Why Functional Form Matters
A continuous covariate such as body weight, age, or renal function can be related to a typical PD parameter in several ways. The choice of functional form determines what a change in the covariate means: a constant additive change in the parameter, a constant multiplicative change per covariate unit, or a constant proportional change for a given percentage change in the covariate. These choices can produce quite different predictions, especially toward the ends of the observed range.
The aim is to describe how the typical parameter changes with the covariate. A fitted relationship does not by itself establish that the covariate causes the parameter to change. Its predictions are most directly supported where covariate values were observed; behavior outside that range depends on the form's mathematical assumptions.
Three Common Functional Forms
A linear relationship adds or subtracts the same amount from the typical parameter for each one-unit increase in the covariate. A log-linear relationship makes the logarithm of the parameter linear in the covariate, so equal covariate increments produce equal multiplicative changes. A power relationship makes the logarithm of the parameter linear in the logarithm of the covariate, so equal proportional changes in the covariate produce equal proportional changes in the parameter.
Here \(\beta\) is the covariate-effect coefficient, but its interpretation depends on the form. In the linear form, \(\beta\) has units of parameter per unit of covariate. In the log-linear form, \(\beta\) has inverse-covariate units. In the power form, \(\beta\) is dimensionless, and both \(x\) and \(x_{\mathrm{ref}}\) must be positive. For the log-linear and power forms, \(\theta_{\mathrm{ref}}\) must be positive, and predictions remain positive. A linear form does not guarantee positive predictions, which matters when \(\theta\) is constrained to be positive.
The reference value anchors the interpretation: setting \(x=x_{\mathrm{ref}}\) gives \(\theta_{\mathrm{typ}}(x)=\theta_{\mathrm{ref}}\) in each form. It is also useful to distinguish a log-linear relationship from a power relationship. The former is linear in \(x\) on the log-parameter scale; the latter is linear in \(\log(x)\) on that scale.
Worked Example: A Linear Effect of Age
Suppose a hypothetical PD model relates age to a typical baseline response. The observed ages range from 20 to 80 years. Let the typical response be 40.0 response units at age 50 years, with a linear coefficient of 0.300 response units per year. Calculate the typical response at ages 20 and 80, and interpret the coefficient.
The model is
At age 20 years, \(x-50=-30\) years, so
At age 80 years, \(x-50=30\) years, so
The coefficient means that each additional year of age corresponds to an increase of 0.300 response units in the model's typical response. The predictions remain positive across the specified observed range. This check is important: extending the same line far enough below age 50 would eventually predict a negative response, which could be scientifically or mathematically impossible for the parameter being modeled.
Worked Example: A Log-Linear Effect of Renal Function
Suppose a hypothetical model relates renal function \(x\), measured in specified renal-function units, to a positive PD parameter measured in concentration units. The observed range is 30 to 90 units. At \(x_{\mathrm{ref}}=60\), the typical parameter is 2.00 concentration units, and \(\beta=-0.00600\) per renal-function unit. Find the predictions at 30 and 90, then interpret the change for a 10-unit increase.
The model is
At \(x=30\), the exponent is \(-0.00600(30-60)=0.180\), giving
At \(x=90\), the exponent is \(-0.00600(90-60)=-0.180\), giving
For any increase of 10 units, the parameter is multiplied by \(\exp\{-0.00600(10)\}=\exp(-0.0600)\approx0.9418\). Thus, the model predicts a decrease of about \(100(1-0.9418)=5.82\%\) per 10-unit increase. This is a multiplicative change, not a constant subtraction of 5.82 concentration units. Because the coefficient is negative and the parameter is positive, the relationship decreases monotonically while remaining positive throughout the observed range.
Worked Example: A Power Effect of Body Weight
Consider a hypothetical positive PD parameter whose typical value is 100 response units at a reference weight of 70 kg. A power model has \(\beta=0.750\), and the observed weight range is 50 to 100 kg. Calculate the predictions at 56 kg, 84 kg, and both endpoints of the observed range.
The model is
At 56 kg, \(56/70=0.800\), so
At 84 kg, \(84/70=1.20\), so
The weight increase from 70 to 84 kg is 20%. The parameter multiplier is \(1.20^{0.750}\approx1.1465\), corresponding to an increase of about 14.7%. At the lower endpoint,
at the upper endpoint,
The positive exponent makes the relationship increase with weight. Its percentage response to a given percentage change in weight is constant under the model; its absolute change in response units is not.
What the Coefficients Imply
The different interpretations follow directly from comparing predictions at two covariate values. For the linear form, subtracting the predictions gives \(\theta_{\mathrm{typ}}(x_2)-\theta_{\mathrm{typ}}(x_1)=\beta(x_2-x_1)\). Thus \(\beta\) is a constant absolute change per covariate unit. For the log-linear form, dividing the predictions gives a ratio of \(\exp\{\beta(x_2-x_1)\}\). The parameter's relative change depends on the size of the covariate difference, but not on the starting value of \(x\). For the power form, the ratio is \((x_2/x_1)^\beta\), so it depends on the proportional change in \(x\), rather than its absolute difference.
