The controversy over standardized beta coefficients (created 2009-09-12)

This page is moving to a new website.

I have a client who is working on her dissertation. I always warn people working on dissertations or theses that they should listen more to what their committee members say about statistics than what I say about statistics. If the committee loves the statistical analysis and I hate it, you still get your degree. If I love the statistical analysis and the committee hates it, you get nothing. For this client, a committee member asked if she could produce standardized beta coefficients in her regression models. I helped her write an argument as to why the unstandardized coefficients are better, but the committee member gave a reasonable counter-argument, so there was no point in persisting. Still, it would be helpful here to outline some of the controversy over standardized beta coefficients.

I first talked about standardized beta coefficients (sometimes the name is shortened to just "beta coefficients") on my old website

That page did not give a critical review of the pros and cons.

The Wikipedia page on standardized beta coefficients did have some critical comments.

Advocates of standardized regression coefficients point out that the coefficients are the same regardless of an independent variable's underlying scale of units. They also suggest that this removes the problem of comparing, for example, years with kilograms since each regression coefficient represents the change in response per standard unit (one SD) change in a predictor. However, critics of standardized regression coefficients argue that this is illusory: there is no reason why a change of one SD in one predictor should be equivalent to a change of one SD in another predictor. Some variables are easy to change--the amount of time watching television, for example. Others are more difficult--weight or cholesterol level. Others are impossible--height or age. --

The Wikipedia page cites a webpage by Jerry Dallal

and this webpage elaborates on some of the criticisms in greater detail.

Andrew Gelman has some discussion of standardized beta coefficients on his blog, and he is somewhat supportive. He argues that many coefficients are uninterpretable because the scale is so small.

What bugs me the most is regression coefficients defined on scales that are uninterpretable or nearly so: for example, coefficients for age and age-squared (In a political context, do we really care about the difference between a 51-year-old and a 52-year-old? How are we supposed to understanding the resulting coefficients such as 0.01 and 0.003?) or, worse still, a predictor such as the population of a country (which will give nicely-interpretable coefficients on the order of 10^-8)? I used to deal with these problems by rescaling by hand, for example using age/10 or population in millions. But big problems remained: manual scaling is arbitrary (why not age/20? Should we express income in thousands or tens of thousands of dollars?); still left difficulties in comparing coefficients (if we're typically standardizing by factors of 10, this leaves a lot of play in the system); is difficult to give as general advice. --

He suggests that during the model building phase that every continuous variable be centered and then divided by two standard deviations. This places the variable on the same scale as the indicator (0-1) variables for the categorical data.

In my experience, unitless quantities are easier to work with, but they lack an interpretability that only comes when you discuss quantities that have a unit of measurement. For example, it is easy to get someone to specify a small, medium, or large effect size for a power calculation, but this does not lead to a fruitful consideration of clinical importance. Many meta-analyses will report estimates of continuous effects on a standardized scale, which helps when there is heterogeneity in outcome measures, but the resulting statistics are incapable of addressing the practical impact of a therapy.

I have a joke in my book about a store that displays a large sign saying "Major Sale. All prices reduced by one half a standard deviation."

I would argue that standardized beta coefficients lead to the same problem. If I know that a standard deviation increase in the weight of a car leads to a 0.25 standard deviation decline in mileage, what does that tell me exactly? I'd rather know that an extra thousand pounds leads to a 5 mile per gallon decline on average. (Sorry to my non-U.S. friends for not using metric here).

I'm not saying that you should never use a unitless quantity. I'm just saying that unitless quantities have severe limits on their interpretability.