Outcome-Dependent Sampling For Multinomial Logits

Why the Sampling Rule Matters

When a study samples people at different rates according to their outcome category, the observed category proportions need not match the population proportions. In a binary logistic model, outcome-dependent sampling can change the intercept while preserving the slope under suitable conditions. The same principle extends to a multinomial logit: each nonbaseline category has its own intercept shift, and its slope is preserved only when the relative sampling fraction for that category is constant across covariate values.

The distinction is important. A selected-sample model may provide information about category-specific covariate slopes even when its intercepts do not describe population probabilities. But if selection also depends on covariates, its effects can be mixed with the population slopes. The sampling mechanism must therefore be stated before interpreting the fitted coefficients.

Population Model and Sampling Mechanism

Let the outcome $Y$ take values $0,1,\ldots,K$, with category $0$ as the baseline, and let $X=x$ denote a covariate vector. Write $p_j(x)=P(Y=j\mid X=x)$. The population baseline-category multinomial-logit model is

\log\frac{p_j(x)}{p_0(x)}=\alpha_j+x^\mathsf{T}\beta_j,\qquad j=1,\ldots,K.

Here $\alpha_j$ is the population intercept for category $j$ relative to category $0$, and $\beta_j$ is the corresponding vector of category-specific slopes. Let $S=1$ indicate that an individual is included in the analyzed sample. Denote the inclusion probability for category $j$ by $c_j(x)=P(S=1\mid Y=j,X=x)$. The central case is outcome-dependent sampling with constant within-category fractions: $c_j(x)=c_j$, where $0<c_j\leq 1$.

Definition: Under constant category-specific sampling, the selected-sample category probability is $q_j(x)=P(Y=j\mid X=x,S=1)$, and the sampling fraction $c_j=P(S=1\mid Y=j,X=x)$ is positive and does not depend on $x$. Sampling may differ between outcome categories, but within a category it does not favor one covariate value over another.

How Constant Sampling Fractions Change the Logits

Conditioning on selection reweights each population category probability by its inclusion fraction. Normalization is needed to obtain probabilities, but it cancels in a category-to-baseline odds ratio.

Theorem: Suppose $c_j=P(S=1\mid Y=j,X=x)$ is constant in $x$ and positive for every category. If the population model is a baseline-category multinomial logit, then the selected-sample model is also a baseline-category multinomial logit, with $$ \log\frac{q_j(x)}{q_0(x)} =\left(\alpha_j+\log\frac{c_j}{c_0}\right)+x^\mathsf{T}\beta_j,\qquad j=1,\ldots,K. $$ Thus the selected-sample intercept is $\alpha_j+\log(c_j/c_0)$, while the category-specific slope $\beta_j$ is unchanged.

Proof. By the definition of conditional probability, $q_j(x)=P(Y=j,S=1\mid X=x)/P(S=1\mid X=x)$. The numerator is $p_j(x)c_j$, and the denominator is $\sum_{\ell=0}^{K}p_\ell(x)c_\ell$. Therefore

q_j(x)=\frac{c_jp_j(x)}{\sum_{\ell=0}^{K}c_\ell p_\ell(x)}.

Taking the ratio of this expression for category $j$ and for category $0$ cancels the common denominator:

\frac{q_j(x)}{q_0(x)} =\frac{c_jp_j(x)}{c_0p_0(x)} =\frac{c_j}{c_0}\exp\!\left(\alpha_j+x^\mathsf{T}\beta_j\right).

Taking logarithms gives the stated intercept and slope. Positivity of all $c_j$ ensures these odds are defined. $\square$

The shift is relative to the baseline category, not simply the sampling fraction for category $j$ alone. If $c_j<c_0$, category $j$ is underrepresented relative to the baseline and its selected-sample intercept is lower. If all category fractions are multiplied by the same positive constant, their ratios—and therefore all these intercept shifts—stay the same.

