Monthly Archives: February 2014

Why a high R Square is not necessarily better

Often I encounter academics thinking that a high proportion of explained variance is the ideal outcome of a statistical analysis. The idea is that in regression analyses a high R Square is better than a low R Square. In my view, the emphasis on a high R2 should be reduced. A high R2 should not be a goal in itself. The reason is that a higher R2 can easily be obtained by using procedures that actually lower the external validity of coefficients.

It is possible to increase the proportion of variance explained in regression analyses in several ways that do not in fact our ability to ‘understand’ the behavior we are seeking to ‘explain’ or ‘predict’. One way to increase the R2 is to remove anomalous observations, such as ‘outliers’ or people who say they ‘don’t know’ and treat them like the average respondent. Replacing missing data by mean scores or using multiple imputation procedures often increases the Rsquare. I have used this procedure in several papers myself, including some of my dissertation chapters.

But in fact outliers can be true values. I have seen quite a few of them that destroyed correlations and lowered R squares while being valid observations. E.g., a widower donating a large amount of money to a charity after the death of his wife. A rare case of exceptional behavior for very specific reasons that seldom occur. In larger samples these outliers may become more frequent, affecting the R2 less strongly.

Also ‘Don’t Know’ respondents are often systematically different from the average respondent. Treating them as average respondents eliminates some of the real variance that would otherwise be hard to predict.

Finally, it is often possible to increase the proportion of variance explained by including more variables. This is particularly problematic if variables that are the result of the dependent variable are included as predictors. For instance if network size is added to the prediction of volunteering the R Square will increase. But a larger network not only increases volunteering; it is also a result of volunteering. Especially if the network questions refer to the present (do you know…) while the volunteering questions refer to the past (in the past year, have you…) it is dubious to ‘predict’ volunteering in the past by a measure of current network size.

As a reviewer, I give authors reporting an R2 exceeding 40% a treatment of high-level scrutiny for dubious decisions in data handling and inclusion of variables.

As a rule, R Squares tend to be higher at higher levels of aggregation, e.g. when analyzing cross-situational tendencies in behavior rather than specific behaviors in specific contexts; or when analyzing time-series data or macro-level data about countries rather than individuals. Why people do the things they do is often just very hard to predict, especially if you try to predict behavior in a specific case.

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