Confidence Intervals for Conditional Means

What a Confidence Interval for a Conditional Mean Describes

A regression model can estimate the mean outcome for cases with a particular set of predictor values. For example, it can estimate the mean test score for students with a specified baseline score and age, or the mean crop yield at a specified irrigation level. That estimated mean is not the outcome for any one individual at that profile. Individual outcomes can vary around the conditional mean.

A confidence interval describes uncertainty in the estimated conditional mean. It uses the fitted mean as its centre and extends in both directions by a margin based on the fitted mean’s standard error and a critical value. The interval is expressed in the outcome’s units, and its interpretation must name the predictor profile to which it applies.

Model, Assumptions, and the Interval Formula

For the standard t-based interval, write the regression model for the \(n\) observed outcomes as \(\mathbf{Y}=\mathbf{X}\boldsymbol{\beta}+\boldsymbol{\varepsilon}\), where \(\mathbf{X}\) contains the predictor values and the model’s columns, and \(\boldsymbol{\beta}\) is the vector of coefficients. Assume the mean function is correctly specified, the columns of \(\mathbf{X}\) are linearly independent, and, conditional on the predictors, the errors are independent with mean zero and common variance \(\sigma^2\). For an exact t interval, also assume the errors are normally distributed. In this setting, the predictor values are treated as fixed for the inference.

Let \(p\) be the number of estimated coefficients, including an intercept if the model has one. The residual degrees of freedom are \(n-p\). Estimate \(\sigma\) with the residual standard error \(s\), calculated from the residual sum of squares divided by \(n-p\). At profile \(\mathbf{x}_0\), the fitted mean’s standard error is \(SE(\hat{y}_0)=s\sqrt{\mathbf{x}_0^\mathsf{T}(\mathbf{X}^\mathsf{T}\mathbf{X})^{-1}\mathbf{x}_0}\). This is uncertainty in the estimated mean, not the standard deviation of individual outcomes at that profile.

The t critical value accounts for estimating the error variance from the regression residuals. The confidence level determines \(\alpha\): for a 95% interval, \(\alpha=0.05\), so the required quantile is \(t_{0.975,\ n-p}\). The degrees of freedom depend on the number of fitted coefficients, not just on the number of predictors named in a verbal description.

Why the t Interval Has This Form

For a fixed profile, \(\hat{y}_0\) is a linear combination of the estimated regression coefficients. Under the model assumptions, it is normally distributed around the true conditional mean \(\mathbf{x}_0^\mathsf{T}\boldsymbol{\beta}\). Its variance is \(\sigma^2\mathbf{x}_0^\mathsf{T}(\mathbf{X}^\mathsf{T}\mathbf{X})^{-1}\mathbf{x}_0\), so its standard deviation is the standard error formula with \(\sigma\) in place of \(s\).

When \(\sigma\) is unknown, it is estimated by \(s\). Under the normal-error model, the standardized estimate using \(s\) has a t distribution with \(n-p\) degrees of freedom. In symbols,

The central \(1-\alpha\) portion of this t distribution lies between \(-t_{1-\alpha/2,\ n-p}\) and \(t_{1-\alpha/2,\ n-p}\). Therefore, the standardized quantity falls between these limits with probability \(1-\alpha\). Rearranging those inequalities to isolate the fixed conditional mean gives the stated interval. Repeated samples under the model assumptions produce intervals that contain that mean at the stated rate; after an interval is calculated, the parameter is not treated as randomly moving within it.

Worked Intervals at Different Predictor Profiles

Interpretation and Common Pitfalls

A complete report identifies the confidence level, the predictor profile, and the outcome units. For example: “The 95% confidence interval for the mean yield at 50 mm of irrigation is 38.8 to 46.4 kg per plot.” The interval concerns a model-based mean for cases at that profile. It does not describe the range of likely individual yields.

Do not confuse this interval with a prediction interval for a future individual outcome. A prediction interval must account both for uncertainty in the estimated mean and for individual variation around that mean, so its purpose and calculation differ. Here, the target is only the conditional mean. Also avoid interpreting a 95% interval as having a 95% probability of containing the fixed mean after it has been computed. The 95% statement refers to the long-run performance of the interval procedure under the model assumptions.

• Wrong: “The endpoints are the likely range for 95% of individual outcomes.” Correction: They estimate uncertainty about the mean outcome at the named profile.
• Wrong: “There is a 95% probability that the true mean lies in this particular interval.” Correction: The procedure has 95% coverage over repeated samples under its assumptions.
• Wrong: “Use the normal critical value for every regression interval.” Correction: In the usual model with unknown error variance, use a t critical value with \(n-p\) degrees of freedom.
• Wrong: “The interval applies to the outcome generally.” Correction: State the predictor profile and use the outcome’s units.

Before reporting an interval, verify the model and its assumptions, identify the complete profile, use the standard error for the fitted mean at that profile, and match the critical value to both the confidence level and residual degrees of freedom. These checks make the resulting range interpretable rather than merely a pair of numbers.