Re: r-Squared Question

Let's assume some observed data, which I hope makes my question
clearer:

X Y
1 101
2 102
3 103
4 104
5 105
6 106
7 107
8 108
9 109
10 110

The relationship here is obvious, but bare with me. Assume that some
regression procedure (obviously not least squares) produces a linear
model, YHat:

X Y YHat
1 101 97
2 102 99
3 103 101
4 104 103
5 105 105
6 106 107
7 107 109
8 108 111
9 109 113
10 110 115

YHat has a correlation ( r ) with Y of 1.0. r-squared is hence 1.0.
What I'm getting at is: the r-squared is at its best possible value,
yet the model is obviously suboptimal. Have I gone wrong somewhere, or
is this a fundamental weakness of r-squared?


Thanks very much,
Will


Radford Neal wrote:
> In article <1121175957.796245.150510@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> Predictor <predictr@xxxxxxxxxxxxxxxx> wrote:
>
> >I am trying to undertand r-squared (the coefficient of determination)
> >of regression lines. If r, which is squared to obtain r-squared, is
> >the correlation between the predicted Y and the observed Y, then
> >doesn't that mean that any regression line whose predicted Y is a
> >perfect linear function of the observed Y has an r (and thus r-squared)
> >of 1?
>
> That's true. You may be a bit confused, however. The only way that
> the predicted and observed Y can be related by a linear function is if
> the predicted and observed Y are identical (ie, the linear function is
> observed=predicted). You may be confusing "predicted value" with
> "predictor" (also known as a covariate or explanatory variable). The
> predicted value is the linear function of the covariates in which the
> coefficients are those found by fitting to the data.
>
> ----------------------------------------------------------------------------
> Radford M. Neal radford@xxxxxxxxxxxxxx
> Dept. of Statistics and Dept. of Computer Science radford@xxxxxxxxxxxxxxxxxx
> University of Toronto http://www.cs.utoronto.ca/~radford
> ----------------------------------------------------------------------------

.

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