Why Look at Residual Patterns?
A fitted effect-compartment model can reproduce a response curve reasonably well overall while missing its timing or shape in important ways. Residuals—the differences between observed responses and model predictions—help reveal these structured errors. Their order in time and their relation to plasma concentration can indicate whether the model is tracking the delayed response appropriately or whether its equilibration rate or response relation may be inadequate.
A residual pattern is a diagnostic clue, not a diagnosis. It can point toward a misspecified effect-site model, but similar patterns can arise from incorrect observation times, an unsuitable error model, unmodeled covariates, or measurement problems. Interpretation therefore begins by defining the prediction and residual consistently, then checking the pattern against the dosing and concentration history.
In an effect-compartment model, the predicted response is commonly obtained in two stages. Plasma concentration \(C_p(t)\) drives effect-site concentration \(C_e(t)\), and a response relation maps \(C_e(t)\) to the PD prediction. For a first-order effect-site model with equilibration rate constant \(k_{e0}\), a constant plasma concentration over an interval of length \(\Delta t\) gives
Here \(k_{e0}\) has units of inverse time. The response relation might be increasing or decreasing; the examples below use an increasing response so that a larger effect-site concentration predicts a larger response. If the modeled response decreases with concentration, the sign interpretation of a residual pattern must be adjusted accordingly.
Time-Ordered Residuals
A residual-versus-time plot preserves the order in which observations occurred. Under an adequate mean model, residuals should not show a persistent trend or a sequence of runs that the model systematically misses. They need not alternate randomly: measurement error, repeated observations within a subject, and the assumed residual-error structure affect their variation. But a long run of residuals on one side of zero is a reason to investigate.
For a rising concentration and an increasing response, a fitted \(k_{e0}\) that is too small makes the modeled effect-site concentration equilibrate too slowly. During the rise, the model tends to lag behind the true effect-site concentration, underpredicting response and producing positive residuals. During the decline, that slowly changing modeled concentration tends to remain too high, overpredicting response and producing negative residuals. If the fitted \(k_{e0}\) is too large, the typical signs reverse: the model responds too quickly, tending to overpredict on the rising phase and underpredict on the falling phase.
This reasoning is conditional on the concentration history and response relation being otherwise suitable. A sequence of positive residuals followed by negative residuals does not, by itself, prove that \(k_{e0}\) is too small. A timing offset, an incorrect baseline, or another time-varying process can also create runs of signs.
Worked Example: A Run of Residuals Suggests Slow Fitted Equilibration
Suppose a hypothetical subject has an increasing response, and a fitted effect-compartment model produces the following predictions. The plasma concentration rises through hour 3 and then falls. Calculate the residuals and interpret their time order.
| Time (h) | Observed response (score units) | Predicted response (score units) |
|---|---|---|
| 0 | 10.2 | 10.0 |
| 1 | 13.1 | 12.0 |
| 2 | 17.0 | 15.0 |
| 3 | 19.2 | 18.0 |
| 4 | 15.0 | 16.0 |
| 5 | 11.8 | 13.0 |
Using \(r_i=Y_i-\hat{Y}_i\), the residuals are \(10.2-10.0=0.2\), \(13.1-12.0=1.1\), \(17.0-15.0=2.0\), \(19.2-18.0=1.2\), \(15.0-16.0=-1.0\), and \(11.8-13.0=-1.2\) score units. The residuals are mostly positive during the rise and negative during the decline.
For an increasing response, this pattern is consistent with a fitted \(k_{e0}\) that is too small: the model may be lagging during the rise and retaining too much effect during the fall. It is not conclusive. The same sequence could reflect a time offset, a changing baseline, or a model error unrelated to equilibration. The next diagnostic step is to compare residuals with concentration and phase, and to verify that the input concentration history and observation times are correct.
Residuals Against Plasma Concentration
Plotting residuals against \(C_p\) can show whether errors depend on concentration. However, plasma concentration is not generally the model’s direct driver of response when an effect compartment is present. The same \(C_p\) can occur once on the rising phase and again on the falling phase, while \(C_e\)—and thus the predicted response—differs between those times. A useful plot therefore identifies phase or time, for example with different symbols for rising and falling observations.
At the same plasma concentration, a systematic difference in residuals between rising and falling observations can indicate that the model’s delay is wrong or that some delay is missing. The direction depends on the fitted equilibration rate and whether larger concentrations increase or decrease the modeled response. If rising and falling observations are pooled without marking phase, their opposing patterns can cancel and conceal the problem.
Worked Example: Phase-Specific Residuals at Matched Concentrations
Consider hypothetical observations from a single concentration rise and fall. The fitted response is increasing with effect-site concentration. Calculate the residuals and compare the two observations made at each repeated plasma concentration.
| Time (h) | Phase | \(C_p\) (mg/L) | Observed response (score units) | Predicted response (score units) |
|---|---|---|---|---|
| 1 | Rising | 2 | 12 | 13 |
| 2 | Rising | 4 | 14 | 16 |
| 3 | Near peak | 6 | 18 | 18 |
| 4 | Falling | 4 | 17 | 15 |
| 5 | Falling | 2 | 13 | 12 |
The residuals are \(12-13=-1\), \(14-16=-2\), \(18-18=0\), \(17-15=2\), and \(13-12=1\) score units. At \(C_p=2\) mg/L, the rising-phase residual is \(-1\) and the falling-phase residual is \(+1\) score unit. At \(C_p=4\) mg/L, they are \(-2\) and \(+2\) score units, respectively.
