* Evils of standardization:
* This program illustrates problems with the use of standardized
* variables and standardized coefficients.

* Hypothetical data: a sample of 1200 blacks and 1200 whites.  Assume
* that equal-sized samples of blacks and whites have been drawn, but
* that in this population, whites outnumber blacks by 8 to 1.  Hence,
* blacks have been oversampled by a factor of 8, and when the groups
* are analyzed together, each black should therefore be weighted only
* 1/8 as heavily as each white.  In this hypothetical example,
* all individuals conveniently have incomes of $3000, $3500, $4000, 
* $4500, $5000, or $5500, hence cases are grouped together.

* Or, as an alternative, assume that 2 populations have been sampled
* from.  In one population, whites outnumber blacks by 8 to 1, whereas
* in the other population it is a 50-50 split.

* The vars are:
* Ncases - the number of cases having a particular combination of
      values
* Inc - Income
* Race - 0 = Black, 1 = White
* Inc2 - Income with random error.  The change in income conveniently 
*     works out to be $1100 either way.  The variability in income
*     increases as a result.

DATA LIST FREE / Ncases Inc Race Inc2.
BEGIN DATA.
200   3000  0  1900
200   3000  0  4100
200   3500  1  2400
200   3500  1  4600
200   4000  0  2900
200   4000  0  5100
200   4500  1  3400
200   4500  1  5600
200   5000  0  3900
200   5000  0  6100
200   5500  1  4400
200   5500  1  6600
END DATA.
List   Variables = Ncases Inc Race Inc2.

* Compute the correct weight.  We weight each case by Ncases;
* then, weight the Black cases only 1/8 as heavily as each white
* case; then, since sample size has been deflated, inflate the weights
* for all cases by a constant factor so that the total sample size
* is correct.
Compute     WgtRight = Ncases.
If          (Race = 0) WgtRight = Ncases/8.
Compute     WgtRight = WgtRight * 2400/(1200 + 1200/8).

* The incorrect weight ignores the oversampling of blacks, and weights
* all cases equally.
Compute     WgtWrong = Ncases.

*               E X A M P L E   1
* What effect does improper weighting of cases have on
* coefficients in a perfectly specified model? .

* First, use the correct weight.
WEIGHT BY WgtRight.
REGRESSION /VARIABLES Inc Race /DESCRIPTIVES DEFAULT
           /DEPENDENT Inc /METHOD ENTER Race.

* Now, repeat the analysis using the incorrect weight.  What differences
* occur?.

Weight by WgtWrong.
REGRESSION /VARIABLES Inc Race /DESCRIPTIVES DEFAULT
           /DEPENDENT Inc /METHOD ENTER Race.



*               E X A M P L E   2
* Suppose variability in the dependent variable increases on a random
* basis; that is, the residual variance that cannot be explained by
* other variables goes up.  What effect does an increase in residual
* variance have on metric and standardized coefficients?

* To examine this, we compare Inc and Inc2. Inc2 adds random variation 
* to Inc.

* Use the correct weight.
Weight by WgtRight.
REGRESSION /VARIABLES Inc Race /DESCRIPTIVES DEFAULT
           /DEPENDENT Inc /METHOD ENTER Race.
REGRESSION /VARIABLES Inc2 Race /DESCRIPTIVES DEFAULT
            /DEPENDENT Inc2 /Method enter race.