Why the Parameter Table Matters
A final model equation specifies how parameters enter the model, but readers also need to find the estimated values without reconstructing them from prose or software output. A clear parameter estimate table links each model quantity to its meaning, symbol, unit, estimate, and a concise measure of estimation uncertainty. It should be possible to read a row and understand both what was estimated and on what scale.
This tutorial focuses on the table for fixed-effect or typical parameter estimates. It does not use that table to describe between-individual variability or residual error; those are distinct model components. Nor should a table substitute for the final model equations: its symbols and parameter definitions must agree with those equations.
Designing the Rows and Columns
A useful table is compact, but it should not make the reader guess what a number represents. For a typical parameter, a practical set of columns is: parameter name, model symbol, estimate, unit, and uncertainty. Depending on the reporting convention, uncertainty may occupy one or more columns for standard error (SE), relative standard error (RSE), or a confidence interval (CI). The column headings must identify which quantity is shown; a bare column labelled “Error” is not sufficient.
Use symbols consistent with the final model equations. For example, distinguish apparent oral parameters \(CL/F\) and \(V/F\) from \(CL\) and \(V\). State the parameter meaning in words at least once, and give units that match the quantity: clearance has units of volume per time, volume has units of volume, and a first-order rate constant has units of inverse time. If a parameter is dimensionless, say so rather than leaving its unit blank.
Consider a hypothetical one-compartment oral model with first-order absorption and apparent parameters. A table might report typical \(CL/F\), \(V/F\), and \(k_a\). The model equations, parameter symbols, and table entries must all refer to the same parameterization. If the equation defines \(k_a\) in \(h^{-1}\), reporting its estimate in \(day^{-1}\) without an explicit conversion invites misinterpretation.
Worked Example: Building a Table for Typical Values
Suppose a hypothetical final model estimates typical apparent clearance of \(4.80\) L/h, apparent volume of \(52.0\) L, and absorption rate constant of \(1.25\) \(h^{-1}\). The corresponding SEs are \(0.42\) L/h, \(5.8\) L, and \(0.19\) \(h^{-1}\). Construct a table that also reports RSE and 95% CIs.
Relative standard errors. For a positive estimate \(\hat{\theta}\), define \(RSE=100(SE/\hat{\theta})\%\). Thus, for apparent clearance, \(100(0.42/4.80)=8.75\%\). For apparent volume, \(100(5.8/52.0)=11.2\%\). For the absorption rate constant, \(100(0.19/1.25)=15.2\%\).
Confidence intervals. For this hypothetical example, suppose the intervals were calculated on the log scale, using log-scale standard errors of \(0.0875\), \(0.1115\), and \(0.152\), respectively, and then exponentiated. For clearance, the calculation is \(\exp[\ln(4.80)\pm1.96(0.0875)]\), which gives approximately \(4.04\) to \(5.70\) L/h. Applying the same calculation gives \(41.8\) to \(64.7\) L for apparent volume, and \(0.928\) to \(1.68\) \(h^{-1}\) for \(k_a\).
| Parameter | Symbol | Typical estimate | Unit | SE | RSE | 95% CI |
|---|---|---|---|---|---|---|
| Apparent clearance | \(CL/F\) | 4.80 | L/h | 0.42 | 8.75% | 4.04–5.70 |
| Apparent volume | \(V/F\) | 52.0 | L | 5.8 | 11.2% | 41.8–64.7 |
| Absorption rate constant | \(k_a\) | 1.25 | \(h^{-1}\) | 0.19 | 15.2% | 0.928–1.68 |
The intervals are on the same original scale and in the same units as their estimates. Their asymmetry around the point estimates is expected when a symmetric interval is formed on the log scale and then transformed back. In a real report, identify the method used to obtain intervals rather than implying that all CIs use this construction.
What the Uncertainty Columns Mean
The standard error describes the estimated sampling variability of an estimator under the stated model and estimation procedure. It has the same units as the estimate when both are reported on the original scale. RSE expresses that SE relative to the estimate and is useful for comparing the relative precision of estimates with different magnitudes or units. It is not a measure of between-individual variability, and a small RSE alone does not establish that a model is appropriate.
A 95% confidence interval is an interval produced by a procedure that, under its assumptions and in repeated comparable samples, covers the fixed parameter 95% of the time. It is not correct to say that there is a 95% probability that the fixed parameter lies in this particular interval. A confidence interval also does not summarize all sources of uncertainty in a model or in predictions.
When estimates are obtained on a transformed scale, state that fact. For example, if a positive parameter is estimated as \(\beta=\ln(\theta)\), the reported original-scale estimate is \(\exp(\hat{\beta})\). A standard error for \(\hat{\beta}\) is on the log scale, not in the original parameter’s units. To form a log-scale Wald interval and return it to the original scale, use:
This construction follows because the interval on the log scale is centered at \(\hat{\beta}\) with half-width \(1.96SE_{\hat{\beta}}\), and exponentiation is increasing, so it preserves the interval’s ordering. The approximate 95% coverage relies on the sampling distribution of \(\hat{\beta}\) being adequately approximated by a normal distribution. If another method, such as a profile-likelihood interval, was used, name that method instead.
