Why Compare an Interval With a Bound?
A DDI exposure ratio is often summarized by a geometric mean ratio (GMR), together with a confidence interval. The ratio describes the estimated relative exposure under two conditions; the interval describes uncertainty in that estimate. A prespecified bound supplies a separate reference: it marks a range of ratios treated as having no important effect for a particular purpose, or a value at which a clinical decision may change. These are distinct pieces of information. A ratio can be close to one but imprecisely estimated, or precisely estimated yet far from one.
The task here is to interpret the estimate and interval in relation to bounds that have already been chosen for the relevant endpoint and decision. The bounds are not universal constants. Their rationale depends on the drug, endpoint, and decision context. In particular, an interval comparison does not by itself establish the clinical consequences of an exposure change.
Because exposure is positive and ratios are multiplicative, analysis is commonly performed on the log scale. A GMR of 1.20 represents a 20% higher geometric mean in \(T\) than in \(R\); a GMR of 0.80 represents a 20% lower geometric mean. These descriptions concern the ratio of geometric means, not a claim that every participant’s exposure changes by that percentage.
Constructing and Reading a Confidence Interval
For a simple paired analysis, suppose the subject-level log differences \(d_i\) are independent across subjects and are adequately described by a normal distribution with mean \(\mu_d\) and variance \(\sigma_d^2\). Then \(\bar d\) estimates \(\mu_d\), and its estimated standard error is \(SE=s_d/\sqrt{n}\), where \(s_d\) is the sample standard deviation of the log differences. A two-sided \(100(1-\alpha)\%\) t confidence interval for \(\mu_d\) is \(\bar d\pm t_{1-\alpha/2,n-1}SE\). Exponentiating both limits gives the confidence interval for the GMR.
This simple formula assumes the paired log differences are independent between subjects and that their distribution is sufficiently close to normal for the t-based interval to have suitable coverage. It also presumes that the endpoint, treatment contrast, and analysis population have been defined appropriately and that the study design supports the comparison. For a crossover study, for example, carryover or period effects not handled by the analysis can undermine the interpretation. An adjusted model may instead provide the estimated log contrast and its standard error; the same basic comparison with bounds applies, but the interval must come from a model appropriate to that design.
A frequentist 95% confidence interval is produced by a procedure that, under its assumptions, covers the fixed population log contrast in 95% of repeated samples. It does not mean there is a 95% probability that the fixed GMR lies in this particular interval. The interval’s width reflects statistical precision; the position of the interval relative to the bounds concerns the range of values compatible with the estimate and its uncertainty.
Worked Example: An Interval Crossing an Upper Bound
In a hypothetical paired DDI study, the estimated mean log AUC difference is \(\bar d=\ln(1.32)=0.2776\), with \(SE=0.0800\) and \(n=24\). Assume the paired log differences meet the t-interval assumptions. Use a 95% interval and \(t_{0.975,23}=2.069\). Suppose the prespecified no-effect range for this illustration is 0.80 to 1.25.
The margin on the log scale is \(2.069(0.0800)=0.1655\). Thus the log-scale interval is \(0.2776\pm0.1655=[0.1121,0.4431]\). Exponentiating the limits gives a GMR estimate of \(1.32\) and a 95% confidence interval of approximately \([1.119,1.558]\).
The lower limit is below 1.25 and the upper limit is above it. The interval therefore crosses the upper no-effect bound rather than lying wholly inside or wholly above the stated range. The point estimate exceeds 1.25, but the interval includes values on both sides of that bound. The result does not establish that the ratio is within the no-effect range, nor does this interval alone establish that the population ratio is above 1.25.
