Please show your work, include all your results in the write-up and...

Please show your work, include all your results in the write-up and submit the R code:

Problem 1. (Dataset: Dielectric.csv)
Electromagnetic technologies offer effective nondestructive sensing techniques for determining characteristics of pavement. The propagation of electromagnetic waves through the material depends on its dielectric properties. The following data, kindly provided by the authors of the article “Dielectric Modeling of Asphalt Mixtures and Relationship with Density” (J. of Transp. Engr., 2011: 104–111), was used to relate y
= diaelectric constant to x = air void (%) for 18 samples having 5 % asphalt content.

a) Identify the outcome/response variable and the independent/predictor variable
b) Graphical Analysis:
Before you begin building the regression model, it is a good practice to analyze and understand the variables and to understand these variables graphically. Typically, for each of the independent variables (predictors), the following plots are drawn to visualize the following behavior:
• Scatter plot: Visualize the linear relationship between the predictor and response
• Box plot: To spot any outlier observations in the variable. Having outliers in your predictor can drastically affect the predictions as they can easily affect the direction/slope of the line of best fit.
• Density plot: To see the distribution of the predictor variable. Ideally, a close to normal distribution (a bell shaped curve), without being skewed to the left or right is preferred.

Construct a Scatter plot, a box plot, and a density plot and comment briefly on them.

c) Obtain a linear correlation measurement to initially investigate the linear relationship between these two variables. Are these variables linearly related to each other? Explain.

d) Build the linear regression model.

e) Comment on the test (i.e. HO: Regression is not useful vs HA: Regression is useful) to formalize your initial investigation of these variables. What is your decision for this test? Write a conclusion in everyday language for this test.

f) Comment on the test to ensure that the slope of our regression line is not zero (i.e. HO: bMin/G = 0 vs HA: bMin/G ≠ 0). What is your decision for this test? Write a conclusion using everyday language for this test.

g) Now that you have built the linear model, and established the relationship between the predictor and response write down the mathematical formula for the outcome as a function the independent variable.

h) What is the R-Square value for this analysis? In the context of this problem, carefully explain what this number is measuring.

i) In the context of this problem, carefully interpret the y-intercept and slope of your estimated regression line. Again, carefully explain what these numbers are measuring. (You need to do more than say they are the y-intercept and slope of the line.)

j) Discuss whether or not the assumptions for this procedure are being meet. Checking the assumptions:
> Model Appropriate: Make sure no existing trends remain in the residual plot.
> Constant Variance: Make sure there is no megaphone patterns in the residual plot
> Normality: Make a histogram of the residuals and make sure they follow a normal distribution
> Outliers: Any observations that fall outside ±2*RMSE are considered possible outliers. Who are the outlying players?

k) What value of dielectric would you predict when air void is 5.70, and what is the value of the corresponding residual? What is the 95% prediction interval?

l) What is the 95% Confidence interval for the slope estimate? Interpret it.

Problem 2.
Consider the Prestige dataset from the car package. Build a regression model to model the relationship between income and the predictors taking into account possible interaction terms. Make sure to summarize the data properly (numerically and graphically), check multi-collinearty, assess model assumptions and comment on and interpret the results of the final model.

Use all the following cross validation techniques discussed in class to validate your model and discuss your results: The Validation set Approach, Leave one out cross validation – LOOCV, K-fold cross-validation, Repeated K-fold cross-validation and discuss your results.

Problem 3.
Consider the dataset PimaIndiansDiabetes2 [in mlbench package]. Our goal is to predict the probability of being diabetes positive based on multiple clinical variables.

Use the following code to read the data and remove NAs

data("PimaIndiansDiabetes2", package = "mlbench") PimaIndiansDiabetes2 <- na.omit(PimaIndiansDiabetes2)

1. Summarize the data graphically and numerically.
2. Construct a pairwise correlation matrix
3. Build a model to predict the probability of being diabetes-positive based on the plasma glucose concentration:
a. Comment on this model results and interpret the coefficients
b. Write the logistic equation and make predictions for values of glucose of 30 and 90.
c. The logistic function gives an s-shaped probability curve, plot it and comment on the plot.
4. Build a multiple logistic regression model that includes all the predictors
a. Comment on this model results and interpret the coefficients
b. Write the logistic equation
c. Predict the class membership probabilities of observations based on predictor variables
d. Assign the observations to the class with highest probability score (i.e above 0.5)
e. Assess model accuracy.