Interpreting Negative Out-of-Sample R-Squared

What a Negative Score Compares

An out-of-sample \(R^2\) is a comparison, not a guarantee that a regression model predicts well. It compares a model’s prediction errors on held-out cases with the errors from a simple baseline prediction. A negative value means that, on those cases and under squared-error loss, the model did worse than the baseline.

The baseline used here predicts the mean outcome in the training data for every test case. That mean is available before the test outcomes are examined. The model and baseline must be evaluated on the same held-out cases, using the same outcome units and the same squared-error calculation.

Both sums have squared outcome units, so their ratio—and therefore \(R^2_{\text{out}}\)—has no units. The score depends on the specified baseline. In particular, this definition uses the training mean, not a mean calculated from the held-out outcomes. Using the test outcomes to choose the baseline would compare against information that would not have been available when making genuine future predictions.

Why the Score Can Be Negative

Write \(SSE_{\text{model}}\) for the model’s sum of squared test errors and \(SSE_{\text{base}}\) for the baseline’s sum of squared test errors. When the baseline error is positive, the score is negative exactly when the model’s error is larger:

This is the central interpretation. A negative score does not mean that the calculation has failed, that the model is impossible, or that a proportion of variance has become negative. It says that the model’s squared errors on this evaluation set exceed the squared errors from predicting the training mean for everyone.

The score also gives the relative size of the error comparison. Rearranging its definition gives \(SSE_{\text{model}}/SSE_{\text{base}}=1-R^2_{\text{out}}\). Thus, a score of \(-0.30\) means the model’s test sum of squared errors is \(1.30\) times the baseline’s: it is 30% greater. A score of \(-1.00\) means the model’s sum of squared errors is twice the baseline’s. The percentage refers to the baseline error, not to variance explained.

Interpreting the Size of a Negative Value

A score just below zero indicates a relatively small disadvantage to the baseline, while a more negative score indicates a larger error ratio. However, the size is always relative to the baseline’s test error. It is not an absolute measure of how many outcome units the model missed by, and it is not a universal measure of how bad a model is. The underlying error sums and the test context matter.

Negative scores have no fixed lower bound. If the model makes very large errors while the baseline errors are small, the ratio \(SSE_{\text{model}}/SSE_{\text{base}}\) can be very large, making \(1-SSE_{\text{model}}/SSE_{\text{base}}\) strongly negative. A large negative value should therefore prompt a check of both the model errors and the baseline denominator rather than an automatic conclusion about the model’s performance in other settings.

Negative, Zero, and Positive Values

The sign can be read directly by comparing the two error sums. This table assumes \(SSE_{\text{base}}>0\), so the ratio is defined.

A positive value is an improvement over this baseline by the proportion of baseline squared error removed. For example, \(R^2_{\text{out}}=0.20\) corresponds to a model sum of squared errors that is \(0.80\) times the baseline sum. A zero value is a tie under this metric; it does not mean that every model prediction equals every outcome. Different errors can produce the same total.

If \(SSE_{\text{base}}=0\), the displayed ratio is undefined. That happens when every test outcome equals the training mean, so the baseline predicts each outcome exactly. In that special case, the usual formula cannot provide a finite score; report the model and baseline errors directly instead of assigning an out-of-sample \(R^2\) through division by zero.

Conditions for a Useful Interpretation

The arithmetic is simple, but the comparison is meaningful only when the evaluation is set up correctly. The model must produce predictions for cases that were held out from the fitting process, and the mean used by the baseline must come from the training data. If the test outcomes influence model fitting, model selection, or the baseline mean, the result no longer represents the same clean held-out comparison.

The model and baseline must also be scored on the same test cases. If one prediction is missing, exclude that case from both sums or specify another consistent rule; comparing sums from different cases can mislead. Both sets of predictions and observed outcomes must be on the same outcome scale. Finally, this metric uses squared errors, so large misses have a particularly strong effect because each error is squared.

The out-of-sample score describes performance on the evaluation set under the chosen comparison. It is not a causal statement, a probability, or a guarantee about future cases. If the test set is small or unusual, the score may not represent the performance expected in a different group of cases. The direction of the comparison remains precise: for this set, did the model’s squared-error sum exceed the baseline’s?

Common Mistake

A frequent error is to say, “An \(R^2\) of \(-0.30\) means the model explains negative 30% of the outcome variation,” or to assume that a negative result must be a calculation mistake. Neither conclusion follows. The correct reading is that, with the stated baseline and test cases, the model’s squared error is \(1-(-0.30)=1.30\) times the baseline’s, so it is 30% greater.

Another mistake is to treat every negative value as equally poor. A score of \(-0.05\) means the model’s squared-error sum is 1.05 times the baseline’s; \(-2\) means it is 3 times the baseline’s. In both cases the model loses to the baseline, but by different relative amounts. Always translate the score back to the error ratio and identify the baseline being used.