Why Separate Biological and Assay Parameters?
A QSP model predicts biological quantities such as a biomarker concentration or receptor response. An assay does not necessarily report that quantity on the same scale or with the same baseline. Its readout may include a constant offset, a multiplicative scale factor, and measurement error. If these observation features are absorbed into mechanistic parameters, the fitted model can suggest a biological change where the data support only a difference in measurement scale or baseline.
The distinction is especially important when multiple assays measure the same biological samples. Paired measurements can reveal how the assays relate, because both are exposed to the same underlying biological conditions. But pairing alone does not guarantee that the biological scale, baseline, or every assay parameter can be identified. Calibration anchors and variation across conditions may still be needed.
A Joint Biological and Observation Model
Let \(u_i\) denote the experimental condition for sample \(i\), and let \(f(u_i;\theta)\) be the model-predicted biological quantity at that condition. When sample \(i\) is measured by assay \(s\), with replicate index \(r\), consider
Here \(y_{isr}\) is the observed assay readout, \(a_s\) has the units of that readout, and \(c_s\) converts the model prediction to the assay scale. The biological parameters \(\theta\) are shared across assays; \(a_s\), \(c_s\), and potentially \(\sigma_s^2\) are source-specific. This is an observation model: the assay parameters map a biological prediction to an expected measurement.
A common starting assumption is that, conditional on the model prediction and parameters, errors are independent with \(\varepsilon_{isr}\sim N(0,\sigma_s^2)\). That assumption means errors have mean zero, constant variance within assay \(s\), and no dependence across observations. It should be checked against the design and residual patterns; repeated measurements on the same sample may require a correlated-error model rather than independence. If an assay reports on a log scale, the model and interpretation of its offset and scale must instead be specified on that log scale.
The normal negative log-likelihood, omitting constants that do not depend on the parameters, is
where \(N_s\) is the number of observations from assay \(s\). The weighted residual-sum-of-squares term alone is proportional to the negative log-likelihood only when the variances are fixed. If \(\sigma_s^2\) is estimated, the \(N_s\log(\sigma_s^2)\) terms are part of the likelihood and cannot be omitted: otherwise, increasing a variance could reduce the residual penalty without paying the likelihood cost associated with that variance.
Worked Example: Reading Paired Assay Measurements
Suppose two hypothetical assays measure the same samples at three conditions. Assay A has been independently calibrated to report the model quantity directly, so \(a_A=0\) and \(c_A=1\). The paired measurements are:
| Condition | Assay A | Assay B |
|---|---|---|
| 1 | 2.0 | 4.9 |
| 2 | 4.0 | 8.1 |
| 3 | 6.0 | 11.0 |
For illustration, take the Assay A readings as the biological-scale values \(f_i\). Fit the simple observation relationship for Assay B, \(y_{iB}=a_B+c_B f_i+\varepsilon_{iB}\), by least squares. The means are \(\bar f=4.0\) and \(\bar y_B=8.0\). The slope and intercept are
The fitted Assay B expectations are \(4.950,\ 8.000,\ 11.050\), giving residuals \(-0.050,\ 0.100,\ -0.050\) and a residual sum of squares of \(0.0150\) in squared Assay B units. The estimated relationship is therefore an offset of \(1.900\) Assay B units and a scale factor of \(1.525\) Assay B units per Assay A unit. These are observation-scale features; this small example does not establish that either assay is biologically correct without the calibration information assumed for Assay A.
What Pairing Can and Cannot Identify
With paired assays measured at the same conditions, their expected readings are linked through the same \(f(u_i;\theta)\). For two assays with nonzero scale factors,
where \(f_i=f(u_i;\theta)\). Thus, across conditions that span a range of biological predictions, the paired data can inform the relative offset and relative scale between assays. Variation matters: if all samples have effectively the same \(f_i\), the paired data provide little information about a slope and intercept relationship.
A simple derivation uses changes between two conditions. Subtracting expected readings within assay \(s\) eliminates its offset:
The offset does not affect the change, but the scale factor does. This is why within-assay changes can carry information about biological change while still being measured in assay-specific units. If paired errors are independent across assays, the variance of the difference \(y_{i2}-y_{i1}\) is \(\sigma_2^2+\sigma_1^2\). If the measurements share correlated errors, the variance is instead \(\sigma_2^2+\sigma_1^2-2\operatorname{Cov}(\varepsilon_{i2},\varepsilon_{i1})\); assuming independence without justification can misstate uncertainty.
Pairing generally does not determine an absolute biological scale by itself. For example, if the model prediction has an unknown baseline component \(\theta_0\), so \(f_i=\theta_0+g_i\), then replacing \(\theta_0\) by \(\theta_0+d\) and each \(a_s\) by \(a_s-c_s d\) leaves every expected observation unchanged. The data cannot distinguish these parameter combinations without an anchor or a constraint. Similarly, a multiplicative rescaling of the biological prediction can be offset by a corresponding change in assay scale factors if no calibrated scale is available. This is structural confounding, not a numerical optimizer failure.
