Why Parameterization Audits Matter
A model can run successfully and produce plausible estimates while still answering a different scientific question from the one described in its report. A parameter may be named as clearance in the output but represent a rate constant in the code; a value may be expressed per minute in one place and per hour in another; or an estimate on a log scale may be reported as though it were on the original scale. These mismatches can change the scientific meaning without producing an obvious software error.
An independent parameterization audit follows each important quantity across four representations: its scientific definition, its mathematical role in the model, its implementation in code, and its label and interpretation in output. The earlier material on checking structural equations against model code concerns whether the equations are translated correctly. Here the emphasis is whether the parameters used in those equations are defined, scaled, transformed, and reported consistently.
The basic assumptions of the checks below are explicit: the scientific definition and units are stated, the code expression can be inspected, and the output value can be linked to the estimated code parameter. Dimensional analysis can reveal inconsistencies, but it cannot establish that the chosen model is scientifically adequate or that the data support precise estimation. Those are separate questions.
A Traceable Parameter Map
For every parameter selected for review, record the symbol and meaning in the specification, the code variable and equation in which it appears, its units and scale, and the corresponding output field. Include derived quantities separately from parameters estimated directly. A value calculated from other estimates is not interchangeable with an independently estimated value just because both have the same name.
A useful first check is dimensional analysis. In a valid equation, terms that are added must have the same units, and the exponent of an exponential must be dimensionless. For example, a rate constant \(k\) with units \(\mathrm{h}^{-1}\) can multiply time \(t\) in hours to form the dimensionless product \(kt\). A value with units \(\mathrm{L/h}\) cannot replace \(k\) in that product unless another quantity supplies the necessary units.
Consider a one-compartment elimination model in which concentration declines exponentially. If \(C_0\) is a concentration and \(t\) is elapsed time, the elimination-rate form is \(C(t)=C_0\exp(-kt)\). The clearance–volume form uses clearance \(CL\) and volume \(V\), related by \(k=CL/V\). These forms are equivalent when the parameters and time use consistent units:
The equivalence follows by substituting \(k=CL/V\) into the rate-constant form. Then \(-kt=-(CL/V)t\), so the exponential terms, and therefore the predicted concentrations for the same \(C_0\) and \(t\), are identical. This algebraic equivalence is a useful audit check; it does not mean that the parameter labels or numerical estimates can be copied between forms without conversion.
Worked Example: Checking Clearance and Rate-Constant Forms
Suppose a hypothetical model specification defines \(CL=5.00\ \mathrm{L/h}\) and \(V=50.0\ \mathrm{L}\). The code is described as using \(k\) in \(C(t)=C_0\exp(-kt)\), while a report presents a rate estimate of \(0.100\ \mathrm{h}^{-1}\). Check whether the reported rate is consistent with the stated clearance and volume.
Calculate the rate implied by the stated parameters.
The units reduce to inverse hours, as required for a rate multiplying time in hours. The calculation agrees with the reported \(0.100\ \mathrm{h}^{-1}\).
Check the model expression. At \(t=8.00\ \mathrm{h}\), the exponent from the reported rate is \(-(0.100\ \mathrm{h}^{-1})(8.00\ \mathrm{h})=-0.800\). The clearance–volume expression gives \(-[(5.00\ \mathrm{L/h})/(50.0\ \mathrm{L})](8.00\ \mathrm{h})=-0.800\) as well. Both produce the same concentration fraction, \(\exp(-0.800)=0.4493\) of \(C_0\).
The parameterization is internally consistent for these quantities. The audit record should still state whether the code estimates \(CL\) and \(V\) and derives \(k\), or estimates \(k\) directly. That distinction matters when reporting the estimates and their meanings.
Units and Transformations in Code and Output
Unit mismatches often arise when time is represented differently in input data and model code. A rate value does not have a complete interpretation without its time unit. If code multiplies a rate by time, inspect the units on both operands rather than relying on variable names such as RATE, K, or TIME. The numerical value of a rate changes when its time unit changes, even though the physical rate is the same.
Worked Example: Detecting a Time-Unit Mismatch
In a hypothetical model, elapsed time is supplied in minutes and code uses \(\exp(-kt)\). The code parameter is \(k=0.0200\ \mathrm{min}^{-1}\), but the output table labels it \(0.0200\ \mathrm{h}^{-1}\). Determine whether the table label is consistent with the code.
Convert the code value to inverse hours. One hour contains 60 minutes, so a rate of \(0.0200\ \mathrm{min}^{-1}\) corresponds to:
Thus the code value is \(1.20\ \mathrm{h}^{-1}\), not \(0.0200\ \mathrm{h}^{-1}\). Conversely, \(0.0200\ \mathrm{h}^{-1}\) would equal \(0.0200/60=0.0003333\ \mathrm{min}^{-1}\).
Check the effect over one hour. With time in minutes, the code exponent at 60 minutes is \(-(0.0200\ \mathrm{min}^{-1})(60\ \mathrm{min})=-1.20\), giving a concentration fraction of \(\exp(-1.20)=0.3010\). If the reported value \(0.0200\ \mathrm{h}^{-1}\) were instead used with time in hours, the exponent at one hour would be \(-0.0200\), giving \(\exp(-0.0200)=0.9802\). These are materially different model implications. The output's number and unit cannot both correctly describe the code parameter; the definition, code time scale, and output conversion must be reconciled before interpretation.
