Ed230B/C

Linear Statistical Models: Regression

b vs β

Example

use http://www.philender.com/courses/data/hsbdemo, clear

regress write read math science female

      Source |       SS       df       MS              Number of obs =     200
-------------+------------------------------           F(  4,   195) =   62.80
       Model |   10065.716     4  2516.42899           Prob > F      =  0.0000
    Residual |  7813.15904   195  40.0674822           R-squared     =  0.5630
-------------+------------------------------           Adj R-squared =  0.5540
       Total |   17878.875   199   89.843593           Root MSE      =  6.3299

------------------------------------------------------------------------------
       write |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        read |   .2327584   .0627504     3.71   0.000     .1090018    .3565151
        math |   .2939597   .0688252     4.27   0.000     .1582223    .4296971
     science |   .2570204   .0633098     4.06   0.000     .1321606    .3818802
      female |   5.936716   .9082913     6.54   0.000      4.14538    7.728052
       _cons |   8.580501   2.874498     2.99   0.003     2.911404     14.2496
------------------------------------------------------------------------------

regress write read math science female, beta

      Source |       SS       df       MS              Number of obs =     200
-------------+------------------------------           F(  4,   195) =   62.80
       Model |   10065.716     4  2516.42899           Prob > F      =  0.0000
    Residual |  7813.15904   195  40.0674822           R-squared     =  0.5630
-------------+------------------------------           Adj R-squared =  0.5540
       Total |   17878.875   199   89.843593           Root MSE      =  6.3299

------------------------------------------------------------------------------
       write |      Coef.   Std. Err.      t    P>|t|                     Beta
-------------+----------------------------------------------------------------
        read |   .2327584   .0627504     3.71   0.000                 .2517736
        math |   .2939597   .0688252     4.27   0.000                  .290544
     science |   .2570204   .0633098     4.06   0.000                 .2684715
      female |   5.936716   .9082913     6.54   0.000                 .3126764
       _cons |   8.580501   2.874498     2.99   0.003                        .
------------------------------------------------------------------------------

listcoef  /* listcoef can be downloaded from Stata via the Internet */

regress (N=200): Unstandardized and Standardized Estimates 

 Observed SD: 9.478586
 SD of Error: 6.329888

---------------------------------------------------------------------------
   write |      b         t     P>|t|    bStdX    bStdY   bStdXY      SDofX
---------+-----------------------------------------------------------------
    read |   0.23276    3.709   0.000   2.3865   0.0246   0.2518    10.2529
    math |   0.29396    4.271   0.000   2.7539   0.0310   0.2905     9.3684
 science |   0.25702    4.060   0.000   2.5447   0.0271   0.2685     9.9009
  female |   5.93672    6.536   0.000   2.9637   0.6263   0.3127     0.4992
---------------------------------------------------------------------------

First Thoughts

  • Interpretation of β is analogous to the interpretation of b, except that β expresses change in standard scores.
  • β's are scale free.
  • Some researchers use the relative magnitude of β to indicate relative importance of the independent variables.
  • The magnitude of β reflects the presumed effect of the variable but also the variances and covariances of other variables in the model -
  • As well as variances of variables not in the model.

    Note:

  • β is sample specific.
  • It cannot be used for the purpose of generalizations across settings and populations.
  • b, on the other hand, remains fairly stable despite differences in the variances and covariances of variables in different settings or populations.
  • Unstandardized (raw) coefficients translate more directly into guides for policy decisions.
  • Beware though of changing the unit of measure, and
  • the fact that many independent variables are not on an interval scale.

    Recall the formula for the standardized coefficient.

    Note that the value of β is dependent of the ratio of the standard deviations. So if two researchers were to collect data on the same predictors with one collecting over a smaller range and the other over a larger range, they may end up with βs that are very different in size. And they may come to different conclusions about the relative contributions of the variables.

    Linear Statistical Models Course

    Phil Ender, 29Jan98