Homework 4 St 708

 A real data example 
 Questions 

/*      
(1.)  These are the data from chapter 5.   Please read
through chapter 5, which describes one analysis of this 
(real) data set.  
      Page xi of the preface tells you that the data are
located at
     
 http://www.stat.ncsu.edu/publications/ 
Go there, click on Rawlings, and find the data set 
        LINTH-5.DTA

A. Get the data into the SAS program editor window then 
create the appropriate data step to read it.  I will 
use the names as given on page 163.  Using PROC CORR, 
compute the correlation matrix as on page 164 of the text. 
If there is any discrepancy, we will go with the data
off the internet. 

B. Regress BIO (this is Y) on zinc Zn.  Is the slope 
significant?  Compute R(pH|X0, Zn) and R(Zn|X0, pH)
where X0 stands for the intercept.  Which of these 
are significant?  (use the MSE from the regression 
with both of these variables)  How is the R-square 
from the simple linear regression of BIO on Zn  
related to an entry of your correlation matrix? 

C. Regress BIO on all the other available numeric 
variables. Construct a 95% confidence interval for 
the coefficient of Zn in this full regression. 
(see Rawlings et al pg. 164-5)

D. Rawlings suggests (pg. 170) that BIO can 
be modeled using just pH and K. By subtraction, 
test the full model from part C against this reduced 
model.  
   Repeat the same test in a single PROC REG by 
using a test statement.  Write down the K and M 
matrices as in H0: K Beta - M = 0 (K is a matrix 
here, not a symbol for potassium) that are being 
used in this test assuming the variables are 
entered in the order in the text: 
  intercept, Sal, pH, K (potassium), Na, and Zn. 
    Do the test a third time by appropriately rearranging
variables in your full MODEL statement and requesting the 
type I sums of squares.  Explain how the F test can be 
computed from these Type I SS.  

E. In the full model, table 5.2 gives the 5 estimated
coefficients.  Suppose (just for illustration) I want 
to test that the true values of these coefficients are, 
respectively, 
    -35.0000, 300.0000, -1.0000, 0, and -18.0000
Compute the K and M matrices for this test as in 
H0: K Beta - M = 0.  (K is a matrix here, not a 
symbol for potassium)  Issue a test statement in 
PROC REG to test this hypothesis. 

F. Suppose (just for illustration) I want to test the
joint null hypothesis that the true coefficients of SAL 
and Zn are the same as each other, the coefficient 
of Na is 0, and the coefficient of K is -.5.  

	(i) Regress BIO + .5K on pH and some linear 
          combination of SAL and Zn with an intercept
          so that this gives your reduced model.  Compute
          the F test by comparing this to the full model.

	(ii) Compute the F test by writing your K and M 
          matrices, then issuing a test statement in
          the full model to do the job. 

G.  Using the variable names as above, this program 
creates the principal components of your data.  Do prin1 
and prin2 have interpretations (I did not see any neat 
ones - do you?  no right or wrong answer) 

Plot prin1 vs prin2 using location as a plot symbol. 

proc princomp out=outlin; var  bio sal ph K Na Zn ;
proc gplot; plot prin1*prin2=loc/legend;
symbol1 v=dot i=none c=red;
symbol2 v=dot i=none c=green;
symbol3 v=dot i=none c=cyan;

Plot the first 3 principal components in a 3-D plot again 
using color to denote location

DATA OUTLIN; SET OUTLIN;
  IF LOC = "OI" THEN C = "RED  ";
  IF LOC = "SI" THEN C = "GREEN";
  if loc = "SM" THEN c = "CYAN";
proc g3d; scatter prin3*prin2=prin1/
   shape="balloon" size = .75 COLOR=c;

Turn in the plots.  Any comment about whether the locations
form clusters in these plots? (again, no right or wrong, just
note that this is one kind of thing you might do with principal
components) 

Optional:  use SAS INSIGHT or JMP to create a rotating 3-D 
plot. (nothing to hand in)  . 

  (postponed prob. 4.3 to next time) 

(2.) Practice with quadratic forms:
Suppose E is a column vector of 5 independent N(0,1)
random variables, that is, these are normally distributed
with mean 0 and variance 1.

Suppose also that V is this matrix:

                              13   6  -1  -1  - 1
                               6   4   2   2    2
                V =           -1   2   5   5    5
                              -1   2   5   5    5
                              -1   2   5   5    5

Is V symmetric?  Find, if possible,  a scalar multiple 
m of V so that A = mV is idempotent.
(hint:  Compute VV and compare it to V).

Let Y be the quadratic form Y = E'VE.
Find the expected value of Y.
Find a value c such that Pr{Y > c} = .05.
(hint: Express E'VE as Z'AZ where A=mV is idempotent)
*/