Marginal Effects Across Ordered Categories

Why Category-Specific Effects Matter

An ordered-response model describes outcomes such as low, medium, and high satisfaction. A covariate can shift the distribution toward higher outcomes, but that does not mean it raises the probability of every higher category. It may reduce the probability of a middle category while increasing the probabilities of the lowest and highest categories, or have other patterns depending on where the index lies relative to the thresholds.

The relevant quantities are therefore the effects on the probabilities of the individual categories, not just the effect on a latent index. These probability effects can be calculated as derivatives for continuous covariates or as finite differences for a discrete change. The two calculations answer different questions and need not have the same numerical value.

Model and Probability Effects

Let \(Y\) take one of the ordered values \(1,\ldots,J\). In the standard ordered-response specification, a latent index is \(\eta=x'\beta\), and the observed category is determined by whether that index plus a continuously distributed error falls between successive thresholds. Write the thresholds as \(\kappa_1<\cdots<\kappa_{J-1}\), with \(\kappa_0=-\infty\) and \(\kappa_J=+\infty\). Let \(F\) and \(f\) be the cumulative distribution function and density of the error, respectively. For ordered logit, \(F\) is the logistic CDF; for ordered probit, it is the standard normal CDF. The scale convention for the error is fixed as part of the model.

The formulas below require that the thresholds are ordered and that \(F\) is differentiable with density \(f\). They also presume the ordered-response probability specification is appropriate for the covariate values being considered. Given these conditions, the probability of category \(j\), conditional on \(x\), is the probability that the latent error falls between \(\kappa_{j-1}-\eta\) and \(\kappa_j-\eta\).

Definition: A continuous marginal effect of covariate \(x_r\) on category \(j\) is the partial derivative \(\partial P(Y=j\mid x)/\partial x_r\), holding other covariates fixed. A discrete change is the difference in category probabilities between two specified covariate values, holding other covariates fixed. A discrete change is not a derivative.

p_j(x)=P(Y=j\mid x) =F(\kappa_j-\eta)-F(\kappa_{j-1}-\eta), \qquad j=1,\ldots,J.

This expression includes both endpoint categories because \(F(-\infty)=0\) and \(F(+\infty)=1\). Thus \(p_1(x)=F(\kappa_1-\eta)\) and \(p_J(x)=1-F(\kappa_{J-1}-\eta)\).

Deriving the Continuous Marginal Effects

For a continuous covariate \(x_r\), the index derivative is \(\partial\eta/\partial x_r=\beta_r\), since \(\eta=x'\beta\). Applying the chain rule to the category probability gives the effect for every category, including the endpoints:

\frac{\partial p_j(x)}{\partial x_r} = \beta_r\left[ f(\kappa_{j-1}-\eta)-f(\kappa_j-\eta) \right].

At the lowest category, this reduces to \(-\beta_r f(\kappa_1-\eta)\); at the highest, it is \(\beta_r f(\kappa_{J-1}-\eta)\). The endpoint effects therefore have opposite signs when \(\beta_r\) is nonzero. For an interior category, the sign depends on the difference between the density at the two shifted thresholds. It is not determined by the sign of \(\beta_r\) alone.

The derivative is a local rate of change. For example, if \(x_r\) is measured in years, the derivative is the change in category probability per additional year, evaluated at the specified covariate values. It is not generally a constant effect: the density terms depend on \(\eta\), and so the effect may differ across observations.

Worked Example: Continuous Effects in a Four-Category Ordered Logit

Consider a hypothetical ordered-logit model with thresholds \(\kappa_1=-1,\ \kappa_2=0,\ \kappa_3=1\). At the covariate values of interest, suppose \(\eta=0.2\), and let \(\beta_r=0.8\), where \(x_r\) is measured in years. The logistic CDF is \(F(z)=1/(1+\exp(-z))\), and its density is \(f(z)=F(z)[1-F(z)]\). The shifted thresholds are \(-1.2,-0.2,0.8\).

The category probabilities are \(p_1=F(-1.2)=0.2315\), \(p_2=F(-0.2)-F(-1.2)=0.2187\), \(p_3=F(0.8)-F(-0.2)=0.2398\), and \(p_4=1-F(0.8)=0.3100\). They sum to 1, up to rounding.

