Why Diagnostic Axes Matter
A diagnostic plot can make one kind of model discrepancy visible while making another harder to see. On a linear concentration axis, a fixed vertical distance represents a fixed absolute difference in concentration. On a logarithmic concentration axis, equal distances represent equal ratios. These are different visual questions: are the errors similar in absolute size, or similar in proportion to concentration?
This distinction matters in population pharmacokinetics because concentrations can span a broad range, and measurement variability may increase with concentration. A linear plot can then be dominated by the largest values, while a logarithmic plot can clarify relative discrepancies. Neither scale is inherently superior; the appropriate choice follows from the diagnostic target and the error structure being considered.
For concentration plots, use the same units for observed concentration \(C_{\mathrm{obs}}\) and predicted concentration \(C_{\mathrm{pred}}\). A log-scaled axis requires positive values. The plotted numerical positions depend on the selected log base, but the interpretation in terms of ratios does not. In contrast, signed residuals can be zero or negative, so a logarithmic residual axis is generally not defined.
What a Logarithmic Scale Changes
Suppose an observation is compared with a positive prediction. On the original concentration scale, its signed difference is \(C_{\mathrm{obs}}-C_{\mathrm{pred}}\), with concentration units. On a log scale, the difference between the logged values is
where \(c_0\) is a positive reference concentration in the same units. The reference cancels, leaving the log of the observed-to-predicted ratio. Thus a log-scale difference is dimensionless and represents a multiplicative discrepancy: a ratio of 2 is as far above the prediction as a ratio of \(1/2\) is below it, with opposite signs.
For a small relative difference, write \(C_{\mathrm{obs}}=C_{\mathrm{pred}}(1+r)\), where \(r=(C_{\mathrm{obs}}-C_{\mathrm{pred}})/C_{\mathrm{pred}}\). Then
This approximation explains why log differences can be read approximately as relative errors near the prediction. The exact interpretation, however, is the log ratio, not the percentage difference. For example, a ratio of 1.2 is a 20% excess, while a ratio of 0.8 is a 20% deficit; their log differences are not equal in magnitude.
Worked Example: The Same Absolute Error at Different Concentrations
Consider three hypothetical observed–predicted pairs, all measured in mg/L: \((2,3)\), \((20,21)\), and \((200,220)\), where each pair is written as (prediction, observation). Compare their absolute and proportional discrepancies.
Linear-scale differences. The signed differences are \(3-2=1\) mg/L, \(21-20=1\) mg/L, and \(220-200=20\) mg/L. The first two points are equally far from their predictions on a linear concentration scale.
Relative differences. Dividing each difference by its prediction gives \(1/2=0.50\), \(1/20=0.050\), and \(20/200=0.10\). These correspond to excesses of 50%, 5%, and 10%, respectively. Equal absolute deviations do not imply equal relative deviations.
Log-scale differences. The log ratios are \(\ln(3/2)=0.4055\), \(\ln(21/20)=0.04879\), and \(\ln(220/200)=\ln(1.1)=0.09531\). The first pair is farthest from the identity relationship in relative terms. A plot with both concentration axes logarithmic would show this distinction more clearly than the absolute difference of 1 mg/L for the first two pairs.
The log ratios are dimensionless. They describe multiplicative discrepancy, while the linear differences retain units of mg/L. The choice between them depends on which aspect of model fit is relevant.
Reading Observed-Versus-Predicted Plots
In an observed-versus-predicted plot, each point compares \(C_{\mathrm{obs}}\) with a model prediction. The identity line represents exact agreement: observed and predicted concentrations are equal. With both axes linear, points above the identity line have \(C_{\mathrm{obs}}>C_{\mathrm{pred}}\), and points below it have \(C_{\mathrm{obs}}<C_{\mathrm{pred}}\). The vertical separation is an absolute error in concentration units.
If both axes are logarithmic and use the same units and base, the identity relationship is still equality: equal observed and predicted concentrations lie on the identity line. Distance from that line now reflects ratios. A point twice as high as its predicted value and a point at half its predicted value have equal-magnitude log discrepancies with opposite signs. A log scale can therefore make low concentrations visible when a few high concentrations otherwise dominate the plot.
