Unit-weighted regression

In statistics, unit-weighted regression is perhaps the easiest form of multiple regression analysis, a method in which two or more variables are used to predict the value of an outcome.

At a conceptual level, the example of weight loss can illustrate the idea of multiple regression. If a group of people join a weight loss program, we might wish to predict who would lose weight. The outcome is weight loss. We might find that those who lost weight were likely to increase their fruit intake, to exercise more, and to substitute low-calorie drinks for sugary drinks. The point is that several variables can be considered at the same time for their effect on an outcome of interest.

Beta weights

In the standard form of multiple regression, each predictor is multiplied by a number that is called the beta weight. The prediction is obtained by adding these products (and usually by adding a constant, as well). In the weight loss example above, suppose that reducing sugary drinks led to twice as much weight loss as did the other variables. If that were the case, then the beta weight for weight loss would be twice as big as the weights for the other variables.

When the weights are chosen to give the best prediction by some criterion, the model is called a proper linear model. Therefore, multiple regression is a proper linear model. By contrast, unit-weighted regression is called an improper linear model.

Model specification

Standard multiple regression has a major assumption: it assumes that all the important predictors are in the equation. This assumption is called model specification. A model is specified when all the predictors are in the equation, and no irrelevant predictors are in the equation.

However, in the social sciences, it is rare for a study to be able to know all the important predictors of a behavioral outcome. Therefore, most models are not specified. When the model is not specified, the estimates for the beta weights are not accurate. Because the inclusion of one variable can cause the beta weights to fluctuate wildly, this fluctuation is sometimes called the problem of the bouncing betas. It is this problem with bouncing betas that makes unit-weighted regression a useful method.

Unit weights

Unit-weighted regression proceeds in three steps. First, predictors for the outcome of interest are selected; ideally, there should be good empirical or theoretical reasons for the selection. Second, the predictors are converted to a standard form. Finally, the predictors are added together, and this sum is called the variate, which is used as the predictor of the outcome.

The Burgess Method

The Burgess method was first presented by the sociologist Ernest W. Burgess in a 1928 study to determine success or failure of inmates placed on parole. In this method, the predictors are converted to the standard form of zero or one (Burgess, 1928). When predictors have two values, the value associated with the target outcome is coded as one; for example, if the target outcome is failure on parole, then a predictor such as a history of theft would be coded as “no” = 0 and “yes” = 1. For predictors with more than two values, a cutoff score is selected based on subjective judgment. For example, one study measured the number of complaints for delinquent behavior and coded as follows: “zero to two complaints” = 0, and “three or more complaints” = 1 (Gottfredson & Snyder, 2005. p. 18).

The Kerby Method

The Kerby method is similar to the Burgess method, but differs in the manner of selecting the cutoff score for predictors with more than two values (Kerby, 2003). While the Burgess method relies on subjective judgment for predictors with more than two values, the Kerby method applies classification and regression tree (CART) analysis to the predictor. CART analysis will select a cutoff score based on a statistical criterion, such as selecting the point where the t-value is the greatest. Because CART analysis is not only binary, but also recursive, the result can be that a variable will be divided again, yielding two cutoff scores. Two cutoff scores emerged in a study using neuroticism scores to predict a measure of suicidal thinking; higher neuroticism scores predicted more suicidal thinking, so these two cutoff scores led to the following coding: “low Neuroticism” = 0, “moderate Neuroticism” = 1, “high Neuroticism” = 2 (Kerby, 2003).

The z-score Method

Another method can be applied when the predictors are measured on a continuous scale. In such a case, each predictor can be converted into a standard score, or z-score, so that all the predictors have a mean of zero and a standard deviation of one. With this method of unit-weighted regression, the variate is a sum of the z-scores (Bobko, Roth, & Buster, 2007).

Literature review

The first empirical study using unit-weighted regression is widely considered to be a 1928 study by sociologist Ernest W. Burgess. He reported that unit weights were a useful tool in making decisions about which inmates to parole (Burgess, 1928).

The mathematical issues involved in unit-weighted regression were first discussed in 1938 by Samuel Stanley Wilks, a leading statistician who had a special interest in multivariate analysis. Wilks described how unit weights could be used in practical settings, when data were not available to estimate beta weights. For example, a small college may want to select good students for admission. But the school may have no money to gather data and conduct a standard multiple regression analysis. In this case, the school could use several predictors—high school grades, SAT scores, teacher ratings. Wilks (1938) showed mathematically why unit weights should work well in practice.

