R2y.12...k1 has all of the same variables as R2y.12...k2 plus more additional variables. Thus, R2y.12...k1 can be said to be nested in R2y.12...k2. The denominator always contains (1 - R2y.12...k1) for the model with more variables.
An Example Using hsbdemo
First model includes read math science socst female & ses.
use http://www.philender.com/courses/data/hsbdemo, clear
regress write read math science socst i.female i.ses
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 7, 192) = 41.70
Model | 10784.7713 7 1540.68161 Prob > F = 0.0000
Residual | 7094.10375 192 36.948457 R-squared = 0.6032
-------------+------------------------------ Adj R-squared = 0.5887
Total | 17878.875 199 89.843593 Root MSE = 6.0785
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
read | .1237858 .0652878 1.90 0.059 -.0049876 .2525593
math | .2383564 .0673903 3.54 0.001 .105436 .3712768
science | .2453025 .0610984 4.01 0.000 .1247923 .3658127
socst | .2348613 .0540357 4.35 0.000 .1282815 .341441
1.female | 5.386751 .888778 6.06 0.000 3.633728 7.139774
|
ses |
2 | -.9670662 1.119495 -0.86 0.389 -3.175154 1.241022
3 | -.6221902 1.292749 -0.48 0.631 -3.172004 1.927623
|
_cons | 6.438565 2.883858 2.23 0.027 .7504525 12.12668
------------------------------------------------------------------------------Second model includes all of the above variables except for read female & ses.
regress write math science socst
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 3, 196) = 69.36
Model | 9206.56411 3 3068.8547 Prob > F = 0.0000
Residual | 8672.31089 196 44.2464841 R-squared = 0.5149
-------------+------------------------------ Adj R-squared = 0.5075
Total | 17878.875 199 89.843593 Root MSE = 6.6518
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
math | .2887075 .0696839 4.14 0.000 .151281 .4261339
science | .221637 .0624744 3.55 0.000 .0984286 .3448454
socst | .3017268 .0533039 5.66 0.000 .1966039 .4068496
_cons | 10.27213 3.002846 3.42 0.001 4.35009 16.19416
------------------------------------------------------------------------------Manual Arithmetic
(R2y.12...k1 - R2y.12...k2)/(k1 - k2)
F = -----------------------------------
(1 - R2y.12...k1)/(N - k1 - 1)
(.6032 - .5149)/(7-3) .0883/4 .022075
= ------------------------ = --------- = ---------- = 10.681452
(1 - .6032)/(200 - 7 -1) .3968/192 .00206667
with df = (k1 -k2) & (N - k1 -1) = 3 & 193
Using Stata
regress write read math science socst i.female i.ses
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 7, 192) = 41.70
Model | 10784.7713 7 1540.68161 Prob > F = 0.0000
Residual | 7094.10375 192 36.948457 R-squared = 0.6032
-------------+------------------------------ Adj R-squared = 0.5887
Total | 17878.875 199 89.843593 Root MSE = 6.0785
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
read | .1237858 .0652878 1.90 0.059 -.0049876 .2525593
math | .2383564 .0673903 3.54 0.001 .105436 .3712768
science | .2453025 .0610984 4.01 0.000 .1247923 .3658127
socst | .2348613 .0540357 4.35 0.000 .1282815 .341441
1.female | 5.386751 .888778 6.06 0.000 3.633728 7.139774
|
ses |
2 | -.9670662 1.119495 -0.86 0.389 -3.175154 1.241022
3 | -.6221902 1.292749 -0.48 0.631 -3.172004 1.927623
|
_cons | 6.438565 2.883858 2.23 0.027 .7504525 12.12668
------------------------------------------------------------------------------
testparm read i.female i.ses
( 1) read = 0
( 2) 1.female = 0
( 3) 2.ses = 0
( 4) 3.ses = 0
F( 4, 192) = 10.68
Prob > F = 0.0000
Linear Statistical Models Course
Phil Ender, 24sep10, 14jan00