Alternative Link Structures For Ordinal Imputation

Why the Link Structure Matters

An ordinal response has categories with a meaningful order, such as mild, moderate, and severe. A model for such a response can use that order in different ways. It might model the probability of being at or below each threshold, compare the odds of two neighboring categories, or model the chance of progressing past a category among people who have reached it. These are distinct conditional probability structures, not merely different labels for the same calculation.

In MICE, the ordinal model is used to draw plausible values for a variable with missing entries, conditional on other variables in the imputation model. The link structure affects those draws. A useful choice therefore reflects both the ordering of the categories and the process the model is intended to represent.

The categories should be mutually exclusive and collectively exhaustive for the response being modeled, and their order must be substantively meaningful. Each of the structures below also assumes that its chosen link and predictor effects adequately describe the relevant conditional probabilities. These assumptions concern the response model; choosing a link does not by itself establish the missingness assumptions required for valid multiple imputation.

Cumulative Models: Probabilities at Ordered Thresholds

A cumulative model describes the probability of being at or below each cut point. For \(j=1,\ldots,K-1\), define \(F_j(x)=P(Y\leq j\mid x)\). A cumulative logit model may be written

Here \(\alpha_j\) is the intercept for cut point \(j\). In the proportional-odds version, the slope vector \(\beta\) is shared across cut points. The minus sign in this convention means that a positive coefficient for a predictor is associated with lower cumulative odds \(P(Y\leq j\mid x)/P(Y>j\mid x)\), holding the other predictors fixed. This is the proportional-odds structure treated in the earlier material on proportional-odds models.

The category probabilities are obtained by differencing consecutive cumulative probabilities: \(P(Y=1\mid x)=F_1(x)\), \(P(Y=j\mid x)=F_j(x)-F_{j-1}(x)\) for interior categories, and \(P(Y=K\mid x)=1-F_{K-1}(x)\). Thus the fitted cumulative probabilities must be nondecreasing in \(j\); otherwise differencing could produce a negative category probability.

Adjacent-Category Models: Comparing Neighbors

An adjacent-category model directly compares the probabilities of neighboring categories. One common form is

The slope may vary by pair of neighboring categories. If a common slope \(\beta\) is imposed, a one-unit predictor increase multiplies each adjacent-category odds \(P(Y=j+1\mid x)/P(Y=j\mid x)\) by the same factor, conditional on the model. Unlike a cumulative model, the basic comparison is between a pair of categories, not between one side of a threshold and the other.

The adjacent logits are not themselves the category probabilities. To obtain probabilities, set \(r_j=\exp(\alpha_j+x^\mathsf{T}\beta_j)\), the modeled ratio of category \(j+1\) to category \(j\). Set \(w_1=1\), then recursively define \(w_{j+1}=w_jr_j\). The resulting probabilities are \(w_k/\sum_{\ell=1}^{K}w_\ell\). This normalization ensures that the probabilities sum to one.

Continuation-Ratio Models: Progression Through Categories

A forward continuation-ratio model treats the categories as stages in an ordered progression. At stage \(j\), it models the probability of moving beyond category \(j\), conditional on having reached category \(j\) or higher:

The condition \(Y\geq j\) matters: for \(j>1\), \(q_j(x)\) is generally not the unconditional probability \(P(Y>j\mid x)\). At \(j=1\), however, all possible categories satisfy \(Y\geq1\), so \(q_1(x)=P(Y>1\mid x)\). A common-slope version uses the same \(\delta\) at every stage; stage-specific slopes allow predictor associations to differ as the response progresses.

For this forward form, category 1 has probability \(1-q_1(x)\). For an interior category \(j\), the response must first progress beyond stages \(1,\ldots,j-1\), then stop at \(j\). The final category requires progression beyond every earlier stage.

Why the Probability Constructions Work

The adjacent-category normalization and the continuation-ratio construction each produce a proper probability distribution when their component quantities are finite and the resulting category weights are positive. The normalization argument for adjacent categories follows directly from the modeled ratios.

Thus normalization preserves each adjacent odds ratio while making the probabilities sum to one. For continuation ratios, define the probability of reaching stage \(j\) as \(S_j=\prod_{h=1}^{j-1}q_h(x)\), with \(S_1=1\). Then the probability of stopping at \(j<K\) is \(S_j(1-q_j)=S_j-S_{j+1}\), while the final category has probability \(S_K\). Summing these terms telescopes:

These identities justify the probability calculations; they do not show that any particular link structure fits a real response well. That is a modeling question requiring substantive judgment and diagnostics.

Choosing Among the Structures

The models are most useful when their conditioning matches the question being asked. A cumulative model is natural when interest concerns crossing ordered thresholds, such as being at least moderate rather than mild. An adjacent-category model is useful when the substantive comparisons are between neighboring levels. A continuation-ratio model is appropriate when categories can be understood as sequential stages, with the possibility of stopping at each stage.

An ordinal scale alone does not establish a progression process. For example, categories that are severity ratings may support threshold comparisons without implying that every person passes through each rating. Conversely, if categories genuinely represent sequential decisions or stages, cumulative or adjacent comparisons may not express the process of interest as directly. The nominal multinomial model is an alternative when the ordering is not meaningful, but it does not use the order in the same way.

Diagnostics, Limitations, and Common Mistakes

Check that fitted probabilities are valid across the predictor values used for imputation: they must be between zero and one and sum to one. For a cumulative model, also check that the fitted cumulative probabilities are nondecreasing across cut points. Compare observed and predicted category frequencies across relevant predictor patterns, and examine whether important groups or ranges of predictors are systematically under- or over-predicted. These checks can reveal poor fit, but no single diagnostic proves a link structure is correct.

A model can also be unstable when data contain little information about some category comparisons or stages. Warning signs include very large coefficient estimates, convergence problems, and fitted probabilities close to zero or one for many records. Such problems can make imputed draws unreliable. The chosen model also remains an approximation: a simple common-slope restriction may be inappropriate if associations differ by threshold, neighboring pair, or stage.

• Mistake: treating a continuation ratio as an unconditional probability. The claim \(q_j(x)=P(Y>j\mid x)\) generally ignores the condition \(Y\geq j\). The correction is to retain the stage condition; at \(j=1\), that condition covers the full support, but at later stages it generally does not.
• Mistake: using adjacent logits as category probabilities. A log odds for categories \(j+1\) and \(j\) is a ratio, not either category's probability. Recover all category weights and normalize them.
• Mistake: choosing a model solely because the response is ordered. Order permits ordinal modeling but does not say whether thresholds, neighboring comparisons, or progression stages are the right structure. Choose based on the response process and check predictive behavior.

In MICE, the model must also be useful as a conditional imputation model: it should include relevant predictors and preserve the relationships needed for the planned analysis. A better-fitting link cannot compensate for omitted predictors or inappropriate imputation-model assumptions, and link choice alone does not resolve uncertainty about the missingness mechanism.