The log-scale view is appropriate only when the relevant observations and predictions are positive. It also should not be mistaken for a diagnostic of absolute error: the same 1 mg/L difference can be visually large at a low concentration and small at a high concentration. When both questions matter, inspect both scales rather than trying to make one plot answer both.
Worked Example: Comparing Linear and Log Views of Proportional Scatter
Suppose a hypothetical model produces predictions of 1, 10, and 100 mg/L, with two observations at each level: 0.8 and 1.2 mg/L near the first prediction, 8 and 12 mg/L near the second, and 80 and 120 mg/L near the third. Assume all values are positive and compare the scatter around prediction.
On a linear scale, the signed differences at the three levels are \(-0.2\) and \(+0.2\) mg/L, \(-2\) and \(+2\) mg/L, and \(-20\) and \(+20\) mg/L. The absolute deviations grow from 0.2 to 20 mg/L. If the plot shows this pattern, scatter is not constant in absolute units.
On a relative scale, each lower observation is \(0.8\) times its prediction and each higher observation is \(1.2\) times its prediction. The corresponding log ratios at every level are \(\ln(0.8)=-0.2231\) and \(\ln(1.2)=0.1823\). Thus, the relative pattern is the same at low, medium, and high concentrations, even though the linear deviations differ by a factor of 100.
The values support describing this hypothetical scatter as proportional to prediction rather than constant in mg/L. A log-scale concentration plot would make the repeated ratio pattern easier to compare across levels. This example does not establish that a lognormal error model is correct; it only shows which scale makes the stated pattern visually comparable.
Residual Diagnostics Need a Different Decision
Residual plots have a signed quantity on at least one axis. A residual may be negative, zero, or positive. Taking its logarithm is therefore impossible for many ordinary residual values and would remove the central reference at zero. Keep raw, weighted, and standardized residual values on a linear vertical axis when their signs and distance from zero are being assessed.
A horizontal axis can still be logarithmic when its variable is positive. For example, a residual-versus-prediction plot may use a log-scaled concentration axis and a linear residual axis. That choice spreads low predicted concentrations apart while preserving the residual's signed value and zero line. State clearly which axis is logarithmic; “a log residual plot” is ambiguous and can imply an invalid transformation.
Worked Example: Choosing Axes for Signed Weighted Residuals
A hypothetical diagnostic set contains predicted concentrations of 0.2, 2, and 20 mg/L, paired with conditional weighted residuals of \(-1.1\), \(0.1\), and \(1.0\). The residuals are dimensionless and can take either sign. Choose axes to inspect whether their distribution changes across the concentration range.
A log-scaled horizontal axis is possible because all three predictions are positive. The values 0.2, 2, and 20 are separated by equal multiplicative factors of 10, so they have equal spacing on a base-10 logarithmic axis. Plot the residual values \(-1.1\), \(0.1\), and \(1.0\) on a linear vertical axis, with a horizontal reference at zero.
Do not take \(\ln(-1.1)\), \(\ln(0.1)\), or \(\ln(1.0)\) to make a “log residual” axis: the first is not real-valued, and the transformation would not preserve the signed residual scale. If predicted concentration includes zero, a logarithmic horizontal axis cannot display that point either; use a linear axis or a separately justified alternative display.
Assumptions, Diagnostics, and Failure Modes
Choosing an axis is a display decision, not a model-fitting procedure. It does not make a model fit better, correct bias, or establish an error distribution. The interpretation is most useful when the plotted quantities are correctly defined, observed and predicted concentrations use the same units, and the chosen scale matches the discrepancy of interest. A log-scale comparison additionally requires strictly positive values.
Look for patterns that indicate the scale is not answering the question you intended:
- Curvature or systematic displacement from the identity line can remain visible on either scale and may indicate prediction bias. A log axis does not remove such bias; it changes how deviations are spaced.
- Unequal absolute scatter on a linear plot may be expected if variability grows with concentration. Compare the relative pattern on a log scale before concluding that the model has a problem.
- Apparent constant scatter on a log plot does not prove constant relative error or a lognormal observation model. Check the model assumptions and other diagnostics rather than inferring a distribution from appearance alone.
