Biostatistics 2 nd year Comprehensive Examination. Due: May 31 st, 2013 by 5pm. Instructions:

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1 Biostatistics 2 nd year Comprehensive Examination Due: May 31 st, 2013 by 5pm. Instructions: 1. The exam is divided into two parts. There are 6 questions in section I and 2 questions in section II. 2. Answer each question to the best of your ability. 3. Be as specific as possible and write as clearly as possible. 4. This is a take- home examination. You may consult books, notes, papers, and you may use the Internet. However, you may not consult or discuss this exam with another human being or statistical oracle, nor may you seek help from another individual on the internet (e.g., no posting questions to chat rooms or message boards). 5. If you have any questions, please contact Professor Blume by or by phone (cell: ). Do not worry about being polite; Professor Blume as needed and call for emergencies. 6. Turn in your exam by ing it to Professor Blume at j.blume@vanderbilt.edu AND Linda Wilson at linda.l.wilson@vanderbilt.edu. You must get confirmation from Professor Blume or Ms. Wilson that your exam was received before 5pm on Friday 5/31. Alternatively, you may turn in a hard copy to either person by the deadline. 7. Vanderbilt s academic honor code applies; be sure to adhere to the spirit of this code. Question Points Score Comments Section II pts. Part I; 60 pts. Part II Total 240

2 Section I 1. Let Ω be a finite set and let F = PP Ω be the σσ- algebra of all subsets of Ω. For AA Ω, let AA be the number of elements of AA. a. Prove that the formula PP AA = AA / Ω defines a probability on Ω, F. b. Prove that PP BB is the uniform distribution on BB, where BB Ω. c. Suppose now that Ω is a countably infinite set. Prove that a probability PP on Ω, F cannot be the uniform distribution. 2. Let XX, XX, XX, be iiiiii random variables each with zero expectation and common finite variance σσ. Define SS = XX. a. What is the limiting distribution (convergence in distribution) of the sequence (justify your answer): SS σσ nn b. Show that there cannot exist a limiting random variable SS such that SS.. SS σσ nn 3. Let h be an absolutely continuous function on 0,1 with 0 h xx 1 for all xx. We wish to estimate the integral II h h xx dddd. a. Let UU, UU,, UU be an iiiiii sample from a Uniform 0,1 distribution. Using this sample, argue that II h = 1 nn is the standard Monte Carlo estimator. h UU 1 of 7

3 b. The method of antithetic variables leads to an alternative estimator, namely II h = 1 2nn h UU + h 1 UU i. Show that both II and II are consistent. Specifically, that II.. II h and.. II II h as nn. ii. Show there exist positive constants σσ h and σσ h such that nn II h II h NN 0, σσ h nn II h II h NN 0, σσ h iii. For a given h, determine if one method is to be preferred over the other in terms of minimizing σσ h. 4. Consider the squared- error loss function LL yy, ff xx = yy ff xx. Given data TT = xx, yy : ii = 1,, nn and a fitted model yy = ff xx, we define the in- sample error to be EEEErr = 1 nn EE LL YY, ff xx TT where the expectation is over a future observation YY drawn from the conditional distribution YY XX = xx. The training error is defined as eeeeee = 1 nn LL yy, ff xx and model optimism is defined as oooo = EEEErr eeeeee. Note that when xx : ii = 1,, nn are viewed as fixed, oooo is a function of the random variables YY = yy,, yy. Under these conditions, show that EE oooo = cov yy, yy. 2 of 7

