Two of the more common measures of effect size for regression analysis are eta2 and
partial eta2. Eta2 is the proportion of the total variance that is
attributed to an effect or set of effects. Partial eta2 is the proportion of effect + error
variance that is attributable to the effect. The formula differs from the eta squared formula in
that the denominator includes the SSEffect plus the SSError rather than the SSTotal. The Stata regress postestimation command estat esize can be used to estimate eta2 for the model and partial eta2 for each effect in the model.
Below, we run a linear regression analysis the hsbdemo dataset.
use https://stats.idre.ucla.edu/stat/data/hsbdemo, clear
regress write i.female read math i.prog
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 5, 194) = 45.01
Model | 9602.28627 5 1920.45725 Prob > F = 0.0000
Residual | 8276.58873 194 42.6628285 R-squared = 0.5371
-------------+------------------------------ Adj R-squared = 0.5251
Total | 17878.875 199 89.843593 Root MSE = 6.5317
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
1.female | 5.384982 .929572 5.79 0.000 3.551617 7.218346
read | .3069424 .0611262 5.02 0.000 .1863852 .4274996
math | .3603705 .0690064 5.22 0.000 .2242715 .4964695
|
prog |
2 | .436372 1.230379 0.35 0.723 -1.990265 2.863009
3 | -2.219748 1.359353 -1.63 0.104 -4.900756 .4612603
|
_cons | 15.16272 3.225088 4.70 0.000 8.801985 21.52346
------------------------------------------------------------------------------
We follow the regress command with estat esize, which displays estimates and confidence intervals for eta2 for the model and partial eta2 for each effect in the model.
estat esize
Effect sizes for linear models
-------------------------------------------------------------------
Source | Eta-Squared df [95% Conf. Interval]
--------------------+----------------------------------------------
Model | .5370744 5 .433662 .6003297
|
female | .1474719 1 .0667184 .2382202
read | .115024 1 .043701 .2017348
math | .1232518 1 .0493088 .2111699
prog | .0232415 2 . .0732192
-------------------------------------------------------------------
An anova table of this regression allows us to see how eta2 and partial eta2 are calculated.
anova write i.female c.read c.math i.prog
Effect sizes for linear models
Number of obs = 200 R-squared = 0.5371
Root MSE = 6.53168 Adj R-squared = 0.5251
Source | Partial SS df MS F Prob>F
-----------+----------------------------------------------------
Model | 9602.2863 5 1920.4573 45.01 0.0000
|
female | 1431.7 1 1431.7 33.56 0.0000
read | 1075.743 1 1075.743 25.21 0.0000
math | 1163.5091 1 1163.5091 27.27 0.0000
prog | 196.93763 2 98.468814 2.31 0.1022
|
Residual | 8276.5887 194 42.662829
-----------+----------------------------------------------------
Total | 17878.875 199 89.843593
The model eta2 is SSModel/SStotal = 9602.2863/17878.875 = 0.53707441. This matches the estimated for R-squared. The familiar interpretation is that the model explains 53.71% of the total variance of write.
Each partial eta2 is SSEffect/(SSEffect+SSError). The SSError for all of these terms is SSResidual. We can thus calculate partial eta2 for female = SSEffect/(SSEffect+SSError) = 1431.7/(1431.7+8276.5887) = 0.14747192. We can interpret this to mean that about 14.75% of the variance unexplained by effects other than female is explained by the female effect. If we need estimates of eta2 for each effect, it is simply SSEffect/SSTotal.