- Missing low values or excluded zero values can alter the apparent pattern. Do not silently omit observations that cannot be shown on a log axis; explain the limitation and inspect an appropriate linear-scale view as well.
- A changing residual pattern can be concealed if only one scale is inspected. In particular, a wide range of concentrations may compress important low-concentration behavior on a linear plot.
A useful comparison is to ask what each display preserves. Linear observed-versus-predicted plots preserve absolute concentration differences and include zero values. Logarithmic observed-versus-predicted plots compare multiplicative differences but require positive concentrations. Residual-versus-prediction plots can combine a log-scaled concentration axis with a linear residual axis, preserving residual sign while spreading prediction levels. These are complementary displays, not competing tests.
| Display | What distance or pattern emphasizes | Main limitation |
|---|---|---|
| Observed versus predicted, both axes linear | Absolute concentration differences | High concentrations may dominate the visible range |
| Observed versus predicted, both axes logarithmic | Observed-to-predicted ratios | Zero and negative concentrations cannot be shown |
| Signed residual versus prediction; linear residual axis | Residual trends and sign across predictions | Linear prediction scale may compress low values |
| Signed residual versus prediction; log prediction axis and linear residual axis | Residual trends across multiplicative prediction ranges | Predictions must be positive; residuals must not be log-transformed |
Common Pitfall: Calling Every Log Plot a Test of Relative Error
A common mistake is to see a log axis and conclude that the plotted error is a percentage error. That is only justified when comparing positions through log differences, such as the difference between \(\ln(C_{\mathrm{obs}}/c_0)\) and \(\ln(C_{\mathrm{pred}}/c_0)\). Merely placing an axis on a logarithmic scale changes coordinate spacing; it does not replace a raw residual with a relative residual.
A related mistake is to put signed residuals on a log scale to “match” a log concentration axis. The residual scale has a meaningful zero and both signs, so this transformation is not generally defined. Instead, use a log concentration axis where appropriate and retain a linear residual axis. If relative residuals are explicitly calculated, define them and interpret them as a separate quantity rather than relabelling raw residuals.
Practical Choice of Scale
Decide whether the question concerns absolute concentration differences, multiplicative discrepancies, or signed residual behavior.
Every value on a logarithmic axis must be positive. If zero or negative values occur, do not omit them without explanation.
Use a linear residual axis with a zero reference. A positive concentration predictor axis may be logarithmic if that helps display its range.
Use linear and logarithmic concentration displays when both absolute and proportional discrepancies matter, and describe what each reveals.
The purpose is not to select the plot that looks most favorable. It is to make the scale explicit, align it with the diagnostic question, and check whether the interpretation is robust to a complementary view. An axis choice can reveal a pattern; it cannot by itself explain its cause.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- For a prediction of 5 mg/L and an observation of 6 mg/L, calculate the absolute difference, relative difference, and natural-log ratio.
- Why can equal vertical distances on a log concentration axis represent equal multiplicative discrepancies rather than equal mg/L differences?
- A residual is \(-0.7\). Can it be placed on a logarithmic residual axis? Give an appropriate alternative display if predictions are positive.
- Predictions and observations are all positive, and the scientific question concerns proportional disagreement across a wide concentration range. Which observed-versus-predicted view is especially informative, and what does it not prove?
Solutions. (1) The absolute difference is \(6-5=1\) mg/L. The relative difference is \((6-5)/5=0.20\), or 20%. The log ratio is \(\ln(6/5)=\ln(1.2)=0.1823\), a dimensionless quantity. (2) A logarithmic position is proportional to the log of the value, so the distance between \(a\) and \(ka\) is \(\ln(ka)-\ln(a)=\ln(k)\), independent of \(a\). The spacing therefore represents a ratio, not a fixed concentration increment. (3) No. The logarithm of a negative residual is not real-valued, and a log transformation would not preserve the residual's sign and zero reference. Plot the residual on a linear vertical axis against prediction; use a log-scaled horizontal axis only if all predictions are positive. (4) Plot observed versus predicted concentrations with both axes logarithmic, retaining the identity relationship. Distance from that line emphasizes observed-to-predicted ratios. This view does not prove a lognormal error model or establish that the model is adequate.