Verifying Software Intervals for a Response

Why Reconstruct a Software Interval?

Regression software can report an interval with a few clicks, but the output is only useful if its numbers and label are understood. A confidence interval for the mean response describes uncertainty about the average response at a specified predictor value. A prediction interval describes uncertainty about one future individual response at that value. They use the same fitted value, but they do not use the same standard error.

Reconstructing an interval is a practical audit. It can reveal whether the correct predictor value, residual standard error, degrees of freedom, and interval type were used. It also gives a direct way to assess whether software output is consistent with the regression summaries.

The Model and the Required Summaries

Use the simple linear regression model \(Y_i=\beta_0+\beta_1x_i+\varepsilon_i\), where \(x_i\) is the predictor value and \(Y_i\) is the response for observation \(i\). For the usual exact t intervals, the errors are independent, have mean zero and common variance \(\sigma^2\), and are normally distributed. The mean response is linear in \(x_i\). The interval is conditional on the predictor values; a future individual response is assumed to follow the same model at the chosen predictor value.

In an application, these conditions need to be plausible: the observations should be independent, the relationship should be adequately linear, and residual variation should be reasonably constant and approximately normal for small-sample exact inference. The predictor value should also be within a range supported by the observed data. The calculations below assume these conditions have been checked or are stipulated for the hypothetical examples.

Here \(b_0\) and \(b_1\) are the fitted intercept and slope, and SSE is the sum of squared residuals. Some software labels \(s\) as the residual standard error, residual standard deviation, or root mean square error. Check how the output defines it: the formulas here require the estimate with denominator \(n-2\). If the software supplies \(s_x\) rather than \(S_{xx}\), convert it using \((n-1)s_x^2\), not \(ns_x^2\).

The Two Interval Formulas

For a two-sided interval with confidence level \(1-\alpha\), let \(t^*\) be the positive t critical value with \(n-2\) degrees of freedom that leaves probability \(\alpha/2\) in each tail. Both intervals are centered at the fitted response \(\hat{y}_0\). Their half-widths differ because the uncertainty being described differs.

The quantity multiplying \(t^*\) is the interval’s standard error. To reconstruct an interval, compute \(S_{xx}\) if needed, substitute into the appropriate standard error, multiply by \(t^*\) to obtain the margin of error, and subtract and add that margin to \(\hat{y}_0\). Keep the response units throughout: if \(s\) is measured in points, both the standard error and margin are in points.

Why the Standard Errors Have These Forms

A short variance argument explains the two formulas. In simple linear regression, the fitted value can be written as \(\hat{y}_0=\bar{Y}+b_1(x_0-\bar{x})\), where \(\bar{Y}\) is the sample mean response. Under the model, \(\operatorname{Var}(\bar{Y})=\sigma^2/n\), \(\operatorname{Var}(b_1)=\sigma^2/S_{xx}\), and \(\bar{Y}\) and \(b_1\) have covariance zero. Therefore,

For a future response \(Y_{\text{new}}\) at \(x_0\), its error around the fitted value includes both uncertainty in the estimated mean and the new individual error. The latter has variance \(\sigma^2\) and is independent of the fitted estimation error. Adding the variances gives

Replacing \(\sigma\) with the residual standard error \(s\) estimates these standard errors. With normal errors, the resulting standardized errors use a t critical value with \(n-2\) degrees of freedom. The extra \(1\) in the prediction formula represents the variation of an individual response around its mean; it is not part of the uncertainty in the mean response.

Reconstructing the Mean-Response Interval

Reconstructing an Individual Prediction Interval

Auditing Both Intervals in Software Output

A Reliable Verification Process

If the endpoints disagree substantially, check the interval label first, then confirm the predictor value, residual standard error, sample size, \(s_x\), confidence level, and degrees of freedom. If the discrepancy is only in the last displayed digit, rounding of the reported summaries may explain it.

Common Mistake

A frequent error is to use the mean-response formula for an individual prediction because both intervals are centered at the same fitted value. For example, using \(s\sqrt{1/n+(x_0-\bar{x})^2/S_{xx}}\) for a future individual omits the individual-response variation. The resulting interval is too narrow and answers a different question: uncertainty about a mean, not uncertainty about one response. Conversely, adding the leading 1 when the target is a mean response makes that interval unnecessarily wide and mislabels the target.