Worked Example With a Spline Analysis

Why a spline analysis needs a compatibility check

Suppose an outcome regression relates a continuous outcome \(Y\) to a continuous covariate \(X\), but some values of \(X\) are missing. If the planned regression allows a nonlinear relationship using a restricted cubic spline, the imputation model must be capable of preserving that relationship. Including \(Y\) as a predictor of \(X\) is important, but does not by itself ensure compatibility with the analysis model.

The key question is not whether the imputation model contains a variable called \(X\), nor whether it predicts \(X\) well on average. It is whether the imputation procedure can reproduce the joint distribution of \(X\), \(Y\), and the other covariates that is consistent with the planned spline regression. The audit below focuses on a continuous outcome with normally distributed errors. The same general principle applies to other outcome models, although their likelihoods differ.

Write down the spline analysis model first

A restricted cubic spline represents a smooth relationship that is linear beyond the outer knots. With \(K\) ordered knots \(k_1<k_2<\cdots<k_K\), one common basis contains \(X\) and \(K-2\) nonlinear terms. Define \(a_+=\max(a,0)\) and, for \(j=1,\ldots,K-2\), define

The regression can be written using the basis \(x,d_1(x),\ldots,d_{K-2}(x)\). Rescaling or otherwise changing the basis can make coefficients easier to read, without changing the fitted curve when the transformation is full rank. The spline terms are functions of \(X\), not additional observed covariates.

For the main derivation, assume \(Y\) is continuous and that, conditional on \(X\) and \(Z\), the outcome follows a normal linear model with constant variance:

where \(B_1(x)=x\), the remaining \(B_j(x)\) are the restricted cubic spline terms, and \(Z\) denotes the other predictors in the analysis. This model assumes that the conditional mean has the stated spline form, the residual variance is constant, and the conditional outcome distribution is normal. If any of these assumptions is not appropriate, the implied imputation conditional below also changes.

Derive the conditional distribution used for imputation

The analysis model specifies \(Y\) given \(X\) and \(Z\), not \(X\) given \(Y\) and \(Z\). To obtain the latter, also specify a covariate distribution \(f(x\mid z)\). The probability factorization rule gives

For the normal outcome model above, the outcome likelihood as a function of \(x\) is proportional to the exponential term below. Consequently, the conditional distribution that an exact imputation would need to represent is

This result follows directly from the normal density: its factor \(1/\sqrt{2\pi\sigma^2}\) does not depend on \(x\), so it cancels when the conditional density is normalized. The remaining likelihood factor depends on the spline-shaped mean \(\eta(x,z)\), and therefore generally has a nonlinear dependence on \(x\). The covariate distribution \(f(x\mid z)\) also contributes to the conditional shape.

This is why a normal linear imputation regression \(X\) on \(Y\) and \(Z\) is not automatically compatible with a spline outcome model. Even though \(Y\) appears in that imputation regression, the conditional distribution it specifies may not match the distribution implied by the spline likelihood and \(f(x\mid z)\). Compatibility concerns the whole conditional distribution, not merely which predictors appear in a formula.

A practical compatibility checklist

A useful audit begins with the substantive analysis specification and then checks whether the imputation procedure can represent its implications. For a missing spline covariate, use the following sequence.

The checklist assumes that the variables required to model missingness and the analysis relationship have been handled appropriately. In particular, the imputation model needs relevant observed predictors of the missing covariate and outcome, and the missingness assumption must be defensible given the variables included. Compatibility cannot repair a poor missingness assumption, an incorrect substantive outcome model, or a covariate model with inadequate information.

Diagnostics, alternatives, and limitations

Compatibility is a modeling property, not a diagnostic plot that can be checked once and then treated as proven. Still, several checks can reveal serious problems. Compare observed and imputed covariate distributions overall and within important outcome or covariate groups. Examine whether imputations are plausible and remain within defensible ranges. After fitting the planned spline model in completed data sets, inspect the estimated curve and its uncertainty across the observed support, rather than checking only a single linear coefficient. Large changes in the curve across imputation specifications, implausible tails, or poor mixing of chains warrant investigation.

The main approaches have different strengths and assumptions. A conventional conditional regression is relatively straightforward, but a simple linear-normal form may not represent the spline-implied conditional. Adding flexible terms to a conditional imputation model can improve approximation, but does not automatically make the conditional exactly compatible. Substantive-model-compatible fully conditional specification targets the conditional implied by the analysis model together with a covariate model; it is more tailored, but depends on correct specification and stable estimation of those components. Complete-case analysis avoids specifying an imputation model, but discards cases with missing covariates and may be inefficient or biased under relevant missingness mechanisms.

The derivation also has limits. It assumes the stated outcome model and a suitable \(f(x\mid z)\); neither is known with certainty. Sparse data near the tails or knots can make both spline estimation and imputation unstable. A fitted spline curve does not establish that the true relationship is spline-shaped. And no compatibility construction alone addresses missingness that depends on unobserved values after conditioning on observed data. These issues call for substantive judgment and sensitivity analysis, not merely a more elaborate imputation formula.

Common pitfall

Wrong reasoning: “The outcome is included as a predictor of the missing age, so the imputation model preserves the spline.”

Correction: Including the outcome helps retain outcome–covariate association, but it does not establish that the conditional distribution of age matches the one implied by the spline likelihood and the age distribution given other covariates. Write down both models and compare their implied conditionals. If they differ, describe the imputation as an approximation or use a method designed to target the substantive model.