Ordinal Logistic Regression for Genotype Association

Why Model an Ordered Phenotype?

Some genetic association outcomes have categories with a natural order but no defensible numerical spacing. A hypothetical clinical phenotype might be classified as mild, moderate, or severe. These labels contain more information than an unordered set of groups, but coding them as 1, 2, and 3 and fitting a linear regression would treat the distance from mild to moderate as equal to the distance from moderate to severe. That assumption is often unwarranted.

Ordinal logistic regression models the probabilities of being at or below successive category thresholds. Its proportional-odds form uses one genotype slope across all those thresholds. This gives a compact association measure while preserving the ordering of the phenotype.

Model Definition and Interpretation

Definition: Let $Y$ be an ordered phenotype with $K$ categories, ranked from 1 (lowest) to $K$ (highest), and let $G$ be a single-variant genotype dosage coded 0, 1, or 2. The proportional-odds model is $$ \operatorname{logit}\{P(Y\leq j\mid G)\}=\alpha_j-\beta G,\qquad j=1,\ldots,K-1, $$ where $\operatorname{logit}(q)=\log\{q/(1-q)\}$, the thresholds satisfy $\alpha_1<\cdots<\alpha_{K-1}$, and the same slope $\beta$ applies at every threshold. The model contains an association for this dosage coding; it does not, by itself, establish causation or another inheritance pattern.

The minus sign in this parameterization makes the direction of $\beta$ convenient to state. For any threshold $j$, the odds of being above that threshold rather than at or below it are

\frac{P(Y>j\mid G)}{P(Y\leq j\mid G)} =\exp(\beta G-\alpha_j).

Thus, increasing $G$ by one unit multiplies those odds by $\exp(\beta)$, at every threshold. This is the cumulative odds ratio per dosage unit for being in a higher rather than a lower part of the ordered outcome. It is not a ratio of probabilities, and it is not the odds ratio for any one specific category versus another.

A positive $\beta$ corresponds to higher cumulative odds of a higher phenotype category as dosage increases. A negative $\beta$ corresponds to lower cumulative odds. The proportional-odds assumption is the claim that this same change in log odds applies whether the division is, for example, severe versus mild or moderate, or moderate/severe versus mild.

How the Model Uses the Observed Categories

The model specifies cumulative probabilities, but the likelihood is built from probabilities of the observed individual categories. Let $F(t)=1/\{1+\exp(-t)\}$ be the logistic cumulative distribution function. Set $\alpha_0=-\infty$ and $\alpha_K=+\infty$. Then the probability of category $k$, conditional on genotype $G$, is

P(Y=k\mid G)=F(\alpha_k-\beta G)-F(\alpha_{k-1}-\beta G),\qquad k=1,\ldots,K.

For independent participants, the likelihood is the product of the probability assigned to each participant's observed category. Equivalently, the log-likelihood is the sum of the logarithms of those probabilities. Maximum likelihood estimation chooses the thresholds and slope that maximize this log-likelihood, subject to the thresholds remaining ordered. In practice, numerical optimization is used; the estimates are not generally obtained by a simple closed-form calculation.

Worked Example: Interpreting a Fitted Genotype Effect

In a hypothetical sample of 800 participants, a three-category phenotype is ordered as mild, moderate, and severe. Genotype is coded as dosage $G=0,1,2$. A proportional-odds model fit to these data gives $\hat{\beta}=0.55$ with $SE=0.18$. Interpret and summarize the genotype association.

The estimated cumulative odds ratio per additional dosage unit is

\exp(\hat{\beta})=\exp(0.55)=1.733.

For each one-unit increase in dosage, the estimated odds of being above either phenotype threshold, rather than at or below that threshold, are multiplied by about 1.73. This is a 73.3% increase in the cumulative odds, with the same estimated ratio for severe versus mild/moderate and for moderate/severe versus mild. The estimate does not say that the probability of severe disease increases by 73.3%.

