StATS: What is a beta coefficient? (April 4, 2006)

When you are examining the relative impact of several independent variables on an outcome variable, the estimated slopes may be deceptive. A variable with a wide range might have a very flat slope compared to a variable with a large range, even though the former may be a much more powerful predictor. You can see this intuitively by drawing a graph with a large aspect ratio (much wider than it is tall) and comparing it with the same graph with a smaller aspect ratio (closer to square). The slope looks so much bigger in the square graph, but nothing has fundamentally changed. The statistics community has developed "beta coefficients" that are the regression coefficients using a standardized variables. When you standardize, you allow for a "fair" comparison of the predictive power of variables measured on disparate ranges or even expressed in noncomparable units of measurement.

The great value of the beta-coefficient is that it expresses the "effect" of one variable on another without regard to how differently the variables are scaled.

The interpretation of a beta coefficient is slightly different than the interpretation of a slope coefficient. The slope coefficient represents the estimated average change in the outcome variable when the independent variable increases by one unit. The beta coefficient represents the estimated average change in standard deviation units. So a beta coefficient of 0.5 means that every time the independent variable changes by one standard deviation, the estimated outcome variable changes by half a standard deviation, on average.

It's impossible for a beta coefficient to be greater than one, and if you think about this long enough, that is a good thing. If the outcome variable increased by two standard deviations every time the independent variable increased by a single standard deviation, that would eventually lead to the explosion of the universe.

In a simple linear regression model, the beta coefficient is just the correlation between the independent variable and the outcome variable.

WARNING! In Finance, there is an alternate use of the term "beta coefficient" to represent the level of risk associated with an individual stock or of a  portfolio of stocks. This is not the same thing as the beta coefficient described above.

This page was written by Steve Simon while working at Children's Mercy Hospital. Although I do not hold the copyright for this material, I am reproducing it here as a service, as it is no longer available on the Children's Mercy Hospital website. Need more information? I have a page with general help resources. You can also browse for pages similar to this one at Category: Linear regression.