I guess I read it differently, as indicating that predicted values
were generated by fitting the model to the whole dataset, and that
the within-class R^2s were simply the squares of the bivariate
correlations of those predicted values with the observed values.

Ray Koopman kirjoitti:


I guess I read it differently, as indicating that predicted values
were generated by fitting the model to the whole dataset, and that
the within-class R^2s were simply the squares of the bivariate
correlations of those predicted values with the observed values.

Yes, this is what I meant. Just to make it sure:

Measured values in three different classes A, B and C

y_A = {y1_a, y2_a, ..., yn_a}
y_B = {y1_b, y2_b, ..., yn_b}
y_C = {y1_c, y2_c, ..., yn_c}

I fitted the model y = d0 + d1*x1 + d2*x2 for the whole data set
{A,B,C} and got R^2 = 0.72. Then I divided yhats to corresponding
classes

yhat_A = {yhat1_a, yhat2_a, ..., yhatn_a}
yhat_B = {yhat1_b, yhat2_b, ..., yhatn_b}
yhat_C = {yhat1_c, yhat2_c, ..., yhatn_c}

Then I plotted y_A vs yhat_A, y_B vs yhat_B and y_C vs yhat_C and
calculated corr(y_A,yhat_A) = 39.8 %, corr(y_B,yhat_B) = 46.6 % and
corr(y_C,yhat_C) = 40.1 % which are also R^2 values for lines joining
measured vs predicted value points in class A, class B and class C
respectively.

- - -

Still, I guess I might have created a class variable which indicates to
which class each case belongs (1 <-> A, 2 <-> B, 3 <-> C) and put it as
a "fixed factor" in General Linear Model (GLM) Univariate analysis.
Then class depency would have its own effect on the whole model as
intercept would get different values for different classes.

-PJ

David Jones wrote:
Ray Koopman wrote:
Pekka Jarvela wrote:
I have dataset consisting of three different classes, {A, B, C},
and a linear model for the dataset with R^2 = 0.72. Plotting
measured values (y) against predicted values (yhat) for different
classes separately and fitting a line to each gets:

A => R^2 = 0.398
B => R^2 = 0.466
C => R^2 = 0.401

QUESTION: Can I conclude that the model which is able to explain
72 % of variation in the whole dataset {A, B, C}, is able to
explain only
39.8 % of variation in class A, 46.6 % in class B and 40.1 % in
class C?

Cheers.

-PJ

Yes. R^2 for the whole dataset is larger than the within-class
R^2s
because the whole dataset contains between-class variability that
the model can explain but that by definition is absent from the
within- class data.

Or no. The OP needs to be careful about "the model". The
description
indicates that the model is being refitted within each class, which
is not (or may not be) quite the same as fitting the model to all
the
data and then working out R^2 for each class separately for this
model ... which would give different (certainly no better) R^2
values than the separately-fitted sub-models.

David Jones

I guess I read it differently, as indicating that predicted values
were generated by fitting the model to the whole dataset, and that
the within-class R^2s were simply the squares of the bivariate
correlations of those predicted values with the observed values.

I was also reading between the lines, interpreting the question as
"why is the overall R^2 bigger than the within-class R^2s?".