This exercise uses NHANES (National Health and Nutrition Examination Survey) data to explore relationships between health variables. Complete all questions, showing your code, output, and interpretations where requested.
# Load required packageslibrary(NHANES)library(tidyverse)library(car)library(GGally)# Load and prepare datadata(NHANES)# Create a clean subset for analysisnhanes_clean <- NHANES |>filter(Age >=18, Age <=65) |># Adults onlyselect(Age, Gender, BMI, BPSysAve, BPDiaAve, DirectChol, TotChol, Diabetes, PhysActive) |>na.omit() |>distinct()# Preview datahead(nhanes_clean)
# A tibble: 6 × 9
Age Gender BMI BPSysAve BPDiaAve DirectChol TotChol Diabetes PhysActive
<int> <fct> <dbl> <int> <int> <dbl> <dbl> <fct> <fct>
1 34 male 32.2 113 85 1.29 3.49 No No
2 49 female 30.6 112 75 1.16 6.7 No No
3 45 female 27.2 118 64 2.12 5.82 No Yes
4 58 male 23.7 104 74 0.96 4.24 No Yes
5 54 male 26.0 134 85 1.16 6.41 No Yes
6 58 female 26.2 127 83 1.14 4.78 No Yes
Exercise 1: Correlation Analysis
We want to explore the relationship between systolic blood pressure (BPSysAve) and age.
(a)
Calculate the Pearson correlation coefficient between systolic blood pressure and age. Interpret the value including direction and strength.
# type code here
[type response here]
(b)
Calculate the Spearman correlation coefficient for the same variables. How does it differ from the Pearson correlation? When might Spearman be preferred?
# type code here
[type response here]
(c)
Create a scatterplot with a fitted line to visualize this relationship.
# type code here
Exercise 2: Simple Linear Regression
In this exercise, you will fit a simple linear regression model predicting systolic blood pressure from age.
(a)
Write out the mathematical model for this regression, clearly defining all terms. (Use LaTeX here. Knowing how to write something in LaTeX won’t be on the exam, but knowing how to write out a model in symbols will be.)
[type response here]
(b)
Fit the model in R and display the summary output using tidy() and kable().
# type code here
(c)
Interpret the slope coefficient. What does the p-value for age tell us? State the null and alternative hypotheses being tested.
[type response here]
Exercise 3: Model Assumptions
In the following exercise, we’ll check all four assumptions of linear regression for the model from Question 2.
(a)
Create diagnostic plots to check: (1) Linearity, (2) Normality of residuals, (3) Homoscedasticity, (4) Independence
# Your code here
(b)
For each assumption, state whether it appears to be met or violated based on your plots. Explain your reasoning.
In this question, we’ll fit a multiple linear regression model predicting systolic blood pressure from age, BMI, and gender.
(a)
Fit the model and display the estimated coefficients using tidy() and kable(). Round to 3 digits.
# Your code here
(b)
Interpret each coefficient in the model. How does the interpretation of the age coefficient differ from Question 3?
[type response here]
(c)
What is the \(R^2\) and adjusted \(R^2\)? Interpret the adjusted \(R^2\). Why might we prefer adjusted \(R^2\) versus regular \(R^2\)?
[type response here]
(d)
What is the F-statistic testing? State the null and alternative hypotheses. What is the distribution of the test statistic (include name of distribution and degrees of freedom)? Interpret the p-value.
[type response here]
(e)
Predict the systolic blood pressure for a 45-year-old male with BMI of 28. (First create a new data frame, then use the predict() function.) Write a 1-sentence conclusion about the predicted value.