STA 106 Discussion 6
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
STA 106 Discussion 6
Problem I
1. Assume a study on the effects of various treatments for reducing blood pressure was recorded. The options available were:
Factor A: Drug dosage, with levels High, Low, and None.
Factor B: Biofeedback, with levels Guided (someone guides them through exercises), Individual (they do exercises on their own), Control (no biofeedback).
The result of sample means follows (sample sizes are given in parentheses):
|
Con (j = 1) Guided (j = 2) Ind (j = 3) |
Group means |
|
(i = 2) Low (i = 3) None |
157.07 (15) 180.73 (15) 191.2 (15) |
|
|
|
(a) Does there appear to be an interaction effect? Explain your answer.
Yes, there does. For example, consider the effect of comparing High to None medication (regardless of Factor B). The largest difference in average is 34.13. However, if we consider the difference when comparing High to Low for Guided, Individual, and Control, we have a largest difference of: 18 (Control), 43.4 (Guided), and 41 (Individual). Since the difference changes significantly by group, it would suggest an interaction effect.
(b) What could you do in order to assess if there is an interaction effect? Explain your answer.
We could conduct a hypothesis tests on all (γδ)ij to test if they are all zero, and we could calculate the conditional R1 value of R1 {AB|(A + B)}.
(c) Does there appear to be a significant difference in all averages for Factor A alone? What extra information would be useful in making this assessment?
There does, as there is a large difference between all pairwise means. However, it would be useful to have the estimated standard deviations by group in order to make a more meaningful assessment.
(d) Name a plot (or test) we use to assess the assumptions of the error terms in ANOVA, and explain the intuition behind it.
There are many options, but one is the Normal Probability Plot. This compares the actual percentiles of our centered data to what they should be if the data was normal, so the closer the scatter plot is to the line y = x the more normal our data would be.
Problem II
2. Continue with the previous problem. Assume the SSE for the interactions model is appropriate, and the value is: 4490.
(a) If you were asked to find all confidence intervals for each individual factor A mean, what corrected multiplier/s
could you potentially use? Explain your answer.
Only Bonferroni. Scheffe is for all contrasts, and Tukey is for all pairwise intervals, and a confidence interval for a single mean is not a contrast or a pairwise comparison.
(b) Find the 95% corrected confidence interval for comparing the average blood pressure of the Guided group to the
Individual group, assuming your choice of corrected multipliers is : 2.511 (Bonferroni), 2.444 (Tukey), 2.553 (Scheffe), and interpret the interval in terms of the problem.
Notice here we have equally weighted means, since nij = 5 for all i, j. The confidence interval is:
j cj 
●j 土 (2.444) ← 
j cj(1) i nξó(〇)
Where, c〇 = 0, c1 = 1, c2 = -1, 
●1 = 〇3＋.3′〇5
.8′〇82 .4 = 164, 
●1 = 〇44 .1′〇92(1) .3′〇95 .1 = 181.27.
MSE = 
= 
= 124.7222, j cj(1) i nξó(〇) = (11 ) * (1/5 + 1/5 + 1/5) + (-1)1 (1/5 + 1/5 + 1/5) = 
Thus, the CI is : (164 - 181.27) 土(2.444) * ← 
( 
) = (-52.1742676, -32.2414924).
We are overall 95% confident that the true average blood pressure for the Guided group is lower than that of the Individual group by between 32.2414924, and 52.1742676 mmHg (ignoring interaction effects).
(c) Find the 95% confidence interval for comparing the average blood pressure of High vs. Low medication, for those subjects in the Guided level of factor B, and interpret the interval in terms of the problem.
The confidence interval is: 
〇1 ● - 
11 ● 土(2.444) ←MSE ( n12(〇) + n22(〇) ), or (140.4 - 167.8) 土(2.444) ← 124.7222( 4(〇) +4(〇))
or (-44.6625, -10.1375).
We are overall 95% confident that the true average blood pressure for those in the Guided group and with High dosage of medication is lower than those in the Guided group with Low dosage by between 10.1375 and 44.6625 mmHg.
(d) What is the largest difference in blood pressure we would expect when comparing two subjects, one of which is in the Guided group, and the other is in the Individual group?
Since there are interaction effects, it depends on what level of factor B they were at. If we ignore interaction effects, then it could be considered to be 52.1742676 mmHg.
2022-05-25