Re: Stationary means
- From: Richard Ulrich <Rich.Ulrich@xxxxxxxxxxx>
- Date: Sun, 03 Sep 2006 21:10:51 -0400
On 3 Sep 2006 16:37:01 -0700, "Jason Foster" <retsofaj@xxxxxxxxx>
wrote:
I don't think that this is a FAQ, but if it is I apologize for the
noise...
Discussions about "regression to the mean" tend to focus on fallacies
relating to heights, grades, sickness, etc. Something I have not seen
discussed is when/whether it is appropriate to assume a stationary mean
(or, I think alternatively, a fixed distribution)?
Assuming a constant distribution and repeated sampling, I can
intuitively understand regression to the mean. However, I can imagine
situations where the distribution is changing over time. For example
(and here I'm talking outside of my area of expertise) the mean height
in North America is increasing over time (ostensibly due to dietary and
health factors). If this is the case, then what would "regression to
the mean" mean? Towards which mean would the regression take place?
"Regression to the mean" is more general than you describe.
The X and Y variable are not necessarily the same. The
predictions based on X-deviations-from-X-mean are in
terms of Y-deviations-from-Y-means.
Thus, the child-population predictions regress to the child-mean.
How would an observer know that a regression analysis is appropriate?
Any thoughts (or pointers to resources) on the issues would be
gratefully appreciated.
The difficult version of problem arises when
(a) groups are selected by initial "ability"; and
(b) the outcome is later "ability"; and
(c) the critic argues that initial ability will predict growth.
Thus, the critic argues that *untreated* groups will grow
further apart. The naive (null) hypothesis would be that
the untreated groups will tend to converge.
That illustrates the problem of covariance when the covariates
are not equal at the start -- The data on hand, concerning a
treatment where groups were not matched, cannot tell you
which of these two circumstances was true. The case has
to be argued from other data.
That was the situation in the early evaluation of the efficacy
of U.S. preschool Headstart" programs for disadvantaged
children. The disadvantaged children did not fall further behind.
The regression model would have "expected" convergence.
Was it a realistic expectation that these children would have
fallen further behind, without a program?
(I think that is the argument that won, though the technical
debate may have been pushed aside by new data. I could not
find much on the discussion when I looked for it a few years
ago. Harvard Educational Review comes to mind.)
Hope this helps.
--
Rich Ulrich, wpilib@xxxxxxxx
http://www.pitt.edu/~wpilib/index.html
.
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