Empirical Cumulative-Logit Plots for Assumption Checking

Why Plot Cumulative Logits?

An ordinal response has several ordered cutpoints, so the proportional-odds assumption makes a coordinated claim: the covariate effect is the same at every cutpoint. A grouped empirical cumulative-logit plot gives a direct visual way to examine that claim. It does not replace a formal test or model fit. Its purpose is to reveal patterns that deserve attention, such as roughly parallel trends, changing distances between cutpoint curves, or estimates dominated by sparse data.

The plot starts from observed category counts rather than fitted probabilities. This makes it useful for seeing how the data behave across selected covariate patterns. But grouped proportions are noisy estimates, especially when a group is small or a cumulative proportion is near zero or one.

Constructing the Empirical Cumulative Logits

Definition: Let an ordinal response $Y$ have ordered categories $1,\ldots,K$. For covariate pattern $g$, let $n_{gk}$ be the number of observations in category $k$, and let $n_g=\sum_{k=1}^K n_{gk}$. At cutpoint $j$, where $j=1,\ldots,K-1$, define $$ A_{gj}=\sum_{k=1}^j n_{gk}, \qquad B_{gj}=\sum_{k=j+1}^K n_{gk}. $$ When both counts are positive, the empirical cumulative proportion and empirical cumulative logit are $$ \widehat p_{gj}=\frac{A_{gj}}{n_g}, \qquad \widehat L_{gj}=\log\left(\frac{\widehat p_{gj}}{1-\widehat p_{gj}}\right) =\log\left(\frac{A_{gj}}{B_{gj}}\right). $$

The numerator counts observations at or below the cutpoint; the denominator counts observations above it. There are $K-1$ cumulative logits for each covariate pattern. To make a plot, put the pattern’s value of a selected covariate on the horizontal axis and $\widehat L_{gj}$ on the vertical axis, using a different symbol or line for each cutpoint. With several predictors, the patterns being compared should be chosen so that the plotted covariate changes are interpretable; differences in other predictors can otherwise obscure the comparison.

What Parallel Curves Mean

Use the cumulative-logit convention $$ \operatorname{logit}\{P(Y\leq j\mid x)\}=\alpha_j-x\beta, \qquad j=1,\ldots,K-1, $$ for a scalar covariate $x$. Here $\alpha_j$ is the threshold-specific intercept and $\beta$ is the common slope. Under this model, the conditional cumulative-logit curves have the same slope at every cutpoint. Their vertical separation can vary by cutpoint, but it does not vary with $x$.

Theorem: Under the stated cumulative-logit model, the difference between the cumulative logits at two covariate values $x_a$ and $x_b$ is $-\beta(x_b-x_a)$ at every cutpoint. Thus the model’s population curves are parallel as functions of $x$.

Proof. At cutpoint $j$, subtract the logit at $x_a$ from the logit at $x_b$:

(\alpha_j-x_b\beta)-(\alpha_j-x_a\beta) =-\beta(x_b-x_a).

The threshold $\alpha_j$ cancels, so this difference is the same for every $j$. This proves the claim. $\square$

An empirical plot will not normally produce exactly parallel curves, even when the model is appropriate: observed proportions vary from their population values. The useful question is whether the differences in trend are small relative to the visible uncertainty and sampling variation, not whether every plotted point lies on a perfectly straight line.

Worked Example: Calculating and Comparing Two Cutpoint Curves

Suppose a three-category response is recorded for 100 observations at each of three covariate values. The category counts are as follows.

Covariate value $x$	Category 1	Category 2	Category 3
0	18	64	18
1	13	62	25
2	9	60	31

For the first cutpoint, $Y\leq1$, the cumulative counts are 18, 13, and 9. The corresponding logits are

\log(18/82)=-1.516,\qquad \log(13/87)=-1.901,\qquad \log(9/91)=-2.314.

For the second cutpoint, $Y\leq2$, the cumulative counts are 82, 75, and 69. The logits are

\log(82/18)=1.516,\qquad \log(75/25)=1.099,\qquad \log(69/31)=0.800.

Across the two unit increases in $x$, the first curve changes by $-0.385$ and then $-0.413$; the second changes by $-0.417$ and then $-0.299$. The curves have similar downward trends, although the second increment differs more. These grouped observations are broadly compatible with a parallel-trends description, but the plot alone cannot establish that the common-slope model is correct.

Changing Gaps as a Warning

For a fixed covariate pattern, higher cutpoints include all the observations counted at lower cutpoints, plus additional categories. Thus cumulative proportions and their logits are ordered across cutpoints whenever they are finite. The vertical gap between two cutpoint curves, however, may change as the covariate changes. A marked change in that gap indicates that the empirical cumulative logits are not shifting together in the way the parallel-slopes model describes.

Result: If $0<A_{gj}<n_g$, then $\widehat L_{gj}$ is nondecreasing in $j$. This ordering is a property of cumulative counts, not evidence that the proportional-odds assumption holds.

