Homework 1 
 Review of Basic Regression Computations 


   /* ---------------------------------------Due Sep. 2 2003-----------
   |  This exercise provides a brief review of simple linear regression |
   |  (EXER 1.19) and a review of simple matrix computations.           |
    --------------------------------------------------------------------




 Questions 

Look through sections 1.1-1.6, digesting any information you do not already 
know about simple linear regression, then...

Ex 1.19  The following data relate soybean seed weight Y, to solar
radiation X at two ozone levels.  The program steps that follow
create a SAS data set with which you can do the necessary computations.
Note the use of the '.' to obtain a predicted value for x = 400.
For these data, do the following:



(A) Answer the questions on page 34. 

(B) For high ozone and radiation 400, we have used the "missing Y trick"
   to compute a prediction.  Explain how this prediction can be computed 
   from the regression coefficients for the high ozone data.

(C) Show how the standard error for this prediction is computed
   from the inverse of X'X and the error mean square.

(D) Explain the difference between a prediction interval for an
   individual (CLI) and the confidence interval for a mean (CLM).  
   Show the hand computations for our radiation 400, high ozone
   prediction. You can assume everyone knows how to compute the 
   slope, intercept, error mean square, etc.  Just show how 
   these enter the confidence interval formula.


          SAS Code for part 1 

 

Work exercises 2.1 and 2.3 on matrix operations. You can  
use this IML code to help you, but should be able to do everything
by hand as well.  
 
For your answer to 2.1 (e) explain what SAS is doing and note that
this operation would usually be considered "illegal".     


       SAS IML Code for part 2


Vectors A=(1,8,2)' and B = (-4,1,5)' are obviously in three dimensional
space.  We can get a three dimensional plot of these as follows 

DATA THREED ; INPUT Y1 Y2 Y3; CARDS;
 0 0 0
 1 8 2
-4 1 5
PROC G3D; SCATTER Y1*Y2=Y3/NONEEDLE SHAPE ='BALLOON';
RUN;

(A) Use the points on your 3-D plot and pencil in arrows emanating
from the origin to indicate the vectors A and B.
(B) Note on your plot the length of each vector and the angle between
them.  Be sure to indicate whether the angle is in radians or degrees.
(C) Compute A+B and A-B for the given vectors A and B.
(optional: Add A+B to your plot, connecting the ends of vectors A
and B to A+B, and noting the parallelogram that is formed in
the plane containing A and B).
(D) Is the vector (5,7,-3)' in the plane defined by vectors A and B?
How do you know?
(E) Regress the vector Y = (1,2,3)' on A and B (no intercept) getting
the error sum of squares and the column of residuals R.  For example
if you had column vectors A, B, and Y already defined in PROC IML
you could say:

    X = A||B;   * concatenation;
    XPX = X`*X;  XPY = X`*Y; B = inv(XPX)*XPY;
    SSE = Y`*Y - B`*X`*Y;
    R = Y - X*B;
    PRINT X Y B SSE R;

(F) Compute the vector of predicted values from the regression in E.
Express the vector of predicted values as a linear combination of the 
vectors A and B (this implies it is in the plane formed by A and B).  
This is, in fact, the closest vector in that plane to the original 
Y vector. 
Is the vector (1,2,3)' in the same plane as A and B?  How do you know?
What is the length of the residual vector?  Have you seen the squared
length of the residual vector in your previous computations?


  Output