LOG concerning input of Data on Consumption & Price of 
Beef and Pork, and correlation analyses in SAS.
============================================== 10/8/08

The data, including explanation of variables, can be 
found at :

http://lib.stat.cmu.edu/DASL/Datafiles/agecondat.html

(When you download, you will find that the numeric data 
fields are separated by tabs not blanks, and you will 
have to replace them by blanks using word-processor 
commands in order to read into SAS.)

 libname home ".";
 data home.meat;
    infile "ASCdata/Meat2.dat";  
    input   yr pbe cbe ppo cbo pfo dinc cfo rdinc rfp ;
    run;
proc corr; run;
/* Gives all paiwise correlations plus some simple 
statistics like means and variances. */

*  ## Now we do some more targeted analyses. ;

  options linesize = 70 nodate nocenter;
  proc corr data=home.meat ;
     var pbe cbe;
     with ppo cbo pfo dinc cfo rdinc rfp;
   run;

Pearson Correlation Coefficients, N = 17
       Prob > |r| under H0: Rho=0

                pbe           cbe

ppo         0.32677       0.08341
             0.2005        0.7503

cbo        -0.34615      -0.29150
             0.1735        0.2563

pfo        -0.40552       0.13072
             0.1063        0.6170

dinc       -0.65276       0.32024
             0.0045        0.2102

cfo        -0.77336       0.50682
             0.0003        0.0379

rdinc      -0.64062       0.46816
             0.0056        0.0581

rfp        -0.34021       0.29144
             0.1815        0.2564

* This suggests that the "strong" predictive 
variables for either the price (pbe) or
 consumption amounts (cbe) of beef are:

6. DINC = Disposable income per capita index (1947-1949 = 100)
7. CFO = Food consumption per capita index (1947-1949 = 100)
8. RDINC = Index of real disposable income per capita (1947-1949 = 100)

proc corr;
  var pbe cbe;  /* CORR=-0.75244,  p-val= 0.0005 */

proc corr ;     /* CORR=-0.65961,  p-val= 0.0054 */
  var pbe cbe;
  partial cfo;

proc corr ;     /* CORR=-0.69838,  p-val= 0.0038 */
  var pbe cbe;
  partial cfo dinc;

proc corr ;     /* CORR=-0.87163,  p-val= 0.0001 */
  var pbe cbe;
  partial cfo dinc rdinc; run;

### This succession of partial correlations says that
   "removing" the effect of these other important 
   variables only makes the negative relationship between
   PBE and CBE stronger !!!

### The linear algebra meaning of these results can be 
seen as follows:

data meatmat;
   set home.meat (keep=pbe cbe dinc cfo rdinc);

/*
meatmat = 17 x 5 matrix with columns pbe, cbe, dinc, cfo, rdinc
*/

> cor(pbe, meatmat[,2:5])
            cbe        cfo      dinc      rdinc
[1,] -0.7524356 -0.7733555 -0.652759 -0.6406248


> cor(cbe, meatmat[,c(1,3:5)])
            pbe       cfo      dinc    rdinc
[1,] -0.7524356 0.5068163 0.3202374 0.468159


meatmat2 = 17 x 4 matrix with columns pbe, cbe, dinc, rdinc
           obtained by replacing each column of meatmat1 by its 
           projection orthogonal to the meatmat1 cfo column

## Now the partial correlation of pbe with the other variables
   after "removing the effect of cfo" is 

> cor(meatmat2[,1], meatmat2[,2:4])
           cbe       dinc     rdinc
[1,] -0.659605 -0.3016153 0.1461794


COMPARE with:

proc corr data = meatmat;
   var pbe; with cbe dinc rdinc; partial cfo;
run;

                pbe

cbe        -0.65961
             0.0054

dinc       -0.30162
             0.2563

rdinc       0.14618
             0.5891


meatmat3 = 17 x 3 matrix with columns pbe, cbe, rdinc
           obtained by replacing each column of meatmat2 by its 
           projection orthogonal to the meatmat2 dinc column

> cor(meatmat3[,1], meatmat3[,2:3])
           cbe    rdinc
[1,] -0.698382 0.515713

COMPARE WITH:

proc corr data=meatmat;
     var pbe; with cbe rdinc; partial cfo dinc ;
run;
                pbe

cbe        -0.69838
             0.0038

rdinc       0.51571
             0.0491

meatmat4 = 17 x 2 matrix with columns pbe, cbe 
           obtained by replacing each column of meatmat3 by its 
           projection orthogonal to the meatmat3 rdinc column

> cor(meatmat4)
           pbe        cbe
pbe  1.0000000 -0.8716332
cbe -0.8716332  1.0000000

COMPARE WITH:

proc corr data=meatmat;
     var pbe; with cbe;  partial cfo dinc rdinc;
run;
              pbe

cbe      -0.87163
           <.0001


## All of this shows the meaning of "partial correlations" as
   a results of removing successive projections

### Here is how to do the successive "removing of projections"
###   directly using SAS.

proc reg data=Meat;
    model pbe cbe dinc rdinc = cfo  ;
    output out= meatfil1
        r = pbe1 cbe1 dinc1 rdinc1;
        run;

  options nocenter linesize=70 nodate;
proc corr data = meatfil1;
      var pbe1 cbe1 dinc1 rdinc1;   run;

           Pearson Correlation Coefficients, N = 17
                   Prob > |r| under H0: Rho=0

                  pbe1          cbe1         dinc1        rdinc1

pbe1           1.00000      -0.65961      -0.30162       0.14618
Residual                      0.0040        0.2394        0.5756

cbe1          -0.65961       1.00000      -0.02028       0.04962
Residual        0.0040                      0.9384        0.8500

dinc1         -0.30162      -0.02028       1.00000       0.69207
Residual        0.2394        0.9384                      0.0021

rdinc1         0.14618       0.04962       0.69207       1.00000
Residual        0.5756        0.8500        0.0021


### In particular, note that the correlation here between
pbe1 and cbe1 exactly matches the "partial correlation of 
pbe and cbe with cfo removed" previously calculated.

## To go one step farther in partial correlations, we do:

proc reg data=meatfil1;
    model pbe1 cbe1 = dinc1  ;
    output out= meatfil2
        r = pbe2 cbe2;
proc corr data=meatfil2;
    var pbe2 cbe2; run;

/* THINK OF HOW LABORIOUS IT WOULD BE TO DO THESE CALCULATIONS
    "BY HAND" IN SAS WITHOUT PROC REG. 
    CAN YOU THINK HOW YOU WOULD DO THE CALCULATIONS USING DATA-STEP
    AND PROC MEANS ? */


## and we get 

Pearson Correlation Coefficients, N = 17
       Prob > |r| under H0: Rho=0

                  pbe2          cbe2

pbe2           1.00000      -0.69838  ## matching partial corr of pbe, cbe
Residual                     0.0018  ## after removing cfo ,dinc