From Model Output to a Decision-Ready Table
A model output is not yet a regulatory result. It may show a treatment contrast on the log scale, its standard error, degrees of freedom, and a confidence interval. A reader still needs to know which endpoint was analyzed, which population contributed data, what acceptance limits apply, and whether that endpoint meets its criterion. A clear results table makes those connections visible without hiding how the decision was reached.
This tutorial concerns the reporting step: translating a specified analysis output into a transparent table. The conversion does not validate the underlying model or change the prespecified estimand. Before preparing a table, verify that the output corresponds to the intended endpoint, treatment contrast, analysis population, and regulatory rule. The worked examples use hypothetical outputs and conventional limits solely to illustrate the reporting process.
What the Table Must Preserve
For an endpoint analyzed on the natural-log scale, let \(\hat{\theta}\) be the estimated treatment contrast, defined here as test minus reference on that scale. Let \(SE\) denote its standard error and \(df\) its degrees of freedom. When the stated model supports a \(t\)-based interval, the two-sided 90% confidence interval on the log scale is
Exponentiating the estimate gives the test-to-reference geometric mean ratio, \(\exp(\hat{\theta})\). Exponentiating both confidence limits gives the ratio-scale interval. When the prespecified rule uses ratio limits \([L,U]\), the endpoint meets the interval criterion if and only if the entire ratio-scale confidence interval is within those limits: its lower limit is at least \(L\), and its upper limit is at most \(U\). A point estimate inside the limits is not sufficient.
These calculations rely on the analysis being appropriate for the specified endpoint and design. In particular, the treatment contrast, standard error, and degrees of freedom must come from the intended model and analysis population; the interval method must be the one prespecified or justified for that analysis; and the endpoint must have been transformed and parameterized as stated. The table cannot repair a mismatched estimand, an incorrect analysis set, or an unsuitable model. For endpoints with specialized criteria, such as replicate-design or narrow-therapeutic-index analyses, report the applicable endpoint-specific rule rather than automatically substituting conventional ratio limits.
A concise table typically includes the endpoint, treatment estimate, confidence interval, acceptance limits, analysis population, and decision. State whether estimates and intervals are ratios or percentages, and make the confidence level explicit. A population label should be specific enough to identify the analysis set, such as “PK-evaluable subjects with valid endpoint data,” and should agree with the analysis actually used. If the protocol uses a different set or defines additional criteria, report that accurately rather than relying on a generic label.
Worked Examples: Calculating and Reporting Endpoint Results
Worked Example: AUC Endpoint Meets Its Limits
Suppose a hypothetical analysis of log-transformed AUC reports a test-minus-reference estimate of \(0.018\), \(SE=0.030\), and \(df=28\). Assume the specified analysis supports a \(t\)-based 90% confidence interval, and the endpoint uses illustrative ratio limits of 0.80 to 1.25. The analysis population is the protocol-defined PK-evaluable set with valid AUC data.
For a 90% two-sided interval, the critical value is \(t_{0.95,28}\approx1.701\). The margin is \(1.701(0.030)=0.05103\). Thus, the log-scale interval is \(0.018\pm0.05103=[-0.03303,0.06903]\).
The estimated ratio is \(\exp(0.018)=1.018\). Exponentiating the interval limits gives \([\exp(-0.03303),\exp(0.06903)]=[0.9675,1.0715]\), to four significant figures. Both limits are within 0.80 to 1.25, so this endpoint meets the stated interval criterion.
| Endpoint | Test/reference estimate | 90% CI | Acceptance limits | Analysis population | Endpoint decision |
|---|---|---|---|---|---|
| AUC | 1.018 | 0.9675 to 1.0715 | 0.80 to 1.25 | PK-evaluable subjects with valid AUC data | Meets limits |
The table states the ratio scale and confidence level in its headers, so the numbers are not mistaken for log-scale estimates. In a submission, the endpoint name should also match the prespecified AUC definition, such as AUC over a specified interval.
Worked Example: Cmax Endpoint Does Not Meet Its Limits
Consider a separate hypothetical Cmax analysis with log-scale treatment estimate \(0.190\), \(SE=0.050\), and \(df=24\). Assume a valid \(t\)-based 90% interval and illustrative acceptance limits of 0.80 to 1.25. The analysis population is the protocol-defined PK-evaluable set with valid Cmax data.
Here \(t_{0.95,24}\approx1.711\), so the margin is \(1.711(0.050)=0.08555\). The log-scale interval is \(0.190\pm0.08555=[0.10445,0.27555]\).
The estimated ratio is \(\exp(0.190)=1.209\). The exponentiated interval is \([\exp(0.10445),\exp(0.27555)]=[1.110,1.317]\), to four significant figures. The upper limit, 1.317, exceeds 1.25. Therefore, despite the point estimate being within the acceptance limits, this endpoint does not meet the interval criterion.
| Endpoint | Test/reference estimate | 90% CI | Acceptance limits | Analysis population | Endpoint decision |
|---|---|---|---|---|---|
| Cmax | 1.209 | 1.110 to 1.317 | 0.80 to 1.25 | PK-evaluable subjects with valid Cmax data | Does not meet limits; upper CI limit exceeds 1.25 |
Stating why the criterion is not met is more informative than recording only “No.” It also prevents the estimate of 1.209 from being mistaken for the decision statistic.
Worked Example: Checking the Scale and Rounding for a Third Endpoint
Suppose a hypothetical endpoint has \(\hat{\theta}=-0.040\), \(SE=0.035\), and \(df=32\). Assume the same interval method and illustrative 0.80 to 1.25 limits. The relevant critical value is \(t_{0.95,32}\approx1.694\). The margin is \(1.694(0.035)=0.05929\), giving the log interval \([-0.040-0.05929,-0.040+0.05929]=[-0.09929,0.01929]\).
