Worked Ordinal-Variable Imputation

From an Ordinal Model to an Imputed Rating

A symptom rating such as none, mild, moderate, or severe has a genuine order, but the distance between adjacent levels is not necessarily equal. Imputing a missing rating therefore requires more than predicting a numerical score and rounding it. An ordinal regression model instead predicts probabilities for the ordered categories, and a random draw from those probabilities supplies an imputed category.

In MICE, the ordinal variable is modeled conditional on the other variables available for that record. At an iteration, those predictors may include observed values and current imputations of other incomplete variables. The model must represent the ordering sensibly, fit the observed data adequately, and allow variation in both the model parameters and the imputed category across completed data sets.

Definition: Let $Y$ be an ordinal variable with ordered categories $1,\ldots,K$, and let $x$ denote a vector of predictors. A proportional-odds cumulative-logit model is $$ \operatorname{logit}\{P(Y\leq k\mid x)\}=\theta_k-x^\mathsf{T}\beta,\qquad k=1,\ldots,K-1, $$ where $\theta_1<\theta_2<\cdots<\theta_{K-1}$ are ordered thresholds and $\beta$ is a vector of predictor coefficients. The model uses a common $\beta$ at every cumulative cutpoint.

The sign convention matters. With a positive coefficient for a predictor, larger predictor values reduce the cumulative probability of being at or below any given category. Equivalently, they shift probability toward higher categories. The proportional-odds assumption is that this predictor effect is the same at each cutpoint; the thresholds differ, but the slope vector does not.

Before using this model for imputation, check that the response categories have a defensible order, the cumulative-logit relationship is plausible, and the predictor effects are reasonably represented by the included terms. The model also needs enough observed information across the relevant categories and predictor patterns to estimate its parameters. As with other conditional imputation models, the predictors should contain information relevant to the missing response; inference generally relies on a missing-at-random assumption conditional on the variables included in the imputation model, together with an adequate model specification.

Deriving the Category Probabilities

Write $F_k(x)=P(Y\leq k\mid x)$, with $F_0(x)=0$ and $F_K(x)=1$. The model gives $F_k(x)$ by applying the inverse-logit function to each cumulative linear predictor. The probability of an individual category is then the difference between two adjacent cumulative probabilities.

F_k(x)=\frac{1}{1+\exp\{-(\theta_k-x^\mathsf{T}\beta)\}}, \qquad P(Y=k\mid x)=F_k(x)-F_{k-1}(x).

For example, $P(Y=1\mid x)=F_1(x)$, an interior category has probability $F_k(x)-F_{k-1}(x)$, and the highest category has probability $1-F_{K-1}(x)$. Ordered thresholds and the monotonicity of the logistic function imply $F_1(x)\leq\cdots\leq F_{K-1}(x)$. Thus the differences are nonnegative, and they sum to one:

\sum_{k=1}^{K}P(Y=k\mid x) =F_1(x)+\sum_{k=2}^{K-1}\{F_k(x)-F_{k-1}(x)\} +\{1-F_{K-1}(x)\}=1.

Worked Example: Calculating Four Symptom Probabilities

Suppose the ordered symptom rating is none, mild, moderate, or severe. For illustration, assume the fitted thresholds are $\theta_1=-1.2$, $\theta_2=0.3$, and $\theta_3=1.4$, and the fitted predictor contribution for a particular record is $x^\mathsf{T}\beta=0.4$. These are hypothetical fitted values, not estimates from a real study.

The cumulative linear predictors are $-1.2-0.4=-1.6$, $0.3-0.4=-0.1$, and $1.4-0.4=1.0$. Applying the inverse-logit gives:

F_1=\operatorname{logit}^{-1}(-1.6)=0.167982,\quad F_2=\operatorname{logit}^{-1}(-0.1)=0.475021,\quad F_3=\operatorname{logit}^{-1}(1.0)=0.731059.

