ST512 - Spring, 2005
Activity using Multiple Linear Regression to look at cigarette data
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 This is a study of various cigarettes in which Carbon
mOnoxide is the response and is to be modeled in terms
of WeighT , TAR, and NICotine.

   1. Run PROC CORR; VAR CO; WITH TAR NIC WT; and plot CO against each
      of the 3 variables. Do this in SAS/GRAPH using the following as
      one of them: 

         PROC GPLOT; PLOT CO*TAR; SYMBOL1 V=dot I=R C=red; 

      (V is the symbol, I is interpolation I=R is regression line) 
      Write the correlation on each plot with a pencil. Which variable is 
      most highly related to CO? Also try this 3-D plot 

          PROC G3D; SCATTER TAR*NIC=CO; 

      [A comment: When the explanatory variables are highly
      correlated, we refer to this as "multicollinearity" With the
      fitted surface being a plane, it is as though the 3D plot shows
      the legs for a table. Notice how unstable the fitted surface would
      be here.]

   2. Run a multiple regression of CO on the three explanatory
      variables. Which of the three is most significant according to the
      t statistics? Compute a 95% confidence interval for the
      coefficient of TAR. Give a verbal interpretation of this coefficient.
   3. Run a simple linear regression of CO on TAR. Compare this to the
      full model of part 2 as follows:
          * (A) Compute the difference in model sums of squares. This is
            the sum of squares for NIC and WT adjusted for TAR.
          * (B) Compute the difference in model degrees of freedom.
          * (C) Compute the mean square associated with (NIC,WT) by
            dividing your A answer by your B answer.
          * (D) Now compute F as the mean square from part C divided by
            the error mean square from the full model.
          * (E) What is this F testing and is it significant? 
   4. For your simple model in part 3, compute a 95% confidence interval
      for the mean CO of all possible cigarettes having TAR content 10.
      Also compute a 95% prediction interval for the CO of an individual
      future cigarette with TAR content 10.
   5. Run the multiple regression as follows: PROC REG; MODEL CO = TAR
      NIC WT/ss1; nic_wt: test NIC=0, WT=0; Explain how the answer to 3A
      can be computed from the list of Type I sums of squares. What did
      the TEST statement do for you? Try a different order, say PROC
      REG; MODEL CO = NIC WT TAR/ss1; Do the Type I SS change? How about
      the t-tests? Model F test? Fitted parameters?
   6. Add to the dataset an observation with CO missing (.) and TAR =
      10. WT and NIC can be anything you like. Now run PROC REG; MODEL
      CO = TAR/P CLM; and compare to part 4. Repeat, changing CLM to
      CLI. 

(For help, see "cigarettes.sas")