Worked Example: Calculating Selected Probabilities and Intercepts

Suppose $Y\in\{0,1,2\}$, and for a scalar covariate $x$ the population model is $\log(p_1/p_0)=-1+0.8x$ and $\log(p_2/p_0)=0.5-0.4x$. The category inclusion fractions are $c_0=0.5,\ c_1=0.1,\ c_2=0.25$. Find the selected-sample logits, then calculate the population and selected probabilities at $x=0$.

The relative fractions are $c_1/c_0=0.2$ and $c_2/c_0=0.5$. Hence the selected logits are $\log(q_1/q_0)=-1+\log(0.2)+0.8x$ and $\log(q_2/q_0)=0.5+\log(0.5)-0.4x$. Since $\log(0.2)\approx-1.609438$ and $\log(0.5)\approx-0.693147$, the selected intercepts are approximately $-2.609438$ and $-0.193147$; the slopes remain $0.8$ and $-0.4$.

At $x=0$, the population probabilities are proportional to $1,e^{-1},e^{0.5}$, whose sum is $1+0.3678794+1.6487213=3.0166007$. Thus $p_0(0)\approx0.3315,\ p_1(0)\approx0.1220,\ p_2(0)\approx0.5465$. The selected odds relative to category $0$ are $q_1/q_0=0.2e^{-1}\approx0.07358$ and $q_2/q_0=0.5e^{0.5}\approx0.82436$. The corresponding normalizing sum of relative weights is $1+0.0735759+0.8243606=1.8979365$, giving $q_0(0)\approx0.5269,\ q_1(0)\approx0.0388,\ q_2(0)\approx0.4343$. The selected category proportions differ substantially from the population proportions, although the slopes in the logits are unchanged.

When Are Slopes Preserved and Estimable?

The preservation result depends on constancy of the relative category sampling fractions across covariate values. A precise extension shows what happens when this condition fails. Suppose the inclusion probabilities remain positive but depend on $x$. Define the relative selection effect $r_j(x)=\log\{c_j(x)/c_0(x)\}$. The same cancellation of the common normalizer gives

\log\frac{q_j(x)}{q_0(x)} =\alpha_j+x^\mathsf{T}\beta_j+r_j(x).

If $r_j(x)$ is constant in $x$, it shifts only the intercept. If it is affine, say $r_j(x)=\delta_j+x^\mathsf{T}\gamma_j$, the selected model has intercept $\alpha_j+\delta_j$ and slope $\beta_j+\gamma_j$. In that case the population slope can be recovered from the selected slope only if $\gamma_j$ is known or otherwise identified. With an arbitrary unknown $r_j(x)$, the selected data reveal only the sum of the population log-odds and the relative selection effect; they do not in general separate the two.

Result: Under positive selection probabilities, category-specific population slopes are preserved in the selected conditional logits when each relative selection effect $c_j(x)/c_0(x)$ is constant over the covariate support. If the relative selection effect is known and covariate-dependent, its log can be included as a known offset and subtracted to recover the population slope. If it is unknown and covariate-dependent, the population slope is generally confounded with selection.

For example, with a binary outcome and a scalar binary covariate $x\in\{0,1\}$, let the population logit be $\alpha+0.7x$. Suppose the baseline inclusion probability is $c_0(x)=0.2$ at both covariate values, while $c_1(0)=0.2$ and $c_1(1)=0.4$. Then $c_1(x)/c_0(x)=\exp(x\log 2)$, so the selected logit is $\alpha+(0.7+\log 2)x$, with slope approximately $1.3931$. Without knowing the selection effect $\log 2$, the selected slope alone cannot distinguish it from a different population slope.

Worked Example: Recovering Slopes With Known Covariate-Dependent Selection

Suppose a two-category population model has logit $\log\{p_1(x)/p_0(x)\}=-0.3+0.6x$. The known relative inclusion probability is $c_1(x)/c_0(x)=\exp(0.2+0.1x)$. Find the selected logit and recover the population slope.