The ratio estimate is \(\exp(-0.040)=0.9608\). The ratio interval is \([\exp(-0.09929),\exp(0.01929)]=[0.9055,1.019]\). Since 0.9055 is above 0.80 and 1.019 is below 1.25, this endpoint meets the stated limits.
For reporting, use enough digits to preserve the comparison with the boundary, and use a consistent rounding rule across table rows. Here the displayed upper bound, 1.019, is clearly below 1.25. If a result is very close to a limit, retain additional digits in the underlying comparison and avoid rounding a value across the boundary; document the decision using the unrounded calculation.
Why the Conversion Works—and How to Audit It
The confidence interval is formed on the scale on which the model contrast and its uncertainty are defined. Under the stated \(t\)-based procedure, a two-sided 90% interval uses the 0.95 quantile because 5% probability is allocated to each tail. Exponentiation is a strictly increasing function, so it preserves the ordering of the lower and upper limits while converting a difference of log means into a ratio of geometric means.
Check endpoint, contrast direction, scale, model, analysis population, standard error, degrees of freedom, and confidence level against the analysis specification.
For the stated \(t\)-based interval, calculate \(\hat{\theta}\pm t_{0.95,df}SE\) for a two-sided 90% interval.
Exponentiate the estimate and both interval limits when reporting a ratio; do not exponentiate only the estimate or mix log-scale and ratio-scale values.
Compare both unrounded interval limits with the applicable acceptance limits. Record the analysis population and state the endpoint-specific outcome and, when useful, which boundary determines it.
This procedure assumes the confidence interval and acceptance limits are defined on compatible scales. For example, a ratio interval must be compared with ratio limits, not with log-transformed limits. It also assumes that the reported interval corresponds to the relevant BE criterion. Some analyses use different confidence levels, limits, or variance-based rules; their table entries must reflect those specific procedures rather than the illustrative calculation above.
A table is useful alongside, not instead of, a fuller account of the analysis. A narrative or analysis report may be needed to describe the model, protocol deviations, missing observations, exclusions, or sensitivity analyses. The table’s role is to make the central endpoint results quickly comparable while retaining enough information to trace each decision to its population and rule.
Diagnostics, Alternatives, and Common Pitfalls
Audit each row by recalculating the interval from the reported estimate, \(SE\), and \(df\), then checking that exponentiation reproduces the stated ratio interval. Check that the treatment contrast is consistently test/reference; reversing it changes the ratio to its reciprocal and requires corresponding transformation of the limits. Confirm that each endpoint’s population and acceptance limits are correct, especially when endpoints differ in analysis set or regulatory rule.
A compact table of ratios and confidence intervals is generally easier to scan than a long paragraph of model coefficients. However, reporting only ratio-scale results can conceal how they were derived. Where traceability is important, include a separate model-output table or supplementary columns for the log contrast, standard error, and degrees of freedom. Conversely, putting every model diagnostic in the main results table can make its decision columns difficult to use. Keep the central table focused and provide technical details in a clearly linked analysis summary.
- Using the wrong critical value. A 90% two-sided interval uses the 0.95 quantile, not the 0.975 quantile. The latter produces a wider, 95% two-sided interval. Verify both the confidence level and degrees of freedom.
- Comparing only the point estimate. A point estimate within the acceptance limits does not establish that the interval is within them. Apply the rule to both confidence limits.
- Mixing scales. Do not compare a log-scale interval with ratio limits, or place a log-scale estimate under a ratio estimate heading. Exponentiate all three quantities—estimate and both limits—for ratio-scale reporting.
- Omitting the population. “All subjects” or “evaluable” may be ambiguous. Identify the analysis set precisely and make sure it matches the model output.
- Reporting a bare pass/fail label. State the endpoint-specific outcome and, if a limit is missed, identify which boundary was exceeded. Keep rounding from obscuring a borderline comparison.
The central reporting principle is that a result is interpretable only when the endpoint, scale, interval, limits, population, and decision are aligned. A polished table cannot compensate for a wrong model output or an incorrect rule, but a carefully checked table can expose inconsistencies before they propagate into a regulatory summary.
Check Your Understanding
Work each question, then compare your reasoning with the solutions below.
- A hypothetical log-scale estimate is \(0.020\), with \(SE=0.025\) and \(df=18\). The two-sided confidence level is 90%, and \(t_{0.95,18}\approx1.734\). Calculate the log interval and ratio estimate and interval.
- A ratio estimate is 1.06 and its 90% confidence interval is 0.79 to 1.18. With acceptance limits of 0.80 to 1.25, does the endpoint meet the interval criterion? Why?
- Why must a table identify the analysis population rather than list only the endpoint and confidence interval?
- A report gives a ratio estimate of 0.95 but shows log-scale confidence limits under a column labeled “90% CI, test/reference ratio.” What is wrong, and how should it be corrected?
Solutions. (1) The margin is \(1.734(0.025)=0.04335\), so the log interval is \(0.020\pm0.04335=[-0.02335,0.06335]\). The ratio estimate is \(\exp(0.020)=1.020\); the ratio interval is \([\exp(-0.02335),\exp(0.06335)]\approx[0.9769,1.065]\). (2) No. Although the upper limit is below 1.25, the lower limit 0.79 is below 0.80, so the full interval is not contained within the acceptance limits. (3) The population identifies which observations contributed to the estimate and uncertainty; an unlabeled or inaccurate population can make the table inconsistent with the specified analysis and prevent an informed review. (4) The column label and values are on different scales. Exponentiate both log-scale limits and report those ratio limits alongside the ratio estimate, or relabel the column and report the log-scale estimate and limits consistently.