b vs β (2024)

Linear Statistical Models: Regression

b vs β

Example

use http://www.philender.com/courses/data/hsbdemo, clearregress 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 variablebut 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 variancesand 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 tworesearchers were to collect data on the same predictors with one collecting over a smaller rangeand the other over a larger range, they may end up with βs that are very different in size. Andthey may come to different conclusions about the relative contributions of the variables.

Linear Statistical Models Course

Phil Ender, 29Jan98