Derivation. For $j<K-1$, $A_{g,j+1}=A_{gj}+n_{g,j+1}\geq A_{gj}$, with the total $n_g$ unchanged. Therefore $\widehat p_{g,j+1}\geq\widehat p_{gj}$. The logit function $\log\{p/(1-p)\}$ is increasing for $0<p<1$, since its derivative is $1/\{p(1-p)\}>0$. Hence the finite empirical logits are nondecreasing across cutpoints. $\square$

Worked Example: A Gap That Shrinks Across Covariate Values

Consider three groups of 100 observations, again with a three-category response.

$x$	Category 1	Category 2	Category 3
0	12	61	27
1	18	37	45
2	27	8	65

At $x=0$, the cumulative proportions are $0.12$ and $0.73$, giving logits $\log(12/88)=-1.992$ and $\log(73/27)=0.995$. Their gap is $0.995-(-1.992)=2.987$.

At $x=1$, the cumulative proportions are $0.18$ and $0.55$. The logits are $\log(18/82)=-1.516$ and $\log(55/45)=0.201$, a gap of $1.717$.

At $x=2$, the cumulative proportions are $0.27$ and $0.35$. The logits are $\log(27/73)=-0.995$ and $\log(35/65)=-0.619$, a gap of $0.376$. Both cumulative logits increase with $x$, but the first rises much more than the second and the gap contracts sharply. That changing separation is a visual warning that the cutpoint trends may not be parallel. It is not, by itself, a formal test or proof of a particular alternative model.

Extreme Proportions and Unstable Points

If every observation in a group is at or below a cutpoint, then $B_{gj}=0$ and the empirical logit is $+\infty$. If none is at or below it, then $A_{gj}=0$ and the logit is $-\infty$. These are not ordinary finite plot points. Even when neither count is zero, a very small count can make the point highly sensitive to one or two observations.

For a rough indication of precision when both counts are positive, the binomial plug-in standard error of a cumulative logit is $$ \widehat{\operatorname{SE}}(\widehat L_{gj}) \approx \sqrt{\frac{1}{A_{gj}}+\frac{1}{B_{gj}}}. $$ This approximation grows when either side of the cutpoint has few observations. It also emphasizes why a point based on, for example, one observation in a tail should not be interpreted as a stable trend estimate. Logits from different cutpoints within the same group are dependent because their cumulative counts overlap.

Worked Example: A Zero Cumulative Count

In a group of 20 observations, suppose none is in category 1. At the cutpoint $Y\leq1$, $A=0$ and $B=20$, so the raw empirical logit is $\log(0/20)=-\infty$. It cannot be plotted as a finite value.

A common display-only adjustment adds one half to each side: $$ \log\left(\frac{A+0.5}{B+0.5}\right) =\log\left(\frac{0.5}{20.5}\right) =\log(1/41) \approx -3.714. $$ If one observation instead fell in category 1, the raw logit would be $\log(1/19)\approx-2.944$; the adjusted value would be $\log(1.5/19.5)=\log(1/13)\approx-2.565$. One observation therefore changes the raw logit from infinite to a finite value and changes the adjusted logit by about $1.149$. The adjustment makes a point drawable, but does not make the underlying information abundant or the estimate precise.

Using the Plot Responsibly

A useful plot labels the response cutpoints, shows the covariate patterns and group sizes, and makes zero or extreme proportions visible rather than silently hiding them. If a continuity correction is used, state it and distinguish corrected display values from raw empirical logits. When groups are small, uncertainty intervals or standard-error indicators can help prevent over-reading apparent separation. Because the cumulative logits within a group are dependent, their plotted points should not be treated as independent replications.

The plot is descriptive. Grouping continuous covariates discards within-group information, and group boundaries can affect the visual pattern. With multiple predictors, an apparent trend may also reflect changes in other covariates across the chosen groups. Conversely, a smooth-looking plot may conceal departures that a formal comparison can detect. Use it to identify patterns and data limitations, then interpret it alongside the model specification and suitable inferential checks.

Common pitfall: Treating every plotted curve as a precise estimate and declaring the assumption satisfied because the lines look nearly parallel. The correction is to inspect group sizes and extreme counts, consider sampling variability and dependence, and regard approximate parallelism only as descriptive evidence—not a validation of the model.

Check Your Understanding

Use the definitions and examples above to answer the following.

For a four-category response, how many empirical cumulative logits are calculated for each covariate pattern?
Why does the proportional-odds model imply the same change in cumulative logit between two covariate values at every cutpoint?
What does a gap that steadily changes across covariate values suggest, and why is that suggestion not a formal test result?
What happens to the raw empirical cumulative logit when all observations lie on one side of a cutpoint?
Why should cumulative logits for different cutpoints within the same group not be treated as independent observations?