The Simple-Regression Prediction-Interval Formula

Why Predicting One New Outcome Needs Its Own Interval

A fitted regression line gives a point prediction at a chosen predictor value. But a future response will generally differ from that fitted value: the line estimates the conditional mean, while an individual outcome also contains its own random error. A prediction interval accounts for both sources of uncertainty. This tutorial develops the two-sided t prediction-interval formula for simple linear regression and shows how to use its components.

The interval is for one future response at a specified predictor value \(x_0\). It is not an interval for the mean response at \(x_0\). The extra variation associated with the new outcome is reflected in the formula’s leading \(1\) under the square root.

Model, Conditions, and Notation

Suppose the observed responses \(Y_1,\ldots,Y_n\) are related to predictor values \(x_1,\ldots,x_n\) by the simple linear regression model

For the usual t-based prediction interval, assume the mean response is linear in \(x_i\), the errors are independent with mean \(0\) and common variance \(\sigma^2\), and the errors are normally distributed. For predicting a new response, also assume the future error is independent of the errors used to fit the line and has the same variance. These conditions make the formula appropriate for predicting a new outcome from the same process at \(x_0\).

Let \(\hat{y}_0=\hat{\beta}_0+\hat{\beta}_1x_0\) be the fitted value at \(x_0\), and define

Here \(s\) is the residual standard error, an estimate of the standard deviation \(\sigma\) of the regression errors. The quantity

is the predictor-dependent leverage term for estimating the mean at \(x_0\). Its value depends on where \(x_0\) lies relative to the observed predictor values and their spread.

The Simple-Regression Prediction-Interval Formula

With \(n-2\) degrees of freedom, let \(t_{1-\alpha/2,n-2}\) denote the positive t critical value with upper-tail probability \(\alpha/2\). The two-sided prediction interval is

Equivalently, the lower and upper endpoints are

The \(1\) inside the square root accounts for the error in the new outcome. The remaining terms, \(1/n+(x_0-\bar{x})^2/S_{xx}\), account for uncertainty in the fitted mean at that predictor value. The residual standard error \(s\) scales both contributions into the response’s units.

Why the Prediction Standard Error Has This Form

Write the future response as \(Y_{\mathrm{new}}=\beta_0+\beta_1x_0+\varepsilon_{\mathrm{new}}\). Its prediction error is the difference between that response and the fitted value:

The first term is the new outcome’s random error. The second is the error from estimating the conditional mean with the fitted line. Under the model conditions, the new error is independent of the estimation errors. The variance of the new error is \(\sigma^2\), and the variance of the fitted-mean estimation error at \(x_0\) is \(\sigma^2h_0\). Therefore the prediction-error variance is the sum:

Replacing the unknown \(\sigma\) by the residual standard error \(s\) gives the estimated prediction standard error \(s\sqrt{1+h_0}\). The t multiplier calibrates this estimated standard error for the interval. In simple linear regression, fitting the intercept and slope leaves \(n-2\) residual degrees of freedom. The details of t-based calibration are a separate topic; here, the critical value is used as part of the formula.

Working Through the Predictor-Dependent Term

Away from \(\bar{x}\), the squared-distance term increases \(h_0\). It is important to use the predictor values in the same units as the fitted regression and to square the difference \(x_0-\bar{x}\). The next example makes those steps explicit.

A Second Calculation and a Reporting Pitfall

The formula has several components, so it helps to keep the sequence consistent: find \(h_0\), add \(1\), multiply its square root by \(s\), and then multiply by the t critical value. The response units belong on the prediction standard error, margin of error, and interval endpoints.

Common Mistakes and a Reliable Procedure

A frequent error is to use only \(s\sqrt{h_0}\) as the prediction standard error. That expression leaves out the new outcome’s own variation. For an individual future response, use \(s\sqrt{1+h_0}\). Another mistake is to treat the fitted value as the entire prediction: \(\hat{y}_0\) is the interval’s center, not a guarantee of the future response.

A useful calculation sequence is:

Do not describe a 95% interval as having a 95% probability of containing a particular fixed future value after the interval has been calculated. The confidence level describes the long-run performance of the interval procedure under its assumptions. For a specific prediction, report the calculated range and the case to which it applies.