On Fri, 19 May 2000, Magill, Brett wrote, inter alia:
> Nonetheless, regression and correlation are the same in the bivariate
> case with the exception of scale.
This is a partial truth, and therefore likely to be misleading.
> In a bivariate regression, the standardized Beta coefficient is equal
> to the Pearson r.
This is true: _within the sample in hand_.
> As with any standardization, it removes the scale of the variation
This is another partial truth. It does not so much REMOVE the
scale of the variation (in both variables) as it CHANGES each scale to
units of the SAMPLE standard deviation. Since the sample s.d. depends on
any constraints that may be restricting the range (or for that matter
extending the range) of either variable, the standardized covariance and
the standardized regression coefficient are not invariant over all
possible ranges of X and/or Y.
> and the result is that the slope describes the relationship or B = r.
Yes, for the sample. This is as good a reason as any for the
standard advice that standardized regression coefficients are not so
useful as raw regression coefficients -- which is part of what Herman was
trying to convey. To repeat: if E(Y|X) = a + bx is correct and true
for the variables X and Y, the [raw] regression slope b will be
constant even if the range of X (say) is strongly restricted.
(Restricting the range of X will cause r to be reduced, compared to the
value arising from an unrestricted, or at any rate less restricted,
range. Beta will (as you have pointed out) necessarily be reduced also.)
The relationship between Y and X, of course, is unaffected by any
constraints in sampling, such as restriction of the ranges of X and Y.
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
184 Nashua Road, Bedford, NH 03110 603-471-7128
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