Why Functional-Form Mismatch Matters
An analysis model may describe an outcome using a curve, a transformed predictor, or a spline, while the imputation model uses a straight-line conditional mean. That difference is not automatically a source of bias. It becomes consequential when the imputation model fails to preserve information about the incomplete values that the analysis needs to estimate its parameters. The key question is therefore not whether the formulas look different, but whether the imputation distribution represents the relevant conditional relationships adequately.
For example, if the planned analysis is a regression of \(Y\) on \(X\) and \(X^2\), imputing missing \(Y\) from a linear regression on \(X\) may erase curvature. If \(X\) is incomplete instead, a linear model for \(X\) conditional on \(Y\) and other variables may fail to preserve the conditional distribution implied by the nonlinear outcome model. These are different conditional modeling problems; checking only the outcome-on-predictor curve does not settle both.
Compatibility is a model property; consequences for an estimate depend also on which variables are incomplete, how much information is missing, the distribution of the observed data, and the target parameter. Standard missing-at-random assumptions and adequate overlap do not, on their own, make incompatible functional forms compatible. Nor does a good fit to marginal distributions establish that the conditional relationships required by the analysis have been preserved.
What the Mismatch Can Do
Suppose the substantive analysis is \(E(Y\mid X,Z)=\beta_0+\beta_1X+\beta_2X^2+\gamma^\mathsf{T}Z\), where \(Z\) denotes other predictors. If \(Y\) is incomplete and the imputation model uses only a linear mean in \(X\), it cannot represent a nonzero quadratic component unless other included terms happen to reproduce it. Imputed outcomes can then be systematically too low in one region of \(X\) and too high in another. The analysis may estimate a distorted \(\beta_2\), or distort \(\beta_1\) and other coefficients as well.
When \(X\) is incomplete, the relevant imputation distribution is \(X\mid Y,Z\), not \(Y\mid X,Z\). Even if the analysis specifies a simple conditional mean for \(Y\), reversing the conditioning generally does not give a linear conditional model for \(X\). The reverse conditional distribution depends on the distribution of \(X\) as well as on the analysis model. For a nonlinear outcome relationship, that reverse distribution can change shape or spread with \(Y\); a linear imputation with constant residual spread may miss those features.
The concern is about estimating the analysis parameters, not simply reproducing the observed data’s average or variance. If the imputation model smooths away curvature, draws that are plausible on average can still weaken or alter the association that the analysis is intended to estimate. Multiple imputation represents uncertainty under the chosen imputation model; it does not make that model correctly specified.
Worked Example: A Linear Fit Hides Curvature
Consider hypothetical observations with predictor \(X\) and outcome \(Y\): \((0,10)\), \((1,13)\), and \((2,19)\). Suppose the planned analysis includes a quadratic term, but an imputation model for missing \(Y\) uses a straight line in \(X\). Compare the linear fit with the observed pattern.
The sample means are \(\bar{x}=1\) and \(\bar{y}=14\). The ordinary least-squares slope is
The intercept is \(\hat{\beta}_0=\bar{y}-\hat{\beta}_1\bar{x}=14-4.5=9.5\). The linear fitted values at \(X=0,1,2\) are therefore \(9.5,14,18.5\), with residuals \(0.5,-1,0.5\). This pattern shows that the straight line does not pass through all three observations.
The middle point is not on the line joining the endpoints. The OLS slope is still 4.5: because the predictor values are equally spaced, the middle observation contributes zero to the centered slope numerator, so the slope equals \((19-10)/(2-0)\). The quadratic curve \(Y=10+1.5X+1.5X^2\) passes through all three points: it gives \(10\), \(13\), and \(19\), respectively.
If this pattern persists in a larger hypothetical data set and some \(Y\) values are missing, a linear conditional mean for imputing \(Y\) will miss the curvature. The calculation does not establish the size of any bias in a particular study; it shows precisely what feature the linear specification fails to represent. Residual variation and the other predictors would also need to be modeled appropriately.
