Sta113: Lab 9 (Friday, April 11, 2003)

In this lab, we will use MatLab to do multiple linear regression and residual analysis.

Items that we will investigate using MatLab include:

• 1) Extending simple linear regression to include multiple independent variables.
• 2) Interpreting and doing hypothesis tests in this context.
• 3) Coefficient of multiple determination.
• 4) Residual plots: Residuals vs. fitted values, and residuals vs. predictor variables.

Use the data from EXAMPLE 12.13 on page 677 in the text. The dependent variable is Carbon Monoxide (CM) content of different cigarette brands. We want to investigate the relationship of three predictor or independent variables with CM, these variables are Tar (T), Nicotine (N), and Weight (W). (see page 677 for more details).

Read in the data and call the first column T, the second N, third W, and the fourth column CM.

smoke = load('lab9.dat');
T = smoke(:,1);
N = smoke(:,2);
W = smoke(:,3);
X = [ones(length(smoke),1) T N W ]; % X needs to be a
                                    % matrix with a leading 
                                    % column of ones.
CM = smoke(:,4);
Y = CM;

Let's investigate the the items above:

• 1) Extending simple linear regression to include multiple independent variables. Finding the least squares line: CM = b0 + b1*T + b2*N + b3*W.

   
  % The predictor variables are in the matrix X along with a column of
  % intercepts (1's) and the response variable (Y) is CM.
  % Now find the least squares regression line
  
  [b,bint,r,rint,stats] = regress(Y,X,0.05);
  
  b % = estimates for the regression coefficients
  
  
  2) Interpreting and doing hypothesis tests in this context.
  
   The hypothesis Ho: bi = 0, vs. Ha:bi != 0 can be tested in the
  same way that we tested for b1 = 0 last week (see Lab 8).  What are we testing here?
  
  
  Now that there are more than one predictor variables we can test
  the hypothesis that at least one of the coefficients b1,..., b3 is not
  zero.  That is we test Ho: b1 = b2 = b3 = 0, vs. Ha: at least one is
  not zero.  To perform this hypothesis test we need to look at the
  accompanying F-statistic and p-value recorded in stats:
  
  stats % = R-squared, F-statistic, p-value
  
  
  3) Coefficient of multiple determination.
  
   The multiple coefficient of determination is the R-squared in
  stats.  What does it signify?
  stats % = R-squared, F-statistic, p-value
  
  
  4) Residual plots: Residuals vs. fitted values, and residuals
  vs. predictor variables. 
  
   Examination of the residuals can tell us whether the assumptions
  we make to do regression are reasonably met. Recall the assumptions:
  
  a) The mean of the errors is 0.
  b) The variance of the errors is constant.
  c) The probability distribution of the errors is normal.
  d) The errors are independent.
  
   We can use the residuals to determine if these assumptions are
  met. First plot the residuals against the fitted values. What types of
  patterns should you expect to see if the assumptions are met? What can
  you conclude about the propriety of using multiple regression
  on this dataset? We can also look at a histogram of the residuals to
  determine if they are normal (see Lab 2 if
  you don't remember how).
    
  
  plot(X*b,r,'.') %add appropriate title and labels, etc.
  
  You can get a pretty good idea of whether a variable should be
  included in the model by looking at the residuals vs. predictor variable.
  plot(T,r,'.') % OR
  plot(N,r,'.') % OR
  plot(W,r,'.') 
  
  
  Do any of these independent variables look like they should not be
  included as linear terms? If you don't know what to look for consult
  chapter 12 in the book.