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ReMA | Quantitative Foundations | Fall 2023 Unit 10 Homework POSTED: 12/01/2023 DUE DATE: 12/12/2023 at 11:59 PM  Answers should be typed. Work presented in an appendix will not be accepted. 10 points will be deducted from homework where final answers are presented separately from the work.  Please use the equation editor in Word to SHOW ALL YOUR WORK for problems requiring hand calculations (see Canvas for helpful equation editor shortcuts). You will receive partial credit for showing the steps along the way. A final answer with no work shown is not enough for full credit. o For all questions requiring calculations, use 4 decimal points during computations and round to two decimal points at the LAST step.  Some hints on making the most of homework as a learning opportunity: o You can work in groups or discuss the problems with your classmates, but only in a spirit of learning. Do not simply “cut and paste” from others’ work. Your final submission must be strictly your own, though informed by collaborative group work. o If you do join a group to work on homework assignments, be sure to try all the homework problems on your own first, before meeting with your group. This way, you will have the opportunity to try to devise solutions on your own, without input from others. Then, when you get together, you can compare approaches For TA use (DO NOT DELETE): P1 P2 P3 Total Points Percent Honest Attempt Content Mastery Problem 1 Researchers in NYC assessed whether the average time it takes between receiving a referral to a specialty clinic and the appointment date of that specialty appointment differs based on the clinic and the specialist type patients are referred to. To assess this, data was collected from 3,500 patients attending a specific primary care network who were referred to one of three clinics (Clinic 1, 2, and 3, with Clinic 1 as the reference clinic in the STATA output below). For each patient, information was also collected on the type of specialist the patient was referred to (variable “specialist” in the STATA output; 1 = gastroenterology, 2 = dermatology, 3 = neurology, and 4 = other), insurance type (variable “insurance”; 0 = other, 1 = private, where 0 is the reference category), and time from referral to specialty appointment (variable “days.”) Use the Stata outputs below to answer the following questions. OUTPUT A 1

OUTPUT B a) Write out the full multiple linear regression equation for OUTPUT B . Be sure to define all of your variables (your dependent variable (Y) and your independent variables (Xs)) and how they are coded. Note: you do not need to plug in numbers from the output in your model (i.e. you can write it as ^ y = b 0 + … ) so long as you define what each variable and coefficient refers to. ^ y = 83.4121 − ( 37.3265 ∗ insurance ) + ( 2.1239 ∗ clinic 2 ) − ( 1.6431 ∗ clinic 3 ) − ( 3.4472 ∗ specialist 1 ) +( 36.5539 ∗ speci Insurance : 0 = if other Insurance = 1if private Clinic 2 = 1 if attended clinic 2 and = 0 if attended clinic 1 or 3 Clinic 3= 1 if attended clinic 3 and = 0 if attended clinic 1 or 2 Specialist 1 = 1 if gastroenterology Specialist 2= 1 if dermatology Specialist 3 =1 if neurology b) The researchers hypothesized that both specialist type and insurance type may confound the association between clinic referred to (variable clinic) and length of time between referral and appointment (variable days). Does it? Use the results in Output A and Output B above to assess whether the data are consistent with the hypothesis that specialist type and insurance type confound the causal relationship between clinic and days . Explain your reasoning by writing down exactly which 2

hypothesizes that injection drug use may confound the average association between vaccine receipt and risk of HIV infection. b) Use the stratified data below to answer the following questions. Non-users HIV+ HIV- Total Vaccine 35 160 195 No Vaccine 45 161 206 Total 80 321 401 i. Calculate and interpret the estimated risk ratio for HIV infection among non-injection drug users, comparing those who received the vaccine to those who did not receive the vaccine. Interpret this assuming that you’d like to make inference on some population larger than those in your sample. RR = IE IE + ¿ CE CE + CN = 35 35 + 160 45 45 + 161 = . 8217 Those who do not receive the vaccine have an estimated risk of HIV infection within nonusers 82.17% higher for than those who did receive the vaccine based on the data we found from the study. ii. Calculate and interpret the estimated risk ratio for HIV infection among injection drug users, comparing those who received the vaccine to those who did not receive the vaccine. Interpret this assuming that you’d like to make inference on some population larger than those in your sample. RR = IE IE + ¿ CE CE + CN = 7 7 + 273 59 59 + 10 = . 0292 Those who do not receive the vaccine have an estimated risk of HIV infection within users 2.92% higher for than those who did receive the vaccine based on the data we found from the study. iii. You also calculate the Cochran-Mantel-Haenszel adjusted risk ratio to be 0.28. Based on this summary measure of association and all of the other measures you calculated in the previous parts of this question, what role might injection drug use play in the estimated effectiveness of the vaccination on HIV infection? Is it an effect measure modifier? Why or why not? Is it a confounder? Why or why not? For both explanations, be sure to use the measures you reported in the previous questions to support your conclusion. The average of the adjusted is .28 The crude risk ratio is .2338 Since the Cochran-Mantel-Haenszel is similar it indicates there may be a confounder at play but it would be difficult to determine without further testing. There is enough evidence that there is a confounder because 4 Injection Drug Users HIV+ HIV- Total Vaccine 7 273 280 No Vaccine 59 10 69 Total 66 283 349

there is a difference more than 10% between the two. The users vs non users indicate there is a difference between the groups since it is .8217 for non users and .0292 for users. These numbers indicate it would be an effect measure modifier since they are different. Problem 3 You and your colleagues hypothesize that exposure to adverse childhood experiences (ACEs) is a cause of adult-onset multiple sclerosis (MS). To test this hypothesis, you conduct a case-control study by recruiting 250 cases who have been recently diagnosed with MS and 250 controls who do not have MS. Past exposure history is measured via self-report using a structured interview. You also measure whether each participant had a previous adult-onset autoimmune disease other than MS, because previous adult-onset autoimmune diseases are a hypothesized cause of MS. Use the Stata output below to answer the following questions. Variable Name Coding ms 0 = No multiple sclerosis, 1 = Multiple sclerosis ace 0 = No adverse childhood experiences, 1 = At least one adverse childhood experience autoimmune 0 = No other autoimmune diseases, 1 = At least one other autoimmune disease that developed in adulthood prior to MS onset a) Provide an interpretation for the odds ratio for the “ace” variable along with its 95% confidence interval. Interpret this assuming that you’d like to make inference on some population larger than those in your sample. The estimated odds of developing adult-onset MS given that there was at least one ACE is 2.5338 times higher than those who did not have an ACE. We are 95% confident that the true odds ratio of those who have adult-onset of MS given that there was at least one ACE is between 1.651 and 3.8887. b) Suppose in a separate analysis (Stata output not shown) you estimate the crude odds ratio for the “ace” variable to be 4.45. When comparing this to the adjusted odds ratio in the Stata output above, is this sufficient to conclude that ‘autoimmune’ is acting to confound the relationship between ‘ace’ and ‘ms’? Why or why not? Explain clearly. 5