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PM 513: Experimental Designs

Homework 5

1.  (8 pts. each) A clinical trial was conducted to test the effects of three drugs—A, B, and C—in order to see how they affect reaction time to a stimulus. This was studied with a three-factor CRF-pqr ANOVA design.  Each drug was either given or not given (i. e ., p = q = r = 2).  A total of 32 subjects were randomized with four assigned to each of the possible combinations of treatment. The results for reaction time in seconds for each subject are contained on the sheet Problem 1 in HW 5 data .xlsx and are also shown below.

“Control” group: A, B, C absent:   4, 2, 7, 3

A given; B and C absent:               6, 6, 8, 8

B given; A and C absent:                2, 9, 7, 6

C given; A and B absent:                1, 5, 10, 4

A and B given; C absent:               3, 9, 6, 6

A and C given; B absent:                7, 7, 5, 9

B and C given; A absent:               9, 11, 7, 5

A, B, C all given:                           8, 10, 8, 10

(a) Assume a fixed-effects model. Perform a standard ANOVA on these data and test for all main

effects and interaction effects, testing each at an α = 0.05 level. Provide your conclusions from this type of analysis regarding the effects on reaction time.

(b) Provide point estimates for all the parameters in the model. Give 95% confidence intervals

for each of the following: α 1 , (αβ)11 , and (αβγ)111 . (Do not adjust for multiple testing.)

(c) Suppose that the drugs are thought to have non-overlapping modes of action and that the effects of the individual drugs are expected to be of similar magnitude. This would imply that it is just the number of drugs being given that will affect reaction time. We want to develop a test statistic for seeing if there is a “linear trend” across the number of drugs that are given. In other words,

• Patients who receive one drug would have a slower reaction time than those who receive none; patients who receive two drugs would have a slower reaction time than those who receive one; etc.

• For those who receive one drug it doesn’t matter which one; for those who receive two it doesn’t matter which pair; etc.

• The increases in reaction time as you go from no drug to one drug, from one drug to two drugs, or from two drugs to three drugs are all the same.

Develop a contrast in the cell means for examining this pattern. Describe its distribution in the case where all cell means are the same and in the case where the cell means have the pattern described above. Then apply it to the data and test at an α = 0.05 level. 

(d) Suppose we wanted to test whether any interaction parameters are necessary at all. This is really just asking whether a simple linear additive model in the main effects is all that is needed, which is equivalent to asking whether we can model the data as

Yijk  = µ + αj + βk + γ+ εijk

or whether we need the full model given by

Yijk  = µ + αj + βk + γ+ (αβ)jk + (αγ)j+ (βγ)k+ (αβγ)jk+ εijk This is the same as testing

H0 : (αβ)jk  = 0 and (αγ)jℓ  = 0 and (βγ)kℓ  = 0 and (αβγ)jkℓ  = 0 for all j,k,ℓ

vs.

Ha : at least one 2- or 3-factor interaction term is not zero

Perform a single test using your basic ANOVA sums of squares components which will test whether a simple linear model is sufficient based on these data. Test at an α = 0.05 level.

2.  (10 pts.) Chanda, et al., fed carotene to goats over consecutive two-day periods and used two methods to measure the digestibility of the carotene. Data were collected for four goats over five two-day periods and are contained on the sheet Problem 2 in HW 5 data .xlsx and are also shown below. In each cell, the first number given is the apparent digestibility measured by method 1, and the second number is the measurement by method 2. Pick a model and analyze the data. Provide statistical assessments of the hypotheses:

(1) There is no significant difference between the methods of measuring carotene digestibility; and

(2) There is no significant difference between goats in their ability to digest carotene.

Justify your choice of model by explaining how the mathematical model fits the experimental setup. Provide a reason why you chose the particular cutoff for “evidence” (that is, the p-value).

Period

Goat

I

II

III

IV

V

51.2, 46.7

62.9, 62.1

62.9, 55.3

56.6, 49.5

 

3.  (5 pts. each) In an experiment to study recall decay with three different questionnaires (A, B, C), nine subjects were questioned at three different times three months apart about the number of trips to a shopping center during the preceding three months. Each time a different questionnaire was used. The Latin square design shown below was used to determine the questionnaire order for each subject, with three subjects assigned randomly to each of the three treatment order patterns. The results for number of shopping trips reported are contained on the sheet Problem 3 in HW 5 data .xlsx and are also shown below.

Pattern           Subject                      Time Period (j)                

(i)                  (m)                 1                   2                   3

 

1

40 (C)

18 (A)

30 (B)

1

2

35 (C)

25 (A)

37 (B)

 

3

31 (C)

22 (A)

28 (B)

 

1

10 (B)

43 (C)

33 (A)

2

2

18 (B)

49 (C)

37 (A)

 

3

15 (B)

48 (C)

29 (A)

 

1

7 (A)

19 (B)

59 (C)

3

2

11 (A)

24 (B)

51 (C)

 

3

19 (A)

21 (B)

62 (C)

(a) Test for the presence of treatment order pattern, time period, and questionnaire effects. Use

α = 0.05 and no correction for multiple testing.

