Standardized coefficient

Standardization of the coefficient is usually done to answer the question of which of the independent variables have a greater effect on the dependent variable in a multiple regression analysis where the variables are measured in different units of measurement (for example, income measured in dollars and family size measured in number of individuals).

It may also be considered a general measure of effect size, quantifying the "magnitude" of the effect of one variable on another.

For simple linear regression with orthogonal predictors, the standardized regression coefficient equals the correlation between the independent and dependent variables.

A regression carried out on original (unstandardized) variables produces unstandardized coefficients. A regression carried out on standardized variables produces standardized coefficients. Values for standardized and unstandardized coefficients can also be re-scaled to one another subsequent to either type of analysis.

Suppose that $\backslash beta$ is the regression coefficient resulting from a linear regression (predicting $y$ by $x$). The standardized coefficient simply results as $\backslash beta^\backslash ast\; =\; \backslash frac\{s\_x\}\{s\_y\}\; \backslash beta$, where $s\_x$ and $s\_y$ are the (estimated) standard deviations of $x$ and $y$, respectively.

Sometimes, standardization is done only without respect to the standard deviation of the regressor (the independent variable $x$).{{cite journal|last1=Greenland|first1=S.|last2=Schlesselman|first2=J. J.|last3=Criqui|first3=M. H.|title=The fallacy of employing standardized regression coefficients and correlations as measures of effect|journal=American Journal of Epidemiology|volume=123|issue=2|year=1986|pages=203–208|doi=10.1093/oxfordjournals.aje.a114229|pmid=3946370 |doi-access=free}}{{cite journal|last1=Newman|first1=T. B.|last2=Browner|first2=W. S.|title=In defense of standardized regression coefficients|journal=Epidemiology|volume=2|issue=5|year=1991|pages=383–386|doi=10.1097/00001648-199109000-00014|pmid=1742391 |doi-access=free}}

Standardized coefficients' advocates note that the coefficients are independent of the involved variables' units of measurement (i.e., standardized coefficients are unitless), which makes comparisons easy.

Critics voice concerns that such a standardization can be very misleading.{{cite journal|last1=Greenland|first1=S.|last2=Maclure|first2=M.|last3=Schlesselman|first3=J. J.|last4=Poole|first4=C.|last5=Morgenstern|first5=H.|title=Standardized regression coefficients: A further critique and review of some alternatives|journal=Epidemiology|volume=2|issue=5|year=1991|pages=387–392|doi=10.1097/00001648-199109000-00016|pmid=1742393 |doi-access=free}}

Due to the re-scaling based on sample standard deviations, any effect apparent in the standardized coefficient may be due to confounding with the particularities (especially: variability) of the involved data sample(s).

Also, the interpretation or meaning of a "one standard deviation change" in the regressor $x$ may vary markedly between non-normal distributions (e.g., when skewed, asymmetric or multimodal).

Some statistical software packages like PSPP, SPSS and SYSTAT label the standardized regression coefficients as "Beta" while the unstandardized coefficients are labeled "B". Others, like DAP/SAS label them "Standardized Coefficient". Sometimes the unstandardized variables are also labeled as "b".

• Linear regression
• Correlation coefficient
• Effect size
• Unit-weighted regression

{{Reflist}}

• {{cite book |first1=Larry D. |last1=Schroeder |first2=David L. |last2=Sjoquist |first3=Paula E. |last3=Stephan |year=1986 |title=Understanding Regression Analysis |publisher=Sage Publications |isbn=0-8039-2758-4 |pages=[https://archive.org/details/understandingreg00larr/page/31 31–32] |url=https://archive.org/details/understandingreg00larr/page/31 }}
• {{cite book |first1=Eric |last1=Vittinghoff |first2=David V. |last2=Glidden |first3=Stephen C. |last3=Shiboski |first4=Charles E. |last4=McCulloch |year=2005 |title=Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models |publisher=Springer |pages=75–76 |isbn=0-387-20275-7 }}
• {{cite book |first1=J.|last1=Neter|first2=M. H.|last2=Kutner|first3=C.J.|last3=Nachtsheim|first4=W.|last4=Wasserman|year=1996|title=Applied Linear Statistical Models|publisher=McGraw-Hill|section=7.5 Standardized multiple regression model|pages=281–284|edition=4th|isbn=0-256-11736-5}}

• [http://www.jerrydallal.com/LHSP/importnt.htm Which Predictors Are More Important?] - why standardized coefficients are used

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Category:Regression analysis