A Compatibility Audit for Nonlinear Regression

Why Nonlinear Regression Needs a Targeted Compatibility Audit

When an analysis regression allows the mean outcome to curve with a predictor, an imputation model can produce plausible-looking values while weakening or changing that curve. The audit question is not simply whether imputed values resemble observed values. It is whether the conditional imputation models can preserve the relationship specified for the analysis.

For example, suppose the planned analysis relates an outcome \(Y\) to an incomplete predictor \(X\) through both \(X\) and \(X^2\). A conditional imputation model for \(X\) that is linear and normal in \(Y\) may be convenient, but convenience does not establish compatibility. The implied conditional distribution of \(X\) can have a shape that a linear normal regression cannot represent.

This audit concerns the conditional structure of the models. It does not establish that the missingness mechanism is ignorable, that the analysis model is substantively correct, or that all other aspects of an imputation procedure are adequate. Those are separate questions.

State the Analysis Model, Then Derive the Imputation Target

Let \(Z\) denote fully observed covariates. A common quadratic analysis model is

This specification assumes that, conditional on \(X\) and \(Z\), the outcome has the stated mean and constant variance \(\sigma^2\), with normally distributed errors. The quadratic effect is present when \(\beta_2\ne 0\). If \(X\) is incomplete and \(Y,Z\) are observed for a case, the imputation model must represent \(X\mid Y,Z\), not just the marginal distribution of \(X\).

The conditional distribution required by a joint model follows from Bayes' rule. Writing \(p(x\mid z)\) for the conditional distribution of the predictor, the exact relationship is

This expression is the central audit reference. If \(\beta_2\ne0\), the squared residual contains an \(x^4\) term. Even if \(X\mid Z\) is normal, the resulting conditional distribution \(X\mid Y,Z\) is generally not a normal distribution with a mean linear in \(Y\). Depending on the values of \(Y\) and the other parameters, it can be asymmetric or have more than one mode.

A Short Derivation: Why a Quadratic Mean Can Require a Multimodal Conditional

The shape issue can be seen without fitting a complicated model. Suppose \(Z\) is absent and \(Y=10+X^2+\varepsilon\), where \(\varepsilon\sim N(0,1)\). At the outcome value \(y=14\), the squared residual is minimized when \(14=10+x^2\), which has two solutions, \(x=-2\) and \(x=2\). If the predictor distribution gives appreciable probability to both regions, then \(X\mid Y=14\) can have mass near both values.

In contrast, a single normal regression \(X\mid Y\sim N(a+bY,\tau^2)\) has one mode for each \(Y\). It cannot represent two separated modes at \(-2\) and \(2\) for the same outcome. This does not mean every quadratic model produces a visibly bimodal conditional: the predictor distribution, residual variation, and covariates all matter. It does show why a linear normal conditional is not generally guaranteed to be compatible.

Audit the Terms That Define the Analysis Curve

For a quadratic regression, the analysis relationship is defined jointly by its linear and squared terms. For a spline regression, it is defined by the specified basis functions and knots. A useful audit therefore follows the exact terms used in the analysis, rather than checking only whether the raw predictor and outcome are associated.

If the outcome is incomplete, its imputation conditional should represent the planned nonlinear mean structure, such as \(X\), \(X^2\), or the analysis spline basis. If the predictor is incomplete, the relevant conditional is the distribution of that predictor given the outcome and other variables; it is not generally obtained simply by reversing the outcome regression. This asymmetry is why an audit must begin with the direction of each imputation conditional.

The following checks are practical evidence, not proofs of compatibility. They should be interpreted alongside the assumed data-generating model, the missingness pattern, and the amount of information available in the observed cases.

• Write down every analysis term, including transformations, polynomial terms, and spline basis functions.
• For each incomplete variable, identify the conditional distribution being imputed and ask whether its form can preserve the analysis relationship.
• Compare the nonlinear analysis curve in observed data with the same curve across completed data sets, using the same scale and covariate values.
• Inspect residual patterns and the distribution of imputed predictors, including whether distinct regions or branches have been smoothed away.
• Check whether conclusions about the curve depend strongly on the imputation specification, rather than treating one favorable diagnostic as decisive.

Check Spline Structure Around Knots

The same logic applies to a spline analysis. A spline basis can represent a change in slope or other prescribed curvature over regions of \(X\). An audit should therefore examine the fitted relationship on both sides of each knot and near the knot, where a change in shape is expected. A model that preserves a broad association but smooths over the knot-specific pattern may still distort the analysis relationship.

Assumptions, Diagnostics, and Limits

An audit is conditional on a stated target. In the quadratic derivation, the outcome model assumes a normal conditional distribution with the displayed mean and constant variance. In the spline example, the assumed mean is the specified piecewise-linear function. If these are not defensible descriptions of the intended analysis, the audit should still preserve the chosen analysis terms while recognizing that compatibility with a model does not make that model scientifically correct.

Missingness assumptions are separate. Under a missing at random assumption, the probability of missingness may depend on observed information but not on the missing value after conditioning on that information. Such an assumption does not make incompatible conditional models compatible. Nor does a compatibility audit establish that missing at random is plausible.

Several warning signs deserve attention: completed-data curves that are flatter than observed-data curves; systematic residual patterns around a knot or across the range of \(X\); implausible concentrations of imputed values; and markedly different nonlinear coefficients under reasonable changes to the imputation specification. These are clues, not automatic diagnoses. The observed subset may be small or unrepresentative, and apparent curve differences can reflect sampling variability.

The main alternative to a simple parametric conditional is a richer conditional model capable of representing the required shape. Greater flexibility is not automatically better: sparse data can make complex models unstable, and a flexible fit may extrapolate erratically. The audit should assess whether the chosen model represents the target terms while remaining estimable and credible in the available data. Compatibility is also not the only quality criterion; valid uncertainty quantification, adequate fit, and appropriate handling of the missingness process remain necessary.

A Practical Audit Sequence

Common Mistake

Wrong reasoning: “The imputation model includes \(Y\), so it must preserve the regression of \(Y\) on \(X\).” Including the outcome as a predictor is important, but it does not determine the conditional distribution's shape. For a quadratic mean, the implied \(X\mid Y,Z\) can be nonnormal or multimodal, while a linear normal imputation has only a single conditional mode.

Correction: Start from the exact analysis terms and derive or inspect the conditional structure required for each incomplete variable. Then check whether the proposed imputation model can represent that structure and whether completed-data diagnostics retain the target curve. Do not infer compatibility from the predictor list alone.