Therefore the category probabilities are:

\begin{aligned} P(\text{none}\mid x)&=0.167982,\\ P(\text{mild}\mid x)&=0.475021-0.167982=0.307039,\\ P(\text{moderate}\mid x)&=0.731059-0.475021=0.256038,\\ P(\text{severe}\mid x)&=1-0.731059=0.268941. \end{aligned}

The probabilities sum to $1.000000$, apart from any numerical rounding. In this profile, severe is not the most probable individual category; mild has the largest probability. Nevertheless, the model assigns substantial probability to all four levels, so choosing the most probable category alone would discard meaningful uncertainty.

Drawing Categories for Missing Records

A category imputation is stochastic, not a deterministic assignment to the category with the largest probability. Given category probabilities $p_1,\ldots,p_K$, draw $U$ from the uniform distribution on $[0,1)$. Assign category $k$ if $U$ falls between the cumulative sums $p_1+\cdots+p_{k-1}$ and $p_1+\cdots+p_k$. This gives each category its specified probability over repeated draws.

Worked Example: Turning Uniform Draws into Ratings

Use the probabilities from the previous example for four hypothetical records whose predictor contributions are all $0.4$. Suppose the independent uniform draws for these records are $0.10$, $0.32$, $0.62$, and $0.90$. The cumulative category boundaries are $0.167982$, $0.475021$, $0.731059$, and $1$.

For $U=0.10$, the draw is below $0.167982$, so the imputed rating is none. For $U=0.32$, it lies between $0.167982$ and $0.475021$, so the rating is mild. For $U=0.62$, it lies between $0.475021$ and $0.731059$, so the rating is moderate. For $U=0.90$, it exceeds $0.731059$, so the rating is severe.

The four imputations differ even though their modeled probabilities are identical. That is expected: an imputation is a random draw from a conditional distribution, not a claim that the selected category is known to be the record’s true value. In actual MICE, the probabilities and the random category draw are generated at each relevant update.

For multiple imputation, the model parameters should also vary across completed data sets. A practical proper-imputation procedure draws a plausible parameter vector, including ordered thresholds, from an appropriate parameter distribution based on the observed data; it then calculates the probabilities using that draw and samples a category. The precise parameter-drawing method depends on the fitting procedure. Reusing one fixed coefficient and threshold estimate in every imputation omits parameter uncertainty and can make completed-data results too certain.

Fit the conditional ordinal model.
Use records with observed symptom ratings and the predictors in the current MICE iteration to estimate thresholds and slopes.

Draw model parameters.
For each imputation, generate a plausible parameter draw that respects the ordering of the thresholds.

Calculate each missing record’s probabilities.
Evaluate the cumulative probabilities at that record’s predictor values, then take adjacent differences.

Draw and update the missing rating.
Use a uniform draw and the cumulative category boundaries to select one ordered category.

Worked Example: Interpreting a Predictor Effect

Suppose a hypothetical model includes age measured in 10-year increments, with coefficient $\beta_{\text{age}}=0.4$. Holding the other predictors fixed, a one-unit increase in this predictor means an increase of 10 years. Under the stated sign convention, the cumulative log-odds of a rating at or below any cutpoint decrease by $0.4$.

For any cutpoint $k$, the odds of being above that cutpoint rather than at or below it are the reciprocal of the cumulative odds. Thus increasing age by 10 years multiplies those upper-category odds by:

\exp(0.4)=1.492.

In context, conditional on the other included predictors, the model associates a 10-year age increase with 1.492 times the odds of a rating above rather than at or below each cutpoint. This does not mean the probability of severe symptoms rises by 49.2 percentage points, and it does not establish that age causes a change in symptoms.

The direction can also be checked numerically. With the thresholds in the first example, changing the predictor contribution from $0.4$ to $-0.2$ gives cumulative probabilities $\operatorname{logit}^{-1}(-1.0)=0.268941$, $\operatorname{logit}^{-1}(0.5)=0.622459$, and $\operatorname{logit}^{-1}(1.6)=0.832018$. The resulting probabilities are none $0.268941$, mild $0.622459-0.268941=0.353518$, moderate $0.832018-0.622459=0.209559$, and severe $1-0.832018=0.167982$. Compared with the profile at $0.4$, this lower predictor contribution has more probability in the lower categories, as the positive coefficient implies.

