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MATH 4330 — Assignment 1

Question 1: A researcher wants to study whether an existing drug causes a particular kind of skin reaction. She collects data from an existing medical database; she samplesindividuals who showed the skin reaction as well as individuals without a reaction. She then examines how many individuals had taken the drug of interest.

(a) [2 points] What kind of study is this? Be as specific as possible, and explain your answer.

(b) [2 points] Suppose the researcher finds that those who took the drug had a higher chance of developing a skin reaction. Do you believe this result? Can you think ofother specific factors that might a↵ect the conclusion? Explain your answers.

Question 2: [2 points] A researcher runs a randomized experiment where study partic ipants are randomized to be given either a drug or a placebo. Another researcher wants to perform an additional study on this same study group. He asks participants whether they get more or less than 3 hours of exercise per week. He concludes that individuals with more than 3 hours of exercise per week have lower blood pressure than those who get less than 3 hours, on average. Are you worried about confounders in this conclusion?
Explain.

Question 3: We want to study the genetics of colour distribution in a population of unicorns. Suppose that in unicorns there is a single gene that determines colour; there are two variants of the gene, one called A and the other called a. Each unicorn has two copies of the gene (one from each parent). The combination of the gene variants for copy 1 and copy 2 of the gene determines colour as follows:

Copy 1 Copy 2 Colour

A A Red A a Pink aa Aa WhitePink

We want to determine if “random mating” is happening in this unicorn population. This would mean that the probability that a newly born unicorn inherits variant A or a with probability equal to the prevalence of each variant in the overall population.

1

(a) [3 points] Let p be the proportion of the A variant in the population, and q be the proportion of the variant a, so that q = 1 p. Under random mating, each newborn unicorn inherits A with probability p and a with probability q and the two copies inherited in each unicorn are independent of one another. Calculate the expected proportions of unicorn colours under random mating.
(b) [5 points] Suppose you know that p = 0.75 and q = 0.25, and that you collect the following data from a sample of unicorns:
Colour Number of unicorns

Red 45 Pink 49 White 12

Use the appropriate statistical test to determine whether the random mating as sumption holds in this population. Write out your calculations for this part; using R for this part will not result in credit.

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Question 1
A researcher wants to study whether an existing drug causes a particular kind of skin reaction. She collects data from an existing medical database; she samples individuals who showed the skin reaction as well as individuals without a reaction. She then examines how many individuals had taken the drug of interest.

(a) What kind of study is this? Be as specific as possible, and explain your answer.
(b) Suppose the researcher finds that those who took the drug had a higher chance of developing a skin reaction. Do you believe this result? Can you think of other specific factors that might affect the conclusion?
Explain your answers.

Solution
(a) This study would be considered a retrospective study. The reason would be that the researcher is looking into an already existing database system, hence she is looking into the past for her subjects. Based on this medical database system, she is sampling individuals who already showed a skin reaction vs. individuals that did not have a reaction.

Question 2
A researcher runs a randomized experiment where study participants are randomized to be given either a drug or a placebo. Another researcher wants to perform an additional study on this same study group. He asks participants whether they get more or less than 3 hours of exercise per week. He concludes that individuals with more than 3 hours of exercise per week have lower blood pressure than those who get less than 3 hours, on average. Are you worried about confounders in this conclusion? Explain.

Solution

Question 3
Question 3: We want to study the genetics of colour distribution in a pop ulation of unicorns. Suppose that in unicorns there is a single gene that determines colour; there are two variants of the gene, one called A and the other called a. Each unicorn has two copies of the gene (one from each par ent). The combination of the gene variants for copy 1 and copy 2 of the gene determines colour as follows:

Copy 1 Copy 2 Colour
A A Red A a Pink a A Pink
a a White

We want to determine if “random mating” is happening in this unicorn pop ulation. This would mean that the probability that a newly born unicorn inherits variant A or a with probability equal to the prevalence of each variant in the overall population.
(a) Let p be the proportion of the A variant in the population, and q be the proportion of the variant a, so that q = 1 p . Under random mating, each newborn unicorn inherits A with probability p and a with probability q and the two copies inherited in each unicorn are independent of one another. Calculate the expected proportions of unicorn colours under random mating.

(b) Suppose you know that p = 0.75 and q = 0.25, and that you collect the following data from a sample of unicorns:

Colour Number of unicorns
PinkRed 4945
White 12

Use the appropriate statistical test to determine whether the random mating assumption holds in this population. Write out your calculations for this part; using R for this part will not result in credit.

Solution
(a)

E(Red Unicorn Colour) = proportion of A ⇥ proportion of A
= (p2 ⇥ p)
= p

E(Pink Unicorn Colour) = pproportion of a ⇥ proportion of A
= (q ⇥ p)
= (qp)
= p(1 p)

E(White Unicorn Colour) == (proportion ofq ⇥ q) a ⇥ proportion of a
= q2
= (1 p)2

(b) Let n = 45+49+12 = 106.
E(Red Unicorn colour) = (0.75)2 ⇥106
= 59.625

E(Pink Unicorn Colour) = (0.75)⇥(0.25)⇥(106)
= 19.875

E(White Unicorn Colour) = (0.25)2 ⇥106
= 59.625

2 = (45 59.625)2 + (49 19.875)2 + (12 59.625)2
59.625 19.875 59.625
= 84.3074

H0: Random mating does occur in the unicorn population
Ha: Random mating does not occur
Let ↵ = 0.05 and df = 3 1 = 2
Using R to calculate th test statistic, we get that the p-value is19 which is well below 0.05. Hence we reject 0 4.9304⇥
10is no evidence to show that random mating does occur in the unicornH and say that there population.

Question 1

A) This study would be considered a retrospective study. The reason would be that the researcher is looking into an already existing database system, hence she is looking into the past for her subjects. Based on this medical database system, she is sampling individuals who already showed a skin reaction vs individuals that did not have a reaction.

Question 2