4 5. A lasso regression model is fit to pp predictors using the following pre- specified constraint for the coefficients: ββ tt. The fitted coefficient for one predictor, say XX, is ββ = aa. Consider what happens when one more predictor - a duplicate of XX called XX - is added to the model. The lasso model is refit with pp + 1 predictors using the constraint: ββ + ββ tt, where ββ is the coefficient for XX = XX. a. Describe the set of solutions for the two coefficients ββ and ββ. b. Now suppose a ridge regression model with a pre- specified penalty λλ is used. Show that including the duplicated predictor XX results in equal coefficients for both XX and XX. c. Comment on the implications of these results. 6. Consider the typical linear regression model, yy = ββββ + εε with εε~nn 0, σσ II, where the residual sum of squares RRRRRR ββ = yy ββ xx is minimized to obtain parameter estimates ββ nn, where the subscript is added to emphasize that the parameter estimates are functions of nn data points xx, yy ~FF for ii = 1,, nn. The observed residual sum of squares, say RR = RRRRRR ββ nn, is a random variable. Now suppose that additional data are collected from the same joint distribution, say xx, yy ~FF for jj = 1,, mm (i.e., collect an additional mm data points). The observed fit of the previously established model to these new data is given by the residual fit of the additional data using the previously established parameter estimates, say TT = yy ββ xx. Show that EE RR /nn EE TT /mm and comment on the implication of this result. End Part I. 3 of 7

5 Section II Instructions Prepare a 3 to 6 page statistical analysis report on the Vanderbilt Inpatient Cohort Study (VICS) that addresses the two scientific questions described on page 7. A detailed description of the data and scientific aims are described below. Tables and figures should be included in an appendix; they are not counted against the page total. Be sure to include the following sections in your report: Introduction, Methods, Results, and Conclusions. You may cite this description as a source document for background information. Your Methods section should be comprised of your analytical approach and may be proportionately longer than a typical scientific article. Results and interpretations should be well written, formatted and presented in an accessible manner for a scientific investigator. However, they should also include enough statistical details for a statistical reviewer. Check model assumptions and/or other diagnostics as appropriate. Introduction Following discharge from the hospital 19-23% of patients experience an adverse event, and the majority are due to adverse drug events (ADEs) from medications. ADEs, or harm due to medications, may result from unexplained differences in medication regimens across sites of care and unexplained differences between what medications the patient thinks they should be taking and what is ordered. Some estimates report that 30-50% of patients have a discrepancy between the medications listed at hospital discharge and the patient- reported regimen that persists after discharge. Patients who take cardiovascular medications are at particularly high risk for discrepancies after discharge, due to the multiple types (i.e., classes) of medications prescribed. As a step towards reducing adverse event and rehospitalization rates VUH is interested in identifying social risk factors associated with both medication discrepancies and with overall patient health. There are two parts to this analysis: 1) characterizing the relationship between medication discrepancies (outcome) and educational attainment (exposure), and 2) examining the relationship between self- reported global health status (outcome) and marital status (exposure). The Vanderbilt Inpatient Cohort Study (VICS) Sample The Vanderbilt Inpatient Cohort Study (VICS) is a prospective cohort study of participants admitted to Vanderbilt University Hospital in Nashville, TN with cardiovascular disease. The broad goal of VICS is to determine the impact of social determinants on health outcomes such as medication management, unplanned hospital utilization, and mortality after discharge. The rationale and design of VICS will be detailed in a future publication (so don t look for this). Eligibility screening 4 of 7

6 occurred shortly after admission and sought to identify participants who were at least 18 years of age admitted with an intermediate or high likelihood of acute coronary syndromes (ACS) or acute decompensated heart failure (HF) per a physician s review of the clinical record. Exclusion criteria included: inability to communicate in English or Spanish, severe unstable psychiatric illness, inability to contact after discharge, on hospice, or otherwise too ill. Participants were approached while still hospitalized for written informed consent. The Data The data set VICS.TXT has 14 variables and 473 observations. There are 6 missing values (coded as NA s) and they are noted in the variable descriptions. Primary outcomes and exposures: Medication Discrepancies: Two to eight days following hospital discharge, participants were contacted by phone. They were questioned about their medication regimen and since patients were potentially prescribed multiple medications from multiple classes of medications, they were asked about one randomly selected medication within each class of cardiac medications prescribed. If the patient was on eight cardiac medications from five classes of medications, he/she was asked about five cardiac medications, each from a different cardiac medication class. If the patient did not fully understand the medication indication, the dose he/she was prescribed, or frequency with which he/she was to take the medication, it was considered a cardiac medication discrepancy. Data regarding medication discrepancies are stored as the number of discrepancies (ndiscrep) and the number of cardiac medications tested (ncartest). Global health status: Global health status (promismean) was assessed using a 5- item measure from the NIH Participant Reported Outcomes Measurement Information System (PROMIS). The 5 items ask participants about overall health, quality of life, physical health, mental health, and satisfaction with social activities and relationships with each question on a 5- point Likert scale (1, 2, 3, 4, or 5). Responses for the 5 questions are then averaged with higher scores implying better global health. Education: Education (edu) is stored as the number of years of educational attainment. Twelve years of education should be thought of as obtaining a high school degree (or equivalent) but without college. Sixteen years of education should be thought of as obtaining a college degree but without graduate work. The low end of the education distribution (e.g., less than twelve years) is of particular interest to researchers since such patients are believed to be at highest risk for a discrepancy. 5 of 7