Why a Correct Conditional Mean May Still Be Insufficient
A linear imputation model can fail even when its predicted conditional mean is correct. This occurs because an analysis can depend on features of the conditional distribution beyond its mean, such as how its spread or shape varies across predictor values. The following example isolates that issue.
Worked Example: Correct Mean, Wrong Conditional Spread
In a hypothetical population, let \(X\) take values \(-2,-1,1,2\), each with probability \(1/4\), and define \(Y=X^2\). Imagine that \(X\) is incomplete and an imputation model predicts \(X\) from \(Y\). For each possible \(Y\), the two possible \(X\) values have equal probability: \(X=-1\) or \(1\) when \(Y=1\), and \(X=-2\) or \(2\) when \(Y=4\).
Consequently, \(E(X\mid Y=1)=0\) and \(E(X\mid Y=4)=0\). A linear conditional-mean model with prediction zero can reproduce those two conditional means. But the conditional second moments differ:
A linear regression with normally distributed residual draws centered at zero and a single residual variance would instead use the same conditional spread at both \(Y\) values. If that variance were set to the marginal \(E(X^2)=(1+1+4+4)/4=2.5\), the imputed draws would have \(E(X^2\mid Y)=2.5\) in both groups, not 1 and 4. The imputation has matched the conditional mean while failing to preserve the changing conditional spread and the two-point conditional shapes.
An analysis using \(X^2\) could then be affected: among records whose \(X\) is imputed, the imputed \(X^2\) values no longer track \(Y\) as they do in the specified population. This example is deliberately discrete and deterministic; it is a demonstration of the distributional issue, not a claim that a particular continuous-data analysis has a specific bias.
A Short Justification for the Risk to an Analysis Parameter
The link between a missed conditional mean and a regression coefficient can be seen from the population normal equations. Consider an analysis with an intercept and a nonlinear basis term \(Q(X)\), such as \(X^2\). Let \(R\) be the part of \(Q(X)\) left after its linear projection on the intercept and any other analysis predictors. In a correctly specified analysis, the population coefficient for the nonlinear term is identified using the relationship between \(R\) and the outcome after those other terms are accounted for.
If the imputation changes the conditional mean of \(Y\) by \(\delta(X,Z)\), the population score contribution for that nonlinear term changes by \(E\{R\delta(X,Z)\}\). This follows by substituting the altered conditional mean into the residual in the normal equation:
At the target parameter, the first term on the right is zero under the analysis model. If \(E\{R\delta(X,Z)\}\ne 0\), the altered data do not satisfy the same population score equation, so the fitted nonlinear coefficient can shift. If this expectation is zero, this particular first-order score perturbation vanishes; that does not prove the entire imputation procedure is valid, since the distribution, uncertainty, or other parameters may still be misspecified.
Worked Example: When the Difference May Not Create Curvature Bias
Suppose the true conditional mean is linear, \(E(Y\mid X)=\beta_0+\beta_1X\), with mean-zero errors conditional on \(X\). An analysis nevertheless includes both \(X\) and \(X^2\). Assume the conditional mean model is correct, the missingness assumptions required for the imputation are appropriate, and the imputation represents the outcome distribution sufficiently well for the analysis.
For a symmetric distribution of \(X\), the nonlinear term after removing its intercept component is \(R=X^2-E(X^2)\); it is also orthogonal to \(X\) because \(E(X^3)=0\). The residual from the true linear mean is \(\varepsilon=Y-\beta_0-\beta_1X\), with \(E(\varepsilon\mid X)=0\). Therefore,
Thus, in this setting, there is no population residual association that would force a nonzero quadratic coefficient. A linear imputation specification can be adequate for the conditional mean even though the analysis allows a quadratic term. This is a limited illustration, not a general compatibility result: an incorrect error distribution, missing nonlinear predictors of missingness, or an actually curved conditional mean could change the conclusion.
Assumptions, Diagnostics, and Alternatives
Before using a linear imputation model alongside a nonlinear analysis, make its assumptions explicit. A common linear regression imputation assumes the conditional mean is linear in the included predictors; a normal-regression version additionally uses an appropriate residual distribution and a specified variance structure. The analysis model also needs its own assumptions, including a suitable conditional mean form and adequate information in the observed data. These assumptions must be judged in context, rather than inferred from the fact that an algorithm converged.
