Description

Assignment 4

1. Materials scientists working for a cement mixing company explored the ingredients in the cement impacted the heat of the cement during hardening. They investigated the impact of 5 composition variables (see below) on the heat during hardening (calories/gram). The data can be found in the file cement.csv on Canvas.

Regressor Description
x1 % tricalcium aluminate
x2 % tricalcium silicate
x3 % tetracalcium alumino ferrite
x4 % dicalcium silicate
x5 % sodium oxide

Use only the lm() command in R for this question. Do not use any variable selection functions or downloaded packages in R (e.g., step()).

a. Use forward selection to identify the “best model”. Use a (conditional) t-test with aIN=0.10 as the cutoff to decide whether or not to include a variable in the model. At each step in the forward selection, state the variable that was added. At the end of the procedure, state the final model.
b. Use backward elimination to identify the “best model”. Use a (conditional) t-test with aout=0.10 as the cutoff to decide whether or not to remove a variable from the model. At each step in the procedure, state the variable that was removed from the model. At the end of the procedure, state the final model.
c. In the first step of the forward procedure in part a, both x4 and x5 has small pvalues. However, in the second step of the forward selection procedure, when both x4 and x5 were considered together in the model, neither of the variables had a p-value less than aIN. Why do you suppose this happened?
d. Use stepwise regression to identify the “best model” using t-tests with aIN=aout=0.10. At the end of each step, state the variable that was removed or added from the model. At the end of the procedure, state the final model.
e. What is the AIC for the models in a, b and d?

2. Consider the multiple linear regression model: y = b0 + b1 x1 + b2 x2 + b3 x3 + b4 x4 + b5 x5 + e.

State the appropriate coditional or extra sums of squares (e.g., see section 3.3.2) and F-statistic (with degrees of freedom) for testing the following hypotheses:

a. H0 : b1 = b2 = b3 = b4 = b5 = 0
b. H0 : b4 = b5 = 0, given that the first 3 variables are in the model

3. A tube flow reactor is a chemical process in which materials in typically a series of tubes create desired chemical products. A experiment was conducted to study the relationship between the concentration of a desired product, NbOCl3 (y), and the concentration of COCl2 (x1), reaction time (x2), molar desnity (x3), mole fraction (x4), tube temperture (x5), ramp-up time (x6) and product density (x7). The data can be found in reactor.csv on Canvas.

a. For each p = 2, …, 5, find the maximum R2. Plot the maximum R2 versus p. Which model would you choose for these data and why?
b. For each p = 2, …, 5, find the minimum MSRes. Plot the minimum MSRes versus p. Which model would you choose for these data and why?
c. For each p = 2, …, 5, find best model using Mallow’s Cp. Plot the best Cp versus
p. Which model would you choose for these data and why?