F-test Comparing Two Models
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 hsb2
First model includes read math science socst female & ses.
use http://www.gseis.ucla.edu/courses/data/hsb2
regress write read math science socst female ses
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 6, 193) = 48.66
Model | 10763.6571 6 1793.94285 Prob > F = 0.0000
Residual | 7115.21791 193 36.866414 R-squared = 0.6020
-------------+------------------------------ Adj R-squared = 0.5897
Total | 17878.875 199 89.843593 Root MSE = 6.0718
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write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
read | .1263026 .0651304 1.94 0.054 -.0021563 .2547614
math | .2390707 .0673088 3.55 0.000 .1063154 .371826
science | .2439102 .0610028 4.00 0.000 .1235925 .3642278
socst | .2336959 .0539537 4.33 0.000 .1272814 .3401104
female | 5.444119 .8845485 6.15 0.000 3.699496 7.188742
ses | -.2751714 .6439092 -0.43 0.670 -1.545174 .994831
_cons | 6.297198 2.838673 2.22 0.028 .6983924 11.896
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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
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Manual Arithmetic
(R2y.12...k1 - R2y.12...k2)/(k1 - k2)
F = -----------------------------------
(1 - R2y.12...k1)/(N - k1 - 1)
(.6020 - .5149)/(6-3) .0871/3 .0290333333333
= ------------------------ = -------- = ---------------- = 14.078
(1 - .6020)/(200 - 6 -1) .398/193 .0020621761658
with df = (k1 -k2) & (N - k1 -1) = 3 & 193
Using Stata
regress write read math science socst female ses
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 6, 193) = 48.66
Model | 10763.6571 6 1793.94285 Prob > F = 0.0000
Residual | 7115.21791 193 36.866414 R-squared = 0.6020
-------------+------------------------------ Adj R-squared = 0.5897
Total | 17878.875 199 89.843593 Root MSE = 6.0718
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
read | .1263026 .0651304 1.94 0.054 -.0021563 .2547614
math | .2390707 .0673088 3.55 0.000 .1063154 .371826
science | .2439102 .0610028 4.00 0.000 .1235925 .3642278
socst | .2336959 .0539537 4.33 0.000 .1272814 .3401104
female | 5.444119 .8845485 6.15 0.000 3.699496 7.188742
ses | -.2751714 .6439092 -0.43 0.670 -1.545174 .994831
_cons | 6.297198 2.838673 2.22 0.028 .6983924 11.896
------------------------------------------------------------------------------
test read gender ses
( 1) read = 0.0
( 2) female = 0.0
( 3) ses = 0.0
F( 3, 193) = 14.08
Prob > F = 0.0000
Phil Ender, 14jan00