Description
1. This question involves the use of multiple linear regression on the Auto data set from the
course webpage (https://scads.eecs.wsu.edu/index.php/datasets/). Ensure that you remove
missing values from the dataframe, and that values are represented in the appropriate
types.
a. Produce a scatterplot matrix which includes all the variables in the data set.
b. Compute the matrix of correlations between the variables. You will need to
exclude the name variable, which is qualitative.
c. Perform a multiple linear regression with mpg as the response and all other
variables except name as the predictors. Show a printout of the result (including
coefficient, error and t values for each predictor). Comment on the output:
i. Which predictors appear to have a statistically significant relationship to
the response, and how do you determine this?
ii. What does the coefficient for the displacement variable suggest, in simple
terms?
d. Produce diagnostic plots of the linear regression fit. Comment on any problems
you see with the fit. Do the residual plots suggest any unusually large outliers?
Does the leverage plot identify any observations with unusually high leverage?
e. Fit linear regression models with interaction effects. Do any interactions appear to
be statistically significant?
f. Try transformations of the variables with X3 and log(X). Comment on your
findings.
2. This problem involves the Boston data set, which we saw in the lab. We will now try to
predict per capita crime rate using the other variables in this data set. In other words, per
capita crime rate is the response, and the other variables are the predictors.
a. For each predictor, fit a simple linear regression model to predict the response.
Include the code, but not the output for all models in your solution. In which of
the models is there a statistically significant association between the predictor and
the response? Considering the meaning of each variable, discuss the relationship
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between crim and nox, chas, medv and dis in particular. How do these
relationships differ?
b. Fit a multiple regression model to predict the response using all the predictors.
Describe your results. For which predictors can we reject the null hypothesis H0 :
βj = 0?
c. How do your results from (a) compare to your results from (b)? Create a plot
displaying the univariate regression coefficients from (a) on the x-axis, and the
multiple regression coefficients from (b) on the y-axis. That is, each predictor is
displayed as a single point in the plot. Its coefficient in a simple linear regression
model is shown on the x-axis, and its coefficient estimate in the multiple linear
regression model is shown on the y-axis. What does this plot tell you about the
various predictors?
d. Is there evidence of non-linear association between any of the predictors and the
response? To answer this question, for each predictor X, fit a model of the form
Y = β0 + β1X + β2X2 + β3X3
+ ε
Hint: use the poly() function in R. Again, include the code, but not the output for
each model in your solution, and instead describe any non-linear trends you
uncover.
3. An important assumption of the linear regression model is that the error terms are
uncorrelated (independent). But error terms can sometimes be correlated, especially in
time-series data.
a. What are the issues that could arise in using linear regression (via least squares
estimates) when error terms are correlated? Comment in particular with respect to
i) regression coefficients
ii) the standard error of regression coefficients
iii) confidence intervals
b. What methods can be applied to deal with correlated errors? Mention at least one
method.
