Homework 2

  • Problem 1: There is a great interest in comparing different countries in the World based on variety of factors reflecting the country's internal and external international ranking. Use the Political, Economic, Health, and Quality-of-Life Data of 100 Countries to estimate the probabilities below. Let ED=Economic Dynamism of a Country, which is an index of productive growth in US dollars. Use the SOCR Modeler to fit a Normal Distribution Model to the ED variable (column) in this dataset (see this Help page). Onces you obtain estimates for the mean and standard deviation of the normal model (see the Results tab in the Modeler) use the SOCR Normal Distribution Calculator to estimate the likelihoods of these events:
    • P(ED ≤ 46)
    • P(35 ≤ ED ≤ 43)
    • P(48 ≤ ED)
    • P(53 ≤ ED)
    • P(47 ≤ ED ≤ 87)
    • P(15 ≤ ED ≤ 51)
  • Problem 2: During a typical 24-hour shift in the ER, the healthcare providers (doctors, nurses, staff) expect to see 132 emergency visits including 10 traumatic brain injuries (TBIs). Find the probability that the ER team will see over 145 cases in total and the probability that there will be between 8 and 11 TBIs within a given 24-hour period. Recall that the Poisson Distribution can be used as a model.
  • Problem 3: Many clinical and translational studies involve multiple variables (or events) that may be independent of one another or closely associated. Identifying and untangling data dependencies is critical in such situations. We can use the SOCR Coin Die Experiment to simulate dependence between clinical variables. Suppose we have 2 discrete clinical variables, for example, X={stage of melanoma} (categorical) and Y={gender} (dichotomous). We can simulate this situation, specifically simulate event independence between the outcome of a die (event B, representing the cancer stage) and the outcome of a coin (event A, represengint the patient gender). In the SOCR Coin Die Experiment set the probabilities of both dice to be identical. Run 100 experiments and argue that the observed data implies independence between the events A={Coin=Head, say male} and B={Die=3, say stage 3 melanoma}, i.e., P(AB) = P(A) P(B), approximately. You basically need to count the proportion of times each of the tree events (A, B and C={A∩B}) of interest occur in the 100 experiments and validate (or disprove) the equality above. Also, try this with a larger number of experiments (e.g., n=10,000). Next, make the probability distributions of the two dice different (by clicking on the dice and manually changing the die probabilities). Show empirically the dependence of the probabilities, A={Coin=Head} and B={Die=3}. Do we have evidence of independence or association in the outcomes?
  • Problem 4: Using the SOCR Clinical, Genetic and Imaging Data of Alzheimer’s Disease:
    • Using these 2 groups: Group0={CDGLOBAL=0} vs. Group1={CDGLOBAL=1}, compute the correlations between systolic and diastolic blood pressure (VSBPSYS and VSBPDIA) within each group (ro and r1). Then test a hypothesis for the equivalence of these correlations.
    • Fit a simple linear model for VSTEMP and Weight_Kg. Formulate and assess a hypothesis about trivial slope of the regression curve on these two variables. Elaborate on your findings.

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