From a Log-Scale Coefficient to a Ratio
In an NLME parameter model, a covariate may enter the natural logarithm of an individual parameter. The coefficient then describes a change on the log scale, not a direct percentage change on the parameter scale. To communicate the effect in parameter units, translate the coefficient into a ratio for the covariate contrast of interest.
For example, a coefficient associated with body weight could be reported per kilogram, while a clinical question might concern a 10-kilogram difference. The coefficient alone does not answer that question until the contrast is specified and the log-scale change is exponentiated. The resulting ratio is exact under the stated model.
A ratio greater than 1 indicates a larger parameter at \(x_b\) than at \(x_a\); a ratio between 0 and 1 indicates a smaller parameter. The corresponding percentage change from \(x_a\) to \(x_b\) is \(100\{\exp[\beta(x_b-x_a)]-1\}\%\). This percentage is relative to the parameter at \(x_a\), and the direction depends on the order of the contrast.
Why the Ratio Formula Holds
The ratio follows directly from the parameter model. At \(x_b\), with the individual effect held fixed, the parameter is \(P_{\mathrm{ref}}\exp\{\beta(x_b-x_{\mathrm{ref}})+\eta_i\}\). At \(x_a\), it is \(P_{\mathrm{ref}}\exp\{\beta(x_a-x_{\mathrm{ref}})+\eta_i\}\). Dividing cancels the shared reference value and random effect:
Thus, the reference covariate value determines where the typical parameter is anchored, but it does not affect the ratio for a contrast. The coefficient is multiplied by the contrast \(x_b-x_a\) before exponentiation. Exponentiating \(\beta\) alone gives the ratio for a one-unit increase in \(x\), not necessarily for the contrast relevant to a particular question.
This result assumes that the covariate enters linearly on the log scale with a constant coefficient over the contrast being considered. It also assumes that the comparison holds the individual random effect fixed. If two different individuals have different random effects, their parameter ratio includes those differences as well as the covariate contrast. The covariate-only ratio should not be used to explain an observed difference between two people unless their random effects are accounted for.
Worked Example: A 10-Kilogram Increase in Body Weight
Suppose a hypothetical clearance model has a body-weight coefficient \(\beta=-0.0180\) per kilogram. What is the model-implied clearance ratio for a 70-kg individual compared with a 60-kg individual, holding the individual random effect fixed?
The contrast is \(x_b-x_a=70-60=10\) kg. The coefficient is per kilogram, so the log-scale change is \((-0.0180\ \text{kg}^{-1})(10\ \text{kg})=-0.180\). Therefore, $$ \frac{CL(70,\eta)}{CL(60,\eta)}=\exp(-0.180)=0.8353. $$
The clearance at 70 kg is therefore 0.8353 times the clearance at 60 kg under this model, or \(100(0.8353-1)=-16.47\%\) lower. The negative coefficient means the parameter decreases as weight increases in this model. The statement is conditional on the modeled log-linear relationship and a shared \(\eta\); it does not claim that every observed 70-kg person has lower clearance than every observed 60-kg person.
Choosing and Interpreting a Contrast
Always state both covariate values, or their difference, before calculating a ratio. This makes the unit, direction, and reference point explicit. If the question asks about a change from \(x_a\) to \(x_b\), use \(x_b-x_a\); reversing the comparison takes the reciprocal of the ratio. For continuous covariates, the coefficient's units are inverse units of the covariate, so the product \(\beta(x_b-x_a)\) is dimensionless, as required inside an exponential.
Worked Example: A Decrease in Age
In a hypothetical model, a parameter has an age coefficient of \(\beta=0.0350\) per year. Calculate the ratio for age 42 years compared with age 50 years, holding the individual effect fixed.
The contrast is \(42-50=-8\) years, not \(+8\) years. The log-scale change is \((0.0350\ \text{year}^{-1})(-8\ \text{years})=-0.280\). Thus, $$ \frac{P(42,\eta)}{P(50,\eta)}=\exp(-0.280)=0.7558. $$
At age 42, the modeled parameter is 0.7558 times its value at age 50, which is \(100(0.7558-1)=-24.42\%\) lower. Equivalently, the age-50 value is \(1/0.7558=1.323\) times the age-42 value, about 32.3% higher. These percentages differ because each is calculated relative to a different starting value.
The model gives a constant log-scale slope of 0.0350 per year. The calculation is justified for the specified ages only if the model's linear relationship on the log scale is suitable over that range; it should not be extended to ages outside the covariate range without support.
A categorical indicator can be treated in the same way when it is coded as 0 or 1. The contrast from the reference category \(z=0\) to \(z=1\) is one indicator unit, so its ratio is \(\exp(\beta)\). The ratio is relative to the category coded 0; switching the reference category reverses the comparison.
