Delta Adjustments for Ordinal Outcomes in Missing-Data Models

Why Shift Cumulative Logits?

An ordinal outcome has categories with a meaningful order, such as none, mild, moderate, and severe symptoms. The gaps between categories need not be equal, so changing a numerical mean is generally not a complete description of how the outcome distribution changes. In a pattern-mixture sensitivity analysis, a delta adjustment instead modifies the assumed distribution of outcomes that are missing, relative to a reference distribution estimated under missing at random (MAR).

For an ordinal outcome, the cumulative-logit scale provides a direct way to shift that reference distribution. A single delta can move probability toward lower or higher categories while retaining a coherent ordered probability distribution. The shift is a sensitivity assumption: the observed data do not identify the outcomes that are missing or determine the appropriate delta.

Definition: Let $Y$ have ordered categories $1,\ldots,K$, and let $F_k^{\text{MAR}}(x)=P(Y\leq k\mid x,\text{missing};\text{MAR})$, for $k=1,\ldots,K-1$, be the MAR reference cumulative probabilities at covariate value $x$. A shared cumulative-logit delta adjustment is $$ \operatorname{logit}\{F_k^\delta(x)\} = \operatorname{logit}\{F_k^{\text{MAR}}(x)\}+\delta, \qquad k=1,\ldots,K-1, $$ where $\operatorname{logit}(u)=\log\{u/(1-u)\}$ and $\operatorname{expit}(v)=1/(1+e^{-v})$. In this tutorial, larger category numbers are defined to represent more favorable outcomes; therefore, positive $\delta$ shifts the missing-outcome distribution toward lower, less favorable categories. The MAR reference corresponds to $\delta=0$.

The shift is applied to the missing-outcome reference distribution, not to observed outcomes. It can be used at a specified covariate value or within a missingness pattern, with the MAR probabilities supplied by the chosen ordinal model. The proportional-odds form is one common way to obtain those probabilities: it models each cumulative logit using ordered thresholds and shared covariate effects. The delta adjustment adds the same value to each cumulative logit, leaving the covariate effects unchanged.

For every threshold $k$, the adjusted cumulative odds are the MAR cumulative odds multiplied by $e^\delta$: $$ \frac{F_k^\delta(x)}{1-F_k^\delta(x)} = e^\delta \frac{F_k^{\text{MAR}}(x)}{1-F_k^{\text{MAR}}(x)}. $$ Thus, $\delta$ is a log cumulative-odds shift, not a fixed change in probability. The probability change depends on the starting cumulative probability: the same delta does not add the same number of percentage points at every threshold.

From Cumulative Probabilities to Categories

A cumulative probability $F_k=P(Y\leq k)$ combines categories $1$ through $k$. To recover probabilities for individual categories, take differences between adjacent cumulative probabilities. Write $F_0^\delta=0$ and $F_K^\delta=1$; then

P(Y=j\mid x,\text{missing};\delta) = F_j^\delta(x)-F_{j-1}^\delta(x), \qquad j=1,\ldots,K.

The first category has probability $F_1^\delta$, an interior category $j$ has probability $F_j^\delta-F_{j-1}^\delta$, and the last category has probability $1-F_{K-1}^\delta$. A valid cumulative-logit model has ordered cumulative logits, $\ell_1\leq\ell_2\leq\cdots\leq\ell_{K-1}$. Adding the same delta preserves that order, and the expit function is increasing. Therefore the adjusted cumulative probabilities remain ordered, making all category probabilities nonnegative. They also sum to one because the differences telescope.

Worked Calculations

Worked Example: Shifting a Three-Category Distribution

Suppose a hypothetical MAR model gives cumulative logits $-1.0$ and $0.5$ for a covariate profile, so the corresponding cumulative probabilities for categories 1 and 2 are $\operatorname{expit}(-1.0)$ and $\operatorname{expit}(0.5)$. Consider $\delta=0.4$. Since higher categories are more favorable, this positive shift represents a sensitivity assumption of worse outcomes among people whose outcomes are missing.

Add $0.4$ to each MAR cumulative logit. The adjusted logits are $-0.6$ and $0.9$, giving

F_1^\delta=\operatorname{expit}(-0.6)\approx0.3543437, \qquad F_2^\delta=\operatorname{expit}(0.9)\approx0.7109495.

