Why Pooling Requires a Model
Measurements from different experimental protocols can inform the same biological parameters, but they are not automatically interchangeable. Protocols may use different doses, schedules, cell systems, sampling times, assay platforms, or reporting scales. Combining their values as if they were repeated measurements of one identical quantity can obscure the biological relationship the calibration is meant to estimate.
The key distinction is between what differs because the experiment differs and what differs because the biological mechanism differs. A change in administered dose is usually represented as a protocol-specific input to a shared model. A change in assay gain is represented in the observation model. A change in receptor abundance caused by a different biological condition may require a different biological parameter or an explicit condition-dependent model.
Let \(k\) index protocols and \(i\) index observations within protocol \(k\). A useful general representation is
Here \(f\) is the biological model, \(\theta\) is the vector of biological parameters proposed to be shared, \(u_{ki}\) contains the inputs and conditions for that observation, and \(t_{ki}\) denotes its observation time or other design coordinate. The function \(g_k\) maps the modelled biological quantity to the scale reported by protocol \(k\); \(\eta_k\) denotes any protocol-specific observation parameters. The measurement error \(\varepsilon_{ki}\) follows the error model specified for that source. If observations are correlated within a protocol, their joint error distribution must represent that correlation rather than treating them as independent.
A joint calibration is defensible only if the shared parameters have the same biological meaning and units across protocols, protocol-specific inputs and observation mappings are adequately represented, and the data contain enough information to estimate the shared quantities. Differences in protocol are not a nuisance to ignore: they determine the inputs and observation model that make a common biological interpretation possible.
Deciding What to Share
Begin by stating the biological claim behind each proposed shared parameter. For example, an \(EC_{50}\) parameter can be shared when it represents the same concentration scale for the same underlying response mechanism. It should not be shared merely because two model files use a parameter with the same name. Confirm that concentration units, response definitions, species or cell context, and relevant biological conditions are compatible, or represent their differences explicitly.
Then separate the model components into three groups:
- Shared biological parameters: quantities such as a common binding affinity or response sensitivity, when the biological meaning and conditions justify a common value.
- Protocol-specific inputs: administered dose, dosing schedule, measured concentration-time profile, sampling times, or other design conditions. These enter the model for the relevant observations rather than being averaged across protocols.
- Protocol-specific observation terms: quantities such as assay offset, gain, or error scale when the measurement process differs. These describe how a biological quantity is recorded, not necessarily a difference in biology.
Whether a quantity belongs in the input or observation model depends on where the protocol changes the data-generating chain. A different free-drug fraction changes the concentration available to the biological system and therefore belongs in the input to the mechanism. A different instrument scale changes the recorded signal and therefore belongs in the observation mapping. If a protocol changes a biological state itself, a shared parameter may no longer be appropriate without an explicit model for that condition.
The assumptions should be made explicit before fitting. In particular, the proposed shared mechanism must apply to the experimental conditions; protocol inputs must be known or estimated with a stated model; the observation mapping must put each observation in correspondence with the modelled quantity; and the error distribution and dependence structure must be suitable for each source. A joint fit can produce a numerical answer even when these assumptions are wrong, so successful optimization alone does not establish that pooling is justified.
Constructing the Joint Calibration
Suppose the data sources are conditionally independent given the model parameters, and each source has an appropriate likelihood \(L_k\). The joint likelihood is the product of the source likelihoods because all sources share \(\theta\), while each can have its own inputs and observation terms:
Maximizing this product is equivalent to minimizing the negative log-likelihood, which is a sum of source contributions. For independent normal errors with known standard deviations \(\sigma_{ki}\), this gives the weighted sum of squared residuals below, up to a constant:
This expression shows why the model, rather than a raw average, is the basis of pooling: each observation is compared with the prediction generated under its own protocol inputs and observation process. The formula also depends on its assumptions. If the errors are not normally distributed, or observations within a source are correlated, the appropriate likelihood must replace this independent-normal expression.
Sharing parameters can improve precision when the common-mechanism assumption is sound and the protocols contribute informative, complementary data. It cannot create information that the experiments do not contain. If all protocols measure only a narrow, similar part of a response curve, for example, the shared parameters may remain poorly distinguished. Parameters that are not identifiable from the combined design should not be treated as reliably determined merely because more rows of data have been added.
Worked Example: Different Concentration Designs, One Shared Response Curve
Consider two hypothetical protocols measuring the same normalized response in the same cell system. The proposed model is \(E(C)=E_{\max}C/(EC_{50}+C)\), where \(C\) is concentration in nM. Suppose the candidate shared parameters are \(E_{\max}=100\) response units and \(EC_{50}=2\) nM. Protocol A samples \(C=1\) and \(4\) nM and observes 32 and 68 response units. Protocol B samples \(C=2\) and \(8\) nM and observes 49 and 81 response units. Assume, for this illustration, independent normal errors with a common known standard deviation of 2 response units and no protocol-specific offset or gain.