7 Marital Status: Marital status (maritalcat) is stored as a categorical variable with four distinct values: Married/Living with partner, Separated/Divorced, Widowed, Single/Never married. Other Variables in the dataset: studyid: participant identifier diagnosis: All patients in this study had symptoms of ACS or HF. Some had symptoms of both. This is a three category variable with levels: ACS only, HF only, and ACS and HF. age: patient age at admission to the hospital (years). gender: Male or Female racecat: White, Black/AA, or Other. There is 1 missing value. essi6sum: Social support was assessed using 6 of the 7 items from the ENRICHD Social Support Inventory (ESSI). Participants were asked the 6 questions regarding emotional and instrumental support, each of which had a 5- item Likert response scale. The ESSI is ranges from 5 to 30, where higher scores indicate a greater level of social support. stofhlasum: Participants completed the short form of the Test of Functional Health Literacy in Adults (s- TOFHLA), a timed test administered in a maximum of 7 minutes. Scores can range from 0 to 36; there are 5 missing values. This variable is a measure of health literacy or how well patients understand their health and the health care system. snsmean: Numeracy is the degree to which individuals have the capacity to access, process, interpret, communicate, and act on numerical, quantitative, graphical, biostatistical, and probabilistic health information needed to make effective health decisions. We employed a shortened 3- item version of the Subjective Numeracy Scale (SNS), which quantifies the participants perceived quantitative abilities through items about math skills and preferences for numerical information. The SNS is reported as the mean of the 3 items on a Likert scale of 1 to 6, with higher scores reflecting greater subjective numeracy. phqsum: We assessed depression during the 2 weeks prior to the interview using the Participant Health Questionnaire- 8 (PHQ- 8). The 8- item measure is scored on a scale of 0 to 24 with higher scores reflecting more severe depression. ncartest: Number of cardiac medication tested for the follow- up phone interview. 6 of 7

8 The Scientific Questions: Part 1 Researchers would like to characterize the relationship between educational attainment and the risk of medication discrepancies. They are particularly interested in those patients at the low end of the educational attainment distribution since they believe that such subjects are likely to be at greatest risk of a discrepancy. Note that journal referees and readers expect age, gender, race, and marital status to be included in the regression model. Consider the goals of the analysis to determine the other variables that should be included, and provide justification of your decisions. Part 2 Researchers hypothesize that patients who are married or have a partner are more likely to have improved health overall than those who are not married and not living with a partner. They believe that being married or having a partner leads to decreased stress levels and increased security, which in turn leads to improved health. Specifically, researchers want to know the probability that patients who are married or living with a partner have a greater mean PROMIS score than other types of patients. They want to know this probability for comparing married/living with partner vs. each of the following groups of patients: 1) separated/divorced 2) widowed 3) single/never married 4) separated/divorced, widowed, or single/never married (all combined) Use a single linear regression model that can be interpreted appropriately for this question. Consider the goals of the analysis to determine whether other variables should be included in the model, and provide a brief justification of your decisions. End Section II. 7 of 7