The model overpredicts on the rise and underpredicts on the fall. For an increasing response, this is consistent with a fitted \(k_{e0}\) that is too large: the modeled effect may track the rising plasma concentration too quickly and decline too quickly. The repeated-concentration comparison is informative because phase is explicit; a plot that showed only residuals against \(C_p\) without phase could make this structure harder to see. Other explanations, including incorrect timing or a faulty plasma concentration input, still need consideration.
Residual Shape and the Response Relation
Not every structured residual pattern is a timing problem. A response relation can be wrong even when the effect-site concentration trajectory is adequate. For instance, a linear response relation may be unable to represent saturation. Residuals plotted against \(C_e\), or against fitted response, can reveal curvature more directly than a residual-versus-\(C_p\) plot. Residuals against plasma concentration may also show curvature when plasma concentration tracks effect-site concentration closely, but a delay can obscure or distort that pattern. When a delay is plausible, distinguish response-shape clues from phase-specific timing clues rather than interpreting a pooled concentration plot alone.
Worked Example: Curvature From a Misspecified Linear Response
Suppose a hypothetical dataset has little delay over the sampled interval, so \(C_e\) is approximately \(C_p\). The underlying response is \(E=10+20C_e/(5+C_e)\), with response in score units and concentration in mg/L. A fitted linear relation is \(\hat{E}=10+2C_e\). Calculate residuals at four concentrations.
At \(C_e=1\) mg/L, the observed mean response implied by the nonlinear relation is \(10+20(1)/(5+1)=13.333\) score units; the linear prediction is \(10+2(1)=12\), so \(r=13.333-12=1.333\) score units.
At \(C_e=2\) mg/L, the nonlinear response is \(10+40/7=15.714\) score units and the linear prediction is \(14\), giving \(r=1.714\) score units. At \(C_e=5\) mg/L, the nonlinear response and linear prediction are both \(20\), giving \(r=0\). At \(C_e=10\) mg/L, the nonlinear response is \(10+200/15=23.333\) score units, while the linear prediction is \(30\), giving \(r=-6.667\) score units.
The residuals move from positive at low concentrations to zero and then negative at high concentration. This curvature is consistent with a linear relation failing to represent a saturating response. Because the values are illustrative model means, this calculation demonstrates the pattern rather than estimating a response relation from noisy observations. In real data, uncertainty and the full concentration-time history must be considered before changing the model.
Assumptions, Checks, and Alternative Views
Residual interpretation depends on several basic conditions. The response, prediction, and measurement units must be aligned; observation times must correspond to the times used for predictions; and the model must use a sufficiently accurate plasma concentration history. The predicted values must also be of a consistent type across observations. For example, conditional predictions that use individual-level information should not be mixed in the same plot with population predictions without a clear reason.
A short diagnostic sequence helps keep the interpretation grounded:
Confirm that residuals are observed minus predicted, that units match, and that the same prediction type is used throughout.
Look for sustained runs, trends, or a sign change around the concentration peak. Relate them to the known rising and falling phases.
Mark rising, peak, and falling observations so that different residuals at similar \(C_p\) are not hidden by pooling.
Inspect residuals against \(C_e\) or fitted response for curvature, while accounting for the possibility that delay makes \(C_p\) a poor proxy for \(C_e\).
Verify timing, concentration interpolation, baseline and covariates, observations, and the residual-error model before attributing a pattern to \(k_{e0}\) or the response relation.
A time plot emphasizes ordering and runs; a residual-versus-concentration plot emphasizes dependence on concentration and can expose phase separation; and a residual-versus-\(C_e\) or fitted-response plot is often more directly informative about the response relation. These views are complementary, not interchangeable. Formal comparison of models and simulation-based predictive checks answer different questions and should not be replaced by visual residual inspection.
Common Mistake and Limitations
Common mistake: “Residuals are negative while plasma concentration rises and positive while it falls, so the data prove that \(k_{e0}\) is too large.” The correction is that this sign pattern is consistent with an overly fast fitted equilibration rate for an increasing response, but does not prove it. Incorrect observation timing, an incorrect plasma input, a time-varying baseline, or another misspecification can produce a similar pattern. Also, if the response decreases as effect-site concentration increases, the sign logic reverses.
Residuals are themselves noisy, and sparse sampling may miss the transition between rising and falling phases. A few large residuals can dominate visual impressions, while multiple observations from one subject are not independent. Raw residuals can also have changing variability across the prediction range; a pattern in spread is not the same as a pattern in the mean. These plots help locate where a model may fail, but cannot alone establish the correct mechanism, parameter value, or physiological interpretation.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- An increasing response has positive residuals during a concentration rise and negative residuals during the fall. Which direction of \(k_{e0}\) error is consistent with this pattern?
- Why should rising- and falling-phase observations be marked separately on a residual-versus-\(C_p\) plot?
- If a model predicts 24 score units and the observed response is 21 score units, what is the residual, and what does its sign mean?
- What kind of residual pattern might suggest that a linear response relation fails to describe saturation?
- Name two explanations, other than an incorrect \(k_{e0}\), for phase-specific residuals.
Solutions. (1) A fitted \(k_{e0}\) that is too small is consistent with this pattern: it may underpredict on the rise and overpredict on the fall for an increasing response. (2) The same plasma concentration can occur on both phases while effect-site concentration differs; pooling phases can conceal or cancel phase-specific errors. (3) \(r=21-24=-3\) score units. The negative sign means the observation is 3 score units below the prediction. (4) Residuals may show systematic curvature—for example, positive values at lower concentrations and negative values at higher concentrations—rather than fluctuating around zero without structure. (5) Incorrect observation timing and an inaccurate plasma concentration history are two possibilities; a time-varying baseline, unmodeled covariate, or measurement issue could also contribute.