Interval Placement and Its Interpretation
Let the prespecified no-effect range be \([L,U]\), where \(0<L<1<U\) is common but not required. Compare both confidence limits with both bounds. This gives a clear description of the evidence without converting the interval into a yes-or-no statement unsupported by its placement.
| Interval placement | Interpretation relative to the stated bounds |
|---|---|
| Entire interval lies within \([L,U]\) | The estimated ratio and its interval are contained within the stated range. This supports a conclusion of compatibility with that range, subject to the analysis assumptions and the decision procedure. |
| Interval overlaps a bound | The interval includes values on both sides of at least one bound. The data do not resolve whether the ratio is inside or outside that bound at the interval’s confidence level. |
| Entire interval lies above \(U\) | The interval is wholly beyond the upper bound, supporting an increase beyond the stated range at that confidence level. |
| Entire interval lies below \(L\) | The interval is wholly beyond the lower bound, supporting a decrease beyond the stated range at that confidence level. |
These descriptions apply to the bounds that were specified for the particular analysis. If the bounds are clinical decision bounds rather than a no-effect range, the same geometric comparison is useful, but the conclusion should name the decision boundary and the endpoint. For example, an interval wholly above a prespecified AUC decision bound indicates that the AUC ratio is estimated beyond that boundary with the stated interval procedure. It does not, without additional evidence, quantify patient harm or prescribe a dose action.
Worked Example: An Interval Contained Within the Bounds
In a second hypothetical paired study, \(\bar d=\ln(1.04)=0.03922\), \(SE=0.0550\), and \(n=30\). Assume independent subjects and an adequate t approximation for the paired log differences. For a 95% interval, use \(t_{0.975,29}=2.045\). Compare the result with the illustrative range \([0.80,1.25]\).
The log-scale margin is \(2.045(0.0550)=0.1125\). The interval on the log scale is \(0.03922\pm0.1125=[-0.0733,0.1517]\). Exponentiating gives \([0.929,1.164]\), with estimated GMR \(1.04\).
Both limits lie inside 0.80 and 1.25. Thus the entire 95% interval is contained within the illustrative no-effect range. This is stronger evidence of compatibility with those bounds than an interval that merely includes 1. It is not proof that the true ratio equals 1. Nor does it establish that the same bounds are appropriate for another endpoint or decision.
If the objective is a formal equivalence claim, the decision rule and confidence level must match the prespecified testing procedure. For example, a common two-one-sided-test procedure at significance level \(\alpha=0.05\) corresponds to checking whether a 90% confidence interval is entirely within the equivalence bounds. A 95% interval lying inside the same bounds is more stringent in this respect, but the analysis should still follow the planned procedure. Merely noticing after analysis that some interval is inside some range is not a substitute for a prespecified equivalence assessment.
Worked Example: A Point Estimate Near One With a Wide Interval
In a third hypothetical study, \(\bar d=\ln(1.02)=0.01980\), \(SE=0.150\), and \(n=12\). Assume the paired log differences are independent and the t-based method is reasonable. For a 95% interval, use \(t_{0.975,11}=2.201\). Again compare with \([0.80,1.25]\).
The margin is \(2.201(0.150)=0.3302\). The log-scale limits are \(0.01980\pm0.3302=[-0.3104,0.3500]\). Exponentiating gives a GMR interval of approximately \([0.733,1.419]\).
The point estimate, 1.02, is close to one, but the interval extends below 0.80 and above 1.25. The estimate is imprecise relative to these bounds. It would be incorrect to conclude that there is no important interaction simply because the point estimate is near one. The interval includes ratios that are materially lower and higher than the illustrative no-effect range.
Precision, Statistical Evidence, and Clinical Importance
A confidence interval is not itself a measure of clinical importance. Importance is assessed relative to a justified bound and the clinical context of the drug and endpoint. An interval can be narrow but wholly beyond a decision bound, indicating a precise estimate of a ratio outside that range. Conversely, it can be wide and cross both bounds, indicating that the data do not precisely locate the ratio relative to the range.
The estimate’s relation to 1 is also not a replacement for comparison with the bounds. A confidence interval that excludes 1 indicates evidence of a difference in geometric means under the model and confidence procedure. It does not automatically indicate a clinically important difference: the interval might exclude 1 while remaining entirely within a prespecified no-effect range. Similarly, an interval that includes 1 may still include values beyond one or both decision bounds.
The lower and upper bounds need not be symmetric on the arithmetic scale. A range such as \([0.80,1.25]\) is approximately reciprocal, while a different decision context may justify asymmetric bounds. Do not replace the specified bounds with a convenient symmetric percentage rule. Keep the exposure endpoint clear as well: an AUC interval is compared with AUC bounds, and a Cmax interval with Cmax bounds. Bounds for one endpoint cannot automatically be applied to another.