Adding the log relative inclusion probability to the population logit gives \[ \log\{q_1(x)/q_0(x)\} =(-0.3+0.6x)+(0.2+0.1x) =-0.1+0.7x. \] Thus the selected intercept is $-0.1$ and its slope is $0.7$, not the population slope $0.6$. Because the selection effect is known, subtracting $0.1$ from the selected slope recovers $0.6$; subtracting $0.2$ from the selected intercept recovers $-0.3$. This correction requires the relative selection effect, not merely knowledge that sampling depends on outcome.

Preserved slopes are not automatically estimable from every selected data set. First, every category being compared with the baseline must have positive probability of selection; a category never sampled supplies no selected-sample information about its logit. Second, the selected observations must cover covariate values sufficiently to distinguish the model coefficients. For a linear predictor with an intercept and $d$ covariates, a necessary design condition is that the selected covariate design matrix, with columns for the intercept and those covariates, has full column rank. For category-specific slopes, the data must also contain information distinguishing each category from the baseline across that support. These conditions address identifiability; finite-sample separation can still prevent an ordinary maximum-likelihood estimate from being finite.

Worked Example: Checking Whether a Slope Can Be Identified

Consider a three-category outcome with baseline $0$, and a model with an intercept and one covariate $x$. Suppose the sampled observations all have $x=2$. Even if all three categories appear, can the data separately identify each category's intercept and slope?

No. For category $j$, every observed linear predictor is $\alpha_j+2\beta_j$. Replacing $\alpha_j$ by $\alpha_j-2t$ and $\beta_j$ by $\beta_j+t$, for any real $t$, leaves this predictor unchanged: \[ (\alpha_j-2t)+2(\beta_j+t)=\alpha_j+2\beta_j. \] The selected probabilities therefore do not change under this alteration, so $\alpha_j$ and $\beta_j$ cannot be separately identified. The design matrix has a column of ones and a column that is twice that column, and hence does not have full column rank. Observations at multiple covariate values are needed to distinguish intercept from slope; having every outcome category represented is not enough by itself.

Interpretation and a Common Mistake

With constant category-specific sampling fractions, the selected-sample slopes describe the same covariate contrasts in category-versus-baseline log odds as in the population. The intercepts, however, describe selected-sample odds. If the sampling fractions are known, the population intercept is obtained by subtracting $\log(c_j/c_0)$ from the selected intercept. If the fractions are unknown, the selected intercept alone does not determine the population intercept. In either case, fitted selected probabilities $q_j(x)$ should not be interpreted as population probabilities $p_j(x)$ without addressing the sampling design.

Mistake: “Sampling by outcome changes all multinomial-logit coefficients, so the slopes cannot be used.” Correction: With positive fractions constant within each outcome category, the log-odds identity shows that only the category-specific intercepts shift.
Mistake: “Because the slopes are preserved, the fitted probabilities are population risks.” Correction: The selected probabilities use the sampling fractions in their normalization. Preserved slopes do not make selected intercepts or probabilities population-calibrated.
Mistake: “Any outcome-dependent sampling preserves slopes.” Correction: Preservation requires relative sampling fractions constant over covariate values. Covariate-dependent relative fractions contribute their own term to the selected logit.

The practical sequence is therefore to specify the population logits, state how selection depends on outcome and covariates, derive the selected-sample logits, and then check both the design support and the information available about the sampling fractions. This separates a structural slope-preservation result from the separate question of whether the data can estimate the desired coefficients.

Check Your Understanding

Use the definitions and derivations above to answer these questions.

For category $j$, what is the selected-sample intercept shift relative to baseline category $0$ when the sampling fractions are constant within category?
Why does a common normalizing denominator disappear from the selected category-to-baseline odds?
If $\log\{c_j(x)/c_0(x)\}=\delta_j+x^\mathsf{T}\gamma_j$ and the selection effect is known, how do you recover the population slope from the selected-sample slope?
Give a reason why a category-specific slope may fail to be identifiable even when all outcome categories occur in the selected sample.
What part of the selected-sample model is generally confounded with the population slope when relative sampling fractions depend on covariates in an unknown way?