R-squared is a goodness-of-fit measure for linear regression models. This statistic indicates the percentage of the variance in the dependent variable that the independent variables explain collectively. R-squared measures the strength of the relationship between your model and the dependent variable on a convenient 0 – 100% scale.

After fitting a linear regression model, you need to determine how well the model fits the data. Does it do a good job of explaining changes in the dependent variable? There are a several key goodness-of-fit statistics for regression analysis. In this post, we’ll examine R-squared (R^{2 }), highlight some of its limitations, and discover some surprises. For instance, small R-squared values are not always a problem, and high R-squared values are not necessarily good!

**Related post**: When Should I Use Regression Analysis?

## Assessing Goodness-of-Fit in a Regression Model

Linear regression identifies the equation that produces the smallest difference between all of the observed values and their fitted values. To be precise, linear regression finds the smallest sum of squared residuals that is possible for the dataset.

Statisticians say that a regression model fits the data well if the differences between the observations and the predicted values are small and unbiased. Unbiased in this context means that the fitted values are not systematically too high or too low anywhere in the observation space.

However, before assessing numeric measures of goodness-of-fit, like R-squared, you should evaluate the residual plots. Residual plots can expose a biased model far more effectively than the numeric output by displaying problematic patterns in the residuals. If your model is biased, you cannot trust the results. If your residual plots look good, go ahead and assess your R-squared and other statistics.

Read my post about checking the residual plots.

## R-squared and the Goodness-of-Fit

R-squared evaluates the scatter of the data points around the fitted regression line. It is also called the coefficient of determination, or the coefficient of multiple determination for multiple regression. For the same data set, higher R-squared values represent smaller differences between the observed data and the fitted values.

R-squared is the percentage of the dependent variable variation that a linear model explains.

R-squared is always between 0 and 100%:

- 0% represents a model that does not explain any of the variation in the response variable around its mean. The mean of the dependent variable predicts the dependent variable as well as the regression model.
- 100% represents a model that explains all of the variation in the response variable around its mean.

Usually, the larger the R^{2}, the better the regression model fits your observations. However, this guideline has important caveats that I’ll discuss in both this post and the next post.

## Visual Representation of R-squared

To visually demonstrate how R-squared values represent the scatter around the regression line, you can plot the fitted values by observed values.

The R-squared for the regression model on the left is 15%, and for the model on the right it is 85%. When a regression model accounts for more of the variance, the data points are closer to the regression line. In practice, you’ll never see a regression model with an R^{2} of 100%. In that case, the fitted values equal the data values and, consequently, all of the observations fall exactly on the regression line.

## R-squared has Limitations

You cannot use R-squared to determine whether the coefficient estimates and predictions are biased, which is why you must assess the residual plots.

R-squared does not indicate if a regression model provides an adequate fit to your data. A good model can have a low R^{2} value. On the other hand, a biased model can have a high R^{2} value!

## Are Low R-squared Values Always a Problem?

No! Regression models with low R-squared values can be perfectly good models for several reasons.

Some fields of study have an inherently greater amount of unexplainable variation. In these areas, your R^{2} values are bound to be lower. For example, studies that try to explain human behavior generally have R^{2} values less than 50%. People are just harder to predict than things like physical processes.

Fortunately, if you have a low R-squared value but the independent variables are statistically significant, you can still draw important conclusions about the relationships between the variables. Statistically significant coefficients continue to represent the mean change in the dependent variable given a one-unit shift in the independent variable. Clearly, being able to draw conclusions like this is vital.

**Related post**: How to Interpret Regression Models that have Significant Variables but a Low R-squared

There is a scenario where small R-squared values can cause problems. If you need to generate predictions that are relatively precise (narrow prediction intervals), a low R^{2} can be a show stopper.

How high does R-squared need to be for the model produce useful predictions? That depends on the precision that you require and the amount of variation present in your data. A high R^{2} is necessary for precise predictions, but it is not sufficient by itself, as we’ll uncover in the next section.

**Related post**: Understand Precision in Applied Regression to Avoid Costly Mistakes

## Are High R-squared Values Always Great?

No! A regression model with a high R-squared value can have a multitude of problems. You probably expect that a high R^{2} indicates a good model but examine the graphs below. The fitted line plot models the association between electron mobility and density.

The data in the fitted line plot follow a very low noise relationship, and the R-squared is 98.5%, which seems fantastic. However, the regression line consistently under and over-predicts the data along the curve, which is bias. The Residuals versus Fits plot emphasizes this unwanted pattern. An unbiased model has residuals that are randomly scattered around zero. Non-random residual patterns indicate a bad fit despite a high R^{2}. Always check your residual plots!

This type of specification bias occurs when your linear model is underspecified. In other words, it is missing significant independent variables, polynomial terms, and interaction terms. To produce random residuals, try adding terms to the model or fitting a nonlinear model.

**Related post**: Model Specification: Choosing the Correct Regression Model

A variety of other circumstances can artificially inflate your R^{2}. These reasons include overfitting the model and data mining. Either of these can produce a model that looks like it provides an excellent fit to the data but in reality the results can be entirely deceptive.

An overfit model is one where the model fits the random quirks of the sample. Data mining can take advantage of chance correlations. In either case, you can obtain a model with a high R^{2} even for entirely random data!

**Related post**: Five Reasons Why Your R-squared can be Too High

## R-squared Is Not Always Straightforward

At first glance, R-squared seems like an easy to understand statistic that indicates how well a regression model fits a data set. However, it doesn’t tell us the entire story. To get the full picture, you must consider R^{2} values in combination with residual plots, other statistics, and in-depth knowledge of the subject area.

I’ll continue to explore the limitations of R^{2} in my next post and examine two other types of R^{2}: adjusted R-squared and predicted R-squared. These two statistics address particular problems with R-squared. They provide extra information by which you can assess your regression model’s goodness-of-fit.

You can also read about the standard error of the regression, which is a different type of goodness-of-fit measure.

Be sure to read my post where I answer the eternal question: How high does R-squared need to be?

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