(b) Analyze the questionnaire main effects by estimating all pairwise comparisons of treatment

means. Use the Tukey procedure with a 90% family confidence level.

(c) Suppose that you suspected that there might be carryover effects in this experiment. Would this be an appropriate design? Why or why not?

4.  (5 pts. each) In Settlers of Catan, the 5- and 6-player games add an extra wrinkle. You decide you want to also have the finalists from the Lecture Notes play in either 5- or 6-player games to test this additional strategy.

(a) The design given in 13.6 in the Lecture Notes is the unreduced design for v = 8 and k = 6.

Verify the claim in the notes that no smaller BIBD exists for these parameters.

(b) Show that there is no reduced BIBD with v = 8 players and k = 5 players per game. What

are the parameters of the unreduced design?

(c) We have seen that from a practical standpoint, resolvable designs tend to lead to shorter tournaments. What is the smallest number of finalists v that could have resolvable designs for both k1  = 4 and k2  = 5? How about for both k1  = 4 and k2  = 6?

(d) Many different ways to construct BIBDs have been developed over the years. Here’s one due to Bose:

Suppose that 2λ + 1 = p is a prime number. (A similar method works for 2λ + 1 = pn  with n > 1, but the details are more complicated.)  First, find an x ∈ {1, . . . ,p − 1} such that x,x2 , . . . ,xp − 1   (mod p) are exactly the numbers 1, . . . ,p − 1 in some order.  (x is called a

primitive root mod p.) Start with the two generators (0,x0 ,x2 , . . . ,x2λ −2 ) (mod p) and (∞,x0 ,x2 , . . . ,x2λ 2 ) (mod p). Then, add each of 1, . . . ,p − 1 to each component of each of the two sets of numbers and reduce everything mod p. (Arithmetic with the“∞”is simple: adding anything and reducing mod p both simply keep it as ∞ .) When you’re done, add 1 to everything, then replace with p + 1. The resulting 2p = 4λ + 2 sets of numbers (the

generators plus the sets of numbers you generated using them) will represent a non-resolvable BIBD.

If, in addition, λ is odd, then doing the same process starting with the two generators (0,x0 ,x2 , . . . ,x2λ −2 ) (mod p)  and  (∞,x,x3 , . . . ,x2λ − 1 ) (mod p) will lead to  a resolvable

BIBD.

For λ = 5, use the above method to find both the resolvable and non-resolvable BIBDs. What are the resulting parameters?

(Hint: If you want to make sure you’re doing this correctly, using λ = 3 and x = 3 should give the resolvable BIBD from Example 13.3 in the Lecture Notes.)

(e) Verify that the following set of three generators creates a resolvable BIBD. What are the

parameters?

(0, 1, 3, 7), (2, 4, 9, 10), (5, 6, 8, ∞)   (mod 11)

5.  (10 pts.) Use the method of 12.4 to verify the expected mean squares for the three-level nested ANOVA model from 13.1.5. Also get the formulas for the point estimates of each of the parameters µ , αj , βk(j) , γ(jk), and εijk.

6. Researchers were interested in the occurrence of a specific toxicity in children being treated for cancer. They measured GTBL, which gave an indication of how likely a child was to develop the toxicity of interest, and related it to the age and body surface area (BSA) of the child. (BSA—not weight—is commonly used to determine the amount of drug given to children.) The data for each subject are contained on the sheet Problem 6 in HW 5 data .xlsx.

(a)  (3 pts.) Run simple linear regression models for each of age and BSA as the explanatory

variables and GTBL as the response variable. What do you conclude about the univariate relationships of age with GTBL and BSA with GTBL at the α = 0.05 significance level?

(b)  (2 pts.) Run a multiple linear regression without interaction. What do you conclude about

the multivariable relationships between each of age and BSA with GTBL at the α = 0.05 significance level?

(c)  (3 pts.) Explain how the relationships you saw in parts (a) and (b) happened. (Hint: What is the relationship between age and BSA in the data set? What are the relative effects of age and BSA on GTBL in the multivariable model?)

Note: This data set is based on an actual data set that was observed in the Children’s Oncology Group study ADVL06B1:  A  Pharmacokinetic-Pharmacodynamic-Pharmacogenetic Study  of Actinomycin-D and  Vincristine in Children with Cancer.  The original analysis was a logistic regression with the response an indicator variable for whether or not the toxicity of interest actually occurred, but the same contrast between the effects of age and BSA in the univariate and multivariable analyses occurred in that analysis.