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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
parts of the output (the variable name, numeric estimate, test statistic if applicable, etc.) Does confounding by specialist type and insurance type completely explain the observed association between clinic and days ? Why or why not? Since Output A is a crude measurement, the coef of clinic 2 is 12.2674. Output B is adjusting for confounding and the coef goes to 2.1239, meaning that there is confounding because there is a large difference between the numbers. It is also notable that both outputs contain 0 in the confidence interval meaning there is not enough evidence to reject the null hypothesis. Confounding by specialist type and insurance type does completely change explain the observed association between clinic and days. X clinic & Y days c) Provide an interpretation for the regression parameter estimates (“Coef.” In the Stata output) and their 95% Confidence Intervals for clinic 2 and clinic 3 in OUTPUT B. While controlling for insurance type and specialist type, we are 95% confident that the true mean of days shorter between referral and appointment for clinic 2 is between 1.4250 and 5.4168 days shorter compared to individuals from the reference group. While controlling for insurance type and specialist type, we are 95% confident that the true mean of days shorter between referral and appointment for clinic 2 is between 8.2917 and 36.5331 days shorter compared to individuals from the reference group. Problem 2 Your research team has developed a new vaccine that aims to prevent HIV infections. After producing promising results in preliminary trials, they enrolled 750 HIV-negative individuals at high risk of HIV infection and offered this vaccine to all. 475 participants opted to receive the vaccine while the other 275 opted to not to receive the vaccine. All participants were followed for two years. At the end of the study, the researchers determined whether or not participants were HIV-positive or -negative. A table of their crude results is provided below. HIV+ HIV- Total Vaccine 42 433 475 No Vaccine 104 171 275 Total 146 604 750 a) Calculate and interpret the estimated risk ratio for HIV infection in your study, 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 the 750 in your sample (using words like ‘estimated’ or ‘average’). RR = IE IE + ¿ CE CE + CN = 42 42 + 433 104 104 + 171 = .08842 .3782 = .2338 Those who do not receive the vaccine have an estimated risk of HIV infection 23.38% higher for than those who did receive the vaccine based on the data we found from the study. You hypothesize that injection drug usage may act as an effect measure modifier of the relationship between vaccine exposure and risk of HIV infection: you think the vaccine may be most effective at preventing HIV infection among individuals who routinely use injection drugs. However, your colleague disagrees and 3
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