Why Categorizing Renal Function Deserves Scrutiny
A renal-function measure is continuous: two patients can have values that differ slightly, substantially, or anywhere in between. Replacing those values with labels such as “low,” “moderate,” and “normal” may seem to simplify a population pharmacokinetic model. But the simplification changes what the model can represent. It treats people within a category as equivalent for that covariate and can treat people just across a boundary as different.
That may be appropriate if the scientific or clinical effect truly changes in steps at specified boundaries. Otherwise, categories can obscure gradual relationships, make estimates depend heavily on the chosen cut points, and leave within-category differences unexplained. The earlier material on continuous renal-function effects provides the context for comparing those approaches; here the focus is what is lost or altered by categorization.
Clinical categories can be useful for describing a population or applying a decision rule. Commonly encountered eGFR bands include values below 30, from 30 to below 60, from 60 to below 90, and 90 or above, with eGFR conventionally reported in mL/min/1.73 m². Such boundaries are not automatically the right categories for a PK model. The renal-function metric, units, population, intended use, and scientific rationale all matter. In particular, eGFR indexed to body surface area and an absolute clearance measure are not interchangeable without considering their definitions and units.
What the Categories Remove
Once \(R\) has been converted to \(C\), the original value within a category is no longer available to that covariate term. A model that assigns one renal effect to everyone with \(R\) from 30 to below 60 cannot represent a gradual difference between \(R=32\) and \(R=58\), unless other model terms recover some of that variation. It also cannot express a smooth change as a patient moves from one value to another within that band.
This loss can be stated directly for a simple model. Suppose a measured outcome \(Y\), such as an individual clearance-related quantity, follows \(Y=\beta_0+\beta_1R+\varepsilon\), where \(E(\varepsilon\mid R)=0\). If the category model predicts the mean outcome within each category, its prediction is \(E(Y\mid C)=\beta_0+\beta_1E(R\mid C)\). The expected squared prediction error after categorization is therefore:
To see why, subtract the category prediction from the model: \(Y-E(Y\mid C)=\beta_1[R-E(R\mid C)]+\varepsilon\). Squaring and taking expectations gives a renal-function term \(\beta_1^2 E\{[R-E(R\mid C)]^2\}\), the error term \(E(\varepsilon^2)\), and a cross-product. The cross-product has expectation zero under the stated conditional-mean assumption, and the squared deviation within categories has expectation \(E\{\operatorname{Var}(R\mid C)\}\). Thus, when \(\beta_1\ne 0\), variation in \(R\) within categories contributes prediction error that a model using the actual continuous values would not incur under this simple correctly specified relationship. This result does not say categorization always produces biased estimates; it shows why category-based predictions can be less informative when the effect is gradual.
Worked Example: Quantifying Within-Category Information Loss
Consider four hypothetical patients with renal-function values \(R\) in mL/min. Suppose their clearance values follow the exact relationship \(CL=2+0.04R\), with \(CL\) in L/h. For illustration, use two categories: \(R<60\) and \(R\ge 60\).
| Patient | \(R\) (mL/min) | \(CL=2+0.04R\) (L/h) | Category prediction (L/h) |
|---|---|---|---|
| A | 45 | 3.80 | 4.00 |
| B | 55 | 4.20 | 4.00 |
| C | 65 | 4.60 | 4.80 |
| D | 75 | 5.00 | 4.80 |
For the \(R<60\) category, the mean renal-function value is \((45+55)/2=50\) mL/min. Its predicted clearance is \(2+0.04(50)=4.00\) L/h. For \(R\ge60\), the mean is \((65+75)/2=70\) mL/min, giving \(2+0.04(70)=4.80\) L/h. The category predictions leave residuals of \(-0.20\), \(0.20\), \(-0.20\), and \(0.20\) L/h, respectively. The mean squared residual is \((0.04+0.04+0.04+0.04)/4=0.0400\) (L/h)\(^2\). The continuous relationship predicts all four values exactly in this constructed example.
