Baseline-Category and Local Odds

Two Ways to Compare Category Probabilities

A response with several categories can be summarized through many possible comparisons. One approach compares every category with the same chosen reference. Another compares each category only with its neighbor in an ordered list. These choices determine which contrasts are represented directly: all comparisons with a common anchor in the first case, local comparisons along an ordering in the second.

Let \(Y\) take one of \(K\) categories, and write \(p_j=P(Y=j)\) for the probability of category \(j\), where \(j=1,\ldots,K\). The same definitions apply conditionally within a covariate pattern, by replacing \(p_j\) with \(P(Y=j\mid \mathbf{x})\). We assume probabilities in the comparisons are positive when taking their logarithms.

An odds ratio \(p_a/p_b\) compares the probabilities of two particular outcomes. Equivalently, among observations in those two categories, it is the odds of category \(a\) rather than category \(b\): \(P(Y=a\mid Y\in\{a,b\})/P(Y=b\mid Y\in\{a,b\})=p_a/p_b\). This conditional interpretation concerns the two named categories only; it is not itself a probability of category \(a\) in the full set of outcomes.

Baseline-Category Odds: A Common Reference

For nominal categories, there may be no defensible ordering, but a category can still be selected as a reference. Every other category is then contrasted with that same category. The reference is a comparison choice, not a claim that the reference is more important or more common.

For example, if category \(r\) is “no response,” a baseline log odds \(\log(p_j/p_r)\) describes the log odds of response \(j\) rather than “no response.” It does not directly describe the odds of \(j\) rather than some third category \(k\). That latter contrast can be calculated from two baseline log odds, as shown below.

Changing the reference changes the list of baseline coordinates, but it does not change the underlying category probabilities or any actual pairwise odds. The contrast for a chosen pair can be recovered by subtracting the appropriate baseline log odds.

To verify the identity, use the logarithm rule \(\log(a)-\log(b)=\log(a/b)\):

The reference probability cancels. This algebra explains why a baseline parameterization needs only \(K-1\) contrasts to represent all pairwise log odds: every other pairwise contrast is a difference of two of the \(K-1\) baseline contrasts.

Adjacent-Category Odds: Neighboring Comparisons

Adjacent-category odds require categories with a specified order, such as disease stages I, II, III, and IV, or ordered response levels from “strongly disagree” to “strongly agree.” For each neighboring pair, the odds compare the higher-index category with the immediately preceding one. The chosen direction matters: reversing numerator and denominator takes the reciprocal of the odds and changes the sign of the log odds.

Unlike baseline-category odds, adjacent-category odds do not use a single common reference. For \(K\) ordered categories, they directly compare category 2 with 1, category 3 with 2, and so on through category \(K\) with \(K-1\). A comparison between categories that are not neighbors is not one of these local contrasts.

A sequence of adjacent odds can be combined to compare more distant categories. The product cancels intermediate probabilities, and the corresponding log odds add.

For the product, expand the factors:

because every intermediate probability \(p_{a+1},\ldots,p_{b-1}\) appears once in a numerator and once in a denominator. Taking logarithms of both sides gives the sum identity, since the logarithm of a product is the sum of the logarithms. This is a relationship among contrasts; it does not make the distant comparison one of the local comparisons themselves.

What Each Parameterization Models Directly

A parameterization models a particular collection of category contrasts directly when those contrasts are the quantities assigned their own equations or parameters. This distinction is useful even before specifying how a contrast depends on covariates: it clarifies what a coefficient or fitted contrast is intended to compare.

Having the same number of contrasts does not make the two parameterizations equivalent in interpretation. Baseline comparisons are anchored to one selected category; adjacent comparisons are anchored to successive neighbors. The first does not require an ordering, while the second depends on the ordering being substantively meaningful. Reordering labels changes which pairs count as neighbors and therefore changes the adjacent contrasts.

Common Mistake: Treating Odds as Probabilities

A frequent error is to read \(p_j/p_r\) as the probability of category \(j\). It is a ratio of two probabilities, not a probability. For example, an odds ratio of 3 means the numerator category is three times as probable as the denominator category; among those two categories alone, the numerator probability is \(3/(1+3)=0.75\). In the full response distribution, its probability could be different because other categories also have probability.

A related mistake is to describe an adjacent-category contrast as comparing every category with a common reference. It compares only one neighboring pair directly. A distant pair can be compared by multiplying intervening odds, or adding their log odds, but that is a derived contrast. Conversely, a baseline parameterization directly compares all nonreference categories with the same category, not every pair of categories separately.