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MeanMetric

MeanMetric is used for experimental endpoints reported as a mean value with an associated variability measure. It compares the mean of individual-level simulation outputs to a target experimental mean.

The metric assumes that experimental summary statistics are fixed targets and does not attempt to model uncertainty of the experimental mean itself.

See API reference: MeanMetric.

Concept

Many experimental datasets report only a population mean and a standard deviation, without providing individual-level observations. In such cases, model calibration often proceeds by matching simulated population averages to reported mean values.

MeanMetric formalizes this comparison by measuring the discrepancy between the mean of simulated individuals and the experimental target mean, scaled by the reported variability.

The experimental standard deviation is treated as a known scale parameter and is used to normalize the mismatch.

Example

Assume experimental data report a mean biomarker value of 2.5 with a standard deviation of 0.6, based on a cohort of 50 patients.

This can be encoded as a MeanMetric:

using DigiPopData

m_metric = MeanMetric(
    50,     # experimental sample size
    2.5,    # experimental mean
    0.6     # experimental standard deviation
)

Assume a virtual population where each individual produces a single numeric outcome:

sim_values = randn(1000) .* 0.6 .+ 2.5

The mismatch between simulation and experiment is computed as:

loss = mismatch(sim_values, m_metric)

Mathematics

Let simulated individual values be $y_1, \dots, y_N$, with sample mean

\[\mu_{virt} = \frac{1}{N} \sum_{i=1}^{N} y_i.\]

Let the experimental target be a mean value $\mu_{exp}$ with reported standard deviation $\sigma$.

The mismatch is computed as:

\[\Lambda = N \frac{(\mu_{virt} - \mu_{exp})^2}{\sigma^2}.\]

This expression corresponds to a Gaussian (second-order) approximation of the negative log-likelihood under the assumption that individual observations are normally distributed with known variance $\sigma^2$.

Quadratic formulation for binary selection

In virtual population selection problems, each simulated individual is either included or excluded from the selected cohort. This is represented by a binary selection vector $X \in \{0,1\}^{N_{tot}}$, where $X_i = 1$ indicates that individual $i$ is selected.

Let simulated individual outcomes be $y_1, \dots, y_{N_{tot}}$. The mean of the selected virtual cohort of fixed size $N_{virt}$ is

\[\mu_{virt}(X) = \frac{1}{N_{virt}} \sum_{i=1}^{N_{tot}} X_i y_i.\]

The mismatch defined by MeanMetric can then be written as

\[\Lambda(X) = N_{virt} \frac{(\mu_{virt}(X) - \mu_{exp})^2}{\sigma^2}.\]

Substituting the expression for $\mu_{virt}(X)$ yields

\[\Lambda(X) = \frac{1}{\sigma^2 N_{virt}} \left( \sum_{i=1}^{N_{tot}} X_i y_i - N_{virt} \mu_{exp} \right)^2.\]

This expression is a quadratic function of the binary variables $X$. Therefore, virtual population selection based on MeanMetric can be formulated as a Mixed-Integer Quadratic Programming (MIQP) problem, provided that the cohort size $N_{virt}$ is fixed (e.g. enforced via a constraint $\sum_i X_i = N_{virt}$).

Practical notes

  • The reported standard deviation is treated as a fixed scale parameter and is not re-estimated from simulated data.
  • The experimental sample size is stored for consistency with other metrics but is not currently used in the loss computation.
  • The resulting loss is likelihood-based and can be combined with other metric losses.
  • The quadratic form enables use in optimization workflows, including mixed-integer quadratic programming formulations.