Worked Example: Comparing Two Indicator Categories
Suppose a hypothetical model includes an indicator \(z\) for a formulation, with \(z=0\) for the reference formulation and \(z=1\) for the alternative. The coefficient for a parameter is \(\beta=\log(1.25)=0.2231\). Find and interpret the alternative-to-reference ratio, holding the individual effect fixed.
The contrast is \(1-0=1\), so $$ \frac{P(z=1,\eta)}{P(z=0,\eta)} =\exp\{0.2231(1-0)\} =1.250. $$
Under the model, the parameter for the alternative formulation is 1.25 times the parameter for the reference formulation, or 25.0% higher. This is a model-based relative comparison, not by itself evidence that the formulation causes a change. The category coding and reference category must be checked before interpreting the sign or direction.
Assumptions, Diagnostics, and Common Pitfalls
The exact ratio calculation is algebraically valid for the specified model, but its scientific interpretation depends on whether that model is appropriate. In particular, the covariate effect must be linear on the log scale over the contrast, the coefficient must apply to the covariate values being compared, and the covariate must be expressed in the units used when the coefficient was defined. A contrast far outside the observed covariate range is an extrapolation, even though the exponential can still be calculated.
When evaluating a reported ratio, check the following:
- Units: Confirm whether the coefficient is per kilogram, per 10 kilograms, per year, or another scale. A coefficient per 10 kg must be multiplied by the number of 10-kg increments, not by the number of kilograms.
- Contrast order: Confirm that the numerator corresponds to \(x_b\) and the denominator to \(x_a\). Reversing the order changes the ratio to its reciprocal.
- Functional form: Consider whether a constant log-scale slope is plausible across the contrast. Systematic departures from the fitted covariate relationship may indicate that the single-slope model is inadequate.
- Support: Check whether the covariate values are represented in the data. A numerical prediction outside that range can be highly dependent on the assumed functional form.
- Individual variation: Distinguish the covariate-only ratio at fixed \(\eta\) from a comparison of two individuals whose random effects may differ.
A common mistake is to read \(\beta=-0.0180\) as a decrease of 1.8% per kilogram. That is not the exact model interpretation: \(\beta\) is a change in the logarithm of the parameter per kilogram. The exact one-kilogram ratio is \(\exp(-0.0180)=0.9822\), a 1.78% decrease; for a 10-kg contrast the exact ratio is \(\exp(-0.180)=0.8353\), a 16.47% decrease. Multiplying \(\beta\) by 100 is only an approximation to percentage change for a small log-scale change, and can become misleading for larger contrasts.
| Approach | What it gives | When to use it |
|---|---|---|
| Exact exponential transformation | \(\exp(\beta\Delta x)\), the model-implied ratio | Use for the exact relative change under a log-linear parameter model |
| Small-change approximation | Approximately \(100\beta\Delta x\%\) | Use only as a rough approximation when \(|\beta\Delta x|\) is small; report the exact ratio when possible |
| Direct ratio reporting | A ratio for a named covariate contrast | Use to communicate the effect in context, while also giving the contrast and units |
The ratio is a relative effect, not an absolute parameter difference. If the modeled parameter at \(x_a\) is known, an absolute difference can be obtained by multiplying that value by the ratio minus one. Without a starting parameter value, the ratio alone does not specify the difference in units such as liters per hour. Nor does a ratio alone express uncertainty in the estimated coefficient; inferential claims require the uncertainty of that estimate to be carried through the transformation.
Check Your Understanding
For each problem, identify the contrast direction and show how the log-scale coefficient becomes a ratio.
- A parameter has coefficient \(\beta=0.0200\) per kg. What is the ratio at 75 kg compared with 65 kg, and what is the percentage change?
- A coefficient is \(-0.0300\) per year. Calculate the ratio at 30 years compared with 40 years. Is the parameter higher or lower at 30 years under the model?
- A category coefficient is \(\beta=\log(0.80)\), with the alternative category coded 1 and the reference category coded 0. Interpret the alternative-to-reference ratio.
- Why is \(\exp(\beta\Delta x)\) not generally the ratio between two observed individuals with different random effects?
Solutions. (1) The contrast is \(75-65=10\) kg, so the ratio is \(\exp(0.0200\times10)=\exp(0.200)=1.221\). The modeled parameter is 22.1% higher at 75 kg. (2) The contrast is \(30-40=-10\) years, so the ratio is \(\exp[(-0.0300)(-10)]=\exp(0.300)=1.350\). The parameter is 35.0% higher at 30 years than at 40 years. (3) The contrast is one indicator unit, giving \(\exp[\log(0.80)]=0.800\); the alternative-category parameter is 20.0% lower than the reference-category parameter. (4) The formula cancels the random effect only when it is held fixed. For two individuals with effects \(\eta_a\) and \(\eta_b\), the ratio also contains the factor \(\exp(\eta_b-\eta_a)\), so it is not the covariate-only ratio unless those effects are equal.