2. Some simple path analysis.

We take one of the simple pictures from lecture, involving 4 variables
nice drawing, huh? :)

     -----  ------------------------------------------\
   /| SES | -------------\                             \>
  |  ------               >------                       -----
  |                       | nAch  |------------------->| GPA |
  |   -----                -------                      -----
   \ | IQ  |-------------/^                                /^
      ----- \---------------------------------------------/
            

The correlation matrix for these four variables is (n = 100)
      SES     IQ     nAch     GPA
SES          .3       .41     .33
IQ                    .16     .57
nAch                          .50
GPA

Standard deviations for these variables are:
      SES     IQ     nAch     GPA
      2.1     15     3.25     1.25

Write down the regression equation for GPA that is indicated by this
path diagram.

Compute the path coefficients for the nAch equation from the 'model'
depicted above.
Do this first for standardized coefficients.
Repeat for the unstandardized coefficients.
-----------------------------------------------------------------------
-----------------------------------------------------------

2. path analysis 
----------

(a) GPA has three arrows coming into it from SES, nACH and IQ

So the regression equation implied is:

GPA = beta0 + beta1*SES + beta2*IQ + beta3*nACH + error

(b) the nACH variable has two arrows coming in, SES and IQ.

so the model is:

nACH= delta0 + delta1*SES + delta2*IQ + error

(NOTE: deltas are used here to show that the parameters are
 different from the betas in the GPA equation (part a).  
You can call 'em anything you want)

The path coefficients are just (either standardized or not) regression
coefficients...

NWK p 292 has a useful formula for calculating standardized regression
coefficients (aka "beta weights" -- remember, that's a bad term) for a
regression with two predictors (use d to indicate a sample estimate
of a delta parameter; the ' indicates we are in the standardized metric
So the sample standardized path coefficients are

        r(y1)-r(12)*r(y2)
d1'=    ---------------
           1-r(12)^2

        r(y2)-r(12)*r(y1)
d2'=    ---------------
           1-r(12)^2

Here X1=SES, X2=IQ, y=nACH ...

Note the similarity between these formulas and the formulas
for part correlations...they're the same except the above formulas do
not have a square root in the denominator.

Now we plug in values from the correlation matrix

        .41 - .16 * .3          .362
b1'= -------------------- =     ----    = .398
        1- .3^2                 .91

        .16 - .41 * .3          .037
b2'= -------------------- =     ----    = .0407
        1- .3^2                 .91

So I guess we 'conclude' that SES causes motivation & IQ does not;
makes a good case for income redistribution to improve the schools?

(c)  To get the unstandardized path coefficients (i.e. the estimates
of delta1 and delta2), we use the relation:

        sd(y)
d1 = --------- * d1'    NWK p.289
        sd(x1)

remember, d1' = (sd(x1)/sd(y))*d1 --> just standardizing the
                                regression slope by the ratio of sd's.

So:
        3.25
d1 = -------- * .398   =   .616
        2.1

and similarly for d2:

        3.25
d2 = -------- * .0407  =   .00881
        15