Prediction Intervals for Individual Responses

Predicting One New Response

A fitted regression line estimates the mean response associated with a predictor value. Often, however, the question concerns one particular future case: for example, the delivery time for one new order, given its distance, or the blood pressure of one new patient, given an explanatory measurement. A point prediction gives a single fitted value, but actual individual responses vary around the regression line. A prediction interval gives a range of plausible values for one new response while accounting for that variation and for uncertainty in the fitted line.

The target is important. At a specified predictor value \(x^*\), let \(Y_{\text{new}}\) denote the response for one new individual. A prediction interval concerns \(Y_{\text{new}}\), not the unknown mean response at \(x^*\). The interval is conditional on the chosen predictor value and on the regression model being suitable there.

Model and Prediction-Interval Formula

For simple linear regression, suppose the observed responses satisfy \(Y_i=\beta_0+\beta_1x_i+\varepsilon_i\), for \(i=1,\ldots,n\). The predictor values are treated as fixed, or the analysis is understood conditional on them. The errors are independent, have mean zero, have the same variance \(\sigma^2\), and follow a normal distribution. Equivalently, \(Y_i\mid x_i\sim N(\beta_0+\beta_1x_i,\sigma^2)\). A new response at \(x^*\) is assumed to follow this same model and to have an error independent of the data used to fit the line.

In practice, the model conditions are assessed using the data and the study design. The conditional mean should be adequately linear over the predictor range of interest; observations and the new response should be independent; the spread of errors should be reasonably constant; and the error distribution should be approximately normal for a small-sample t interval. The prediction should generally be made within the observed predictor range, since the linear pattern and error behavior may not continue beyond it.

Write \(\hat{y}^*=b_0+b_1x^*\) for the fitted response at \(x^*\), \(\bar{x}\) for the sample mean of the predictor, and \(S_{xx}=\sum_{i=1}^n(x_i-\bar{x})^2\). Let \(s=\sqrt{SSE/(n-2)}\) be the residual standard error, where \(SSE\) is the sum of squared residuals. With \(t^*=t_{n-2,\,1-\alpha/2}\), the critical value having cumulative probability \(1-\alpha/2\) for a t distribution with \(n-2\) degrees of freedom, the prediction interval is:

The square-root factor multiplied by \(s\) is the prediction standard error. Its terms account for uncertainty in estimating the fitted mean at \(x^*\) and for the random variation of a new individual response around that mean. The interval endpoints have the same units as the response. The \(n-2\) degrees of freedom arise because the fitted simple linear regression estimates two coefficients, the intercept and slope.

Why This Standard Error Applies

A brief derivation shows why the formula includes these sources of variability. In the model, the new response is \(Y_{\text{new}}=\beta_0+\beta_1x^*+\varepsilon_{\text{new}}\), while its fitted value is \(b_0+b_1x^*\). Their difference is

The new error is independent of the data used to estimate the line. Its variance is \(\sigma^2\). Under the simple linear regression model, the variance of the estimation error in the fitted mean at \(x^*\) is \(\sigma^2\left(\frac{1}{n}+\frac{(x^*-\bar{x})^2}{S_{xx}}\right)\). Adding the two variances gives:

Replacing \(\sigma\) with \(s\) estimates the standard deviation of this prediction error. Under the stated normal-error model, the standardized prediction error uses the residual estimate \(s\) and has a t distribution with \(n-2\) degrees of freedom. Multiplying its standard error by the corresponding t critical value gives the margin of error in the formula.

Constructing and Interpreting the Interval

The calculation follows a consistent sequence. First identify the new individual response and the predictor value at which it is to be predicted. Then check whether the regression model is appropriate for the data and whether that value is in a suitable range. Finally, obtain the fitted value and the regression summaries, calculate the prediction standard error, choose the t critical value for the desired confidence level, and form the endpoints.

Common Mistakes and Practical Checks

A prediction interval is useful when a decision concerns an individual case rather than a group average. For example, an organization may need a plausible range for one future service time to plan staffing, or a clinician may want a range of outcomes for one new patient. The interval communicates that the fitted value is not a guaranteed outcome. Its reliability depends on whether the model reasonably describes the population and whether the new case is comparable to the observations used to fit it.

• Confusing the target with a mean. A fitted value \(\hat{y}^*\) estimates the model’s mean response at \(x^*\); the prediction interval targets one new response.
• Omitting the individual-error contribution. Using only the uncertainty for an estimated mean response does not give the prediction standard error shown here. For an individual prediction, use the full formula, including the leading \(1\) under the square root.
• Calling the calculated interval a probability statement about fixed endpoints. The 95% refers to the long-run coverage rate of the interval procedure under the model, not a 95% probability that the fixed endpoints contain a response.
• Ignoring the study design or diagnostics. A formula cannot correct for dependent observations, a badly misspecified linear mean, or a new case outside the model’s useful range.
• Mixing units. The fitted value, residual standard error, margin of error, and interval endpoints all use response units. The predictor’s units appear in the location calculation but not in the final endpoints.

Before reporting a prediction interval, identify the individual response being predicted, state its predictor value and response units, document the model checks, and interpret the confidence level as a repeated-procedure coverage statement. These steps make clear both what the interval says and the conditions on which it depends.