Epsilon (Domin) vs. Eta2 (esize) - Dominance analysis/power analysis

[Daniel Weimar]

Hi everyone,

i am trying to estimate the partial effects of my variables regarding the overall variance explanation after a linear Regression, to interpret the strength of the results. There are two possibilities in Stata. The first is to run a dominance Analysis, which computes epsilon values, which reflect the individual contribution of each explanatory variable to the Overall R2. Second, I run esize for applying power analysis, which reports eta squared values (or Omega if desired), which according to literature also reflect the individual contribution. Both epsilon and eta values are "similar" but not equal. Does anyone could explain the difference and what of both approaches is the more valid way?

Best regards
Daniel

[Joseph Luchman]

Hi Daniel,

Must note that the reference to -domin- is to a module on SSC and you presumably are using Stata version 14.1.

The -epsilon- option of -domin- mimics the computation of general dominance statistics which is an average R² contribution by a variable across all possible combinations of independent variables included and not (see -domin-'s help file and references therein). Eta² values are different in 2 ways. First, Eta² values are equivalent to squared partial correlations (see below highlighted red) which makes them dependent on the full model alone. Second, Eta²/squared partial correlations do not know what to do with the shared explained variance and, you will notice, the sum of the Eta² values is not equal to the R², whereas the sum of the epsilon-based dominance statistics do and are hashed out during the averaging process.

In the end, see some of the literature in the -domin- helpfile. Seems like you may be interested in relative importance here. If so, the distinction between squared partial correlations and dominance statistics has been discussed extensively and most examining the topic (including me) favor dominance/RWA and their ilk.

- joe

. sysuse auto
(1978 Automobile Data)


. regress price headroom trunk weight

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(3, 70)        =     10.91
       Model |   202302540         3  67434180.1   Prob > F        =    0.0000
    Residual |   432762856        70  6182326.51   R-squared       =    0.3186
-------------+----------------------------------   Adj R-squared   =    0.2893
       Total |   635065396        73  8699525.97   Root MSE        =    2486.4

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    headroom |  -726.8171   460.0698    -1.58   0.119    -1644.397    190.7632
       trunk |   28.34831   107.5862     0.26   0.793    -186.2256    242.9222
      weight |   2.321676    .506973     4.58   0.000      1.31055    3.332802
       _cons |   940.6103   1292.159     0.73   0.469    -1636.519     3517.74
------------------------------------------------------------------------------

. estat esize

Effect sizes for linear models

-------------------------------------------------------------------
             Source |   Eta-Squared     df     [95% Conf. Interval]
--------------------+----------------------------------------------
              Model |   .3185539         3     .1274273    .4470055
                    |
           headroom |   .0344263         1            .    .1490721
              trunk |   .0009909         1            .     .056701
             weight |   .2305299         1     .0777684    .3816559
-------------------------------------------------------------------

. pcorr price headroom trunk weight
(obs=74)

Partial and semipartial correlations of price with

               Partial   Semipartial      Partial   Semipartial   Significance
   Variable |    Corr.         Corr.      Corr.^2       Corr.^2          Value
------------+-----------------------------------------------------------------
   headroom |  -0.1855       -0.1559       0.0344        0.0243         0.1187
      trunk |   0.0315        0.0260       0.0010        0.0007         0.7929
     weight |   0.4801        0.4518       0.2305        0.2042         0.0000


. domin price headroom trunk weight, epsilon reg(regress)


General dominance statistics: Epsilon-based regress
Number of obs             =                      74
Overall Fit Statistic     =                  0.3186

            |      Dominance      Standardized      Ranking
 price      |      Stat.          Domin. Stat.
------------+------------------------------------------------------------------------
 headroom   |         0.0175      0.0550            3 
 trunk      |         0.0559      0.1756            2 
 weight     |         0.2451      0.7694            1 
-------------------------------------------------------------------------------------

[Daniel Weimar]

Hi Joseph,

thanks for your detailed answer. It completely clarified my questions. Thank you very much!