Why Model Relationships Matter
Two QSP models can tell different biological stories yet predict the same measurements under a given experiment. Or one model can be a restricted version of another, so that every prediction from the simpler model is also available in the more flexible one. These relationships matter because the comparison method must match the relationship between the models. Treating every pair as if it were nested can make a formal test invalid; treating equivalent models as distinguishable can create a false sense that the data identify a mechanism.
The classification is not based only on whether two models have different diagrams, numbers of states, or parameter counts. It depends on the probability distributions they can generate for the observations, given the specified experimental design and observation model. This focus on observable consequences connects model comparison to the earlier material on translating biological claims into discriminating predictions.
These definitions concern model families, not just the best-fitting parameter values. Two non-nested models may still share some predictions, or even fit the observed data equally well. Conversely, two models may have visibly different structures while being observationally equivalent for the measurements currently collected. The design matters: a distinction that is invisible in one experiment may become observable after adding a biomarker, changing a sampling time, or applying a different perturbation.
How to Establish Nesting
A common way to establish nesting is to show that parameter restrictions in the more flexible model recover the simpler model. Let \(M_0\) have parameters \(\theta_0\), and let \(M_1\) have parameters \((\theta_1,\gamma)\). If setting \(\gamma\) to a specified value and mapping \(\theta_0\) to \(\theta_1\) reproduces the same predictions and observation distributions, then \(M_0\) is nested in \(M_1\). The mapping must preserve the full observable model, including the observation and error assumptions, not just one selected output at one parameter value.
Worked Example: An Added Feedback Mechanism
Suppose a hypothetical base QSP model predicts a biomarker mean at concentration \(C\) according to
An expanded model adds a feedback contribution:
where \(h(C,t)\) is a specified function of concentration and time, and \(\gamma\geq 0\) is the feedback strength. Assume both models use the same measurement schedule and the same error distribution, with all shared parameters having the same meanings and permitted values.
If \(\gamma=0\), then \(\mu_1(C,t)=\mu_0(C)\) at every measured concentration and time. The expanded model therefore includes every distribution from the base model: choose the same \(B\), \(E_{\max}\), and \(EC_{50}\), and set \(\gamma=0\). The base model is nested in the expanded model. The restriction is at the boundary of the allowed range \(\gamma\geq 0\), so the classification establishes nesting but does not, on its own, establish that every standard nested-model test has its usual reference distribution.
The likelihood implication of nesting follows directly. Let \(L_0\) and \(L_1\) be likelihoods for the same observed data under the two models. If the parameter mapping reproduces the \(M_0\) distribution inside \(M_1\), then for every allowed \(\theta_0\), the corresponding \(M_1\) parameters have the same likelihood. Maximizing over the larger family cannot produce a smaller maximum:
This is a property of fitting nested model families, not evidence by itself that the added mechanism is biologically real. The larger family can fit at least as well because it has at least the same choices available. A formal likelihood-ratio test is a standard option for nested models, subject to its assumptions; its statistic, reference distribution, and special cases belong to the detailed treatment of likelihood-ratio testing and boundary parameters.
Non-Nested Models and a Comparison Example
If neither model can reproduce all the observable distributions of the other, the models are non-nested. This often occurs when biological hypotheses imply different response patterns rather than a simple “mechanism added” extension. A likelihood-ratio statistic does not automatically have the usual nested-model reference distribution in this setting. Broad alternatives include comparing predictive performance on data not used for fitting, using a suitable information criterion, or using a Bayesian comparison with explicitly specified priors. Which approach is appropriate depends on the scientific goal and assumptions; none turns a poor or uninformative design into discriminating evidence.
Worked Example: Two Non-Nested Biomarker Signatures
Consider two hypothetical mechanisms that predict different proportional changes in two biomarkers. At the specified design, let their mean prediction vectors be
Assume independent normal measurement errors with known standard deviation \(2\) normalized biomarker units for each measurement. Suppose the hypothetical observed values are \((8,5)\) units. Model \(M_A\) predicts vectors on one ray, while \(M_B\) predicts vectors on a different ray. For instance, \((1,2)\) is available under \(M_A\), but no \(b\) makes \((2b,b)=(1,2)\); and \((2,1)\) is available under \(M_B\), but no \(a\) makes \((a,2a)=(2,1)\). Thus neither model contains the other.
Under the stated independent normal error model, maximizing the likelihood is equivalent to minimizing the sum of squared residuals. For \(M_A\), the fitted value of \(a\) minimizes \((8-a)^2+(5-2a)^2\). Differentiating and setting the derivative to zero gives
The residual sum of squares is \((8-3.6)^2+(5-7.2)^2=4.4^2+(-2.2)^2=24.20\) square normalized units. For \(M_B\), minimizing \((8-2b)^2+(5-b)^2\) gives
Its residual sum of squares is \((8-8.4)^2+(5-4.2)^2=(-0.4)^2+0.8^2=0.80\) square normalized units. Under these assumptions, \(M_B\) gives the higher maximized likelihood for these observations. The calculation does not make the models nested, and it does not establish that mechanism \(M_B\) caused the biomarker pattern. A non-nested comparison strategy, and further evidence relevant to the scientific question, are needed to assess whether the difference is persuasive.
