[SOLVED] MATH4330 Assignment 3

Question 1: Hourly carbon monoxide (CO) averages were recorded on summer week- days at a measurement station in Los Angeles. The station was established by the Environmental Protection Agency as part of a larger study to assess the e↵ectiveness of the catalytic converter. It was located about 25 feet from the San Diego Freeway, which in this particular area is located at 145 degrees north. It was located such that winds from 145 to 325 degress (which in the summer are the prevalent wind directions during the daylight hours) transport the CO emissions from the highway toward the measure- ment station. Aggregate measurements were recorded for each hour of the day 1 to 24 and the dataset is available in the file CO2.txt. Note: you can load this file in R using read.table() and setting the argument header=TRUE.

Hour – CO – TD – WS –

hour of the day, from midnight to midnight
average summer weekday CO concentration (parts per million) average weekday trac density (trac count/trac speed) average perpendicular wind-speed component
(wind speed ⇥ cos(wind direction – 235 degrees))

1. (a)  [5 points] Run a linear regression model to examine the e↵ect of weekday trac density and wind-speed component on CO concentration. Report the estimated slope parameters and their confidence intervals.
2. (b)  [3 points] Examine residual plots in the model from part (a). Do you think any of the linear regression assumptions have been violated? Explain.
3. (c)  [5 points] Run a weighted least-squares model using the same outcome and pre- dictors from part (a). Have the estimates and confidence intervals changed much? Explain.

Question 2: Recall the chest pain dataset from Assignment 2. This dataset is from the Duke University Cardiovascular Disease Databank and consists of 2258 patients and 6 variables. The patients were referred to Duke University Medical Center for chest pain. The variables included in the dataset acath2.csv are the following:

– sex: sex of the patient (0=male, 1=female)

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– age: age of the patient

• –  cad.dur: duration of symptoms of coronary artery disease
• –  cholest: cholesterol (in mg)
• –  sigdz: significant coronary disease by cardiac catheterization (defined as 75% diameter narrowing in at least one important coronary artery – 1=yes, 0=no)
• –  tvdlm: severe coronary disease (defined as three vessel or left main disease by cardiac catheterization – 1=yes, 0=no))

1. (a)  [4 points] Run a logistic regression model to see the e↵ect of cholesterol (contin- uous measure) on significant coronary disease (sigdz). Report the odds ratio and interpret. Calculate and interpret a 95% confidence interval for the odds ratio.
2. (b)  [3 points] Calculate the predicted probability of significant coronary disease for an individual with cholesterol equal to 400.
3. (c)  [1 points] Do you think the expression in (b) can be used to accurately predict significant coronary disease in the general population? Explain.
4. (d)  [4 points] Run another logistic regression model to see the e↵ect of cholesterol on sigdz, but this time adjust for age and sex. Report the odds ratio. From this new model fit do you think there is evidence that age and sex are confounders in the cholesterol/coronary disease relationship? Explain. (Hint: it has nothing to do with significance of the predictor variables).
5. (e)  [5 points] Create an ROC curve for each of the models in part (a) and (d). Report the AUC for each one. Which model has better predictive accuracy? Explain.

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