Why Results and Interpretation Need Separate Jobs
A model report has to do two things: state what the analysis produced and explain what those findings mean in context. These tasks are connected, but they are not interchangeable. A numerical result should be traceable to an estimate, diagnostic, or comparison. An interpretation should explain the relevance of that result while staying within what the analysis supports.
When the two are blended, a reader may not be able to tell whether a sentence reports an observed finding or offers an explanation. For example, “The model captured the concentration-time profile well because the elimination process was correctly specified” combines a broad evaluation with a causal-sounding explanation. Even if some diagnostics look satisfactory, they do not by themselves establish that a particular structural assumption is correct.
A useful report makes this relationship visible: first give the result with enough detail to evaluate it, then interpret it. “The estimated typical clearance was 5.0 L/h (bootstrap 95% interval 4.2–5.9 L/h)” is a result. “The estimate is reasonably precise for this analysis, although the interval still includes values differing by roughly 1 L/h from the point estimate” is an interpretation. The second sentence does not claim that the parameter has a 95% probability of lying in the interval.
The earlier material on coherent model evaluation provides the context for deciding which findings matter. Here the emphasis is narrower: report each finding in a way that lets reviewers distinguish evidence from explanation and assess whether the explanation follows from the evidence.
A Practical Sequence for Reporting
A compact reporting sequence is to identify the analysis or quantity, state the numerical finding, describe the pattern relevant to the question, and then offer a bounded interpretation. Include denominators, units, reference values, and uncertainty when they are needed to understand the finding. Avoid giving a conclusion first and leaving the reader to infer which numbers support it.
Specify which model estimate, diagnostic, or validation summary is being reported and the population or data subset it concerns.
Report the value, units, uncertainty or denominator as appropriate, and the direction or pattern without evaluative adjectives standing in for evidence.
Explain what the result supports, identify important qualifications, and do not present an explanation as if it were directly measured.
This sequence is not a requirement to put every result in a separate paragraph. Tables and figures can carry detailed results, while the text highlights the findings needed to follow the evaluation. The key is that the text identifies which displayed numbers support the interpretation; a table alone does not explain their relevance.
Worked Example: Summarizing Goodness-of-Fit Findings
Hypothetical analysis. A population PK model was evaluated using 312 concentration observations from 84 participants. The report summarizes observed-to-individual-prediction ratios and a visual predictive check divided into 10 time bins. The median observed-to-predicted ratio was 1.02; 281 of 312 observations (90.1%) were within a factor of two of their individual predictions. In the predictive check, the observed median lay within the simulated 90% interval in 8 of 10 bins. In the other two bins, both at later sampling times, the observed median was above the interval.
Results. The median ratio was 1.02, close to 1. Of the 312 observations, \(312-281=31\) were outside the factor-of-two range, or \(31/312=0.0994\), approximately 9.94%. The observed median was within the simulated interval in 8 of 10 time bins, with both exceptions at later times and above the interval.
Interpretation. The ratio summary and most time-bin medians are broadly consistent with predictions in this evaluation. However, the two late-time departures indicate a pattern worth noting rather than a uniform fit across time. These summaries do not, by themselves, establish why the later medians are high or prove that the model is adequate for every intended use.
The interpretation is tied to specific findings: the proportion within a stated range and the location of exceptions. “The model fitted well” would discard the late-time pattern, while “the model systematically underpredicted late concentrations” would be too strong based only on two bin medians. Reporting the count, denominator, and location makes the qualification reviewable.
Deriving an Interpretation From a Covariate Estimate
A coefficient in a model is not always directly interpretable as a percentage change. In a power model, the coefficient becomes an exponent, so a change in the covariate must be translated through the model equation. Showing this translation prevents a common reporting error: describing the coefficient itself as though it were the predicted ratio.
The ratio follows by dividing the predicted clearance at \(X\) by the predicted clearance at \(X_{\mathrm{ref}}\). At the reference value, the covariate multiplier is 1. Dividing cancels the reference clearance, leaving the ratio. Multiplying that ratio by \(100\%\) gives the predicted percentage of reference clearance; subtracting it from \(100\%\) gives the predicted percentage reduction when the ratio is below 1.
Worked Example: Reporting a Clearance Covariate Effect
Hypothetical model result. Typical clearance was modeled as \(CL = 5.0(eGFR/90)^{0.36}\) L/h, with eGFR measured in the model’s stated units and 90 as its reference value. The estimated exponent was 0.36, with a bootstrap 95% interval from 0.10 to 0.62. Consider the model prediction at eGFR 45, holding other covariates at their reference values.
Result. The covariate ratio is \((45/90)^{0.36}=0.5^{0.36}=0.779\) (to three significant figures). The predicted clearance is \(5.0 \times 0.779=3.90\) L/h. Thus, at eGFR 45, the model predicts clearance equal to about 77.9% of the reference prediction, or about \(100\%-77.9\%=22.1\%\) lower.
The interval for the exponent can also be translated to ratios at this covariate value. The lower exponent gives \(0.5^{0.62}=0.651\), and the upper exponent gives \(0.5^{0.10}=0.933\). Because the ratio decreases as the positive exponent increases when the base is below 1, the corresponding ratio interval is approximately 0.651 to 0.933. This is a transformed interval based on the reported exponent interval, not a separate bootstrap interval calculated directly from each replicate.
Interpretation. Under the fitted power relationship, the model predicts lower typical clearance at eGFR 45 than at the reference eGFR of 90. The estimate describes a model-based association; it does not establish that changing an individual’s eGFR would cause the predicted change in clearance. The translated interval also indicates uncertainty in the magnitude of the predicted contrast.