For a Wald interval on the log-odds scale, use $\hat{\beta}\pm1.96SE$:

0.55\pm1.96(0.18)=0.55\pm0.3528=[0.1972,0.9028].

Exponentiating the endpoints gives a 95% confidence interval for the cumulative odds ratio of $[\exp(0.1972),\exp(0.9028)]=[1.218,2.466]$. Under the model and the usual large-sample approximation, this interval procedure has 95% coverage over repeated samples. In this hypothetical analysis, the interval excludes 1.

The Wald statistic for testing $H_0:\beta=0$ against $H_a:\beta\ne0$ is $z=0.55/0.18=3.056$. Its two-sided normal-approximation p-value is about 0.0022. This is evidence of an association under the fitted model and its assumptions; it is not the probability that $H_0$ is true, and it does not establish a causal effect.

From Thresholds to Category Probabilities

The slope summarizes a common shift in cumulative odds, while the thresholds determine where the categories sit on the cumulative scale. Together they yield fitted probabilities for each category. These probabilities help show why an odds ratio is not a probability change: the same odds ratio can correspond to different probability changes at different starting probabilities.

Worked Example: Calculating Fitted Probabilities

Suppose a hypothetical three-category phenotype has fitted thresholds $\hat{\alpha}_1=-1.10$ and $\hat{\alpha}_2=0.35$, and fitted genotype slope $\hat{\beta}=0.55$. Calculate the category probabilities for a participant with $G=1$.

For $G=1$, the cumulative logit values are $-1.10-0.55=-1.65$ and $0.35-0.55=-0.20$. Applying the logistic function gives

P(Y\leq1\mid G=1)=F(-1.65)=0.1611,\qquad P(Y\leq2\mid G=1)=F(-0.20)=0.4502.

The first category probability is $0.1611$. The second is the difference between the two cumulative probabilities, $0.4502-0.1611=0.2891$. The third is the remaining probability, $1-0.4502=0.5498$. They sum to $0.1611+0.2891+0.5498=1.0000$, up to rounding.

For comparison, at $G=0$, the first cumulative probability is $F(-1.10)=0.2497$, and the second is $F(0.35)=0.5866$. The category probabilities are therefore $0.2497$, $0.5866-0.2497=0.3369$, and $1-0.5866=0.4134$. These fitted probabilities shift toward the higher category as dosage increases, consistent with the positive slope. They are model-based estimates for these genotype values, not observed proportions necessarily.

A Short Likelihood Calculation

A single participant's contribution to the likelihood is the probability of that participant's observed category. For example, with three categories, a participant in category 2 contributes $F(\alpha_2-\beta G)-F(\alpha_1-\beta G)$. A category-1 observation contributes $F(\alpha_1-\beta G)$, and a category-3 observation contributes $1-F(\alpha_2-\beta G)$. Multiplying these contributions over participants gives the likelihood.

Worked Example: One Observation's Likelihood Contribution

Use the hypothetical fitted thresholds and slope from the previous example. One participant has dosage $G=1$ and is observed in category 2. Calculate this participant's likelihood contribution.

The category-2 probability is

$$ F(0.35-0.55)-F(-1.10-0.55) =F(-0.20)-F(-1.65) =0.4502-0.1611=0.2891. $$

The participant contributes 0.2891 to the likelihood. If a second independent participant contributed probability 0.5498, their two observations together would contribute $0.2891(0.5498)=0.1589$. In an actual fit, the optimizer evaluates such category probabilities for all observations and finds the parameter values that make the full observed data most likely under the model.

Assessing the Proportional-Odds Assumption

The central structural assumption is that $\beta$ is common to every cumulative split. A practical check compares that model with one allowing a separate genotype slope at each threshold. For $K$ categories, the proportional-odds model has one genotype slope, while the more flexible model has $K-1$ slopes. The proportional-odds model is nested within the flexible model when all those slopes are equal.