Worked Example: An Interval Entirely Below the Lower Bound
In a fourth hypothetical paired study, the estimated GMR is 0.680, so \(\bar d=\ln(0.680)=-0.3857\). Let \(SE=0.0600\), \(n=20\), and assume the paired log differences satisfy the t-interval assumptions. For a 95% interval, use \(t_{0.975,19}=2.093\).
The log-scale margin is \(2.093(0.0600)=0.1256\). The log interval is \(-0.3857\pm0.1256=[-0.5113,-0.2601]\). Exponentiating gives an interval of approximately \([0.600,0.771]\).
The entire interval is below the illustrative lower bound of 0.80. Under the stated assumptions, the data support a GMR below that bound at the 95% confidence level. The endpoint is lower exposure in the treatment condition relative to the reference. Whether that exposure decrease has a clinical consequence is a separate question; this comparison alone does not answer it.
Diagnostics, Alternatives, and Common Mistakes
Before interpreting the interval, check that the ratio’s numerator and denominator are stated in the correct order, that both exposure measurements are positive, and that the interval was constructed on a scale appropriate for a ratio. On the log scale, inspect the paired differences for strong skewness or influential observations, especially in a small sample. Confirm that the reported standard error and degrees of freedom correspond to the actual design and model. In a crossover analysis, also check that period, sequence, and carryover considerations are addressed as planned. A well-calculated interval cannot repair a mismatched endpoint, a reversed contrast, or a design problem.
The simple paired t interval is one approach, not the only one. An appropriate mixed-effects or other model may account for design factors and estimate an adjusted log contrast. A bootstrap interval can be useful when its resampling scheme respects the unit of independence and study design. These approaches may produce different intervals; the chosen method should be justified before examining which one gives a more favorable comparison with the bounds. A point estimate alone is less informative about precision, while a p-value for testing a ratio of 1 answers a different question from whether the interval is contained within clinical bounds.
- Mistake: “The point estimate is close to one, so the interaction is negligible.” A point estimate does not show precision. Check whether the full interval lies within the justified bounds.
- Mistake: “The interval includes one, so the ratio is within the no-effect range.” An interval can include one and extend beyond either bound. Compare both limits with \(L\) and \(U\).
- Mistake: “The interval crosses a bound, so there is no effect.” Crossing means the interval includes values on both sides of the boundary; it is inconclusive relative to that boundary at the stated confidence level, not evidence that the effect is absent.
- Mistake: “A ratio beyond a bound proves clinical harm.” The comparison describes exposure relative to the specified boundary. Clinical consequences require context and evidence beyond the ratio alone.
- Mistake: “A 95% interval gives a 95% probability that the parameter is inside.” The confidence level describes the long-run coverage of the procedure under its assumptions, not a probability assigned to a fixed parameter after observing the data.
Check Your Understanding
For each question, use the stated illustrative bounds of 0.80 and 1.25 unless another bound is specified. Solutions follow the questions.
- A 95% GMR interval is \([0.91,1.18]\). What does its placement establish relative to the illustrative no-effect range, and what does it not establish?
- A 95% interval is \([1.19,1.34]\). Does it lie entirely within the range, cross a bound, or lie entirely beyond a bound? State the relevant comparison.
- Why is “the point estimate is 1.01, so there is no important interaction” not a sufficient interpretation when the interval is \([0.72,1.38]\)?
- A 95% interval excludes 1 but is \([1.03,1.12]\). How should it be described relative to the illustrative no-effect range?
- Suppose an AUC ratio interval lies below the AUC lower bound. Can the same conclusion automatically be made about Cmax?
Solutions. (1) The entire interval is within \([0.80,1.25]\), so it is compatible with the stated range at this confidence level. It does not prove that the true ratio equals 1 or determine clinical consequences. (2) The interval crosses the upper bound 1.25: its lower limit is below 1.25 and its upper limit is above it, so it is not wholly within or wholly beyond the range. (3) The wide interval includes ratios below 0.80 and above 1.25; the near-one estimate does not provide precision relative to the bounds. (4) The interval excludes 1, but lies wholly within \([0.80,1.25]\). It indicates a ratio above 1 under the interval procedure while remaining inside the illustrative no-effect range. (5) No. AUC and Cmax are distinct endpoints; compare each interval with bounds specified for that endpoint.