Translating Local and Baseline Odds

Why Translate Between Local and Baseline Odds?

For an ordered response, an adjacent-category logit describes the odds between neighboring categories. A baseline-category logit instead compares each category with one selected reference category. These are different views of the same category probabilities: adjacent odds are local, while baseline odds span the distance from a category to the reference. Translating between them helps interpret a fitted model without mistaking a local contrast for a reference-category contrast.

Let $Y$ take ordered values $1,\ldots,K$. At covariate value $x$, write $p_j(x)=P(Y=j\mid X=x)$, and assume each $p_j(x)>0$. We use the adjacent-logit convention $\log\{p_j(x)/p_{j+1}(x)\}$, and take category $K$ as the baseline. A different convention that reverses the adjacent ratio reverses the corresponding logit signs.

Definition: The adjacent odds at categories $j$ and $j+1$ are $q_j(x)=p_j(x)/p_{j+1}(x)$, for $j=1,\ldots,K-1$. Their log is the adjacent-category logit $a_j(x)=\log q_j(x)$. With category $K$ as reference, the baseline odds for category $j$ are $p_j(x)/p_K(x)$, and the baseline-category logit is $g_j(x)=\log\{p_j(x)/p_K(x)\}$, for $j=1,\ldots,K-1$.

From Adjacent Odds to Baseline Odds

To compare category $j$ with category $K$, move through the intervening neighboring pairs. The probability ratios multiply, so all intermediate probabilities cancel:

\frac{p_j(x)}{p_K(x)} = \frac{p_j(x)}{p_{j+1}(x)} \frac{p_{j+1}(x)}{p_{j+2}(x)} \cdots \frac{p_{K-1}(x)}{p_K(x)} = \prod_{r=j}^{K-1}q_r(x).

Taking logarithms turns this product into a sum. Thus a baseline logit is the cumulative sum of the adjacent logits from category $j$ through the last adjacent pair.

Theorem: For positive category probabilities and reference category $K$, $$ g_j(x)=\sum_{r=j}^{K-1}a_r(x),\qquad j=1,\ldots,K-1. $$ Conversely, $a_j(x)=g_j(x)-g_{j+1}(x)$ for $j=1,\ldots,K-2$, and $a_{K-1}(x)=g_{K-1}(x)$. Therefore, either set of logits determines the other exactly.

The forward identity follows from the telescoping product above. For the converse, subtract the two sums: $g_j-g_{j+1}=\sum_{r=j}^{K-1}a_r-\sum_{r=j+1}^{K-1}a_r=a_j$. For the last pair, the sum defining $g_{K-1}$ has only one term, so $g_{K-1}=a_{K-1}$. This proves both directions. $\square$

Worked Example: Three Categories and a Reference Category

Suppose the adjacent odds at a given covariate value are $p_1/p_2=2$ and $p_2/p_3=3$. Find the baseline odds with category 3 as reference, and recover the probabilities.

The baseline odds for category 2 are $p_2/p_3=3$. For category 1, multiply the two local odds: $p_1/p_3=(p_1/p_2)(p_2/p_3)=2\cdot3=6$. Thus the category probabilities are in the ratio $p_1:p_2:p_3=6:3:1$. The total number of ratio units is $6+3+1=10$, giving $p_1=6/10=0.6$, $p_2=3/10=0.3$, and $p_3=1/10=0.1$.

Check the ratios using these probabilities: $0.6/0.3=2$ and $0.3/0.1=3$. The baseline odds are $0.6/0.1=6$ and $0.3/0.1=3$, as required.

Translating Regression Effects

Suppose an adjacent-category model has linear predictors $a_r(x)=\alpha_r+\beta_r^\mathsf{T}x$, where $\alpha_r$ is an intercept and $\beta_r$ is a coefficient vector. The theorem gives the baseline logits directly:

g_j(x) = \sum_{r=j}^{K-1}\alpha_r + \left(\sum_{r=j}^{K-1}\beta_r\right)^\mathsf{T}x.

Hence the baseline intercept for category $j$ versus $K$ is the sum of the adjacent intercepts from $j$ onward, and its coefficient vector is the corresponding sum of adjacent coefficient vectors. This is a translation of parameterizations, not a claim that the adjacent effects are identical across pairs.

For two covariate values $x$ and $x'$, the adjacent odds ratio for pair $r,r+1$ is $q_r(x)/q_r(x')$. The baseline odds ratio for category $j$ versus $K$ is the product of those adjacent odds ratios along the path from $j$ to $K$. In a linear adjacent model, its log is $\{\sum_{r=j}^{K-1}\beta_r\}^{\mathsf T}(x-x')$. The intercepts cancel in this contrast.

Worked Example: Translating Adjacent Slopes into Baseline Slopes

For four ordered categories, suppose $x$ is a binary covariate and the three adjacent logits are $a_1(x)=0.4+0.5x$, $a_2(x)=-0.2-0.1x$, and $a_3(x)=0.1+0.3x$. Find the logits and odds ratios for categories 1, 2, and 3 against baseline category 4 when $x$ changes from 0 to 1.

Summing the adjacent logits gives $g_1(x)=a_1(x)+a_2(x)+a_3(x)=0.3+0.7x$, $g_2(x)=a_2(x)+a_3(x)=-0.1+0.2x$, and $g_3(x)=a_3(x)=0.1+0.3x$. Thus the baseline log-odds slopes are $0.7$, $0.2$, and $0.3$, respectively.

