Question 1: You are an oncologist testing an experimental treatment for glioblastoma multiforme. You begin an experimental study with 16 late stage patients. Determine a p-value for the null hypothesis that the hazard (probability of death) is the same for drug and placebo.
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Patient
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Outcome
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Treatment
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A
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Survived > 3 years
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Drug
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B
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Died 6 months
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Placebo
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C
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Died 1 year 5 months
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Drug
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D
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Lost contact at 1 year 0 months
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Drug
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E
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Survived > 3 years
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Drug
|
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F
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Died 2 years 2 months
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Placebo
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G
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Died at 6 months
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Drug
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H
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Died of unrelated cause at 9 months
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Placebo
|
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I
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Survived > 3 years
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Placebo
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J
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Died 5 months
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Placebo
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K
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Survived > 3 years
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Drug
|
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L
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Stopped treatment at 1 year 9 months
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Placebo
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M
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Survived > 3 years
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Drug
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N
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Died 11 months
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Placebo
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O
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Died 1 year 5 months
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Placebo
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P
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Survived > 3 years
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Drug
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Question 2: In a larger study of glioblastoma multiforme you determine 2 year survival for drug and placebo-treated patients. You have 56 total cases. 31 get drug and 17 of these survive 2 years. 25 get placebo and 5 of these survive 2 years.
A) Calculate the two proportions of patients that survive 2 years. Also determine the 95% confidence intervals of the two proportions.
B) Determine if there is an effect of treatment on 2 year survival.
C) What is the relative risk in this study? What is the 95% confidence interval?
D) Explain what the relative risk means as if you were explaining to a patient how treatment affects patient outcome.
Question 3: In exam 1, I tested randomly generated Gaussian distributions using normality tests in Prism. In one case, Prism identified 7 false positives with p-value 0.05 in 100 tests.
A) Using what you learned about analyzing counted variables, determine the p-value for this result.
B) In a sample of 100 statistical tests, what is the range of false positive occurrences that would NOT be significantly different from the expected occurance. Assume α = 0.05. In other words between X and Y false positives (p-values less than 0.05) will NOT be significantly different from the null hypothesis, while fewer than X or greater than Y false positives will be statistically significant.
Question 4: Using the information from Question 5 on Exam answers the following. How many animals, in total, would you have to examine to be able to detect an antibiotic effect of 2% increase in growth, with 90% power?
Question 5: Here are poll data from the state of Ohio collected between March 8-13th.
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Pollster
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Voters
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For Clinton
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For Sanders
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Undecided
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ARG
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400
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208
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180
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12
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Monmouth
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302
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163
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121
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18
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PPP (D)
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502
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231
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206
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65
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Quinnipiac
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543
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277
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250
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16
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CBS News
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777
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404
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334
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39
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A) Using this data, determine the p-value for the null hypothesis that support for Clinton and Sanders is equal. Assume that all data were samples from the same parent population.
B) Test the assumption that all the polls were taken from the same parent population.
C) Notice that the undecided seem to vary more than the other numbers. Test the hypothesis that all the polls were taken from the same parent population, when considering only voters that had an opinion.
Question 6: Here are two riddles:
a) One box is labelled "nuts". One box is labelled "bolts" One box is labelled "Nuts and Bolts". All three labels are wrong. You may ask to see only one item randomly pulled from any one box. Can you correctly label all three boxes?
b) Three students go to a pizza restaurant for lunch. They each have $5, so they buy a $15 pizza to share. The manager decides to give them a refund of $5 because they are poor graduate students. He gives $5 to the waiter and tells him to give it back to the students. The waiter can't decide how to fairly split the $5 among three students so he gives the students $1 each, and keeps the remaining $2 for himself. The students initially paid $5 each and then got $1 back. So they have paid $4 each, or $12 total, for their pizza. This plus the $2 kept by the waiter makes $14. What happened to the missing dollar?
(Of course, I am not asking you to solve the riddles. Although you can do so if you want. First, answer the statistical question below).
Question - Fifty college seniors are given 15 minutes to solve riddle
a. Seventeen are successful. Another 75 seniors from the same school are given 15 minutes to solve riddle.
b. Seven are successful. Is there a difference in the difficulty of the two riddles?
Question 7: In the 1990's, a randomized prospective study was begun on a group of 605 survivors of myocardial infections (heart attacks). Some were instructed to follow the post heart attack diet recommended by the American heart association (AHA, control group). Others were instructed to follow the so called 'Mediterranean diet' (Experimental; Group). After 5 years, the following results were obtained.
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Table 3: Number of Events and Risk Ratios
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|
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No. of Events
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Risk Ratio
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95% Confidence Interval
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P
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Control Group (n = 303)
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Experimental Group (n = 302)
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Cancer
|
17
|
7
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0.39
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0.15-1.01
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.05
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Total deaths (including cancer deaths)
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24
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14
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0.44
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0.21-0.94
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.03
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Cardiac death
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19
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6
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0.35
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0.15-0.83
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.01
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Total deaths + nonfatal cancers
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35
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18
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0.44
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0.24-0.83
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.01
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Total deaths + nonfatal cancers + nonfatal myocardial infarctions
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60
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26
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0.38
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0.23-0.61
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<.001
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After adjustment for sex, age, smoking, blood cholesterol level, leukocyte, and aspirin use at baseline
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De Longerill, M., Salen, P., Martin, J., Monjaud, I., Boucher, P., Mamelle, N. (1998). Mediterranean Dietary pattern in a Randomized Trial. Archives of Internal Medicine, 158, 1181-1187.
A) Is there an effect of diet of the proportion of patients who fall into each of five categories: Cancer death, Non-fatal cancer, Cardiac death, Non-fatal cardiac issues, None of the above. I am expecting one answer to this question, not five.
B) Does diet affect the probability of cancer death?
C) Does diet affect the probability of cardiac death?
Question 8: You are comparing coliform bacterial counts in municipal tap water and water from a local river. You spread 0.1 ml of river water on a nutrient agar plate and count 11 total colonies total after overnight growth. You spread 0.1 ml of tap water on a plate and observe 5 colonies.
A) Is there any evidence for the presence of coliform bacteria in the environmental water samples?
B) If you want 90% power to observe a factor of two differences between river and tap water, how many 0.1 ml samples do you need to examine?
C) If you want 90% power to distinguish the counts in tap water from a null hypothesis of zero, how many 0.1 ml samples do you need to examine.
Question 9: Calculate p-values for the following:
A) A proportion of 17/51 versus a proportion of 10/65
B) 0.187 +/- 0.056 (mean +/- SD, N=6) versus Ho=0.
C) A simple count of 120 vs a null hypothesis of 100
D) A simple count of 9 vs a simple count of 19
E) 2.34 +/- 0.51 (mean +/- SE, N=3) versus 2.00 +/- 0.31 (mean +/- SE, N=10)
F) A proportion of 5/12 versus a proportion of 9/11.