The protocols differ in which concentrations they sample, but the proposed biological response and reported response scale are the same. Their concentration values therefore enter as protocol-specific inputs to the shared curve. The four model predictions are
The observed-minus-predicted residuals are \(32-33.33=-1.33\), \(68-66.67=1.33\), \(49-50=-1.00\), and \(81-80=1.00\) response units. At this candidate parameter value, the sum of squared residuals is approximately \(1.78+1.78+1.00+1.00=5.56\) response-units squared. The normal-error objective, omitting its constant term, is therefore \(5.56/2^2=1.39\). This calculation evaluates one candidate; calibration would adjust the shared parameter values to optimize the joint objective. The example shows how distinct concentration designs can contribute to one calibration without treating their concentrations as if they were the same.
Worked Example: Correcting a Protocol-Specific Assay Scale
Suppose the same hypothetical biological model predicts a latent response of 50 units at \(C=2\) nM. Protocol A reports that response directly. Protocol B uses an assay with a known zero offset and a known gain of 2.4, so its reported signal is \(y=2.4z\), where \(z\) is the latent biological response. Assume both protocols measure the same biological condition and their errors are negligible for this arithmetic illustration.
Protocol A reports \(y_A=50\). Protocol B reports \(y_B=120\), which corresponds to a latent response of
After applying the calibrated observation mapping, both observations are consistent with the same latent prediction of 50 units. In contrast, their raw arithmetic mean is \((50+120)/2=85\), which is not an estimate of the shared biological response because the two values are on different scales. If the gain were unknown, it would need suitable calibration data or other identifying information; otherwise, gain and biological response could be confounded.
Worked Example: A Protocol-Specific Input Can Preserve Shared Biology
Consider two hypothetical protocols that both report total drug concentration \(C_{\mathrm{total}}=2\) nM but differ in the fraction \(f_u\) available as unbound drug. Let the shared biological model be \(E(C_u)=100C_u/(1+C_u)\), where \(C_u=f_uC_{\mathrm{total}}\) is unbound concentration in nM. Protocol A has \(f_u=0.5\), while Protocol B has \(f_u=1.0\). Assume the fractions are known and that both protocols measure the same response scale.
For Protocol A, \(C_u=0.5(2)=1\) nM, so the prediction is \(E=100(1)/(1+1)=50.0\) response units. For Protocol B, \(C_u=1.0(2)=2\) nM, so the prediction is \(E=100(2)/(1+2)=66.67\) response units. The response predictions differ even though the total concentrations match. The shared parameters remain plausible because the protocol-specific free fractions are represented in the biological input. Using total concentration as if it were the active concentration would wrongly predict the same response for both protocols.
When Not to Share, and What to Check
If protocols differ in a way that plausibly changes the biological mechanism or state, a common parameter may be scientifically inappropriate. For example, a perturbation that changes receptor abundance may change the maximum response. One option is to fit condition-specific parameters; another is to model receptor abundance as a condition-specific input or predictor if a justified mechanistic relationship is available. The choice should reflect the biology and the information in the data, not a preference for one calibration to be shorter.
A separate fit for each protocol avoids imposing a common value, but it also gives up the information-sharing benefit and may leave parameters weakly estimated. A joint fit with shared parameters directly expresses a common biological hypothesis. A hierarchical model is another possible formulation when variation among sources is itself to be represented, but it requires assumptions and information to estimate that variation. In all cases, raw pooling across incompatible scales is not a substitute for specifying a model.
Before interpreting a pooled calibration, check that units and response definitions have been harmonized, protocol inputs are correctly assigned, observation mappings are supported, and each source contributes information about the shared parameters. Compare predictions with observations in their original protocol-specific units and scales. If a source requires an implausible correction or is systematically poorly represented, the shared structure or its inputs may be inadequate; a good overall fit alone can conceal such a problem. These checks are not a proof that parameters should be shared, but they help reveal when the proposed pooling assumptions are strained.
- Do not equate common variable names with common biology. Verify parameter meaning, units, and experimental conditions.
- Do not average across incompatible scales. Map signals to a common biological quantity using supported observation terms.
- Do not replace protocol inputs with their average. Use the input relevant to each observation or experiment.
- Do not force sharing to gain precision. Precision is meaningful only if the common-parameter assumptions are credible.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Two protocols use different dose schedules but the same biological system and assay scale. What belongs in the shared model, and what differs by protocol?
- A second assay reports a signal that is twice the latent response plus an offset of 5 units. If the latent response is 30 units, what signal should the observation model predict?
- Two protocols have the same total concentration of 4 nM, but known unbound fractions of 0.25 and 0.75. What unbound concentrations enter the biological model?
- Name two conditions that should be checked before treating a biological parameter as shared.
Solutions. (1) The justified biological parameters can be shared; each protocol’s dose schedule and resulting inputs must be supplied for its own observations. (2) The predicted signal is \(2(30)+5=65\) units. (3) The inputs are \(0.25(4)=1\) nM and \(0.75(4)=3\) nM, respectively. (4) For example, check that the parameter has the same biological meaning and units, and that the relevant biological conditions are compatible or explicitly represented. Also check that protocol inputs and observation mappings are adequate.