Nonlinear Terms in the Analysis Model

Why Nonlinear Terms Matter for Imputation

An analysis model may describe an outcome using more than the raw value of a predictor. It might include the predictor squared, its logarithm, or several basis functions that together represent a spline. These terms encode a particular shape: curvature, a transformed scale, or a slope that changes across ranges of the predictor. If the predictor has missing values, imputation must preserve enough of the relevant relationship to make that analysis meaningful.

The central issue is not simply whether the imputation model contains a variable with a similar name. The issue is whether it can reproduce the relevant association between the incomplete predictor, the outcome, and other predictors under the functional form used in the analysis. A model that treats a curved relationship as linear can fill in plausible-looking values while weakening the very curvature the analysis is intended to estimate.

This is a compatibility principle, not a requirement that every imputation equation be a verbatim copy of the final analysis equation. The imputation model must preserve the relevant conditional structure; the form needed to do so depends on which variable is incomplete and on the other variables in the model. As established in the earlier material on outcome inclusion, the outcome generally belongs among the predictors used to impute an incomplete analysis predictor. Here the additional question is whether the functional form is also represented.

Recognizing the Structure to Preserve

Start by reading the analysis model as a list of functions of variables, not just a list of variable names. If the outcome model contains \(X\) and \(X^2\), its mean can bend with \(X\). If it contains \(\log(X)\), the modeled change in the outcome is tied to a change on the log scale. If it contains spline basis functions, the relationship can change slope or curvature across the predictor's range. In each case, a model that includes only a linear \(X\) term imposes a different shape.

When a derived term is a deterministic function of an incomplete variable, its value is not independently missing once the variable is imputed. For example, after imputing \(X\), the corresponding \(X^2\) can be calculated exactly. This prevents inconsistent records such as an imputed \(X=3\) paired with \(X^2=8\). However, deterministic calculation alone does not guarantee that the imputed values of \(X\) preserve the analysis relationship. The imputation model that generates \(X\) still needs to reflect the relevant nonlinear structure.

The term's role also matters. A nonlinear function may appear in the outcome model as a predictor, or a transformed outcome may itself be modeled. The incompleteness pattern and the variables being imputed determine the conditional model that needs to represent the relationship. The safe first step is therefore to inventory the analysis terms and identify which incomplete variables feed into them.

A Justification From the Conditional Distribution

A short derivation shows why matching only a linear trend can be inadequate. Suppose, for illustration, that the analysis model for a continuous outcome is \(Y\mid X \sim N(\beta_0+\beta_1X+\beta_2X^2,\sigma^2)\), with \(\beta_2\ne 0\). Suppose also that \(X\) has a density \(p(x)\). By the definition of a conditional density and Bayes' rule,

The factor \(p(y)\) does not depend on \(x\), so it is absorbed into the proportionality constant. The expression inside the exponential depends on \(x\) through both \(x\) and \(x^2\). In general, the conditional distribution of \(X\) given \(Y\) therefore cannot be represented by a model that assumes only a linear relation in \(y\) and an unchanged, simple distribution for \(X\). The exact conditional form also depends on \(p(x)\); the derivation does not imply one universal imputation equation.

The implication is limited but important: when \(X\) is incomplete, an imputation model must be capable of preserving the relationship implied by the quadratic outcome model. In a chained-equations procedure, conditional models are specified variable by variable, so compatibility must be considered for those conditionals, not inferred solely from the analysis formula. This derivation motivates checking functional form; it does not supply a one-size-fits-all algorithm.

Transformed and Spline Terms

A transformation changes the scale on which a relationship is modeled. For instance, a model \(E(Y\mid X)=5+3\log(X)\) represents a constant change per unit increase in \(\log(X)\), not per unit increase in \(X\). The transformation also imposes a domain restriction: \(\log(X)\) requires \(X>0\). Imputations must respect that domain if the analysis is to be defined.

Spline terms require the same kind of inventory, though there may be several functions rather than one. A spline represents a relationship through basis functions, with knots marking locations where the shape can change. The analysis model's basis functions collectively define the fitted shape. Retaining only the raw predictor, or substituting a single linear term, need not retain the relationship represented by that collection.

Specification Choices and Diagnostics

Several strategies can be considered, but none removes the need to check the conditional relationships. One approach is to include the necessary nonlinear functions in the relevant imputation predictors where appropriate. Another is to impute the underlying variable and calculate its derived terms passively at each imputation. Passive calculation ensures mathematical consistency among the raw value and its functions; it is not, by itself, a model for the distribution of the underlying variable. A richer conditional model may represent curvature more flexibly, but adding terms without regard to sample size, support, or model stability can create a different problem.

Useful checks include comparing observed and imputed predictor distributions, inspecting the relationship between the outcome and the predictor in observed and completed data, and checking that transformed values and basis functions have the expected range and shape. For a squared term, inspect whether imputed values of \(X\) cover the observed range and whether the resulting \(X^2\) values are plausible. For a log term, check for nonpositive imputations. For a spline, check whether imputations provide support on both sides of important knots when the observed data do so.

These checks are diagnostic rather than proof of compatibility. A smooth-looking plot cannot establish that all relevant conditional distributions are correctly specified. Conversely, apparent disagreement may reflect sparse observations, influential values, or genuine uncertainty rather than a single coding error. Investigate the model specification, variable coding, and observed support before interpreting completed-data results.

Common Pitfall and Limits

Wrong reasoning: “I calculated \(X^2\) after imputing \(X\), so the quadratic analysis relationship has automatically been preserved.” Correction: This guarantees that the two columns agree algebraically, but the model used to generate missing \(X\) may still ignore the outcome pattern associated with \(X^2\). Check both deterministic consistency and the conditional model that produced the imputed \(X\) values.

A related mistake is to place every analysis term into every conditional model mechanically. That can create collinearity, separation, sparse cells, or unstable estimates, especially with small samples. The appropriate specification depends on which variables are incomplete, the outcome type, the observed data support, and the substantive model. Multiple imputation cannot recover a shape that is unsupported by the observed data without relying heavily on modeling assumptions. It also does not correct an inappropriate analysis model, resolve structural absence of a predictor value, or make implausible extrapolations safe.