Yes. It is negative when the candidate rule has more error than the baseline under the chosen comparison.

Is PRE always R-squared?

No. R-squared is the standard squared-error PRE for a particular regression baseline and setup.

What if baseline error is zero?

The ratio is undefined because it divides by zero. Report the errors directly or choose a justified alternative comparison.

AI Glossary / Statistics and probability / Intermediate

Proportional reduction in error (PRE)

Definition

Proportional reduction in error, or PRE, measures how much a prediction rule reduces error relative to a specified baseline. Its general form is (baseline error − improved-rule error)/baseline error, provided the baseline error is positive. In the supplied regression context, it describes the reduction in residual squared error after using predictor information. For ordinary least-squares regression with an intercept, compared with predicting the sample mean, the in-sample PRE equals R-squared.

What it means

PRE expresses improvement relative to what would happen without the additional information or model. A value of 0.30 means that the chosen error measure is 30% lower than under the stated baseline. It does not mean that 30% of predictions are correct or that errors are 30% smaller under every other metric.

The baseline and loss function are essential parts of the definition. Counting classification mistakes, summing squared numerical errors, and measuring absolute errors are different comparisons. A meaningful report therefore names the prediction task, evaluation data, baseline rule, and error measure rather than displaying a bare PRE value.

How it works

Choose a baseline that represents the comparison of interest, such as predicting the mean for a numerical outcome or using an existing operational model. Calculate baseline error and candidate-model error on the same cases with the same weighting and loss definition.

Subtract candidate error from baseline error, then divide by baseline error. A value of zero indicates equal error. A positive value indicates improvement, while a negative value indicates deterioration. If the baseline error is zero, the ratio is undefined and another reporting method is needed.

For least-squares training fit with an intercept, the sample-mean baseline has total sum of squares SST, and the fitted model has residual sum of squares SSE. PRE = 1 − SSE/SST. The familiar nonnegative in-sample result depends on that fitting setup; arbitrary predictions or held-out evaluation can produce negative values. Validation is needed before interpreting a training improvement as predictive progress.

Three practical examples

A forecasting team compares a regression's squared errors with those from predicting a baseline mean on specified evaluation cases.
An operations group measures reduction in absolute scheduling error relative to its current planning rule.
A classification team compares mistake counts from a new decision rule and a documented baseline, keeping class costs separate if they differ.

Think of it this way

Think of a baseline error budget. PRE describes the fraction of that budget removed by a new rule. The meaning changes if the budget is measured in squared units, absolute units, or counts of mistakes, so the label on the budget matters as much as the fraction removed.

Worked example

In an original numerical-prediction example, the baseline sum of squared errors on a fixed set of cases is 100. A candidate model's sum of squared errors on the same cases is 64.

PRE = (100 − 64)/100 = 0.36, or a 36% reduction in squared error. The remaining squared-error fraction is 0.64.

If both root mean squared errors use the same number of cases, their ratio is sqrt(64/100) = 0.80. The corresponding RMSE reduction is therefore 20%, not 36%. These are compatible statements using different scales.

Now consider a different candidate whose squared-error total is 121 on those same cases. Its PRE is (100 − 121)/100 = −0.21. It has 21% more squared error than the baseline. A negative value is informative, especially in held-out evaluation; replacing it with zero would hide deterioration. These stipulated errors illustrate the metric without claiming a fitted model or a real improvement.

Illustrative case study

Fictional case study: Cedar Forecasting compares a new AI model with its existing weekly forecast. Its first presentation reports a favorable PRE against a weak constant baseline, while omitting the stronger operational forecast.

The team adds both comparisons, states the squared-error definition, and evaluates all methods over the same held-out weeks. It also reports absolute errors in the units used for planning. This fictional case illustrates why the baseline should be visible and relevant; it does not establish which forecasting method performs best.

Common misconceptions and mistakes

PRE is not a unique metric unless the baseline and definition of error are specified. In regression, R-squared is a particular squared-error PRE, not a universal statement about all prediction quality.

Do not confuse the percentage reduction in squared error with the reduction in root error or absolute error. Another mistake is comparing errors measured on different populations or periods without adjustment. A training improvement can also result from added flexibility rather than a useful generalization gain. Finally, a large PRE does not show that remaining errors are operationally acceptable or that the predictor has a causal effect.

Business use

PRE helps organizations communicate predictive progress relative to a reference rule. It is useful for comparing the value of new information or modeling effort when the evaluation design and error measure match the decision.

Pair the relative value with baseline and candidate error totals, the units, and uncertainty in the comparison. Use a baseline that represents a genuine alternative, and check important subgroups or operating conditions. If errors have unequal business costs, a cost-based loss may be more appropriate than raw mistake counts or squared error. State that choice explicitly.

Check your understanding

A baseline makes 40 classification mistakes and a candidate makes 25 on the same test cases. What is the PRE based on mistake counts?

Answer: It is (40 − 25)/40 = 0.375, or 37.5%. This describes a reduction in mistakes, not a 37.5-percentage-point increase in accuracy; the latter also depends on the test-set size.

Related terms

R-squared
Loss function
Root Mean Squared Error
Proportional improvement in prediction (PIP)
Validation
baseline model
residual error
squared error
proportional improvement in prediction
prediction accuracy

Learn next

Calculate PRE with several clearly stated loss functions and observe how its interpretation changes. Then derive the regression R-squared identity and compare training versus held-out results. Study baseline selection and uncertainty estimation before using small metric improvements to choose a model.

R-squared

Sources

Howell's statistical glossary defines proportional reduction in error in terms of residual error with and without predictor information: https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html

Howell, Statistical Methods for Psychology, section 9.7, discusses the regression squared-error interpretation: https://dspace.sxcjpr.edu.in/jspui/bitstream/123456789/1064/1/Copy%20of%20Statistics-Text.pdf

The error totals, calculations, and fictional case are original. The entry distinguishes the supplied regression context from the broader baseline-relative error principle.

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