Why Check the Predicted Clearance Trajectory?
A pediatric clearance model is intended to describe how clearance changes across age, not merely to produce a plausible value at one age. A prediction can agree well with observations overall while systematically overpredicting in infants and underpredicting in adolescents. Checking the trajectory therefore requires examining prediction errors against age as well as summarizing their overall magnitude.
The comparison is meaningful only when predictions and observations refer to the same clearance quantity. For example, absolute clearance in L/h should be compared with observed absolute clearance in L/h; clearance normalized by body weight should be compared with observations normalized in the same way. The population, age definition, dosing and sampling context, and clearance phenotype must also be aligned. If observations are apparent oral clearance, \(CL/F\), they should not be treated as absolute systemic clearance unless bioavailability is appropriately accounted for.
These checks assume observed clearance estimates are positive, predictions and observations are paired appropriately, and age and units are recorded consistently. If each observation represents a different child, each prediction should use that child’s relevant characteristics when individual-level comparison is intended. If only age-group summaries are available, compare group-level predictions with group-level observations and do not imply that individual errors were assessed. Repeated observations from the same child are not independent replicates; they should be identified as repeated measurements when summarizing or modeling errors.
Quantifying Overall Error
Several summaries answer different questions. Mean prediction error, \(\bar{q}\), describes signed bias in the clearance units used: positive values indicate average overprediction and negative values average underprediction. Mean absolute error summarizes typical error magnitude in those units without allowing positive and negative errors to cancel. Root mean squared error also summarizes error magnitude, but gives relatively greater influence to large errors.
Relative prediction errors express discrepancies as percentages of observed clearance, which can aid comparison across children whose clearances differ greatly. Their arithmetic mean is useful as a descriptive summary, but can be strongly affected by observations with small denominators. Log prediction ratios are often convenient for positive clearance values: a ratio of 1 means exact agreement, a ratio above 1 means overprediction, and a ratio below 1 means underprediction.
The mean log prediction ratio has a direct multiplicative interpretation. If \(\bar{e}\) is the arithmetic mean of the log ratios, then \(\exp(\bar{e})\) is the geometric mean of the prediction-to-observation ratios. A value of 1.10 means the model’s predictions are, on a multiplicative average, 10% above observed clearance. This measure gives reciprocal discrepancies equal and opposite log errors: predicting twice the observation gives \(\ln(2)\), while predicting half gives \(\ln(0.5)=-\ln(2)\).
This multiplicative interpretation follows because the exponential of a sum of logarithms is a product: \(\exp(\bar{e})=\left\{\prod_{i=1}^{n}(CL_{\mathrm{pred},i}/CL_{\mathrm{obs},i})\right\}^{1/n}\). Thus it is the geometric mean ratio, rather than the arithmetic mean of the raw clearances. None of these single summaries indicates whether errors vary systematically with age; that requires age-stratified summaries and plots.
Worked Example: Finding Age-Related Bias Despite a Small Overall Error
Suppose a hypothetical validation set has one observed and predicted absolute clearance value at each of four ages. All clearances are in L/h.
| Age | Observed | Predicted | Relative prediction error |
|---|---|---|---|
| 0.5 y | 0.30 | 0.36 | 20% |
| 2 y | 0.60 | 0.66 | 10% |
| 8 y | 1.20 | 1.08 | -10% |
| 15 y | 2.00 | 1.60 | -20% |
For the 0.5-year-old, \(q=0.36-0.30=0.06\) L/h, and \(100q/CL_{\mathrm{obs}}=100(0.06/0.30)=20\%\). The other relative errors are \(100(0.06/0.60)=10\%\), \(100(-0.12/1.20)=-10\%\), and \(100(-0.40/2.00)=-20\%\).
The arithmetic mean relative error is \((20+10-10-20)/4=0\%\). That summary might suggest no overall relative bias. But the two younger observations are overpredicted, while the two older observations are underpredicted. The age pattern is scientifically important even though the signed percentage errors cancel.