The category model does preserve a difference between the two groups: its predicted contrast is \(4.80-4.00=0.80\) L/h. But it cannot distinguish patients A and B from each other on the basis of renal function, nor C from D. The example isolates information loss; real data also contain residual variability and estimation uncertainty.
Threshold Artifacts and Boundary Dependence
A category model can create a step in predicted clearance at a cut point. If the underlying effect is gradual, that step is a consequence of the coding rather than evidence that physiology changes abruptly at that exact value. Conversely, patients on opposite sides of a boundary may receive different predictions despite nearly identical measured renal function. A different choice of cut points can change who belongs to each group, the estimated group sizes, and the estimated contrasts.
Worked Example: A Prediction Jump at an eGFR Boundary
Suppose a hypothetical model uses two eGFR categories, below 60 and at least 60 mL/min/1.73 m². It estimates a typical clearance of 3.00 L/h for the lower category and 3.70 L/h for the higher category, with all other covariates held fixed. These numbers are invented for illustration, not clinical estimates.
A patient with eGFR 59.9 mL/min/1.73 m² receives the prediction 3.00 L/h, while a patient with eGFR 60.0 receives 3.70 L/h. The model-implied difference is \(3.70-3.00=0.70\) L/h for a 0.1 mL/min/1.73 m² difference in eGFR. That jump is built into the category rule; it does not establish that a real clearance change of 0.70 L/h occurs over that small renal-function difference.
Now move the boundary from 60 to 55 mL/min/1.73 m². A patient at 57 would change category even though the measured value has not changed. The fitted contrast could also change because the patients contributing to each group have changed. If the model estimates separate category effects, both the assignments and the estimated effects may therefore depend on the cut point. This is especially consequential when few patients lie near or within a category.
Measurements themselves are also imperfect. If repeated estimates vary around a threshold, a patient can move between categories because of measurement variability, even when the underlying renal function has not changed meaningfully. The category representation then turns a small change in the measured value into a full change in the coded covariate. This is a reason to examine values near boundaries, not a reason to assume that every change in renal function is measurement error.
How Categorization Can Distort a Covariate Estimate
A category coefficient answers a different question from a continuous renal-function coefficient. For example, a model might represent a lower category with an indicator \(I(R<60)\), equal to 1 for patients below 60 and 0 otherwise:
Here \(CL_{\mathrm{ref}}\) is the typical clearance for the reference category, \(R_i\) is the renal-function value for patient \(i\), and \(\theta\) is the log ratio of typical clearance in the lower category to that in the reference category. The category ratio is \(\exp(\theta)\). This coefficient does not describe the change in clearance per 1 mL/min of renal function. In a model with multiple categories, each coefficient similarly represents a contrast with its specified reference category.
A further risk arises when category membership is associated with another predictor of clearance. If that other covariate is omitted or inadequately represented, a category contrast can partly reflect its distribution rather than a renal effect alone. The following arithmetic illustrates that point. It is a small constructed example, not a general estimator formula for nonlinear mixed-effects models.
Worked Example: An Age Imbalance Changes the Crude Category Contrast
Suppose four hypothetical patients follow the same clearance relationship, \(CL=1+0.04R-0.02(A-50)\), where \(R\) is in mL/min, age \(A\) is in years, and \(CL\) is in L/h. The true renal term in this specified relationship is 0.04 L/h per mL/min, while clearance also decreases by 0.02 L/h per year of age above the reference age of 50.
| Patient | \(R\) (mL/min) | Age (years) | Clearance calculation (L/h) | Clearance (L/h) |
|---|---|---|---|---|
| A | 45 | 70 | \(1+0.04(45)-0.02(70-50)\) | 2.40 |
| B | 55 | 60 | \(1+0.04(55)-0.02(60-50)\) | 3.00 |
| C | 65 | 40 | \(1+0.04(65)-0.02(40-50)\) | 3.80 |
| D | 75 | 30 | \(1+0.04(75)-0.02(30-50)\) | 4.40 |
With a cut point at 60 mL/min, the lower category has mean \(R=(45+55)/2=50\) mL/min, mean age \((70+60)/2=65\) years, and mean clearance \((2.40+3.00)/2=2.70\) L/h. The higher category has mean \(R=70\) mL/min, mean age 35 years, and mean clearance 4.10 L/h. The crude category contrast is \(4.10-2.70=1.40\) L/h.
