Why assess linearity?
A linear concentration-QTc model is useful because its slope summarizes how the mean QTc response changes with exposure. But a single slope can be misleading if the response bends across the concentrations observed in the study. Curvature may change the estimated effect at the upper end of the exposure range, where a safety interpretation can be especially consequential. The question is therefore not whether a biological response is perfectly straight at every possible concentration. It is whether a linear model is an adequate description over the exposures supported by the data.
The outcome and exposure must be defined consistently before comparing shapes. Let \(Y_{ij}\) be the time-matched, baseline-adjusted QTc change, in milliseconds, for participant \(i\) at observation \(j\), and let \(C_{ij}\) be the corresponding concentration. The earlier material on delayed QTc responses and concentration-effect hysteresis is relevant here: if response is delayed, the exposure assigned to an ECG should reflect the intended effect-time relationship. The models below assume that the chosen concentration assignment and any needed time, placebo, or circadian adjustments are held consistent when shape is assessed.
Representing and testing curvature
A common way to examine departures from a straight line is to add a quadratic concentration term. Centering concentration at a meaningful value \(C^*\), often near the mean concentration, makes the linear coefficient easier to interpret and may reduce numerical correlation between the linear and quadratic terms. A mixed-effects version for repeated ECGs can be written as follows:
Here \(\beta_0\) is the population mean response at \(C^*\), \(\beta_1\) is the local slope there, and \(\gamma\) describes curvature. The participant-specific term \(b_i\) represents a random intercept when that structure is appropriate; \(\varepsilon_{ij}\) is residual variation. If the model includes other prespecified terms, such as time or placebo effects, retain them in both models being compared. The linear model is the special case \(\gamma=0\).
The slope implied by the quadratic model at concentration \(C\) is \(\beta_1+2\gamma(C-C^*)\). Thus, a positive \(\gamma\) means the slope increases as concentration moves above \(C^*\), while a negative \(\gamma\) means it decreases. The sign alone does not say whether the overall QTc response rises or falls: that also depends on \(\beta_1\) and the concentration range.
For nested models fitted by maximum likelihood to the same observations, a likelihood-ratio statistic is
where \(\ell\) is the maximized log likelihood. Under \(H_0\), with the usual regularity conditions and one added, unrestricted parameter, \(\Lambda\) is approximately distributed as chi-squared with one degree of freedom. The two fits must use the same outcome data, likelihood specification, and random-effects and residual structure. For mixed-effects models with different fixed-effect terms, use maximum likelihood rather than comparing restricted maximum-likelihood likelihoods. The chi-squared approximation may be unreliable in small or poorly identified data sets.
Worked Example: Interpreting a Centered Quadratic Model
Worked Example: Interpreting a Centered Quadratic Model
Suppose a hypothetical model for time-matched QTc change is
where \(Y\) is in milliseconds and \(C\) is in mg/L. Assume the observed concentration range includes 0 to 10 mg/L, and \(C^*=5\) mg/L. Interpret the model at the center and at the two ends of this range.
At \(C=5\) mg/L, both concentration terms are zero, so the fitted mean change is \(2.00\) ms. The slope there is \(0.600\) ms per mg/L. The coefficient of the squared term has units of ms per \((\text{mg/L})^2\); its positive value indicates that the slope increases with concentration.
At \(C=0\) mg/L, substitute \(C-5=-5\):
At \(C=10\) mg/L, \(C-5=5\), giving
The slope is \(0.600+2(0.0350)(C-5)\). It is \(0.250\) ms per mg/L at 0 mg/L and \(0.950\) ms per mg/L at 10 mg/L. These predictions illustrate the curve implied by the fitted coefficients; they do not, by themselves, show that the curvature is statistically distinguishable from sampling variation or clinically important.
Worked Example: Likelihood-Ratio Test for Curvature
Worked Example: Likelihood-Ratio Test for Curvature
In a hypothetical repeated-ECG analysis, a linear mixed-effects model and a quadratic mixed-effects model are fitted to the same observations. Both use the same random-intercept and residual structures and the same other fixed-effect terms. They are fitted by maximum likelihood. Suppose their maximized log likelihoods are \(-410.2\) and \(-403.8\), respectively.
The linear model is nested within the quadratic model, which adds one fixed-effect parameter, \(\gamma\). The likelihood-ratio statistic is
Under \(H_0:\gamma=0\), the reference distribution is approximately chi-squared with \(df=1\). The upper-tail probability for 12.8 is about \(0.00035\), so the test provides evidence against a linear mean relationship under this model. This result indicates detectable quadratic curvature; it does not establish the direction or clinical importance of that curvature without examining the estimated curve over the observed concentrations.
The comparison is interpretable only if the stated conditions are credible: the two models use the same data and likelihood basis, the quadratic model differs by the one unrestricted curvature term, the likelihood is adequately optimized, and the model’s distributional and covariance assumptions are reasonable. If the model fits poorly or high concentrations are represented by very few observations, the nominal test result alone is not enough to support a dependable conclusion about the exposure-response shape.
Plots, exposure support, and alternative shapes
A curvature test should be read alongside a plot. Plot the observed QTc changes against matched concentrations, together with fitted linear and candidate nonlinear mean curves. Because repeated ECGs from one participant are not independent observations, also consider whether the apparent pattern is driven by a small number of participants or by dense observations at only a few exposure levels. A smooth curve can help reveal a broad bend, but it can also look persuasive where data are sparse; show the data and the exposure distribution rather than relying on the curve alone.
