Why the Ordinal Outcome Must Inform Predictor Imputation
When a predictor is incomplete but the outcome is observed, its imputation model should use the outcome to help predict the missing predictor values. For an ordinal outcome, the categories have an order, but their numerical labels do not generally represent equal distances. Treating categories such as mild, moderate, and severe as the numbers 1, 2, and 3 imposes an extra assumption: that the outcome’s effect on the predictor follows a linear trend across those scores. That assumption need not be appropriate.
The goal here is narrower than building a complete MICE workflow. We focus on the conditional relationship required when the analysis model is ordinal: how the probability of each outcome category depends on a predictor, and how that relationship informs the distribution of the predictor given the observed outcome. This is a specific application of the earlier material on compatibility of conditional imputation models and on why the outcome belongs in predictor imputation.
The Analysis Model and Its Implied Predictor Distribution
For a three-category ordinal outcome \(Y\), coded \(1<2<3\), consider a proportional-odds cumulative-logit model with predictor \(X\) and other fully observed covariates \(Z\):
The cutpoints satisfy \(\alpha_1<\alpha_2\). The common coefficient \(\beta\) means that the same slope applies at both cumulative splits: \(Y\leq1\) versus \(Y>1\), and \(Y\leq2\) versus \(Y>2\). With this sign convention, a positive \(\beta\) shifts probability toward higher outcome categories as \(X\) increases. This model assumes the cumulative-logit form is suitable, the cutpoints are ordered, the proportional-odds assumption holds, and observations are conditionally independent under the specified model. A different analysis model requires its own corresponding conditional likelihood.
Write \(F(t)=1/(1+\exp(-t))\), and define \(q_k(x,z)=F(\alpha_k-\beta x-\gamma^\mathsf{T}z)\), with \(q_0=0\) and \(q_3=1\). The probability of an individual category is the difference between adjacent cumulative probabilities:
For an incomplete predictor, Bayes’ rule gives the key conditional relationship. If \(p(x\mid z)\) is the predictor distribution given the other covariates, then the conditional density of \(X\) given the observed outcome and covariates is proportional to the analysis-model category probability times that predictor distribution:
For discrete \(X\), replace densities by probabilities and normalize across its possible values. For continuous \(X\), the proportional expression is a density kernel that must be normalized by integrating over the possible values of \(x\). This formula shows why simply imputing \(X\) without \(Y\) can lose an important association.
Worked Conditional Calculations
Worked Example: A Binary Predictor Given the Highest Outcome Category
Suppose \(X\) is binary, with \(P(X=1)=0.30\), and an ordinal model has \(\alpha_1=-0.5\), \(\alpha_2=0.8\), and \(\beta=0.8\), with no additional covariates. Assume these values describe the relevant conditional probabilities. Find \(P(X=1\mid Y=3)\).
For \(X=0\), the probability of the highest category is \(P(Y=3\mid X=0)=1-F(0.8)=0.3100255\). For \(X=1\), the upper cumulative probability is \(P(Y\leq2\mid X=1)=F(0.8-0.8)=0.5\), so \(P(Y=3\mid X=1)=0.5\).
Bayes’ rule gives \(P(X=1\mid Y=3)=P(X=1)P(Y=3\mid X=1)/P(Y=3)\). Substituting both possible values of \(X\) into the denominator:
Thus, under these hypothetical probabilities, 40.87% of individuals in category 3 have \(X=1\). This is a conditional probability, not a causal effect. The calculation also illustrates why the imputation model needs outcome-category information: the observed category changes the relative support for the two predictor values.
Worked Example: The Category Likelihood for a Continuous Predictor
Use the same cutpoints, but let \(X\) be continuous with \(X\mid Z\sim N(0,1)\). For a person in outcome category 2, compare the unnormalized conditional density at \(X=1\) with that at \(X=0\). This calculation checks the proportional-odds form and the stated normal predictor model; it is not itself a full draw from the conditional distribution.
