Confidence Intervals Versus Prediction Intervals for Dose Decisions

Two Different Uncertainties in a Dose Decision

A dose-response model can answer two questions that sound similar but concern different quantities. One is, “Which dose is estimated to achieve the target mean response?” The other is, “What response might a future individual have at this dose?” A confidence interval for a model-derived dose addresses uncertainty in the estimated dose. A prediction interval addresses the range of plausible responses for a future individual, accounting for both uncertainty in the fitted mean and individual response variability.

The distinction matters in practice. If a team wants to communicate how precisely a model has located a dose associated with a target average response, an interval for the dose is relevant. If the concern is whether individual patients at a proposed dose may have low efficacy or high toxicity, an interval for future responses is closer to the question. Neither interval automatically answers every dose-selection question: the interval must match the quantity and decision being considered.

What a Dose Confidence Interval Means

A model-derived dose is a quantity calculated from estimated model parameters. For example, a dose may be obtained by solving \(\mu(d;\hat{\theta})=T\), where \(\hat{\theta}\) denotes estimates based on the data. The resulting estimate, \(\hat{d}_T\), can change across repeated studies because the fitted parameters would change. A confidence interval for \(d_T\) is intended to quantify that sampling uncertainty.

In a frequentist interpretation, a 95% confidence-interval procedure has 95% coverage if, under its assumptions and repeated sampling, 95% of intervals constructed by that procedure contain the true \(d_T\). Once one interval has been calculated, it is not correct to say there is a 95% probability that the fixed true dose lies inside it. The interval is evidence about dose uncertainty, not a probability distribution for the true dose.

The dose confidence interval is not the same as an interval for the mean response at one dose. A confidence interval for \(\mu(d)\) concerns the modelled average response at a specified \(d\). A confidence interval for \(d_T\) concerns the dose that solves the target equation. Either may be useful, but they answer different questions. A dose interval also does not state the range of responses expected among individual patients at that dose.

The approximation used to construct an interval for a derived dose depends on how the model and dose are estimated. A symmetric interval such as the example can be reasonable when the estimator is approximately normal and sufficiently far from constraints. If the dose estimate is highly skewed, near a dose boundary, or poorly determined, a symmetric interval may be misleading. The interval’s interpretation does not remove the need to check whether the model identifies a unique target-attaining dose within the dose range of interest.

What a Prediction Interval Means

Suppose responses follow a model \(Y_i=\mu(d_i;\theta)+\epsilon_i\), where \(Y_i\) is a response at dose \(d_i\), and \(\epsilon_i\) represents individual-level deviation from the modelled mean. For a future individual at dose \(d_0\), a prediction interval must allow for two sources of uncertainty: uncertainty about the mean response \(\mu(d_0;\theta)\), and residual variation of individual responses around that mean.

For illustration, assume a correctly specified regression model with independent, normally distributed residuals having constant variance \(\sigma^2\). Let \(\hat{\mu}(d_0)\) be the fitted mean response, \(SE_{\mathrm{mean}}\) its standard error, and \(s\) the estimated residual standard deviation. Under these assumptions, a conventional prediction interval for one future response uses the standard error \(\sqrt{SE_{\mathrm{mean}}^2+s^2}\). A confidence interval for the mean uses only \(SE_{\mathrm{mean}}\). The future individual’s residual is an additional source of variation.

A prediction interval is not always wider than every possible confidence interval from a different analysis, but in this standard setting its added residual-variance term makes it wider than the corresponding mean confidence interval. For a future average of \(m\) independent individuals at the same dose, residual variation in the average is \(\sigma^2/m\), rather than \(\sigma^2\) for one individual. A prediction interval for a future group average is therefore a different quantity from a prediction interval for one person.

Match the Interval to the Dose Question

The practical choice begins by naming the quantity the decision-maker wants to understand. A dose interval is appropriate when the focus is uncertainty in the dose associated with a target mean response. A response prediction interval is appropriate when the focus is the range of outcomes for a future individual at a specified dose. A confidence interval for the mean response at a candidate dose is appropriate when the question concerns the average response there. These are distinct estimands, even when all three arise from the same fitted model.

Assumptions, Diagnostics, and Common Pitfalls

Intervals inherit the assumptions of the model and of the interval procedure. For a regression prediction interval of the form used above, important assumptions include an adequate mean-response model, independent residuals, approximately normal residuals for the usual small-sample \(t\) calibration, and constant residual variance over the relevant doses. The dose at which prediction is made should also be supported by the data; a mathematically computable interval does not guarantee reliable extrapolation.

Diagnostics should be chosen to investigate these assumptions. Residual patterns against fitted values or dose can reveal curvature, changing spread, or systematic lack of fit. Strong outliers or heavy residual tails may undermine normal-based coverage, particularly in a small sample. Dependence between observations—for example, repeated responses from the same individual—can make an independence-based standard error too small. A visibly poor dose-response fit or a dose far outside the observed range calls for caution in interpreting either type of interval.

For a target-dose interval, also check that the target is reached within the modelled dose range and that the target equation has a meaningful, unique solution. If the modelled mean is nearly flat around the target, small changes in the fitted curve can imply large changes in the dose estimate. If the curve never reaches the target, there is no target-attaining dose within the model’s stated range. A narrow interval calculated under an unsuitable model does not resolve either problem.

• Mistake: treating a dose confidence interval as a patient-response range. The interval in the first worked example concerns uncertainty in the dose estimate, not the spread of outcomes at those doses. Use a future-response prediction interval for individual response variability.
• Mistake: using a mean confidence interval to describe individuals. A mean can be estimated precisely even when individual responses vary substantially. The prediction interval includes the residual variation that the mean interval omits.
• Mistake: interpreting “95%” as a probability about a fixed parameter or a particular patient. The frequentist coverage statement concerns the long-run performance of an interval procedure under its assumptions. It does not assign a probability to the fixed true dose after observing the data.

Other uncertainty summaries may be produced using different inferential frameworks or construction methods. They should still be judged by the quantity they summarize and the assumptions they require. The essential choice here remains the estimand: uncertainty in a derived dose, uncertainty in a population mean, or uncertainty in a future individual response.