Baseline-Category Models for Nominal Outcomes

Why Use Baseline-Category Logits?

When an outcome has several possible labels with no natural ordering, a model based on cumulative or sequential comparisons may impose a structure that the scientific question does not support. For example, a patient might choose among three treatments, or a specimen might be classified into one of several nominal subtypes. A baseline-category model compares each nonreference category directly with one selected reference category. It does not treat the labels as steps along an ordered scale.

The model has a separate linear predictor for each nonreference category. Consequently, a covariate can have different effects on different category-versus-reference comparisons. That flexibility is central to the model: it avoids assuming that a one-unit change in a predictor shifts every category comparison by the same amount.

Definition: Let $Y$ take one of $K$ distinct, unordered categories $1,\ldots,K$, and let $x$ be a $p$-dimensional covariate vector. Choose category $K$ as the reference category and write $\pi_j(x)=P(Y=j\mid x)$. A baseline-category logit model specifies, for each $j=1,\ldots,K-1$, $$ \log\left(\frac{\pi_j(x)}{\pi_K(x)}\right)=x^\mathsf{T}\beta_j, $$ where $\beta_j$ is a $p$-dimensional coefficient vector. The reference category has linear predictor zero by convention; it is not assigned a separate estimated coefficient vector.

The response categories must be mutually exclusive and exhaustive for the modeled outcome, so their conditional probabilities sum to one. The model as written has finite linear predictors and hence strictly positive probabilities for every category. Covariates may include an intercept, numerical predictors, indicator columns for categorical predictors, or interactions. The coefficient vector is category-specific: $\beta_j$ belongs to the comparison of category $j$ against category $K$.

From Logits to Probabilities

Write $\eta_j(x)=x^\mathsf{T}\beta_j$ for $j<K$, and set $\eta_K(x)=0$. The probabilities are obtained by exponentiating these predictors and normalizing them. The normalization is necessary because the exponentiated logits are relative weights, not probabilities by themselves.

Theorem: For finite predictors $\eta_1(x),\ldots,\eta_{K-1}(x)$, with $\eta_K(x)=0$, the unique strictly positive probabilities satisfying the baseline-category logits are $$ \pi_j(x)=\frac{\exp\{\eta_j(x)\}}{1+\sum_{h=1}^{K-1}\exp\{\eta_h(x)\}},\quad j=1,\ldots,K-1, \qquad \pi_K(x)=\frac{1}{1+\sum_{h=1}^{K-1}\exp\{\eta_h(x)\}}. $$

To prove the result, first suppose probabilities satisfy the specified logits. Exponentiating gives $\pi_j(x)=\exp\{\eta_j(x)\}\pi_K(x)$ for each $j<K$. Using the requirement that probabilities sum to one,

1=\pi_K(x)+\sum_{j=1}^{K-1}\pi_j(x) =\pi_K(x)\left[1+\sum_{j=1}^{K-1}\exp\{\eta_j(x)\}\right].

The bracket is positive, so this equation uniquely determines $\pi_K(x)$ as the displayed reciprocal. Substitution gives each $\pi_j(x)$. Conversely, the displayed probabilities are positive, sum to one, and satisfy $\pi_j(x)/\pi_K(x)=\exp\{\eta_j(x)\}$; taking logarithms recovers the specified logits. This proves both existence and uniqueness. $\square$

Worked Example: Probabilities for Three Treatment Choices

Suppose the response is one of standard care, alternative treatment, or deferral. Use standard care as the reference category. At a particular covariate value, suppose the two logits are $$ \log\left(\frac{P(\text{alternative treatment})}{P(\text{standard care})}\right)=0.6, \qquad \log\left(\frac{P(\text{deferral})}{P(\text{standard care})}\right)=0.5. $$

The exponentiated logits are $\exp(0.6)\approx1.8221$ and $\exp(0.5)\approx1.6487$. The common normalizing denominator is $1+1.8221+1.6487\approx4.4708$. Therefore,

\begin{aligned} P(\text{standard care})&=\frac{1}{4.4708}\approx0.2237,\\ P(\text{alternative treatment})&=\frac{1.8221}{4.4708}\approx0.4076,\\ P(\text{deferral})&=\frac{1.6487}{4.4708}\approx0.3688. \end{aligned}

The unrounded probabilities sum to 1; the displayed rounded probabilities sum to approximately $0.2237+0.4076+0.3688=1.0001$, allowing for rounding. The alternative-treatment probability is largest, even though both logits are positive: each positive logit says that category has greater odds than standard care, not that its probability is a particular fixed value.

Interpreting Category-Specific Covariate Effects

Each coefficient describes how a covariate changes a log odds comparing one nonreference category with the reference. For example, if $x_r$ is one component of $x$, then holding the other covariates fixed, changing $x_r$ by $\Delta$ changes the $j$-versus-$K$ log odds by $\Delta\beta_{jr}$. This statement concerns an odds ratio, not a constant change in probability.

Result: In the baseline-category model, changing $x_r$ by $\Delta$, with the other covariates fixed, multiplies the odds $\pi_j(x)/\pi_K(x)$ by $\exp(\Delta\beta_{jr})$. In particular, a one-unit increase multiplies those odds by $\exp(\beta_{jr})$.

