})^{2} \)	\( (p-1)(p-2) \)	\( \frac{{SS}_{E}}{{df}_{E}} \)

ls_mod <- lm(c_curve ~ treatment + period + subject, data = bioequivalence)

anova(ls_mod)

Analysis of Variance Table

Response: c_curve
          Df Sum Sq Mean Sq F value   Pr(>F)   
treatment  2 608891  304445  67.733 0.014549 * 
period     2 928006  464003 103.231 0.009594 **
subject    2 261115  130557  29.047 0.033282 * 
Residuals  2   8990    4495                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

bioequivalence <- bioequivalence %>%
  mutate(fitted = fitted(ls_mod), 
         residuals = residuals(ls_mod))

ggplot(bioequivalence, aes(x = fitted, residuals)) +
  geom_point() +
  geom_hline(yintercept = 0, color = "red")

How would you conduct this study?

A plant breeder wishes to study the effects of soil drainage and variety of tulip bulbs on flower production. Twelve 3m by 10m experimental sites are available in the test garden–each is a .5m deep trench. You can manipulate soil drainage by changing the ratio of sand to clay for the soil you put in a trench. After talking to your collaborator, you decided that four different levels of soil drainage would suffice. You'll be testing 15 different types of tulips, and measuring flower production in the spring.

If you suspect a design in a split-plot design, you should be able to answer the following questions:

1. What are the whole plots, that is, what is the blocking factor?
2. What is the between-blocks factor? Is it observational or experimental?
3. What is the within-blocks factor? Is it observational or experimental?

A plant breeder wishes to study the effects of soil drainage and variety of tulip bulbs on flower production. Twelve 3m by 10m experimental sites are available in the test garden–each is a .5m deep trench. You can manipulate soil drainage by changing the ratio of sand to clay for the soil you put in a trench. After talking to your collaborator, you decided that four different levels of soil drainage would suffice. You'll be testing 15 different types of tulips, and measuring flower production in the spring.

1. Crossing: Two sets of treatments are crossed if all possible combinations of treatments occur in the design. The design is called a two-way factorial and has factorial treatment structure.
2. Nesting: One factor is nested within another if each level of the first (“inside”) factor occurs with exactly one level of the second (“outside”) factor.

The disease diabetes affects the rate of turnover of lactic acid in a system of biochemical reactions called the Cori cycle. This experiment compares two methods of using radioactive carbon-14 to measure rate of turnover. Method 1 is injection all at once, and method 2 is infused continuously. 10 dogs were sorted into two groups, 5 were controls and 5 had their pancreas removed (to make it diabetic). The rate of turnover was then measured twice for each dog, once for each method. The order of the two methods was randomly assigned.

Draw the factor diagram for the data on page 263.

• Can use split plot language if blocking is created by sub-dividing blocks (whole plot and subplot factors)
• We can use the repeated measures language if blocking is created by reusing subjects/material (within and between subjects factors)
• We can always use the terms blocks and within-blocks terminology

\[ {y}_{ijk}={\mu}+{\alpha}_{i}+{\beta}_{j(i)}+{\gamma}_{k}+({\alpha\gamma})_{ik}+{e}_{ijk} \]

• \( {\mu} \) is the benchmark
• \( {\alpha}_{i} \) effect of level i of the between-blocks factor, \( i \) from \( 1 \) to \( a \)
• \( {\beta}_{j(i)} \) effect of block \( j \) (for level \( i \) of the between block factor), \( j \) from \( 1 \) to \( n \)
• \( {\gamma}_{k} \) effect of level \( k \) of the within-block factor, \( k \) from \( 1 \) to \( t \)
• \( ({\alpha\gamma})_{ik} \) interaction effect for level \( i \) of the between-blocks factor with level \( k \) of the within-blocks factor

Source	SS	df	MS	F
Between	\( t\frac{N}{a}\sum_{i=1}^{a}(\bar{y}_{i..}-\bar{y}_{...})^{2} \)	\( a-1 \)	\( \frac{{SS}_{A}}{{df}_{A}} \)	\( \frac{{MS}_{A}}{{MS}_{B}} \)
Blocks	\( t\sum_{i=1}^{a}\sum_{j=1}^{n}(\bar{y}_{ij.}-\bar{y}_{i..})^{2} \)	\( N-a \)	\( \frac{{SS}_{B}}{{df}_{B}} \)	\( \frac{{MS}_{B}}{{MS}_{E}} \)
Within	\( Na\sum_{k=1}^{K}(\bar{y}_{..k}-\bar{y}_{...})^{2} \)	\( t-1 \)	\( \frac{{SS}_{T}}{{df}_{T}} \)	\( \frac{{MS}_{T}}{{MS}_{E}} \)
Interaction	\( \sum_{i=1}^{a}\sum_{k=1}^{t}\frac{N}{a}(\bar{y}_{i.k}-\bar{y}_{i..}-\bar{y}_{..k}+\bar{y}_{...})^{2} \)	\( (a-1)(t-1) \)	\( \frac{{SS}_{AT}}{{df}_{AT}} \)	\( \frac{{MS}_{AT}}{{MS}_{E}} \)
Error	\( \sum_{i=1}^{a}\sum_{j=1}^{N}\sum_{k=1}^{t}({y}_{ijk}-\bar{y}_{i.k}-\bar{y}_{.j.}-\bar{y}_{..k}+\bar{y}_{i..})^{2} \)	\( (N-a)(t-1) \)	\( \frac{{SS}_{E}}{{df}_{E}} \)

See r code