Worked Example: Reporting an Estimate From the Log Scale
A hypothetical model estimates \(\beta=\ln(CL/F)=1.569\), with \(SE_{\beta}=0.0875\). Report the original-scale estimate and a log-scale Wald 95% CI.
Back-transform the estimate: \(\exp(1.569)=4.80\) L/h. The reported estimate is therefore \(CL/F=4.80\) L/h, not \(1.569\) L/h.
Calculate the interval: on the log scale, the endpoints are \(1.569-1.96(0.0875)=1.398\) and \(1.569+1.96(0.0875)=1.740\). Exponentiating gives \(\exp(1.398)=4.05\) and \(\exp(1.740)=5.70\). Report the interval as approximately \(4.05\)–\(5.70\) L/h, and label it as a log-scale Wald interval transformed to the original scale.
The table should not place \(0.0875\) under a column labelled “SE (L/h)”: it is the SE of the log estimate and is dimensionless. If the table includes it, the heading should say “SE on log scale.” Alternatively, report an original-scale SE if that quantity has been calculated and clearly identified.
Consistency Checks Before Reporting
A table can look polished and still be misleading if its entries conflict with the model equations or with one another. Before finalizing it, check each parameter’s identity, scale, unit, numerical precision, and uncertainty label. A useful review sequence is:
Confirm that each table symbol and parameter name corresponds exactly to the parameter in the final model equations. Distinguish apparent from non-apparent quantities where relevant.
Verify that the units follow from the parameter’s meaning and model equation. For example, a rate constant has units of inverse time, not time.
Mark an SE as original-scale or transformed-scale, and state the interval method when it is not self-evident.
Check RSE arithmetic and confirm that interval endpoints use the stated scale and method. Round only after calculations are complete.
Ask whether a reader can identify the typical estimate, its unit, and the meaning of each uncertainty entry without consulting unlabelled software output.
Worked Example: Converting Units Consistently
A hypothetical software output reports a typical \(CL/F\) estimate of \(115\) L/day, an SE of \(10.1\) L/day, and a 95% CI of \(97.0\)–\(136.8\) L/day. The report’s other parameters use hours. Convert all entries to L/h.
There are 24 hours in a day, so divide the estimate, SE, and both interval endpoints by 24: \(115/24=4.79\) L/h; \(10.1/24=0.421\) L/h; \(97.0/24=4.04\) L/h; and \(136.8/24=5.70\) L/h.
The table row should therefore show \(CL/F=4.79\) L/h, SE \(=0.421\) L/h, and 95% CI \(4.04\)–\(5.70\) L/h. Converting only the estimate while leaving the SE or interval in L/day would create a row with incompatible units. A positive linear unit conversion changes the numerical values but does not change the RSE.
Choosing a Reporting Format and Recognizing Failure Modes
A table is generally clearer than prose when several parameters must be compared: columns align units and uncertainty measures, while rows make omissions visible. A model equation is better for showing how parameters combine in the model, but it can be difficult to scan for estimates and their precision. Prose is useful for highlighting a result that matters to the interpretation, but it is a poor substitute for a complete, consistently labelled set of estimates. These formats complement one another rather than compete: the equation defines the model, the table organizes estimates, and prose can draw attention to the most relevant implications.
Common signs that a table needs revision include mixed time units, unlabeled transformed-scale SEs, confidence intervals that do not match the stated method, inconsistent symbols, and precision that suggests more certainty than the estimates support. Excessive decimal places are not extra information. Conversely, rounding an estimate and interval too aggressively can obscure useful distinctions. Choose precision that reflects the estimates and uncertainty, and apply it consistently while retaining enough digits to avoid material discrepancies caused by rounding.
A particularly important pitfall is to read a narrow interval as proof that the model is correct. It describes uncertainty in a parameter estimate under the estimation procedure and model assumptions; it does not by itself diagnose model fit, establish clinical importance, or capture every limitation in the analysis. The estimate table should therefore be read alongside the model specification and relevant evaluation results.
Check Your Understanding
Answer each question, then compare your reasoning with the short solutions below.
- A typical volume estimate is \(38.0\) L with SE \(4.0\) L. What is its RSE?
- A table reports an estimate of \(0.70\) \(h^{-1}\) and an SE of \(0.12\) \(h^{-1}\). Is the SE on the same scale and in compatible units?
- Why should an estimate of \(\ln(CL/F)\) not be entered directly into a table column labelled \(CL/F\) in L/h?
- What does a 95% confidence interval mean under its assumptions?
Solutions. (1) \(100(4.0/38.0)=10.5\%\). (2) Yes: both are on the original scale and have units of inverse hours. (3) The log estimate is on a transformed, dimensionless scale; exponentiate it to obtain the estimate of \(CL/F\) on its original scale, and report any transformed-scale uncertainty with an explicit label. (4) The interval-producing procedure covers the fixed parameter in 95% of repeated comparable samples under its assumptions; it does not assign a 95% probability to the parameter being in the particular calculated interval.