From a fitted model to interpretable results
A completed Monolix run does not produce one result that answers every question. Population estimates describe the fitted population model; individual estimates describe model-based values for particular subjects; uncertainty summaries describe precision under specified approximations; and exported tables make results available for inspection or later analysis. Confusing these layers can lead to a plausible-looking but incorrect report—for example, presenting a subject-specific clearance as if it were the population typical clearance.
This tutorial uses a hypothetical one-compartment model with first-order elimination. The earlier material on one-compartment models and interindividual random effects provides the modeling context. The aim here is to read the results and exports in relation to that model, not to re-estimate it.
For a single dose in a one-compartment model, clearance and volume determine the elimination rate constant \(k_i=CL_i/V_i\). This relationship helps check units: if clearance is in L/h and volume is in L, \(k_i\) is in h\(^{-1}\). It does not mean that a result table necessarily reports \(k_i\); the model specification and column labels determine which quantities are actually present.
Population estimates: what the summary represents
The population results summarize estimated model parameters. Depending on the model, these may include typical values such as \(CL_{\mathrm{pop}}\) and \(V_{\mathrm{pop}}\), variability parameters describing the random effects, their correlations, and residual-error parameters. Read every estimate together with its parameter name, parameterization, and unit. An estimate labelled as a standard deviation is not a variance, and a variance is not a coefficient of variation.
Monolix displays results through its interface and makes results available through exports in the run's result directory. Exact labels, columns, and file organization can vary with the software version, run settings, and model. Use the model definition and the headers in the actual result or export to resolve ambiguity; do not infer meaning from a filename alone.
Worked Example: Reading Typical Clearance and Volume
Suppose a hypothetical Monolix population summary reports \(CL_{\mathrm{pop}}=5.00\) L/h and \(V_{\mathrm{pop}}=40.0\) L. The model uses the exponential random-effects form above. What do these numbers imply for the typical elimination rate constant?
The typical values correspond to zero random effects. Thus, for a subject with \(\eta_{CL,i}=0\) and \(\eta_{V,i}=0\), clearance is 5.00 L/h and volume is 40.0 L. The rate constant is $$ k_{\mathrm{typ}}=\frac{CL_{\mathrm{pop}}}{V_{\mathrm{pop}}} =\frac{5.00\ \mathrm{L/h}}{40.0\ \mathrm{L}} =0.125\ \mathrm{h}^{-1}. $$
The units reduce as expected: \((\mathrm{L/h})/\mathrm{L}=\mathrm{h}^{-1}\). The result is a derived typical rate constant, not a claim that every subject has this rate constant. In particular, the ratio of typical clearance to typical volume need not equal the arithmetic mean of the individual rate constants.
Uncertainty summaries and their interpretation
A standard error (SE) summarizes estimated sampling variability of an estimator under the fitted model and the method's assumptions. A relative standard error (RSE) expresses the standard error relative to the estimate, commonly as a percentage: $$ \mathrm{RSE}(\%)=100\frac{SE}{|\mathrm{estimate}|}. $$ An RSE is a scale-free precision summary, not an estimate of between-subject variability. For parameters near zero, it may be very large or unstable, so it should not be interpreted in isolation.
When a standard error is available, a simple approximate 95% Wald interval on the reported parameter scale is estimate \(\pm 1.96SE\). This uses a large-sample normal approximation for the estimator. If the parameter is constrained to be positive, a symmetric interval on the natural scale can extend below zero; a transformation-based interval may be more appropriate when the estimate is obtained on a transformed scale. Check which scale and interval method the output reports.
For many estimation workflows, uncertainty summaries rely on a local approximation to the likelihood around the estimate, with the information matrix used to approximate estimator covariance. Such summaries depend on the model being adequately identified and the local approximation being informative. Treat a missing SE or a warning about a singular or ill-conditioned information matrix as a substantive limitation, not as an invitation to copy the point estimate without qualification.
Worked Example: Calculating RSE and a Wald Interval
A hypothetical results summary reports \(CL_{\mathrm{pop}}=5.00\) L/h and \(SE=0.40\) L/h. Calculate the RSE and the approximate 95% Wald interval.
The RSE is $$ 100\frac{0.40\ \mathrm{L/h}}{|5.00\ \mathrm{L/h}|} =8.00\%. $$ The approximate interval is $$ 5.00\ \mathrm{L/h}\ \pm\ 1.96(0.40\ \mathrm{L/h}) =5.00\ \mathrm{L/h}\ \pm\ 0.784\ \mathrm{L/h} =(4.216,\ 5.784)\ \mathrm{L/h}. $$
Under the approximation used to construct this interval, it is a procedure that would cover the fixed population clearance in about 95% of repeated samples. It is not a statement that there is a 95% probability that this particular fixed parameter lies in this calculated interval. The RSE indicates a standard error equal to 8.00% of the estimate; it does not measure the spread of clearances among subjects.
Individual estimates: conditional on the population model
An individual estimate is calculated using both the subject's data and the estimated population model. It is therefore not simply a directly observed subject characteristic, nor is it generally an independently estimated parameter based only on that subject's observations. In a nonlinear mixed-effects model, sparse or noisy individual data can leave the estimate strongly influenced by the population distribution.
The result may be expressed as an estimated individual parameter, such as \(CL_i\), or as an estimated random effect, such as \(\eta_{CL,i}\). With the exponential parameterization, they are related by $$ \widehat{CL}_i=CL_{\mathrm{pop}}\exp(\widehat{\eta}_{CL,i}). $$ An individual-parameter export may contain one row per subject and columns for model parameters. Other exports may contain random-effect estimates or predictions. Inspect the documentation and column names to identify which quantity is provided. Do not exponentiate a column that already contains \(CL_i\).