The category probabilities, calculated from these unrounded cumulative probabilities, are $P(Y=1)\approx0.3543$, $P(Y=2)\approx0.7109495-0.3543437=0.3566$, and $P(Y=3)\approx1-0.7109495=0.2891$. The displayed values sum to $0.3543+0.3566+0.2891=1.0000$. The shifted cumulative probabilities are larger than their MAR values, which is consistent with more probability accumulating in the lower categories.

Worked Example: Four Categories and Careful Rounding

For a second hypothetical profile, suppose the MAR cumulative logits are $-1.5,-0.3,$ and $0.9$. Apply $\delta=0.4$, producing adjusted logits $-1.1,0.1,$ and $1.3$. The corresponding unrounded cumulative probabilities are approximately $0.2497399,0.5249792,$ and $0.7858350$.

Subtract adjacent unrounded cumulative probabilities to obtain the four category probabilities: $P(Y=1)\approx0.2497399$; $P(Y=2)\approx0.5249792-0.2497399=0.2752393$; $P(Y=3)\approx0.7858350-0.5249792=0.2608558$; and $P(Y=4)\approx1-0.7858350=0.2141650$. Rounded to four decimal places, these are $0.2497,0.2752,0.2609,$ and $0.2142$, which sum to $1.0000$.

This calculation illustrates why category probabilities should be obtained from unrounded cumulative values. Subtracting rounded values such as $0.5250-0.2497$ can produce a displayed difference that does not match the correctly rounded category probability.

Worked Example: Interpreting the Probability Redistribution

Consider a hypothetical three-category symptom outcome, where 1 is severe, 2 is moderate, and 3 is mild. Suppose the MAR cumulative probabilities for a missing participant with a given profile are $F_1^{\text{MAR}}=0.20$ and $F_2^{\text{MAR}}=0.65$. Use $\delta=0.5$, an assumption that shifts toward more severe outcomes.

The MAR cumulative odds are $0.20/0.80=0.25$ and $0.65/0.35\approx1.8571$. Multiplying each by $e^{0.5}\approx1.6487$ gives adjusted odds of approximately $0.4122$ and $3.0619$. Converting odds $o$ to probability $o/(1+o)$ gives $F_1^\delta\approx0.2919$ and $F_2^\delta\approx0.7538$. Therefore the adjusted category probabilities are approximately $0.2919$, $0.7538-0.2919=0.4619$, and $1-0.7538=0.2462$. The displayed values sum to $1.0000$.

The probability of severe symptoms rises from $0.20$ to about $0.2919$, and the probability of severe or moderate symptoms rises from $0.65$ to about $0.7538$. If categories are assigned equally spaced scores $1,2,3$, the adjusted expected score is approximately $1(0.2919)+2(0.4619)+3(0.2462)=1.9543$, compared with the MAR expected score $1(0.20)+2(0.45)+3(0.35)=2.1500$. The lower adjusted score reflects the shift toward worse categories. Because ordinal scores need not be equally spaced in meaning, the category probabilities are the primary interpretation; the expected score depends on the chosen numerical coding.

Why Middle Categories Can Move Either Way

A positive delta increases every cumulative probability $F_k^\delta$, but it does not follow that every individual category probability increases. The first category grows because its probability is $F_1^\delta$. The last category shrinks because its probability is $1-F_{K-1}^\delta$. An interior category is the difference between two cumulative probabilities, and both terms change.

For a concise justification, let $\ell_k=\operatorname{logit}(F_k^{\text{MAR}})$. Since $F_k^\delta=\operatorname{expit}(\ell_k+\delta)$, differentiating with respect to $\delta$ gives $dF_k^\delta/d\delta=F_k^\delta(1-F_k^\delta)$. Hence an interior category has derivative

\frac{d}{d\delta}P(Y=j\mid x,\text{missing};\delta) = F_j^\delta(1-F_j^\delta) - F_{j-1}^\delta(1-F_{j-1}^\delta).

The difference can be positive, zero, or negative, depending on the adjacent cumulative probabilities. The adjustment moves the cumulative distribution toward lower categories, but it need not move every category’s probability in the same direction. This distinction matters when reporting results: describe changes in the relevant category or cumulative probabilities rather than claiming that all less favorable categories necessarily gain probability.

