Interpreting Confidence and Credibility Intervals in Decisions

Intervals Answer Different Questions

Dose-response analyses produce estimates of parameters such as a slope, an \(E_{\max}\), or an \(ED_{50}\). An interval around an estimate communicates uncertainty, but the meaning of its probability depends on how the interval was constructed. A frequentist confidence interval describes the long-run behavior of a procedure under repeated sampling. A Bayesian credible interval describes posterior uncertainty about the parameter conditional on the observed data, model, and prior.

That distinction matters when a parameter is compared with a development threshold. An interval may show whether values on both sides of a threshold remain compatible with an analysis, but an interval does not automatically specify how much evidence is enough to proceed. The decision rule, consequences of alternatives, and relevant model assumptions still need to be stated.

A commonly reported Bayesian interval is the equal-tail credible interval: its lower and upper endpoints leave posterior probability \(\alpha/2\) in each tail. Another option is a highest posterior density interval, which contains the specified posterior mass in regions of greatest density. These constructions need not have the same endpoints, especially for an asymmetric posterior. The interval type should therefore be named when reporting results.

Frequentist Confidence Intervals for a Dose-Response Parameter

Suppose \(\theta\) is a fixed dose-response parameter, and an estimator \(\hat{\theta}\) is approximately normally distributed around \(\theta\) with estimated standard error \(SE\). Under this approximation, a two-sided Wald confidence interval is

where \(z_{1-\alpha/2}\) is the corresponding standard-normal quantile. For a 95% interval, \(z_{0.975}\) is approximately 1.96. This formula is suitable when the estimator's sampling distribution is adequately approximated by a normal distribution and the standard error is reliable. It can be misleading with sparse data, strong skewness, parameters near boundaries, or a poorly estimated standard error.

The repeated-sampling interpretation follows directly from the approximation. If \((\hat{\theta}-\theta)/SE\) has a standard normal distribution, then

Rearranging the inequalities gives

The probability statement is about the interval-producing procedure before the data are observed. After an interval is calculated, the parameter is treated as fixed in this framework; the particular interval either covers it or does not. The nominal coverage is justified only to the extent that the model and approximation assumptions hold.

Bayesian Credible Intervals and Threshold Probabilities

In Bayesian analysis, uncertainty about \(\theta\) is represented by a posterior distribution. Bayes' rule combines a likelihood for the observed data with a prior distribution for the parameter. A 95% equal-tail credible interval has posterior cumulative probability 0.025 below its lower endpoint and 0.025 above its upper endpoint. Conditional on the specified analysis, it is valid to say that the posterior probability \(\theta\) lies in that interval is 0.95.

A decision threshold can also be evaluated directly using the posterior distribution. For a threshold \(\theta^*\), the quantity \(P(\theta>\theta^*\mid \text{data})\) is the posterior probability that the parameter exceeds that threshold. This probability is not the same as a frequentist confidence level. It depends on the likelihood, prior, and model, and it is not itself a complete decision rule unless a decision criterion has been specified.

Intervals on Asymmetric Parameter Scales

Dose-response parameters may have skewed uncertainty, particularly when the parameter is positive or estimated near a boundary. A symmetric interval around an estimate can then be inappropriate. Bayesian intervals can be calculated from posterior quantiles on the parameter's original scale, and the posterior probability relative to a threshold can be calculated from the same distribution. The transformation and interval construction should be reported because they affect the endpoints and their interpretation.

Choosing and Checking an Interval

A Wald interval is convenient, but its symmetry can conflict with the shape of a dose-response parameter's sampling distribution. Frequentist alternatives include profile-likelihood and bootstrap intervals, each with its own assumptions and implementation choices. Bayesian equal-tail and highest posterior density intervals summarize the same posterior differently. No interval construction repairs a misspecified dose-response model or makes an inappropriate decision threshold clinically relevant.

Before using an interval in a decision, check the assumptions behind its construction. For a frequentist interval, examine whether the estimator's sampling approximation and standard error are credible; for a Bayesian interval, examine whether the likelihood, prior, and posterior computation are appropriate. In either framework, assess whether the parameter is well identified by the data and whether the model represents the dose-response relationship over the range relevant to the decision. Implausible endpoints, strong skew, boundary estimates, or large changes under reasonable modeling choices are reasons to investigate rather than to report a single interval without qualification.

A useful decision statement names the parameter, its units, the threshold, the interval type, and the rule applied. For example, “The 95% confidence interval for the slope is 0.918 to 2.682 percentage points per mg; because it crosses the prespecified 1.50 threshold, the criterion requiring the full interval to exceed the threshold is not met.” A Bayesian statement should instead say which posterior distribution and interval were used, and, when relevant, report the posterior probability relative to the threshold.

What an Interval Does Not Decide

An interval is not a substitute for specifying the decision problem. Two decision makers can use the same interval and reach different actions if they have different consequences for a false go decision, a false no-go decision, delay, or additional data collection. Likewise, whether an interval crosses a threshold depends on the chosen confidence or credible level, the parameter scale, and the model. A threshold comparison is informative, but it is not a causal claim and does not establish that selecting a dose will produce a particular clinical outcome.

When communicating results, separate three statements: the parameter estimate and interval; the probability interpretation warranted by the framework; and the decision implied by the prespecified rule. If the decision changes under reasonable interval methods, model assumptions, or priors, report that sensitivity rather than presenting the interval as definitive. This keeps uncertainty visible without overstating what the analysis establishes.