At \(X=0\), the category-2 probability is the difference between the two cumulative probabilities: \(F(0.8)-F(-0.5)=0.6899745-0.3775407=0.3124338\). At \(X=1\), the cumulative logits are \(0\) and \(-1.3\), so \(P(Y=2\mid X=1)=F(0)-F(-1.3)=0.5000000-0.2141650=0.2858350\).
The category likelihood ratio is therefore \(0.2858350/0.3124338\approx0.91486\). The normal-density ratio is \(\phi(1;0,1)/\phi(0;0,1)=\exp(-1/2)\approx0.6065307\). Multiplying the two gives the ratio of unnormalized conditional densities:
This ratio is less than one: at these two points, the conditional density kernel is lower at \(X=1\) than at \(X=0\). It is not a probability ratio for two point values of a continuous variable. The full conditional density is proportional to the product of the normal density and the category-2 likelihood across all \(x\), and requires normalization.
From the Conditional Relationship to a Practical MICE Model
The exact Bayes conditional can be awkward to sample from, particularly for a continuous incomplete predictor. A practical chained-equations model should nevertheless retain the outcome’s category-specific information. For an imputation model that predicts a continuous \(X\), a common starting point is a regression with indicator variables for the outcome categories, rather than a single numeric outcome score. This allows the conditional mean of \(X\) to differ by category without forcing equal changes from one category to the next.
Such a regression is an approximation unless its distributional form matches the conditional density implied by the analysis model and the predictor distribution. For example, normal linear imputation with a category factor assumes normally distributed residuals with the specified variance structure; the exact density proportional to \(P(Y=y\mid X=x,Z=z)p(x\mid Z=z)\) need not be normal. Including the outcome as a factor is therefore an important way to preserve category distinctions, but it does not by itself prove exact compatibility.
Worked Example: Imputing a Continuous Predictor With Outcome Categories as Factors
Suppose the observed hypothetical predictor values are as follows. Each category has three observations, and the imputation model is a normal linear regression of \(X\) on an intercept and indicators for categories 2 and 3.
| Ordinal outcome \(Y\) | Observed \(X\) values | Category mean |
|---|---|---|
| 1 | 2, 4, 6 | 4 |
| 2 | 1, 3, 5 | 3 |
| 3 | 0, 2, 4 | 2 |
The fitted mean for category 1 is the reference mean, so \(\hat{\gamma}_0=4\). The category-2 contrast is \(3-4=-1\), and the category-3 contrast is \(2-4=-2\). Thus the fitted means are \(4\), \(4-1=3\), and \(4-2=2\), respectively. These estimates happen to form an evenly spaced trend, but the factor model did not require that spacing; its two category contrasts are estimated separately.
Within each category, the deviations from the fitted mean are \(-2,0,2\), with a sum of squared deviations \(4+0+4=8\). Across three categories, the residual sum of squares is \(3(8)=24\). There are 9 observations and 3 fitted coefficients, so \(df=9-3=6\), and the estimated residual variance is \(s^2=24/6=4\), giving residual standard deviation \(s=2\).
For an incomplete \(X\) with observed \(Y=2\), the fitted conditional mean is 3. Under this illustrative plug-in normal model, \(X\mid Y=2\) is represented as \(N(3,4)\). A predictive draw can be written \(X=3+2Z^\ast\), where \(Z^\ast\sim N(0,1)\). If one illustrative random draw has \(Z^\ast=0.5\), then \(X=3+2(0.5)=4\). In actual multiple imputation, model parameters and residual variation should be handled so that imputations reflect uncertainty; this single draw only demonstrates the conditional model’s use of the outcome category.
The normal conditional form and equal residual variance are assumptions of this practical imputation model, not consequences of the ordinal analysis model. They should be assessed against the observed predictor distributions. If they are poor approximations, a more suitable conditional distribution or a model that samples from the exact conditional kernel may be needed.