Observational Equivalence: Different Stories, Same Data
Observational equivalence is stronger than two models happening to fit one data set similarly. It means that, for the specified design and observation process, the models allow the same distributions for all the measurements under consideration. One way this can arise is when different interpretations of a latent process produce the same input-output equation. The data then cannot decide which interpretation is correct, even if the biological narratives differ.
Worked Example: Occupancy Versus Mediator Activation
Consider a hypothetical experiment that measures response only at steady state over a range of concentrations \(C>0\). One mechanism interprets the active fraction as target occupancy \(O\); another interprets it as activation of an unmeasured mediator \(Z\). Suppose both specify
with \(K>0\), \(E_{\max}>0\), and the same observation-error model. For every allowed \(C\), \(K\), and \(E_{\max}\), the predicted response is identical. The models therefore give the same observable distributions for this design, although they assign different biological meanings to the unmeasured quantity. A fit to these response measurements alone cannot distinguish the interpretations.
A direct measurement of occupancy or mediator activation, or an intervention that makes their predicted responses diverge, could break this equivalence. Until then, more precise estimates of \(K\) and \(E_{\max}\) do not resolve which hidden mechanism generated the observed response.
Choosing a Comparison Strategy
First specify the design, measured variables, and observation model, then classify the model families. The comparison should use the same data and compatible likelihood assumptions. If one model has a different measurement-error structure or uses a different subset of observations, an apparent advantage may reflect those differences rather than the mechanism. Shared assumptions and comparable data sources are essential for a fair comparison.
| Relationship | What the relationship establishes | Broadly appropriate response |
|---|---|---|
| Nested | The restricted model's observable distributions are available within the larger model. | A likelihood-ratio test may be appropriate when its regularity conditions hold; also consider predictive or information-based comparisons for the scientific goal. |
| Non-nested | Neither model's full family of observable distributions contains the other. | Use a comparison designed for alternatives that are not nested, such as predictive assessment, a suitable information criterion, or a Bayesian comparison with stated priors. |
| Observationally equivalent | The models allow the same distributions for the specified measurements and design. | Do not claim the current data select between them. Seek measurements or interventions that yield different predictions, or report the unresolved mechanistic ambiguity. |
A model pair can also be distinguishable in principle but practically hard to separate with the available sample size, noise level, or sampling times. That is not exact observational equivalence: the predictions differ, but the difference may be too small relative to uncertainty to resolve. Conversely, equivalence under a limited design need not persist under a new design. State the scope of the classification rather than labeling two models equivalent in every possible experiment.
Diagnostics, Pitfalls, and Limitations
A useful classification check is to try to construct an explicit parameter mapping. For a proposed nested pair, identify the restrictions and verify that they reproduce all relevant predictions and the same observation distribution across the design. For a proposed non-nested pair, look for distributions each can produce that the other cannot. For a proposed equivalent pair, verify equality of the observable distributions over the full parameter ranges and design, not only at the fitted values. If the mapping fails for one measured endpoint, time, or intervention, the claimed relationship may be wrong.
- Pitfall: “The larger model has more parameters, so it contains the smaller one.” Parameter count alone is not enough. The added parameter may change the structure without allowing the smaller model's predictions; demonstrate the required restriction and mapping.
- Pitfall: “Both models fit the data equally, so they are equivalent.” Similar fit at one data set is not equality of model families. It may instead reflect limited data, noisy observations, or an intersection of two non-nested families.
- Pitfall: “A lower residual sum of squares proves the mechanism.” Fit is conditional on the chosen model, error assumptions, and measurements. It does not establish causal mechanism or guarantee performance on new data.
- Pitfall: “Different internal states guarantee distinguishable models.” Hidden states matter to a comparison only insofar as they change the observable distributions or support other biological evidence.
These classifications do not decide which biological account is best supported in every sense. A comparison statistic can be sensitive to error assumptions, design choices, and model flexibility; predictive performance addresses prediction rather than proving a mechanism. If the models are observationally equivalent for the available data, the limitation is informational, not a failure of optimization.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A model adds a feedback parameter \(\gamma\geq 0\), and \(\gamma=0\) reproduces every prediction and the same observation model of a simpler model. What is the relationship?
- Two models have different mechanisms, but at one fitted parameter pair they predict the same values. Does this establish observational equivalence?
- Why is a standard nested-model likelihood-ratio test not automatically justified for two non-nested models?
- Two mechanisms predict identical response distributions at steady state but different time courses. Are they observationally equivalent for a study that measures only steady-state responses?
Solutions. (1) The simpler model is nested in the feedback model: the parameter restriction \(\gamma=0\) recovers its full observable model. The boundary location may affect inference, but not this classification. (2) No. Equality at one parameter pair establishes an overlap in predictions, not equality of the two model families. (3) The usual nested-model reference distribution relies on a nested relationship and additional regularity conditions. For non-nested families that justification does not follow, so a comparison designed for non-nested alternatives is needed. (4) Yes, for that study's design and observation model, if the steady-state distributions agree over the models' allowed parameter ranges. A study measuring time courses could distinguish them, so the equivalence is design-specific.