Transformations require a similar distinction between the parameter used for estimation and the scientific quantity reported. Suppose the code estimates \(\eta=\log(CL/(1\ \mathrm{L/h}))\), where \(\log\) is the natural logarithm. The ratio inside the logarithm is dimensionless. The original-scale clearance is then \(CL=(1\ \mathrm{L/h})\exp(\eta)\). If code instead logs a numerical clearance after expressing it in a stated reference unit, the report must use that same convention when converting back.
Worked Example: Back-Transforming a Log-Scale Estimate
Suppose a hypothetical output gives \(\hat{\eta}=1.3863\) for \(\eta=\log(CL/(1\ \mathrm{L/h}))\). A report lists \(1.3863\ \mathrm{L/h}\) as the clearance estimate. Check the reported value and compute the original-scale estimate.
The estimated code parameter is on the log scale, so its numerical value is not itself a clearance. Back-transform it:
The report's value \(1.3863\ \mathrm{L/h}\) is therefore mislabeled if it is intended to report clearance. It is the estimated log-scale parameter's numerical value; the corresponding clearance estimate is approximately \(4.000\ \mathrm{L/h}\). A clear output can show both, with different labels and units.
If an interval is reported for \(\eta\), transform both endpoints to the clearance scale using the same expression. For example, an interval from 1.00 to 1.80 on the log scale maps to \(\exp(1.00)\) to \(\exp(1.80)\ \mathrm{L/h}\), or approximately 2.718 to 6.050 \(\mathrm{L/h}\). Simply attaching \(\mathrm{L/h}\) to the log-scale endpoints does not convert them.
A Practical Review Sequence
A disciplined audit makes discrepancies easier to locate and explain. Keep the exact code expression, units, and output label together so a reviewer can reproduce each conversion. The following sequence is suitable for a parameter or a small group of related parameters.
State what the quantity represents, its units, and whether it is directly estimated or derived. Specify the relevant time, mass, volume, or concentration units.
Find its declaration, transformation, and use in the model equation. Check that any products, ratios, sums, and exponential arguments have compatible dimensions.
Convert units and back-transform values independently, retaining enough digits to identify rounding as distinct from a substantive mismatch.
Check the output name, scale, and unit against the code quantity. Confirm that the scientific sentence describes that quantity rather than a related derived parameter.
Document the evidence, calculation, discrepancy if any, and correction needed. Do not silently infer an intended unit or transformation when the project records do not establish it.
Comparing Parameterizations and Diagnosing Failures
Different parameterizations can represent the same model predictions while producing output values with different meanings. For example, estimating \(CL\) and \(V\) and calculating \(k=CL/V\) differs from estimating \(k\) and \(V\) and calculating \(CL=kV\). The transformation is straightforward, but the directly estimated parameters, units, and output labels must be identified before estimates can be compared.
| Representation | Parameter units | Audit question |
|---|---|---|
| Clearance and volume: \(CL, V\) | \(\mathrm{L/h}, \mathrm{L}\) | Does the code use their ratio when a rate is required? |
| Rate and volume: \(k, V\) | \(\mathrm{h}^{-1}, \mathrm{L}\) | Is clearance reported as the derived value \(kV\), if needed? |
| Log clearance: \(\eta=\log(CL/(1\ \mathrm{L/h}))\) | Dimensionless | Is the reported clearance obtained by the specified exponential back-transformation? |
An audit can also flag likely failure patterns: a dimensioned quantity inside a logarithm without a stated reference unit; a code variable multiplied by time but lacking inverse-time units; an output table that changes units without showing a conversion; or a reported quantity whose value matches a transformed parameter rather than the defined scientific parameter. A large discrepancy is not explained by rounding. Recalculate the transformation, check the units and scale in the source, and identify precisely where the mapping stops agreeing.
A common mistake is to treat a familiar parameter name as proof of a familiar definition: “The output says clearance, so this must be clearance.” The correction is to verify the code equation and units. If the code parameter is \(k\) in \(\mathrm{h}^{-1}\), labeling its numerical estimate as clearance is not justified merely because clearance is related by \(CL=kV\). That relationship requires a volume value and the stated parameterization.
No single representation is universally preferable. A direct scientific parameterization may make estimates easier to communicate; a transformed or derived parameterization may be required by the code's mathematical form. The audit objective is not to force one choice but to ensure that the chosen representation is explicit and the conversions are correct. This review alone does not establish whether the estimates are precise, whether a model is identifiable, or whether dosing records and other model components were implemented correctly. It establishes whether the parameter values being interpreted are the values the model actually defines.
Check Your Understanding
Answer each question, then compare with the solutions.
- A model uses \(k=0.0500\ \mathrm{h}^{-1}\). What is the equivalent value in \(\mathrm{min}^{-1}\)?
- Code uses \(k=CL/V\), with \(CL\) in \(\mathrm{L/h}\) and \(V\) in \(\mathrm{L}\). What are the units of \(k\), and why?
- An estimate is \(\hat{\eta}=0.6931\), where \(\eta=\log(CL/(1\ \mathrm{L/h}))\). What is the corresponding clearance?
- Why is an output value labeled “clearance” not sufficient evidence that the code estimates clearance directly?
Solutions. (1) Divide by 60: \(0.0500/60=0.0008333\ \mathrm{min}^{-1}\). (2) \((\mathrm{L/h})/\mathrm{L}=\mathrm{h}^{-1}\), so the rate can multiply time in hours to form a dimensionless exponent. (3) \(CL=(1\ \mathrm{L/h})\exp(0.6931)\approx2.000\ \mathrm{L/h}\). (4) The code may estimate a related rate or transformed quantity and derive clearance, or the output label may be wrong. Inspect the code definition and conversion, then verify the units and arithmetic before interpreting the output.