MATH4387- Homework 3 Solved

Problem 1
Download the simu_hw3.txt data from canvas and read it in R. The data has four columns x1, x2, x3 and y. Print the summary of the linear regression model

y = β0 + β1x1 + β2x2 + β3x3 +

•     Is there something that surprise you? What it is?

•     Why do you thing it might happend? Justify your answer. (You can use plots or some statistic for justification.)

•     What model do you recommend? Run the recommended model and print the summary.

Problem 2
Fit y = β0 + β1x1 model and populate the following table without using the anova function.

Source              SS
df
MS
SSreg(X1) RSS(X1) TSS
 
 
SS, df, and MS represent the sum of squares, degrees of freedom, and mean sum of squares, respectively. MS = SS/df.

Problem 3
Fit y = β0 + β1x1 + β2x2 model. Now, if you want, you can use the anova function.

Source                     SS
df
MS
SSreg(X1,X2) RSS(X1,X2)

TSS
 
 
Problem 4
Define SSreg(X2|X1) = RSS(X1)−RSS(X1,X2). SSreg(X2|X1) is called the extra sum of squares. Calculate SSreg(X2|X1). Can you write SSreg(X2|X1) in terms of SSreg of the above models?

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Problem 5
The dataset teengamb from faraway package concerns a study of teenage gambling in Britain. Fit a regression model with the expenditure on gambling as the response and the sex, status, income and verbal score as predictors. Present the output.

(a)     What percentage of variation in the response is explained by these predictors?

(b)     Which observation has the largest (positive) residual? Give the case number.

(c)      Compute the mean and median of the residuals.

(d)     Compute the correlation of the residuals with the fitted values.

(e)     Compute the correlation of the residuals with the income.

(f)       For all other predictors held constant, what would be the difference in predicted expenditure on gambling for a male compared to a female?

Problem 6
In this question, we investigate the relative merits of methods for computing the coefficients. Generate some artificial data by:

x<-1:20 y <- x+ rnorm(20)

Fit a polynomial in x for predicting y. Compute βˆ in two ways — by lm() and by using the direct calculation described in the chapter. At what degree of polynomial does the direct calculation method fail? (Note the need for the I() function in fitting the polynomial, that is, lm(y ~ x + I(xˆ2)).

Problem 7
The dataset prostate in the faraway package comes from a study on 97 men with prostate cancer who were due to receive a radical prostatectomy. Fit a model with lpsa as the response and lcavol as the predictor. Record the residual standard error and the R2. Now add lweight, svi, lbph, age, lcp, pgg45 and gleason to the model one at a time. For each model record the residual standard error and the R2. Plot the trends in these two statistics.

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