Diagnostics, Alternatives, and Limits

A fitted model can produce valid probabilities mathematically yet still be a poor imputation model. Check whether observed category frequencies and fitted conditional probabilities are broadly compatible, including across important predictor groups. Examine whether fitted probabilities are extreme, whether some categories are rarely represented for particular predictor patterns, and whether estimation warnings or unstable thresholds occur. The earlier material on sparse ordinal categories is especially relevant when a level has few observed cases.

The proportional-odds assumption can be assessed by checking whether the predictor effects appear reasonably similar across cumulative cutpoints. Strong departures may indicate that the common-slope structure is too restrictive. Also examine whether important nonlinear predictor patterns or interactions are missing. A model that cannot reproduce relationships among the symptom rating and variables used in the substantive analysis can distort those relationships after imputation.

The proportional-odds model is a natural choice when levels are ordered and a common cumulative effect is plausible. Other ordinal link structures, such as adjacent-category or continuation-ratio models, may suit different scientific processes; their choice should reflect the response and the data, rather than be made solely to obtain a convenient fit. Treating an ordinal rating as continuous is another possible approximation, but it imposes numerical spacing and can generate values that are not valid categories unless additional steps are taken. If the response categories have no defensible order, an ordinal model is inappropriate; a nominal-variable model is the relevant alternative.

No one model resolves every limitation. Very sparse levels, separation-like patterns, or weak overlap in predictor values can make probabilities unreliable. Collapsing adjacent categories may sometimes be defensible, but it changes the response scale and should have substantive justification rather than being a purely numerical fix. The imputation model also depends on the included predictors and the missing-at-random condition; random category draws do not correct omitted predictors or departures from that assumption.

Common Mistake

Replacing a random draw with the most likely category. Assigning the category with the largest fitted probability to every missing record creates deterministic, overconfident imputations and can understate variability. Use a draw from the full category distribution, and allow model-parameter uncertainty across imputations.
Subtracting rounded cumulative probabilities and reporting excess precision. If rounded cumulative values are subtracted, the displayed difference may not equal the difference calculated before rounding. Keep adequate precision through the calculation, then round the final probability consistently.
Reading the coefficient as a change in category probability. The coefficient is a change in cumulative log-odds under the model. Calculate predicted category probabilities to describe probability changes for a specified predictor profile.

Key takeaway: An ordinal MICE update maps a record’s predictors to ordered cumulative probabilities, differences those probabilities to obtain category probabilities, and draws one category. A sound implementation also represents parameter uncertainty and checks that the proportional-odds model is plausible for the data.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

For a three-category ordinal response, suppose the fitted cumulative probabilities at a record’s predictors are $F_1=0.25$ and $F_2=0.70$. What are the three category probabilities?
Using the probabilities in Question 1, which category is selected when $U=0.60$?
A predictor coefficient is $-0.3$ under the convention $\operatorname{logit}\{P(Y\leq k\mid x)\}=\theta_k-x^\mathsf{T}\beta$. What is the direction of its association with higher categories as that predictor increases?
Why can imputing the most probable category for every missing rating be problematic?
Name one diagnostic concern that would make a proportional-odds imputation model questionable.

Solutions. (1) The probabilities are $P(Y=1)=F_1=0.25$, $P(Y=2)=F_2-F_1=0.70-0.25=0.45$, and $P(Y=3)=1-F_2=0.30$; they sum to $1$. (2) The cumulative boundaries are $0.25$, $0.70$, and $1$. Since $0.25<0.60\leq0.70$, select category 2. (3) A negative coefficient means that as the predictor increases, $-x^\mathsf{T}\beta$ increases, raising the cumulative log-odds of being at or below each cutpoint. The model therefore shifts probability toward lower, not higher, categories. (4) It removes the stochastic variation represented by the fitted distribution and can understate uncertainty; draw a category using all category probabilities instead. (5) Examples include markedly different predictor effects at different cutpoints, unstable estimates caused by sparse categories, or fitted probabilities that poorly match observed conditional patterns.