Residual Diagnostics for IVIVC Model Fit

Why Diagnose Residuals?

A small overall discrepancy between an IVIVC model and its data can conceal a consequential pattern. A model may systematically underpredict absorption early and overpredict it later, or it may fit most formulations well only because one formulation strongly determines the fitted mapping. Residual diagnostics help locate those failures and suggest whether the dissolution-to-absorption mapping, its time alignment, or the data structure needs closer examination.

The focus here is model fit: how the fitted IVIVC relates to the observations used to estimate it. Diagnostics are not, by themselves, evidence that the model predicts a new formulation accurately. Internal predictability and external validation are separate evaluations.

Residuals are in the response’s units. If the response is a fraction, residuals are fractions; multiplying by 100 expresses them in percentage points. For a model fitted to deconvolved fraction absorbed, observations can share information through common subjects, sampling times, or formulation-level processing. Residuals should therefore be treated as structured diagnostic quantities, not automatically as independent observations from a simple regression model.

What to Plot and What Patterns Mean

Begin with residuals against time, fitted response, the dissolution input used in construction, and formulation. Where data permit, also inspect residuals by subject, batch, study period, and relevant experimental conditions. Use the same response scale as the fitted model, and label axes with units. Plotting points in time order within each formulation can reveal a pattern that is hidden when all observations are pooled.

• Residuals against time: a sustained positive or negative run suggests time-dependent bias. A curved pattern can indicate that a single linear mapping is inadequate, or that time scaling or lag alignment is not capturing the relationship.
• Residuals against fitted values or dissolution input: curvature suggests a nonlinear relationship left unexplained by the chosen mapping. A widening or narrowing spread suggests nonconstant variability.
• Residuals by formulation: a formulation with consistently signed residuals may be poorly represented by the common mapping. A formulation with unusually large residuals may also be influential, but residual size alone does not establish influence.
• Residuals by subject, batch, or study period: clusters or shifts can point to data structure or experimental differences that a formulation-level curve does not represent.

A pattern is a diagnostic signal, not a proof of a specific biological mechanism. For example, late positive residuals could reflect a mapping that rises too slowly at late times, but they could also arise from errors in time alignment, measurement, or deconvolution. Check the data processing and study context before changing the model form.

A Short Justification for Looking for Trends

A useful way to summarize a time plot is to ask whether residuals tend to change with time. If residuals have no systematic time trend, their linear association with time should be absent in the population represented by the model. This motivates a descriptive regression of residuals on time, \(e_i=\gamma_0+\gamma_1t_i+u_i\), where \(t_i\) is time and \(u_i\) is the remaining variation.

The numerator measures whether residuals tend to be positive at later times and negative at earlier times, or vice versa. If residuals are systematically increasing with time, that covariance is positive and the estimated slope tends to be positive. This calculation is a compact description of a plotted pattern; a conventional significance test for \(\gamma_1\) would require assumptions about dependence and variability that may not hold for repeated IVIVC observations. A plot and the scientific structure of the data remain essential.

Influence: Which Formulations Shape the Fit?

An observation is influential when removing or changing it would substantially alter fitted model parameters or fitted values. In ordinary least-squares regression, leverage measures how unusual an observation’s predictor values are relative to the others, while a large residual measures a discrepancy between observed and fitted responses. Cook’s distance combines these ideas to describe the change in the fitted regression when an observation is omitted.

For a linear least-squares model with \(p\) fitted parameters, leverage \(h_i\), and internally standardized residual \(r_i\), a common diagnostic expression is \(D_i=(r_i^2/p)\{h_i/(1-h_i)\}\). These quantities are useful for a simple linear mapping, but their familiar reference rules do not automatically apply to nonlinear, weighted, correlated, or constrained IVIVC fits. In those settings, a direct and interpretable check is to refit the model after omitting each formulation in turn and compare the estimated mapping and fitted curves.

Influence must be assessed at the level that matters. Several time points from one formulation are not several independent formulations. Deleting one time point may diagnose local sensitivity, but it does not answer whether the overall relationship depends on that formulation. Conversely, a formulation can be influential even if its residuals are not the largest, because its dissolution values may cover a part of the predictor range poorly represented by other formulations.

Error Summaries: Useful, but Incomplete

Residual summaries provide compact measures of discrepancy. For \(n\) residuals, mean error describes signed average bias, mean absolute error (MAE) describes average absolute discrepancy, and root mean squared error (RMSE) gives greater weight to larger discrepancies. If \(e_i\) is in fraction units, each summary is also in fraction units.

Mean error can be near zero because positive and negative residuals cancel, even when individual errors are large. MAE avoids that cancellation; RMSE emphasizes large misses. Report summaries alongside plots and, when appropriate, separately by formulation or time region. A single pooled number can conceal opposing patterns.

If variability changes across time or fitted values, consider plotting standardized residuals only when their standardization is justified by a stated variance model. Normalizing errors by a response or reference value can help compare scales, but the denominator must be defined and scientifically sensible. Near-zero denominators can make relative errors unstable. Do not present a normalized summary without stating exactly how it was calculated.

Assumptions, Comparisons, and Failure Modes

Residual plots are useful without assuming that every residual is independent and normally distributed. However, formal tests or reference thresholds for regression diagnostics require a model that specifies the relevant error structure. Repeated times within a formulation can be correlated; estimates of fraction absorbed may also share uncertainty from common concentration data or processing. Treating all points as independent can exaggerate the apparent amount of information.

A residual pattern can also be distorted by the response scale. Fraction absorbed is bounded, commonly between 0 and 1, so variability may change as the fitted fraction approaches a boundary. A straight-line mapping can predict outside the physically meaningful range if used carelessly. Examine whether apparent curvature or changing spread follows from the bounded scale, and assess whether the chosen model’s constraints and response scale are appropriate.

Residual diagnostics complement rather than replace other ways of comparing models. A high \(R^2\) or low pooled RMSE summarizes overall fit but can conceal time structure and formulation-specific failures. A residual plot exposes structure but does not give one summary number. A leave-one-formulation-out refit assesses sensitivity to individual formulations, but it is not the same as a dedicated assessment of internal predictive performance. These approaches answer different questions and should not be treated as interchangeable.

When a diagnostic suggests trouble, proceed in a controlled sequence: verify observations, units, and time alignment; inspect the pattern by formulation and time; assess sensitivity to influential formulations; then consider whether the selected dissolution input or mapping form is inadequate. Avoid adding complexity solely to remove a visible pattern without checking whether the change is scientifically defensible and supported by the available formulations.