Diagnostics should examine conditional relationships relevant to the analysis, especially across the range of predictors where values are missing. Useful checks include comparing observed-data trends with the fitted imputation mean, checking whether residual spread varies with predictors, and examining whether imputed values reproduce nonlinear terms and interactions in the analysis. When feasible, compare analysis estimates across plausible imputation specifications and inspect the estimates of the nonlinear terms, not only overall means and standard deviations.
A stable MICE trace or plausible marginal distribution is not a compatibility check. A chain can converge to draws from its specified conditional models even when those models fail to represent the analysis relationship. Likewise, a favorable diagnostic on observed cases may be weak evidence in regions with little observed information. Severe lack of overlap or sparse data can make the relevant functional form difficult to assess at all.
The main alternatives differ in how directly they preserve the planned analysis structure. A simple linear imputation is easier to estimate and may be adequate when the conditional mean truly is linear or the mismatch is immaterial for the target. Adding analysis-relevant nonlinear terms to the imputation model can represent curvature more directly, but cannot guarantee that all conditional distributions are compatible. An imputation procedure derived from the substantive model can target compatibility more directly, while requiring more modeling and computation. Complete-case analysis avoids imputing values but can lose precision and may be biased depending on the missingness and analysis conditions; it is not an automatic remedy for a functional-form mismatch.
| Approach | Potential strength | Important limitation |
|---|---|---|
| Linear imputation | Simple and potentially adequate when the relevant conditional mean and distribution are well represented | Can smooth away curvature or changing conditional spread |
| Imputation including nonlinear analysis terms | Can preserve specified curvature more directly | Still requires suitable conditional distributions and adequate data support |
| Substantive-model-compatible imputation | Builds the planned analysis structure into the imputation approach | Requires additional modeling choices and implementation care |
| Complete-case analysis | Does not rely on an imputation model | May discard information and can be biased under relevant missingness conditions |
Common Mistake and Key Takeaway
Common mistake: “The linear imputation model predicts the observed mean reasonably well, so it must be safe for the quadratic analysis.” This reasoning checks only one feature of the observed conditional distribution. It overlooks conditional spread, shape, the locations where values are missing, and whether imputed values preserve the nonlinear analysis term.
Correction: Assess whether the imputation model preserves the conditional relationships and distributional features needed for the target analysis parameters. A different functional form is a reason to investigate; its practical importance depends on the data and estimand. Good chain behavior and plausible marginals do not settle the issue, and a mismatch is not automatically proof of bias.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- In the first worked example, why is the OLS slope 4.5 even though the middle observation is not on the line joining the endpoints?
- In the symmetric example, what does the linear imputation reproduce correctly, and what conditional feature does it fail to reproduce?
- Does a converged MICE chain establish that a linear imputation model is compatible with a quadratic analysis? Explain.
- In the derivation, what does a nonzero value of \(E\{R\delta(X,Z)\}\) indicate for the nonlinear-term score equation?
- Give one circumstance in which a linear imputation may not create a systematic curvature problem, and state one limitation of that conclusion.
Solutions. 1. With equally spaced predictor values, the middle observation contributes zero to the centered slope numerator, so the slope is \((19-10)/(2-0)=4.5\); the middle point still has a nonzero residual from the fitted line. 2. It reproduces \(E(X\mid Y)=0\), but not the conditional spread or two-point shape, since \(E(X^2\mid Y)\) is 1 at \(Y=1\) and 4 at \(Y=4\). 3. No. Convergence concerns the specified chain; it does not establish that its conditional models preserve the analysis relationship. 4. The imputation perturbation changes the population score for the nonlinear term, so the target coefficient may no longer solve that score equation. 5. If the true conditional mean is linear and the imputation represents the relevant outcome distribution, an analysis that permits curvature can still have a true quadratic coefficient of zero. That conclusion can fail if the mean or distribution assumptions are wrong, or if missingness-related predictors are omitted.