On Tue, 30 Jan 2001 23:22:51 -0500, "K. Bloom" <[EMAIL PROTECTED]>
wrote:
>If your answer IS the correct answer
>to my question, perhaps you would kindly explain it to me in a more concrete
>way suitable to my simple knowledge of the problem. My means are 2.05,
>6.38, and 12.08 for the three groups respectively. In other words.. what
>does one calculate and how?
I'm still not sure I understand your question completely, but what I
think you're asking is this:
You have 3 groups of observations. You'd graph them with
x-coordinates equal to -1, 0 and +1. The y-coordinates would be the
observed data.
The means for those three groups are 2.05, 6.38, and 12.08, based on
different numbers of observations in each group.
You want the quadratic curve that provides the least-squares fit to
your data.
If that's the case, then the numbers of observations in each group
doesn't matter. There's a quadratic curve that goes exactly through
each group mean, and you can't find a better fit than that.
To find it, do this. Suppose the quadratic curve is A x^2 + B x + C.
Then you've got three equations:
A (-1)^2 + B (-1) + C = 2.05
A (0)^2 + B (0) + C = 6.38
A (1)^2 + B (1) + C = 12.08
There's a unique solution to these equations; it is C = 6.38, A =
(12.08+2.05)/2, B = (12.08-2.05)/2. Those are the least-squares
parameter estimates.
If you have more than three groups, then you won't be able to find an
exact solution like this (you've only got three parameters to play
with), and then the least-squares solution *does* depend on the group
sizes.
In general, you should solve linear least-squares problems using
"multiple linear regression"; there are a ton of texts on that and I'd
suggest you use one of those. In particular, with unequal group sizes
you probably want to use "weighted least squares", with the weights
equal to the group sizes.
Duncan Murdoch
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