If that contrast is divided by the 20 mL/min difference in category means, the resulting descriptive slope is \(1.40/20=0.0700\) L/h per mL/min, rather than the true renal slope of 0.0400 in the constructed relationship. The age distributions differ by 30 years, so the crude contrast combines renal and age differences. This calculation does not show that every category estimate is biased by exactly this amount; it shows why a category contrast cannot automatically be attributed to renal function alone. A model that adjusts for age may address this particular imbalance, but categorization still loses within-category renal variation.
Diagnostics, Alternatives, and Limitations
Before relying on a categorical renal covariate, check whether the data and the scientific question support the coding. Useful diagnostics include:
- Show the observed values, not only the labels. Plot or tabulate the continuous renal-function measure against the relevant PK outcome and indicate the proposed boundaries. Look for a gradual trend, an apparent step, or substantial variation hidden within categories.
- Check category counts and coverage. Report the number of patients and the renal-function range in each category. Sparse groups can yield unstable contrasts; a category spanning a wide range can conceal important within-group variation.
- Inspect values near each cut point. Compare covariate values and model predictions just below and above a boundary. Ask whether the implied jump is scientifically credible and whether modest measurement variation would change assignments.
- Compare reasonable specifications. Where scientifically and statistically supportable, compare the prespecified category model with a suitable continuous form. Examine whether the conclusion depends strongly on the boundaries or on a small number of observations.
- Check balance in other covariates. Compare age, body size, and other relevant predictors across categories. Marked differences do not prove confounding, but they warn that an unadjusted category contrast may combine multiple effects.
A continuous linear or power model retains the order and spacing of measured values and can express a gradual relationship under its chosen form. A flexible continuous form can allow curvature if the data support it, although additional flexibility also requires adequate data and careful assessment. A categorical model is simpler to describe and can be defensible when the scientific question is explicitly about groups, when a genuine step-like effect is supported, or when an externally justified decision boundary is central to the analysis. Even then, the boundary and reference group should be stated, and the categorical contrast should be interpreted as a group contrast.
No approach is guaranteed to be correct solely because it is continuous or categorical. Continuous models can be misspecified, and a category model may be useful for a prespecified clinical comparison. Categorization is particularly limiting when the sample is small, category counts are uneven, boundaries are selected after inspecting outcomes, or the estimated renal relationship changes smoothly across the observed range. A cut point chosen to maximize an apparent difference can exaggerate that difference and make it less reproducible. The choice should be justified by the question and supported by the observed data, not treated as a neutral data-processing step.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A model uses one predicted clearance for all patients with renal function from 30 to below 60 mL/min. What information about renal function has been removed from this covariate term?
- A categorical model predicts 2.8 L/h just below an eGFR boundary and 3.4 L/h just above it. What does the 0.6 L/h difference represent, and what does it not establish?
- In a hypothetical sample, a lower renal-function category is older on average than the reference category. Why might the crude clearance contrast fail to represent only the renal effect?
- Give one circumstance in which a categorical renal-function model could be reasonable, and one diagnostic you would still perform.
Solutions. (1) The model no longer uses the ordering and exact spacing of values within 30 to below 60; patients in that category receive the same category-based renal contribution. (2) It is the model’s predicted contrast between the two categories at the boundary. It does not establish a real abrupt physiological change of that size for an arbitrarily small difference in eGFR. (3) Age may also predict clearance, so the group contrast can combine age and renal differences unless the model appropriately accounts for age. (4) A category can be reasonable for a prespecified, scientifically justified group comparison or a supported step-like effect. One should still inspect category counts and ranges, assess predictions near the boundary, or compare with a suitable continuous specification.