Worked Example: Relating an Individual Estimate to the Population Value
Suppose the hypothetical population estimate is \(CL_{\mathrm{pop}}=5.00\) L/h, and a subject's exported random-effect estimate is \(\widehat{\eta}_{CL,i}=-0.223\). Calculate the corresponding individual clearance under the stated model.
Substituting the values gives $$ \widehat{CL}_i =5.00\ \mathrm{L/h}\times \exp(-0.223) \approx 5.00\ \mathrm{L/h}\times 0.800 =4.00\ \mathrm{L/h}. $$
The estimated clearance is about 4.00 L/h, below the typical value of 5.00 L/h. This is a model-based individual estimate, not proof that the subject's true clearance is exactly 4.00 L/h. Its interpretation depends on the fitted population distribution, the subject's data, and the estimation method. If the export instead reported an individual clearance of 4.00 L/h directly, applying the exponential transformation again would be an error.
Reading exported tables safely
Exports are useful because they allow results to be checked, summarized, or passed into another step of an analysis. Their convenience also makes silent mismatches possible. Before using a table, identify the run and model it came from; inspect the headers and units; check whether there is one row per subject or one row per observation; and confirm how subject identifiers are represented. Identifiers can be stored as text or numbers, and importing software may alter their format.
A population-parameter export and an individual-parameter export answer different questions. A prediction or observation-level export may have repeated rows for a subject because it records multiple times or observations. Joining such a file to a subject-level table requires the correct key and, where needed, additional variables such as time or event. A successful software join does not itself establish that the rows were matched correctly.
Worked Example: Checking a Subject-Level Export
A hypothetical individual-parameter export contains these rows, with clearance in L/h:
| ID | CL estimate (L/h) | V estimate (L) |
|---|---|---|
| 101 | 4.00 | 36.0 |
| 102 | 5.60 | 42.0 |
The population summary reports \(CL_{\mathrm{pop}}=5.00\) L/h and \(V_{\mathrm{pop}}=40.0\) L. Check whether the individual clearances could correspond to the stated exponential model by converting each estimate to its implied random effect: $$ \widehat{\eta}_{CL,i}=\ln\left(\frac{\widehat{CL}_i}{CL_{\mathrm{pop}}}\right). $$ For ID 101, $$ \ln\left(\frac{4.00}{5.00}\right)=\ln(0.800)\approx -0.223. $$ For ID 102, $$ \ln\left(\frac{5.60}{5.00}\right)=\ln(1.12)\approx 0.113. $$
Both are finite and have the expected interpretation relative to the typical value: ID 101 is below it, and ID 102 is above it. This check does not establish that the export is correct, but it can expose a unit mismatch or a mistaken assumption about what a column contains. For example, if clearance were in mL/min rather than L/h, direct comparisons with the population summary would be invalid until units were converted.
Diagnostics, alternatives, and limitations
Read the results as a linked set rather than as isolated numbers. Compare point estimates with their units and parameterization; review the uncertainty summaries and any estimation or information-matrix warnings; then examine whether individual estimates and exports have the expected structure. Patterns such as implausible magnitudes, missing subjects, duplicate identifiers, unexpected numbers of rows, or values that contradict the model's transformations deserve investigation before results are reported.
The main alternatives for inspecting a fit serve complementary purposes. The Monolix results interface is convenient for viewing summaries and model-specific plots. Exported tables support explicit checks and reproducible downstream calculations, but require careful handling of keys, units, and file provenance. A saved report or run record can preserve context and settings for review. None substitutes for the others: a tidy export cannot establish that the model was appropriate, and a graphical summary may not reveal a subtle row-matching error. The earlier material on reading NONMEM estimation results provides a related software-specific perspective; the names and organization of outputs should not be assumed to match across tools.
Population estimates and their uncertainty are conditional on the model and estimation procedure. Individual estimates additionally depend on each subject's information and the population distribution. Consequently, a list of individual estimates should not be read as error-free measurements, and small numerical differences between runs should not automatically be treated as meaningful. When results are to be used in another model or software workflow, first establish exactly what each exported value represents; the subsequent transfer step depends on that distinction.
A practical reading sequence
Confirm which model, data, and estimation run produced the displayed results and exports.
Record each estimate with its parameterization and unit; distinguish typical values, random-effects parameters, and residual-error parameters.
Interpret SE and RSE in context, check the scale of any interval, and note missing uncertainty summaries or information-matrix warnings.
Determine whether the output contains random effects, transformed individual parameters, or another quantity before comparing subjects.
Check identifiers, row structure, units, missing values, and consistency with the model equations and the displayed summary.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A model reports \(CL_{\mathrm{pop}}=6.0\) L/h. Is this necessarily the arithmetic mean clearance of the fitted subjects?
- An estimate is 8.0 L/h with \(SE=1.0\) L/h. Calculate its RSE and approximate 95% Wald interval.
- Under the exponential clearance model, what does \(\widehat{\eta}_{CL,i}=0\) imply about \(\widehat{CL}_i\)?
- An export has several rows per subject. What should you establish before joining it to a table with one row per subject?
- Why should an individual clearance estimate not be described as a directly observed, error-free subject value?
Solutions. (1) No. It is the typical value at zero random effect; with the exponential model, the arithmetic mean can differ from the typical value. (2) \(\mathrm{RSE}=100(1.0/8.0)=12.5\%\). The approximate interval is \(8.0\pm1.96(1.0)=(6.04,\ 9.96)\) L/h. (3) Since \(\exp(0)=1\), the individual estimate equals the population typical value: \(\widehat{CL}_i=CL_{\mathrm{pop}}\). (4) Establish what each row represents and identify the correct matching key, including any necessary observation-level variables such as time. (5) It is estimated using the subject's data and the population model, so it is conditional on both and has uncertainty.