The mean log ratio is \(\{\ln(1.20)+\ln(1.10)+\ln(0.90)+\ln(0.80)\}/4\), approximately \(-0.01272\). Exponentiating gives \(\exp(-0.01272)\approx0.9874\), a geometric mean prediction-to-observation ratio of about 0.987. This is close to 1, but does not remove the age-related pattern. With only four hypothetical observations, the pattern is a diagnostic signal to investigate, not a precise estimate of performance across all children.
Inspecting Errors Across Age
A useful basic diagnostic is a plot of residuals or prediction errors against age. State which sign convention is being plotted: with \(q=CL_{\mathrm{pred}}-CL_{\mathrm{obs}}\), positive values mean overprediction; with residual \(r=CL_{\mathrm{obs}}-CL_{\mathrm{pred}}\), positive values mean underprediction. A satisfactory pattern has errors distributed around zero without a clear age trend, after accounting for the intended prediction target. A smooth drift, a change in sign across age, or a cluster of large errors in a specific age band suggests that overall summaries may be concealing a weakness.
Stratify errors using clinically and physiologically meaningful age bands, while retaining exact age in plots where possible. Within each band, report the number of observations and a signed and magnitude-based summary, such as mean signed error and MAE. Bands with very few observations have unstable summaries; report their counts and avoid treating an apparent difference as definitive. Age is also related to size and developmental capacity. A pattern with age can point to a scaling mismatch, but can also reflect a different mix of body sizes, maturation states, or data quality across ages. The error pattern alone does not identify its cause.
For a log-ratio plot, \(e_i=0\) represents exact agreement, positive values overprediction, and negative values underprediction. A horizontal reference at zero helps reveal systematic departures. Plots of predicted against observed clearance can also show agreement across the range, but high correlation alone is not evidence of accurate predictions: predictions can track the observed ranking while remaining consistently too high or too low.
Worked Example: Calculating Overall Error Summaries
Consider a second hypothetical data set of four matched observations, again in L/h. The observed clearances are 0.40, 0.80, 1.50, and 2.40; the corresponding predictions are 0.50, 0.72, 1.80, and 2.16.
The prediction errors are \(0.10,-0.08,0.30,-0.24\) L/h. Therefore, the mean signed error is \((0.10-0.08+0.30-0.24)/4=0.020\) L/h, indicating modest average overprediction in these four observations. The MAE is \((0.10+0.08+0.30+0.24)/4=0.180\) L/h.
For RMSE, the squared errors sum to \(0.10^2+(-0.08)^2+0.30^2+(-0.24)^2=0.164\). Thus, \(RMSE=\sqrt{0.164/4}=\sqrt{0.041}=0.2025\) L/h. RMSE exceeds MAE because squaring gives the larger discrepancies more influence.
The relative errors are \(25\%,-10\%,20\%,-10\%\), with mean \((25-10+20-10)/4=6.25\%\). This percentage summary and the signed error in L/h answer different questions: the former scales each discrepancy by its observed value, while the latter retains the original clearance scale. Reporting the metric and its units or percentage basis makes the result interpretable.
Comparing Age-Specific Performance
Overall summaries should be paired with age-specific results. A simple table can show age band, number of observations, mean signed error, mean relative error, and MAE. If prediction ratios are summarized, report the chosen summary clearly—for example, the geometric mean prediction-to-observation ratio. These summaries should be viewed alongside the individual points; averages can still hide variation within a band.
Worked Example: Identifying an Age-Specific Error Pattern
Suppose a third hypothetical comparison gives three prediction-to-observation ratios in each of three age bands. The ratios are 1.20, 1.10, and 1.15 for infants; 0.95, 1.00, and 1.05 for older children; and 0.80, 0.85, and 0.90 for adolescents.
