STAT823- Homework 9: Multiple Linear Regression Solved

Directions

Using RMarkdown in RStudio, complete the following questions. Launch RStudio and open a new

RMarkdown file or use the class RMarkdown template provided and save it on your working directory as a .Rmd file. At the end of the activity, save your pdf generated from RMarkdown+Knitr and submit your homework on the Blackboard.

If you have questions, please post them on the lesson discussion board.

All questions are mandatory. Some R-codes and output from the code have been provided for you.


The primary objective of the Study on the E cacy of Nosocomial Infection Control (SENIC Project) was to determine whether infection surveillance and control programs have reduced the rates of nosocomial
 

(hospital-acquired) infection in United States hospitals. The dataset consists of a random sample of 113 hospitals selected from the original 338 hospitals surveyed. Each record in the dataset has an identification number and provides information on 11 other variables for a single hospital. The 12 variables are:

The average length of stay in a hospital (Y) is assumed to be related to infection risk, available facilities and services, and routine chest X-ray ratio.

1.    Run three separate regression models for each of the three potential predictors (i.e., your first model is Y = —0 +—1X1 where X1 = infection risk). Plot the three estimated regression functions over the data in three separate graphs. Does a linear relationship appear to provide a good fit for each of the three predictor variables?

2.    Which predictor leads to the smallest MSE (a.k.a., unexplained (error) variation)? Which predictor variable has the highest R2? So, which of the three accounts for the largest reduction in

variability of the average length of stay?

                                                         Predictor                                     MSE        R2

Infection risk

 Facilities and Services Chest X-ray ratio

3.    Obtain model residuals and use them for model diagnostics. Do you identify any issues with model assumptions? Refer to the last lesson for a review of the approach.

Testing linearity and constant variance by plotting fitted value against against residuals.

4.    Delete cases 47 (X =6.5, Y =19.56) and 112 (X =5.9, Y =17.94) and refit the model for length of stay and infection risk. From this fitted model, obtain prediction intervals for new Y observations at X = 6.5 and X = 5.9. Does what was observed (i.e., Y =19.56,17.94) fall into the bounds of the respective prediction intervals? Discuss the significance of this.

5.    Build the “best" regression model for Y. Begin first with variable selection using the regsubsets function. Justify your final choice of model using criterion-based methods such as BIC and adjusted R2, all of which can be extracted from the regsubsets model object using the summary function.

6.    Once you have identified your final model, check for and comment on any issues with:

–    Multicollinearity between predictors

–    Outliers and influential points

–    Appropriateness of predictors (i.e., is any transformation of predictors necessary?)

–    Normality of residuals

–    Constant variance of residuals. Calculate the variance inflation factor.

# Normality of residuals

par(mfrow = c(1, 2))

shapiro.qqnorm(residuals(lmBM), cex = 2) shapiro.qqnorm(residuals(lmBMsub), cex = 2)

7.    Provide an intuitive interpretation of your final model. In other words, explain your findings to me as if I have a minimal working knowledge of statistics. Extra Plots (Not Required)

library(car)

marginalModelPlots(lmBMsub)

Marginal Model Plots
 

                                                 chestxray                                                             Fitted values

avPlots(lmBMsub, id = list(n = 2, cex = 0.6))

Added−Variable Plots
 

                                          infection | others                                                        facil | others

 

chestxray | others