Worked Example: Estimating an Assay Offset and Scale With Calibration Values
Suppose a hypothetical calibration experiment provides known biological-scale values \(x=(1,2,3)\) and Assay C readings \(y=(2.2,4.1,5.8)\). Fit \(y=a_C+c_Cx+\varepsilon\). The sample means are \(\bar x=2\) and \(\bar y=4.0333\). The least-squares estimates are
The fitted readings are \(2.2333,\ 4.0333,\ 5.8333\). Subtracting fitted values from observed values gives residuals \(-0.0333,\ 0.0667,\ -0.0333\), so the residual sum of squares is approximately \(0.006667\) squared Assay C units. The calibration points identify the assay mapping relative to the stated known values. If those values were not known and the biological scale were also free, the same readings alone would not establish an absolute scale.
Worked Example: Recognizing Offset–Baseline Confounding
Consider a hypothetical single-assay model with expected readout \(\mu_i=a+\theta_0+g_i\), where \(g_i\) is a known condition-dependent component, \(a\) is the assay offset in readout units, and \(\theta_0\) is an unknown biological baseline in the same units. If \(a=2\) and \(\theta_0=5\), the baseline part of the expected readout is \(2+5=7\). Choosing \(a'=4\) and \(\theta_0'=3\) gives the same sum, \(4+3=7\), so every predicted observation remains unchanged for every condition.
More generally, for any \(d\), let \(a'=a+d\) and \(\theta_0'=\theta_0-d\). Then \(a'+\theta_0'=a+\theta_0\), which proves that the data likelihood cannot distinguish the two parameters in this model. Adding replicates can reduce measurement uncertainty around the sum, but cannot resolve this exact confounding. A known offset, a known biological baseline, or another informative measurement is needed to separate them.
Fitting Choices, Diagnostics, and Limitations
A joint calibration model should preserve the distinction between parameters that govern biology and parameters that govern measurement. The earlier material on joint calibration objectives provides the setting for combining sources; here the key modeling choice is which parameters are shared and which are source-specific. Three common choices have different implications:
| Approach | What it assumes | Main risk |
|---|---|---|
| Use one shared observation mapping | Assays report the same expected value on a common scale and baseline. | Assay differences may be forced into the biological parameters or residuals. |
| Use shared biology with source-specific offsets and scales | Sources observe the same biological process through different mappings. | Offsets or scales may be confounded with biological baseline or magnitude. |
| Fit separate biological parameters by assay | Each source may have a distinct biological parameter set. | Measurement differences can be misinterpreted as biological differences, and shared biology is no longer imposed. |
Diagnostics should assess both the observation model and its ability to distinguish parameters. Plot residuals against fitted values and condition, separately by assay. A systematic trend, curvature, or condition-specific pattern suggests that an additive-and-multiplicative mapping or the biological model may be inadequate; it is not automatically evidence for a different biological mechanism. Check whether fitted offsets and scales are stable when the condition range or individual paired samples change. Very broad uncertainty, strong parameter dependence, or a flat objective along combinations of \(a_s\), \(c_s\), and \(\theta\) indicates weak or structural identifiability. Profile and information-based diagnostics are useful, but cannot create information absent from the design.
The model also has limits. A constant offset and scale factor may be inadequate if assay response is nonlinear, depends on concentration range, or changes with sample matrix. The assumed error variance may be unsuitable if variability grows with the expected readout. Replicate correlation and differences in which samples each assay measures affect the information available from pairing. These possibilities motivate model checks, not automatic additions of parameters: added flexibility can make biological and observation parameters harder to distinguish.
Practical Modeling Sequence
State the units and conditions for \(f(u_i;\theta)\), and specify which mechanistic parameters are intended to be shared across sources.
For each assay, state whether an offset, scale factor, and source-specific error variance are justified, and define their units.
Determine whether there are calibration anchors, paired conditions spanning the response range, or constraints that distinguish biological parameters from observation parameters.
Use the full likelihood for estimated error variances, then inspect source-specific residuals and parameter uncertainty for evidence of misspecification or confounding.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In \(y_{isr}=a_s+c_s f(u_i;\theta)+\varepsilon_{isr}\), which quantities are shared across assays, and what units do \(a_s\) and \(c_s\) have?
- Why can paired measurements across several conditions inform relative assay scale, while repeated paired measurements at one unchanging condition may not?
- If assay variances are estimated, what term must accompany the weighted residual sum of squares in the normal negative log-likelihood?
- In the model \(\mu_i=a+\theta_0+g_i\), explain why more replicates alone cannot separate \(a\) from \(\theta_0\).
Solutions. (1) The mechanistic parameter vector \(\theta\) and the biological prediction structure are shared. The offset \(a_s\) is in assay \(s\)'s readout units; \(c_s\) is in readout units per unit of \(f\). (2) Conditions spanning different biological predictions provide changes whose magnitude depends on scale, whereas repeated observations at one condition mainly estimate the combined level and give little information about a slope or relative scale. (3) Include \(\frac{1}{2}\sum_s N_s\log(\sigma_s^2)\), in addition to the variance-weighted residual sum of squares and up to parameter-independent constants. (4) The predictions depend on \(a+\theta_0\) only. Any increase in one parameter can be offset by an equal decrease in the other, leaving all expected observations unchanged, regardless of replicate count.