In the infant band, the mean ratio is \((1.20+1.10+1.15)/3=1.15\). The mean relative prediction error is therefore 15%, showing average overprediction on this arithmetic ratio scale. In the older-child band the mean ratio is \((0.95+1.00+1.05)/3=1.00\), or a mean relative error of 0%. In the adolescent band it is \((0.80+0.85+0.90)/3=0.85\), or a mean relative error of \(-15\%\), showing average underprediction.
The pattern changes from overprediction in infants to underprediction in adolescents, so a single pooled bias could mask an age-related mismatch. These small groups are illustrative only: with three observations per band, the averages are highly sensitive to individual values. The result motivates checking the underlying observations and model assumptions; it does not establish that age itself caused the discrepancy.
Choosing a Comparison and Recognizing Failure Modes
Use the error scale that matches the question. Raw errors in L/h are useful when absolute clearance discrepancies matter. Relative errors are intuitive when proportional discrepancies matter, but become unstable when observed clearance is near zero. Log ratios require positive values and are natural for multiplicative comparisons. MAE is easy to interpret as a typical absolute discrepancy; RMSE emphasizes occasional large errors. No metric is universally best, so a clear evaluation usually reports complementary signed and magnitude summaries and displays errors across age.
| Approach | What it highlights | Main caution |
|---|---|---|
| Signed error in clearance units | Direction and magnitude of average absolute bias | Opposite-sign errors can cancel |
| Relative error | Discrepancy relative to observed clearance | Small observed values can dominate |
| Log prediction ratio | Multiplicative bias with reciprocal errors treated symmetrically | Requires strictly positive values |
| Age-stratified summaries and plots | Systematic differences across the pediatric trajectory | Small or uneven age groups can be unstable |
Observed clearance estimates are not necessarily error-free measurements of a true value. Their uncertainty can arise from sparse sampling, estimation methods, or variability between individuals. When available, retain uncertainty information and avoid interpreting a small discrepancy as decisive if it is comparable to the uncertainty in the observed estimates. Also distinguish validation from calibration: if the same observations were used to tune the model, agreement on those data does not provide an independent assessment of predictive performance.
- Common mistake: relying on near-zero pooled bias. The reasoning is that over- and underpredictions cancel, so the trajectory must be adequate. The correction is to inspect age-specific errors and plots; cancellation can hide a change in error direction across age.
- Common mistake: comparing unlike clearance quantities. Comparing absolute predictions with weight-normalized observations mixes scales. Check that the clearance definition, units, and normalization match before calculating errors.
- Common mistake: treating a high correlation as agreement. Correlation can be high when all predictions are systematically biased. Examine prediction-to-observation differences or ratios in addition to association.
These diagnostics identify where performance is poor; they do not, by themselves, determine whether the explanation is allometry, maturation, covariate imbalance, observation uncertainty, or model structure. A proposed explanation should be checked against the data and assumptions rather than inferred from a residual trend alone.
Check Your Understanding
Use the definitions and sign conventions in this tutorial. Solutions are included after the questions.
- An observed clearance is 0.50 L/h and the prediction is 0.60 L/h. Calculate the prediction error and relative prediction error, and state their direction.
- Why can a mean signed prediction error of zero fail to show that pediatric scaling performs similarly at all ages?
- Predictions are twice observed values for one child and half observed values for another. What are their log prediction ratios, and why are they symmetric?
- Give one reason to report age-band sample sizes alongside age-specific error summaries.
Solutions. (1) \(q=0.60-0.50=0.10\) L/h and \(100(0.10/0.50)=20\%\); both indicate overprediction. (2) Positive and negative errors can cancel in the pooled mean even when one age range is overpredicted and another underpredicted, so errors must also be examined against age. (3) The ratios are 2 and 0.5, giving \(\ln(2)\) and \(\ln(0.5)=-\ln(2)\); reciprocal multiplicative discrepancies have equal and opposite log errors. (4) Small bands produce unstable summaries, and counts show how much information supports each reported age-specific result.