The power coefficient is also the elasticity: the proportional change in the parameter per proportional change in the covariate, locally. Taking logarithms gives \(\log\theta_{\mathrm{typ}}=\log\theta_{\mathrm{ref}}+\beta\log(x/x_{\mathrm{ref}})\); differentiating with respect to \(\log(x)\) gives \(\beta\). Equivalently, the local slope on the original scale is \(\beta\theta_{\mathrm{typ}}(x)/x\). This explains why a power relationship can rise with \(x\) while its absolute slope changes over the range.
| Form | Meaning of \(\beta\) | Key behavior and constraint |
|---|---|---|
| Linear | Absolute parameter change per covariate unit | Constant slope; predictions can become negative |
| Log-linear | Change in log parameter per covariate unit | Constant multiplicative change per fixed covariate increment; requires positive \(\theta_{\mathrm{ref}}\) |
| Power | Elasticity of the parameter with respect to the covariate | Constant multiplicative change per fixed proportional covariate change; requires positive \(x\), \(x_{\mathrm{ref}}\), and \(\theta_{\mathrm{ref}}\) |
Checking Behavior and Choosing a Form
Before interpreting a fitted coefficient, inspect the predictions across the full observed covariate range, not only at the reference value. Check the signs of the slopes or coefficients against the intended direction of association, and calculate predictions at the lowest and highest observed values. For a positive parameter, confirm that the form does not imply impossible values. The log-linear and power forms preserve positivity when their required inputs are positive; a linear form may not. Positivity alone does not guarantee plausibility: a positive prediction can still be far outside a scientifically credible range.
A useful diagnostic is to compare the pattern of the data and model predictions across \(x\). Systematic curvature or a consistent pattern of underprediction in one part of the range and overprediction in another may indicate that a constant-slope linear form is inadequate. Conversely, a log-linear or power form can impose a smooth monotone relationship that is also inappropriate if the observed relationship levels off or changes direction. Visual checks should account for the fact that subject-level observations vary around typical values; an apparent pattern in a small or unevenly distributed sample may not be reliable.
No one form is universally preferable. A linear form is easy to interpret in absolute units and may be suitable over a limited range when a constant absolute change is plausible. A log-linear form is useful when equal covariate increments are expected to produce equal ratios in the parameter. A power form is natural when proportional changes in a positive covariate are expected to have a consistent proportional effect on a positive parameter. The scale, observed range, and scientific interpretation should guide the choice, alongside model fit; numerical convenience alone is not a sufficient rationale.
All three forms impose a particular trend beyond the data. A straight line continues with a constant slope, a log-linear curve continues to change by a constant ratio per covariate unit, and a power curve continues to follow the same elasticity. Therefore, predictions beyond the observed range are extrapolations, not evidence-supported extensions. A functional form that behaves reasonably over the observed range can become implausible outside it.
Common Mistakes
- Reading every coefficient as an absolute change. That interpretation applies to the linear coefficient. In a log-linear model, exponentiating \(\beta\) gives the multiplicative change for a one-unit increment; in a power model, \(\beta\) describes an elasticity.
- Confusing log-linear and power forms. A log-linear relationship uses \(x\) in the exponent, whereas a power relationship raises \(x/x_{\mathrm{ref}}\) to \(\beta\). They generally predict different patterns.
- Checking only the reference prediction. A plausible value at \(x_{\mathrm{ref}}\) does not ensure plausible predictions at the ends of the observed range. Calculate and assess endpoint predictions.
- Assuming a positive prediction is automatically reasonable. Positivity is a mathematical property of some forms, not a guarantee that the magnitude is scientifically credible.
- Treating an association as a causal effect. A covariate relationship describes model predictions conditional on \(x\); the fitted form alone does not demonstrate that changing \(x\) would cause the parameter to change.
Check Your Understanding
For each question, identify the relevant interpretation or calculate the requested prediction. Solutions follow the questions.
- A log-linear model has \(\beta=-0.0400\) per unit of \(x\). What is the predicted parameter multiplier for an increase of 20 units?
- A power model has \(\beta=0.500\). What is the predicted percentage change in the parameter when the covariate increases from 100 to 125 units?
- A linear model predicts a positive parameter at the reference value, but a negative value at the largest observed covariate value. What should be checked before using that form?
- Which form assumes that a 10% increase in a positive covariate has the same proportional effect on the parameter regardless of the covariate's starting value?
Solutions. 1. The multiplier is \(\exp\{(-0.0400)(20)\}=\exp(-0.800)\approx0.4493\). The model predicts about a 55.1% decrease over that 20-unit increase. 2. The covariate ratio is \(125/100=1.25\), so the parameter multiplier is \(1.25^{0.500}=\sqrt{1.25}\approx1.118\), an increase of about 11.8%. 3. Check whether a linear relationship is scientifically and statistically plausible across the entire observed range and whether the parameter must be positive. A negative prediction for a parameter that must be positive signals that this form is unsuitable over that range, or at least requires reconsideration. 4. The power form: its parameter ratio depends on the covariate ratio, so the same percentage increase in \(x\) gives the same multiplier under the model.