Frank Schmidt (1971) conducted a simulation study of unit weights. His results showed that Wilks was indeed correct and that unit weights tend to perform well in simulations of practical studies.

Robyn Dawes (1979) discussed the use of unit weights in applied studies, referring to the robust beauty of unit weighted models. Jacob Cohen (statistician) also discussed the value of unit weights and noted their practical utility. Indeed, he wrote, "As a practical matter, most of the time, we are better off using unit weights" (Cohen, 1990, p. 1306).

Dave Kerby (2003) showed that unit weights compare well with standard regression, doing so with a cross validation study—that is, he derived beta weights in one sample and applied them to a second sample. The outcome of interest was suicidal thinking, and the predictor variables were broad personality traits. In the cross validation sample, the correlation between personality and suicidal thinking was slightly stronger with unit-weighted regression (r = .48) than with standard multiple regression (r = .47).

In a review of the literature on unit weights, Bobko, Roth, and Buster (2007) noted that "unit weights and regression weights perform similarly in terms of the magnitude of cross-validated multiple correlation, and empirical studies have confirmed this result across several decades" (p. 693).

Andreas Graefe applied an equal weighting approach to nine established multiple regression models for forecasting U.S. presidential elections. Across the ten elections from 1976 to 2012, equally weighted predictors reduced the forecast error of the original regression models on average by four percent. An equal-weights model that includes all variables provided well-calibrated forecasts that reduced the error of the most accurate regression model by 29% percent.

Example

An example may clarify how unit weights can be useful in practice.

Brenna Bry and colleagues (1982) addressed the question of what causes drug use in adolescents. Previous research had made use of multiple regression; with this method, it is natural to look for the best predictor, the one with the highest beta weight. One previous study had found that early use of alcohol was the best predictor. Another study had found that alienation from parents was the best predictor. Still another study had found that low grades in school were the best predictor. The failure to replicate was clearly a problem, a problem that could be caused by bouncing betas.

Bry and colleagues suggested a different approach: instead of looking for the best predictor, they looked at the number of predictors. In other words, they gave a unit weight to each predictor. Their study had six predictors: 1) low grades in school, 2) affiliation with religion, 3) age of alcohol use, 4) psychological distress, 5) self-esteem, and 6) alienation from parents. Using the Burgess method, each risk factor was scored as absent (scored as zero) or present (scored as one). For example, the coding for low grades in school were as follows: "C or higher" = 0, "D or F" = 1. The results showed that the number of risk factors was a good predictor of drug use: adolescents with more risk factors were more likely to use drugs.

The model used by Bry and colleagues was that drug users do not differ in any special way from non-drug users. Rather, they differ in the number of problems they must face. "The number of factors an individual must cope with is more important than exactly what those factors are" (p. 277). Given this model, unit-weighted regression is an appropriate method of analysis.

See also

• regression analysis
• linear regression

References

• Bobko, P., Roth, P. L., & Buster, M. A. (2007). The usefulness of unit weights in creating composite scores: A literature review, application to content validity, and meta-analysis. Organizational Research Methods, volume 10, pages 689-709. doi:10.1177/1094428106294734
• Burgess, E. W. (1928). Factors determining success or failure on parole. In A. A. Bruce (Ed.), The Workings of the Indeterminate Sentence Law and Parole in Illinois. Springfield, Illinois: Illinois State Parole Board.
• Cohen, Jacob. (1990). "Things I have learned (so far)". American Psychologist, volume 45, pages 1304-1312. doi:10.1037/0003-066X.45.12.1304
• Dawes, Robyn M. (1979). "The robust beauty of improper linear models in decision making". American Psychologist, volume 34, pages 571-582. doi:10.1037/0003-066X.34.7.571
• Gottfredson, D. M., & Snyder, H. N. (July 2005). The mathematics of risk classification: Changing data into valid instruments for juvenile courts. Pittsburgh, Penn.: National Center for Juvenile Justice. NCJ 209158. http://files.eric.ed.gov/fulltext/ED485849.pdf
• Kerby, Dave S. (2003). "CART analysis with unit-weighted regression to predict suicidal ideation from Big Five traits". Personality and Individual Differences, volume 35, pages 249-261. doi:10.1016/S0191-8869(02)00174-5
• Schmidt, Frank L. (1971). "The relative efficiency of regression and simple unit predictor weights in applied differential psychology". Educational and Psychological Measurement, volume 31, pages 699-714. doi:10.1177/001316447103100310