Model Choice, Diagnostics, and Limitations
The key comparison is between preserving the ordered outcome as a predictor and imposing an unnecessary numerical trend. Including \(Y\) as a numeric score in a linear predictor restricts category effects to a particular trend. Including it as a factor allows separate contrasts. Neither choice should be confused with fitting the analysis model for \(Y\): the imputation model predicts an incomplete predictor, while the ordinal analysis model predicts \(Y\).
| Approach | What it represents | Main caution |
|---|---|---|
| Outcome entered as a numeric score | A constrained trend across category scores | May impose unjustified spacing or linearity |
| Outcome entered as a categorical factor | Separate predictor relationships by observed outcome category | May be less stable with sparse categories; not automatically exact |
| Conditional distribution derived from the analysis model | The category likelihood multiplied by the predictor distribution given other covariates | May require specialized computation and a well-specified predictor distribution |
Check whether imputed values remain plausible and whether their distributions differ appropriately across the observed outcome categories. Compare observed and imputed predictor distributions within each category, inspect category-specific summaries and ranges, and monitor chain behavior using the diagnostic principles established earlier in the course. Refit the ordinal analysis model to the completed data and check whether the intended outcome–predictor relationship has been retained. Sparse outcome categories can make factor effects unstable; extreme imputations, implausible category-specific patterns, or chains that do not settle are signs to investigate the model rather than simply increase the number of imputations.
These checks cannot prove that the imputation model is correct. The exact conditional depends on the distribution \(p(X\mid Z)\), and conclusions also rely on the missingness and model assumptions used for the analysis. When the predictor is missing in patterns that leave little information in some outcome categories, the data may not support reliable category-specific estimates. Sensitivity to reasonable imputation-model choices is then important.
A Short Derivation and a Common Pitfall
The conditional formula follows directly from the definition of conditional probability. For continuous \(X\), the joint density of \(X\) and an observed category \(Y=y\), given \(Z=z\), factors in either order:
Since \(P(y\mid z)\) does not vary with \(x\), rearrangement yields \(p(x\mid y,z)\propto p(y\mid x,z)p(x\mid z)\). For a discrete predictor, the same argument applies to probability masses, and normalization is obtained by summing the product over all possible predictor values. The derivation makes clear that neither the analysis-model likelihood nor the predictor distribution alone is the complete conditional needed for imputation.
Common mistake: “The outcome is ordinal, so I will code its categories 1, 2, and 3 and enter that score linearly in the imputation model. The ordering is preserved, so the model is appropriate.” The ordering is preserved, but equal spacing and a linear predictor effect have also been imposed. Unless those restrictions are justified, use category indicators or another imputation model that represents the relevant category-specific relationships. Then check whether its distributional assumptions are adequate for the predictor being imputed.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In the proportional-odds model given above, what does a positive \(\beta\) imply about the probability of higher outcome categories as \(X\) increases?
- Why is \(P(X\mid Y,Z)\) not generally determined by \(P(Y\mid X,Z)\) alone?
- For an incomplete continuous predictor, what restriction may be imposed by coding a three-category ordinal outcome as the numerical score 1, 2, 3 in a linear imputation model?
- Name one diagnostic that could reveal a poor predictor imputation model when the outcome is ordinal.
Solutions. (1) Under the stated sign convention, increasing \(X\) lowers the cumulative probabilities \(P(Y\leq k\mid X,Z)\), shifting probability toward higher categories. (2) Bayes’ rule also requires the predictor distribution \(p(X\mid Z)\); the conditional predictor density is proportional to the outcome-category likelihood times that distribution. (3) It imposes a linear trend across the numerical scores, which can constrain category effects and treat category spacing as meaningful. Separate category indicators avoid that particular restriction. (4) For example, compare observed and imputed predictor distributions within each observed outcome category; implausible differences or ranges can indicate misspecification.