A report that says “clearance decreases by 36% for each halving of eGFR” would be incorrect: 0.36 is the exponent, not the percentage reduction for a halving. The model-based ratio calculation gives about a 22.1% reduction for this comparison.
Reporting Predictive-Check Results Without Overclaiming
Predictive-check summaries invite interpretation because they compare observed data with a distribution generated under a model. Their meaning depends on what was simulated, what was summarized, and how many observations or groups contribute. A nominal 90% prediction interval is a feature of the predictive procedure; a small observed sample will not necessarily place exactly 90% of observations inside it.
Worked Example: A Small External Prediction Check
Hypothetical analysis. A model was used to predict one prespecified trough concentration for each of 20 new participants. The participants were not used to fit the model, and the reported prediction intervals were calculated before comparing predictions with the observed troughs. Eighteen observed troughs fell within their respective 90% prediction intervals. The two outside values were both above their upper limits.
Result. The observed coverage was \(18/20=0.90\), or 90%. Two of 20 observations (10%) were outside the intervals, and both departures were above rather than below the predicted ranges.
Interpretation. The overall proportion inside the intervals matches the nominal 90% level in this small check, while the direction of both departures suggests that the few misses were not balanced around the predictions. With only 20 participants, this result is limited evidence about predictive performance; it does not establish exact calibration or show that the model will achieve 90% coverage in another population. The direction and context of the misses should be reported alongside the overall proportion.
This example assumes the 20 participants contribute independent prespecified summaries and that the intervals were defined consistently before examining the observed values. If several correlated concentrations per participant were counted as separate independent successes, the simple proportion could exaggerate how much independent information the check contains. The unit of analysis therefore belongs in the result, not only in a methods description.
Selective Emphasis, Alternatives, and Failure Modes
Selective emphasis occurs when a report highlights favorable results while omitting relevant contrary evidence, or presents a precise-looking number without the context needed to judge it. A single summary can be accurate and still create a misleading impression if the choice of summary hides an important pattern. Reporting both the overall result and material exceptions is usually more informative than choosing one as the headline.
| Reporting approach | What it provides | Risk if used alone |
|---|---|---|
| Numerical results in a table | Traceable estimates, units, and uncertainty in a compact form | Readers may not know which entries matter to the evaluation |
| Results plus interpretation in prose | A concise account of relevance and direction | Interpretation can outrun or obscure the displayed evidence |
| Figure with a brief results description | Patterns across time, groups, or ranges | A broad claim may hide exceptions visible only in the figure |
These approaches complement one another. A table is useful for exact values; a figure can reveal patterns that a single statistic compresses; prose can explain why a result matters. The report should make clear which evidence is being interpreted, rather than treating any one format as a substitute for the others.
Several warning signs suggest that the distinction has been lost:
- Conclusion before evidence. “The model is robust” is not a numerical result. State the relevant evaluation findings before drawing that conclusion.
- Selective denominator. “Ninety percent were within range” is hard to assess without the count, denominator, and unit being summarized.
- Unstated reference. A covariate effect needs its reference value and comparison; an exponent alone does not specify the predicted change for every contrast.
- Explanation presented as observation. A discrepancy may be consistent with a hypothesis about model structure, but a diagnostic pattern alone does not prove that explanation.
- Only favorable summaries reported. If exceptions affect interpretation, report their direction and location rather than letting an overall percentage conceal them.
Separating results from interpretation improves transparency, but it does not repair an unsuitable model, incomplete data, or an evaluation that does not address the question. Nor does a neatly worded interpretation make a weak result stronger. The method’s value is that it gives reviewers a clear path from numerical evidence to the conclusion and makes unsupported leaps easier to detect.
Check Your Understanding
For each question, identify the result and keep the interpretation within the evidence. Solutions follow the questions.
- A model predicts clearance of 6.0 L/h at a reference covariate value and 4.8 L/h at a second value. What is the predicted ratio and percentage difference? What can be concluded without further evidence?
- In a hypothetical check, 45 of 50 observations fall within their prediction intervals, and all five outside observations are below the lower limits. State the numerical result and one appropriately bounded interpretation.
- A report states, “The model was unbiased because 80% of observations were within a factor of two of predictions.” Identify the reporting problem and rewrite the claim using a result and a cautious interpretation.
- A covariate exponent is 0.25 in a power model. Is it correct to report that a halving of the covariate reduces the predicted parameter by 25%? Explain how the comparison should be calculated.
Solutions. (1) The ratio is \(4.8/6.0=0.800\), so the second prediction is 80.0% of the reference prediction, or 20.0% lower. This is a model-predicted contrast; without more information it does not establish a causal effect or performance in another population. (2) The observed proportion is \(45/50=0.900\), or 90%; five observations (10%) were outside, all below their lower limits. This suggests that the misses in this check were directionally low, but the result alone does not establish systematic underprediction generally. (3) The proportion within a factor of two does not establish absence of bias; it summarizes distance from predictions and does not show direction. A more careful statement is: “Forty of 50 observations (80%) were within a factor of two of their predictions; this summary describes the size of the deviations but does not by itself establish whether predictions were unbiased.” (4) No. For a halving, the predicted ratio is \(0.5^{0.25}=0.841\), approximately 84.1% of the reference prediction, or about 15.9% lower. The exponent must be applied to the covariate ratio; it is not itself the percentage change for that comparison.