A likelihood-ratio comparison can test that restriction. If $\ell_{\mathrm{PO}}$ is the maximized log-likelihood for the proportional-odds model and $\ell_{\mathrm{flex}}$ is that for the model with separate threshold slopes, the statistic is

\Lambda=2(\ell_{\mathrm{flex}}-\ell_{\mathrm{PO}}).

Under the null hypothesis of equal slopes, and standard regularity conditions including adequate data for both fits, $\Lambda$ is approximately chi-square distributed with $K-2$ degrees of freedom: the flexible model adds $K-2$ independent slope parameters relative to the common-slope model. A large statistic relative to that reference distribution is evidence against proportional odds. This test is one diagnostic, not a guarantee that every aspect of the model is adequate.

Worked Example: Comparing Common and Threshold-Specific Slopes

In a hypothetical analysis with three ordered categories, the proportional-odds fit has log-likelihood $\ell_{\mathrm{PO}}=-312.8$. A model with separate genotype slopes for the two cumulative thresholds has log-likelihood $\ell_{\mathrm{flex}}=-312.1$. Assess the proportional-odds restriction.

There are $K-2=1$ added slope parameters, so the likelihood-ratio statistic is compared with a chi-square distribution with 1 degree of freedom:

\Lambda=2\{-312.1-(-312.8)\}=2(0.7)=1.4.

The upper-tail probability for a chi-square statistic of 1.4 with 1 degree of freedom is about 0.237. At a 0.05 significance level, these data do not provide strong evidence against the common-slope restriction. This does not prove that the slopes are identical; the comparison may have limited power, especially if some categories or genotype groups are sparse.

It is also useful to inspect threshold-specific slope estimates and fitted cumulative probabilities. Marked differences in estimated slopes can indicate that the single common effect hides distinct patterns, even when a formal test is not decisive. Conversely, a non-significant test does not show that every predicted probability is well calibrated. Check that the outcome categories are ordered meaningfully, that each participant belongs to exactly one category, and that the fitted probabilities are plausible for the observed genotype groups.

Why the Model Matters—and a Common Mistake

The proportional-odds model uses the ranking of a phenotype without requiring equally spaced category scores. It can therefore provide a useful single-variant association summary when a phenotype is ordered rather than binary or quantitative. Its compactness comes from a substantive restriction: the genotype effect is shared across cumulative thresholds. The model should be selected because that structure is scientifically and empirically reasonable, not merely because it returns one coefficient.

Mistake: treating the odds ratio as a probability ratio. An odds ratio of 1.73 does not mean that the probability of severe phenotype is 73% higher. The odds and probability are different quantities, and the probability change depends on the starting probability.
Mistake: assuming an ordinal category is a numeric measurement. Replacing categories with scores such as 1, 2, and 3 and fitting a linear model imposes spacing assumptions. The ordinal model instead uses cumulative probabilities.
Mistake: interpreting a non-significant proportional-odds test as proof of the assumption. The test may have low power. Consider the estimates, uncertainty, category counts, and fitted probabilities as well.
Mistake: assuming the common slope proves an inheritance mechanism. A slope for dosage 0, 1, or 2 imposes a per-unit trend in this model. It does not show that this coding is biologically correct or rule out other genotype patterns.

Key takeaway: In the proportional-odds model, $\exp(\beta)$ is the common per-dosage cumulative odds ratio for being above rather than at or below every phenotype threshold. Its interpretation depends on the outcome ordering, genotype coding, and the assumption of a shared slope.

Check Your Understanding

Answer each question by referring to the model and its assumptions.

For a model with $\hat{\beta}<0$, what direction of association does the model estimate between increasing genotype dosage and the ordered phenotype?
In a four-category phenotype, how many cumulative logits are modeled, and how many thresholds are there?
Why is $\exp(\beta)$ not generally the ratio of probabilities for the highest phenotype category?
What restriction does the proportional-odds model impose on genotype slopes across cumulative thresholds?
If a likelihood-ratio test comparing common and threshold-specific slopes is not significant, what can and cannot be concluded?