Assumptions, Diagnostics, and Alternatives

The adjustment has a precise interpretation only when its assumptions are made explicit. Categories must be mutually exclusive, exhaustive, and ordered in the stated direction. The MAR reference model must provide sensible cumulative probabilities for the covariate profiles and missingness pattern to which the delta is applied. If that reference is a proportional-odds model, its common covariate-effect structure must be a reasonable representation of the reference distribution. The shared delta additionally assumes the same log-odds displacement at every cumulative threshold.

The delta itself is not estimated from the observed outcomes: it represents departures from the MAR reference that cannot be determined from those outcomes alone. Analysts should therefore present results over a scientifically justified range of delta values, including $\delta=0$ as the MAR reference, and explain what each value means on the cumulative-odds scale. A positive value here means worse outcomes only because higher category numbers were defined as more favorable. If the ordering is reversed, the sign interpretation reverses as well.

Check category construction. If categories are not genuinely ordered, a cumulative-logit shift is not appropriate. Reversing the order without revising the sign interpretation also reverses what positive delta means.
Check cumulative probabilities. Verify that the MAR cumulative probabilities increase with $k$, and that shifted category probabilities are nonnegative and sum to one. These checks can reveal coding, threshold-order, or subtraction errors.
Check the reference model. Poor fit, sparse categories, or implausible predictions for relevant covariate profiles can make a delta adjustment misleading even when the adjustment arithmetic is correct.
Check the shift assumption. A shared delta presumes a common displacement across thresholds. If that is not clinically credible, a different sensitivity model may be needed; threshold-specific shifts require particular care to maintain ordered cumulative probabilities.

This approach differs from log-odds delta adjustments for binary outcomes because it shifts multiple cumulative logits and then reconstructs all category probabilities. A risk-difference adjustment instead changes a binary probability on the probability scale. Neither binary approach directly describes the redistribution across several ordered categories. Other choices include specifying a different reference distribution or using an alternative ordinal model; those choices change the sensitivity assumptions and should not be treated as interchangeable with a cumulative-logit shift.

Common pitfall: A positive delta does not mean “add delta percentage points to every category probability.” It adds delta to each cumulative logit, multiplying cumulative odds by $e^\delta$. Recover individual category probabilities by differencing the shifted cumulative probabilities, using unrounded values. The resulting middle-category probabilities may rise or fall.

Check Your Understanding

Use the stated category order and show how the cumulative probabilities lead to your answer.

A three-category outcome has MAR cumulative probabilities $0.30$ and $0.70$. With $\delta=-0.2$, what are the adjusted cumulative probabilities and category probabilities, approximately?
Four adjusted cumulative logits are not needed here: for a four-category outcome, the adjusted cumulative logits are $-2,0,$ and $2$. Calculate the four category probabilities to four decimal places.
Why can a positive delta increase the first-category probability but decrease the probability of an interior category?
State two assumptions or checks that matter when applying this adjustment to missing ordinal outcomes.

Solutions. (1) Apply the shift to the cumulative logits, not directly to the probabilities. The MAR logits are $\operatorname{logit}(0.30)\approx-0.8473$ and $\operatorname{logit}(0.70)\approx0.8473$. After adding $-0.2$, they are approximately $-1.0473$ and $0.6473$, so the adjusted cumulative probabilities are approximately $0.2597443$ and $0.6564013$. Calculating the category probabilities from these unrounded values gives approximately $0.2597443$, $0.3966569$, and $0.3435987$; rounded to four decimal places, these are $0.2597$, $0.3967$, and $0.3436$. (2) Applying the expit function gives cumulative probabilities $\operatorname{expit}(-2)\approx0.1192$, $\operatorname{expit}(0)\approx0.5000$, and $\operatorname{expit}(2)\approx0.8808$. The category probabilities are approximately $0.1192,0.3808,0.3808,$ and $0.1192$, which sum to $1.0000$. (3) The first-category probability is $F_1^\delta$, which increases with delta. An interior probability is $F_j^\delta-F_{j-1}^\delta$; the two cumulative probabilities change by potentially different amounts, so their difference may decrease. (4) The categories must be exhaustive and ordered, and the MAR reference model must give credible cumulative probabilities for the profiles being analyzed. Also check that the shared shift is plausible and that the reconstructed category probabilities are nonnegative and sum to one.