The exposure range determines what the study can assess. Check the number and spread of observations across concentration, especially near the low and high ends. A model may be well informed around the center and weakly informed at a high concentration reached by few participants. A lack of visible curvature in the dense central region does not establish linearity at the sparsely observed edge. Conversely, a curve that changes dramatically just beyond the observed range is an extrapolation, not evidence that the observed-range relationship is curved.
A quadratic model is only one alternative. A saturable Emax model may be scientifically plausible when the effect approaches a plateau; a spline or other flexible function can describe bends without imposing a single global quadratic shape. These alternatives answer somewhat different questions and may use different numbers of parameters. Compare their fitted curves, uncertainty, residual patterns, and predictive behavior over the observed range. Information criteria such as AIC can summarize relative fit while penalizing model complexity, but a small AIC difference does not prove one functional form is correct. A formal nested-model test is not automatically available for nonnested models.
Worked Example: Comparing Shapes Within the Supported Range
Consider a separate hypothetical study with 140 participants and observed concentrations from 1 to 12 mg/L. Only 9 participants have concentrations above 10 mg/L. On the same data and likelihood basis, suppose the fitted linear, quadratic, and Emax models have AIC values of 510, 501, and 502, respectively. The Emax model’s fitted mean effect is \(6C/(8+C)\) ms, while the linear model’s fitted concentration effect is \(0.35C\) ms.
At 2 mg/L, the linear model predicts \(0.35(2)=0.700\) ms. The Emax model predicts
At 8 mg/L, the predictions are \(0.35(8)=2.80\) ms and \(6(8)/(8+8)=48/16=3.00\) ms. At 12 mg/L, they are \(4.20\) ms and \(72/20=3.60\) ms, respectively. The two forms therefore differ more at the upper observed end than near 8 mg/L, but only 9 participants contribute concentrations above 10 mg/L. That limited support makes the upper-end shape less certain.
The quadratic model has the lowest AIC, but its advantage over Emax is only one AIC unit; that is not compelling separation between those two alternatives. Both nonlinear models have lower AIC than the linear model in this hypothetical fit, so the evidence favors considering curvature, but it does not identify a unique mechanism or curve. Predictions at 20 mg/L would be outside the observed range and should not be treated as validated by these data.
Justifying the nested comparison
The likelihood-ratio procedure follows from comparing how well each nested model can account for the same data. The quadratic model contains every parameter in the linear model and adds \(\gamma\). Therefore, maximizing the quadratic likelihood cannot produce a smaller maximized likelihood than maximizing the linear likelihood. Under \(H_0\), the added parameter is zero, so the simpler model is sufficient. With regular estimation conditions, twice the log-likelihood improvement has an approximate chi-squared distribution with degrees of freedom equal to the number of added parameters.
The approximation is a reference for the test, not a guarantee. In small samples, with weak concentration variation, or with a nearly unidentified covariance structure, the actual sampling behavior may depart from the chi-squared approximation. A test of one quadratic term also targets only that particular departure from linearity. A curve can be nonlinear in another way and still have a small estimated quadratic component.
Diagnostics, limitations, and common pitfalls
Useful checks include plotting observed data and fitted curves across concentration, checking how many observations support each part of the range, and examining whether residual patterns remain after fitting the candidate shape. A systematic bend in residuals against concentration can signal that a straight line is inadequate, while a few influential high-exposure observations can create an apparent bend. More detailed diagnostics should be considered alongside the model comparison, not replaced by a single p-value.
Interpret findings with the scope of the data in mind. A nonsignificant quadratic-term test means the data did not provide sufficient evidence against that linear model at the chosen level; it does not prove linearity. A significant result can reflect a small but precisely estimated curvature, so assess the size of the predicted difference in QTc milliseconds over concentrations that matter. No curve comparison can compensate for poor exposure assignment, inadequate representation of the range, or a misspecified time and covariance structure.
- Mistake: “The quadratic term is not significant, so the relationship is linear.” Correction: The test may have low power, especially with sparse exposure coverage. Report that curvature was not detected and describe the range and precision available.
- Mistake: Treating a significant quadratic term as proof of a biological quadratic mechanism. Correction: It supports curvature relative to the specified linear model; other forms may fit as well or better.
- Mistake: Choosing a curve because it looks smooth at the highest concentration. Correction: Check how much data support that region, and distinguish within-range inference from extrapolation.
The practical conclusion should combine the estimated shape, uncertainty, exposure support, and comparison with plausible alternatives. A linear model may be adequate for the observed range even if it is not a universal biological law. Where alternatives differ materially in the supported range, report that model dependence rather than hiding it behind one selected curve.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In a centered quadratic model, what does \(\beta_1\) represent when \(C=C^*\), and what does a positive \(\gamma\) imply about the slope above \(C^*\)?
- A quadratic model has log likelihood \(-250.0\), and its nested linear model has log likelihood \(-255.0\). Calculate the likelihood-ratio statistic and degrees of freedom if one curvature term was added.
- Why does a nonsignificant test of \(\gamma=0\) not demonstrate that the concentration-QTc relationship is exactly linear?
- A nonlinear model fits best by AIC, but its predictions differ from the linear model only at concentrations with very few observations. What is the main interpretive limitation?
Solutions. (1) \(\beta_1\) is the local mean-response slope at \(C^*\), in QTc milliseconds per concentration unit. Since the slope is \(\beta_1+2\gamma(C-C^*)\), positive \(\gamma\) makes the slope increase above \(C^*\). (2) \(\Lambda=2\{(-250.0)-(-255.0)\}=10.0\), with \(df=1\). (3) The test concerns one quadratic departure and may have limited power; other nonlinear shapes may also be present. A nonsignificant result is not proof of the null. (4) The model’s difference is weakly supported where the data are sparse, so the shape there is uncertain; the best AIC alone does not validate that part of the curve.