For a direct derivation, let $x'$ differ from $x$ only in component $r$, with $x'_r=x_r+\Delta$. Then $$ \log\left(\frac{\pi_j(x')}{\pi_K(x')}\right) -\log\left(\frac{\pi_j(x)}{\pi_K(x)}\right) =(x'-x)^\mathsf{T}\beta_j =\Delta\beta_{jr}. $$ Exponentiating both sides shows that the ratio of the two odds is $\exp(\Delta\beta_{jr})$, as claimed. This interpretation applies to a numerical covariate under the stated linear predictor and to a binary indicator when its value changes from zero to one. $\square$

Worked Example: Different Effects for Two Categories

Let $Y$ have categories A, B, and C, with C as the reference. For a numerical predictor $x$, specify $$ \log\left(\frac{\pi_A(x)}{\pi_C(x)}\right)=-0.4+0.7x, \qquad \log\left(\frac{\pi_B(x)}{\pi_C(x)}\right)=0.2-0.3x. $$

A one-unit increase in $x$ multiplies the A-versus-C odds by $\exp(0.7)\approx2.014$. It multiplies the B-versus-C odds by $\exp(-0.3)\approx0.741$. Thus the same covariate increases the first comparison's odds and decreases the second's; the model does not impose a common slope across categories.

At $x=0$, the exponentiated logits are $\exp(-0.4)\approx0.6703$ and $\exp(0.2)\approx1.2214$. The denominator is $1+0.6703+1.2214=2.8917$, giving $\pi_C(0)\approx0.3458$, $\pi_A(0)\approx0.2318$, and $\pi_B(0)\approx0.4224$.

At $x=1$, the logits are $0.3$ and $-0.1$; their exponentials are approximately $1.3499$ and $0.9048$. The denominator is $1+1.3499+0.9048=3.2547$, so $\pi_C(1)\approx0.3072$, $\pi_A(1)\approx0.4147$, and $\pi_B(1)\approx0.2780$. Each set sums to approximately one. These probability changes are not equal to the odds-ratio changes: probabilities depend on all category logits through the common denominator.

Specifying Predictors and Reading the Coefficients

For a binary covariate $z$, a category-specific equation might be $\log\{\pi_j(x,z)/\pi_K(x,z)\}=\beta_{j0}+\beta_{j1}x+\beta_{j2}z$. Here $\exp(\beta_{j2})$ is the odds ratio comparing $z=1$ with $z=0$ for category $j$ versus the reference, holding $x$ fixed. If the scientific question suggests that the effect of $z$ varies with $x$, an interaction term $x z$ can be included; then the odds ratio for a one-unit change in $z$ depends on $x$.

A categorical predictor with several levels is represented using indicator columns relative to a chosen predictor coding. Each resulting coefficient still belongs to a particular outcome-category-versus-reference comparison. It is important to keep these two comparisons distinct: the coding of predictor levels determines the covariate contrast, while the outcome reference determines which response-category odds are modeled.

Worked Example: Recovering Category-Specific Slopes From Probabilities

Suppose a binary covariate $x\in\{0,1\}$ is associated with the following conditional probabilities for categories 1, 2, and 3. Choose category 3 as the outcome reference.

$x$	$\pi_1(x)$	$\pi_2(x)$	$\pi_3(x)$
0	0.2	0.3	0.5
1	0.4	0.2	0.4

At $x=0$, the logits are $\log(0.2/0.5)=\log(0.4)$ for category 1 and $\log(0.3/0.5)=\log(0.6)$ for category 2. At $x=1$, they are $\log(0.4/0.4)=0$ and $\log(0.2/0.4)=\log(0.5)$, respectively. Because $x$ takes only zero and one, the intercept is the logit at zero and the slope is the difference between the logits at one and zero. Thus, $$ \begin{aligned} \eta_1(x)&=\log(0.4)+\log(2.5)x,\\ \eta_2(x)&=\log(0.6)+\log(5/6)x. \end{aligned} $$

For category 1, the slope is $\log(2.5)\approx0.9163$, so the odds against category 3 are multiplied by $2.5$ from $x=0$ to $x=1$. For category 2, the slope is $\log(5/6)\approx-0.1823$, and those odds are multiplied by $5/6$. Checking the probability reconstruction: at $x=0$, the relative weights are $0.4,0.6,1$, which normalize by $2$ to $0.2,0.3,0.5$. At $x=1$, the weights are $1,0.5,1$, which normalize by $2.5$ to $0.4,0.2,0.4$. The two outcome comparisons therefore require different slopes.

What the Model Assumes—and a Common Mistake

Baseline-category logits are useful when the outcome labels are genuinely unordered and the question calls for comparisons against a meaningful or convenient reference category. The model gives a coherent probability for every category while allowing separate covariate effects for each comparison. For independent responses $Y_i$ conditional on observed covariates $x_i$, the category probabilities specified by the model also define the usual categorical likelihood; if $Y_i=j$, its likelihood contribution is $\pi_j(x_i)$.

A common mistake is to assume that a single covariate must have one common slope across all nonreference categories. That is not part of the baseline-category model: each $\beta_j$ is its own vector. Imposing equal slopes is an additional restriction that must be stated and justified, not silently presumed. Another mistake is to interpret $\exp(\beta_{jr})$ as a probability ratio or as the change in probability. It is an odds multiplier for category $j$ against the baseline; the probabilities change through the shared normalization across all categories.

Key takeaway: A baseline-category model assigns a separate logit to every nonreference category versus the selected reference. Its category-specific slopes describe changes in those relative odds, and exponentiating and normalizing all linear predictors yields the full set of response probabilities.

Check Your Understanding

Answer each question using the baseline-category model definitions and interpretations above.

For four unordered outcome categories, how many baseline-category logits are specified if one category is chosen as the reference?
If a one-unit increase in a covariate has coefficient $-0.4$ for category 2 versus the reference, what is the corresponding odds multiplier?
Why must the exponentiated linear predictors be normalized before they can be interpreted as probabilities?
In what sense can the same covariate have different effects on two nonreference categories?
For a binary predictor, what covariate contrast does its category-specific coefficient describe?