For a one-unit increase in $x$, the baseline odds ratios are $\exp(0.7)\approx2.014$, $\exp(0.2)\approx1.221$, and $\exp(0.3)\approx1.350$. Equivalently, the adjacent odds ratios are $\exp(0.5)\approx1.649$, $\exp(-0.1)\approx0.905$, and $\exp(0.3)\approx1.350$. Multiplying the three adjacent odds ratios gives $\exp(0.5-0.1+0.3)=\exp(0.7)$, the category-1 baseline odds ratio. Multiplying the last two gives $\exp(0.2)$, the category-2 baseline odds ratio.

As a further check, at $x=0$ the baseline logits are $(0.3,-0.1,0.1)$. The unnormalized category weights, with category 4 assigned weight 1, are $(e^{0.3},e^{-0.1},e^{0.1},1)$. Their sum is approximately $4.359866$, giving fitted probabilities $(0.3096,0.2075,0.2535,0.2294)$. At $x=1$, the logits are $(1.0,0.1,0.4)$, the weights are $(e^1,e^{0.1},e^{0.4},1)$, and their sum is approximately $6.315277$. The fitted probabilities are $(0.4304,0.1750,0.2362,0.1583)$. From these probabilities, $p_1/p_4$ is approximately $1.350$ at $x=0$ and $2.718$ at $x=1$; the ratio of these odds is approximately $2.014$, agreeing with $\exp(0.7)$.

Fitted Probabilities as a Common Basis

A comparison becomes especially transparent when both descriptions are calculated from the same fitted probability vector. Given fitted probabilities $\hat p_1(x),\ldots,\hat p_K(x)$, compute each local odds as $\hat p_r(x)/\hat p_{r+1}(x)$ and each baseline odds as $\hat p_j(x)/\hat p_K(x)$. The translation identity must hold because both quantities are functions of that one vector. This avoids comparing coefficients whose meanings depend on different category contrasts.

For fitted probabilities at two covariate profiles, form the local and baseline odds ratios separately. The baseline odds ratio for category $j$ versus $K$ must equal the product of the local odds ratios for pairs $j,j+1$ through $K-1,K$. This is true whether the probabilities came from an adjacent-category model or another model that supplies valid positive category probabilities.

Worked Example: Checking Local and Baseline Effects From Fitted Probabilities

For three categories, suppose a fitted model gives probabilities $(0.50,0.30,0.20)$ at $x=0$ and $(0.60,0.25,0.15)$ at $x=1$. Compare the changes in local odds with the changes in baseline odds, taking category 3 as reference.

At $x=0$, the adjacent odds are $p_1/p_2=0.50/0.30=5/3$ and $p_2/p_3=0.30/0.20=3/2$. At $x=1$, they are $0.60/0.25=12/5$ and $0.25/0.15=5/3$. The local odds ratios for the change from 0 to 1 are therefore $(12/5)/(5/3)=36/25=1.44$ for the first pair, and $(5/3)/(3/2)=10/9\approx1.1111$ for the second pair.

The baseline odds for category 1 versus 3 change from $0.50/0.20=2.5$ to $0.60/0.15=4$, so their odds ratio is $4/2.5=1.6$. The baseline odds for category 2 versus 3 change from $1.5$ to $5/3$, giving an odds ratio $(5/3)/(3/2)=10/9$. The category-1 baseline odds ratio agrees with the product of the two local odds ratios: $(36/25)(10/9)=360/225=1.6$.

This calculation illustrates why fitted probabilities are a useful common basis. They make clear both the size of each neighboring comparison and how those comparisons accumulate into a reference-category contrast.

Common Mistake and Practical Interpretation

A frequent mistake is to treat one adjacent odds ratio as though it were the baseline odds ratio for the same category. For example, the odds ratio for $p_1/p_2$ compares categories 1 and 2 only; it is not generally the odds ratio for $p_1/p_K$. The latter includes every adjacent pair along the path to category $K$. The correction is to multiply the relevant adjacent odds ratios, or equivalently add their log-odds contrasts.

It is also important to distinguish an odds ratio from a probability difference. The identities here concern ratios of probabilities and their logarithms. A change in one odds or logit does not, by itself, state a fixed change in a category probability; probabilities are normalized across all categories. When comparing interpretations, calculate the fitted probabilities at the covariate values of interest, then derive whichever local or baseline contrasts answer the scientific question.

Key takeaway: With positive category probabilities, baseline log-odds against category $K$ are cumulative sums of adjacent logits. Regression slopes and covariate contrasts translate by the same summation rule, and fitted probabilities let you verify both descriptions on a common scale.

Check Your Understanding

Use the definitions and translation identities to answer the following.

For four categories, write the baseline logit $g_2$ against category 4 in terms of adjacent logits.
If $p_1/p_2=4$ and $p_2/p_3=2$, what is $p_1/p_3$?
In a three-category adjacent model, the slopes are $\beta_1=0.2$ and $\beta_2=-0.5$. What is the slope for category 1 versus baseline category 3?
Explain why a local odds ratio for categories 2 and 3 need not equal the baseline odds ratio for category 2 versus category 4.
Given fitted probabilities at two covariate profiles, describe how to check that the baseline odds ratio for category 1 versus $K$ agrees with the local odds ratios.