Error Model

The [error_model] block defines the residual error structure, specifying how observed data (DV) relates to model predictions.

Syntax

DV ~ ERROR_TYPE(SIGMA_PARAMS)

Available Error Models

Additive

DV ~ additive(SIGMA_NAME)

The residual variance is constant across all predictions:

\[ \text{Var}(DV) = \sigma^2 \]

Use when measurement error is independent of concentration (e.g., assay with fixed precision).

Proportional

DV ~ proportional(SIGMA_NAME)

The residual variance scales with the predicted value:

\[ \text{Var}(DV) = (\sigma \cdot f)^2 \]

where \( f \) is the model prediction. Use when measurement error increases with concentration (most common in PK).

Combined

DV ~ combined(SIGMA_PROP, SIGMA_ADD)

Combines proportional and additive components:

\[ \text{Var}(DV) = (\sigma_1 \cdot f)^2 + \sigma_2^2 \]

Use when both proportional and additive error sources are present. Requires two sigma parameters defined in [parameters].

All sigma parameters are estimated and reported on the standard-deviation scale, not the variance scale. This is true for both proportional and additive components, and for both elements of a combined error model.

In particular:

Error model	Initial value `sigma = X` means …
`proportional`	The residual SD scales as `X · f`, i.e. CV% = `X · 100`
`additive`	The residual SD is `X` in the units of `DV` (no CV interpretation)
`combined`	First sigma is proportional (CV-style), second is additive (units of `DV`)

So sigma PROP_ERR ~ 0.1 for a proportional model is a 10% CV initial value, not 1%. Likewise, the SE on a proportional sigma is on the SD scale — multiply by 100 for an SE in CV-percentage points.

The fitted YAML emits both the SD (estimate) and the variance (variance: estimate²); for proportional components it also emits cv_pct = estimate · 100 so downstream tooling does not have to re-derive it.

Examples

Proportional error (most common):

[parameters]
  sigma PROP_ERR ~ 0.01

[error_model]
  DV ~ proportional(PROP_ERR)

Additive error:

[parameters]
  sigma ADD_ERR ~ 1.0

[error_model]
  DV ~ additive(ADD_ERR)

Combined error:

[parameters]
  sigma PROP_ERR ~ 0.1
  sigma ADD_ERR  ~ 0.5

[error_model]
  DV ~ combined(PROP_ERR, ADD_ERR)

Impact on Estimation

The error model affects:

Individual weighted residuals (IWRES): (DV - IPRED) / sqrt(Var)
Conditional weighted residuals (CWRES): Accounts for uncertainty in random effect estimates
Objective function value (OFV): The likelihood includes log(Var) terms, so the error model structure directly influences parameter estimates

FeRx-NLME Documentation