The required densities are \(f(-1.2)=0.1779\), \(f(-0.2)=0.2475\), and \(f(0.8)=0.2139\). Substitution into the derivative formula gives:

\begin{aligned} \frac{\partial p_1}{\partial x_r} &=-0.8(0.1779)=-0.1423,\\ \frac{\partial p_2}{\partial x_r} &=0.8(0.1779-0.2475)=-0.0557,\\ \frac{\partial p_3}{\partial x_r} &=0.8(0.2475-0.2139)=0.0269,\\ \frac{\partial p_4}{\partial x_r} &=0.8(0.2139)=0.1711. \end{aligned}

At these covariate values, one additional year is associated locally with probability changes of about \(-0.1423\), \(-0.0557\), \(0.0269\), and \(0.1711\) in categories 1 through 4, respectively. In percentage-point terms, the changes are approximately \(-14.23\), \(-5.57\), \(2.69\), and \(17.11\) per year. These are local model-based probability effects, not causal effects by themselves.

Why Effects Can Change Sign Across Categories

A shift in \(\eta\) moves the location of the latent distribution relative to the thresholds. Probability can leave one category and enter another. An interior category receives probability mass through one boundary and loses it through the other. The net effect depends on the density at both boundaries: if the density is larger at the lower boundary, the category probability can rise for a positive \(\beta_r\); if it is larger at the upper boundary, the probability can fall.

The category effects must also add to zero. Since the probabilities of all mutually exclusive categories sum to one, their derivatives with respect to a covariate must sum to zero. Algebraically, summing the derivative formula makes each threshold-density term cancel with its counterpart from the adjacent category:

\sum_{j=1}^J \frac{\partial p_j(x)}{\partial x_r} = \beta_r\sum_{j=1}^J \left[f(\kappa_{j-1}-\eta)-f(\kappa_j-\eta)\right] =0.

The same probability accounting applies to finite changes: if the category probabilities are computed at two covariate values, their differences must sum to zero. This is a useful arithmetic check, but it does not imply that each category effect is small or that effects have a particular sign.

Discrete Covariate Changes

For a discrete covariate, such as an indicator equal to zero or one, a derivative can be difficult to interpret: the covariate does not vary through all intermediate values in the data. Instead, calculate the probability at each specified value, leaving the other covariates fixed, and subtract. The formula works for any finite shift in the index, including a one-unit change in a continuous covariate when the question concerns the actual one-unit probability difference rather than a local approximation.

\Delta p_j = p_j(x_r=b)-p_j(x_r=a),

where all other components of \(x\) are held at the same values in the two evaluations. The full set of changes across categories sums to zero because each set of category probabilities sums to one.

Worked Example: A Finite Index Shift in an Ordered Logit

Consider a hypothetical three-category ordered-logit model with thresholds \(\kappa_1=0.1\) and \(\kappa_2=1.1\). Hold all other covariates fixed and compare an index \(\eta=0\) with \(\eta=0.8\). The middle-category probability is \(F(\kappa_2-\eta)-F(\kappa_1-\eta)\).

At \(\eta=0\), the middle probability is \(F(1.1)-F(0.1)=0.7502601-0.5249792=0.2252809\). At \(\eta=0.8\), it is \(F(0.3)-F(-0.7)=0.5744425-0.3318122=0.2426303\). Its finite change is therefore \(0.2426303-0.2252809=0.0173494\), or \(0.01735\) to four significant figures.

For completeness, the lowest-category change is \(F(-0.7)-F(0.1)=0.3318122-0.5249792=-0.1931670\). The highest-category change is \([1-F(0.3)]-[1-F(1.1)]=0.4255575-0.2497399=0.1758176\). The three changes sum to approximately \(-0.1931670+0.0173494+0.1758176=0\). The positive change in the middle category is possible even though the index rose; the probability change depends on both category boundaries, not on a rule that every middle probability must fall.

Effects at Particular Values and Average Marginal Effects

A continuous marginal effect is conditional on the covariates at which it is evaluated. One summary is the effect at the sample mean of the covariates. Another is the average marginal effect (AME), obtained by calculating the derivative for each observation and then averaging those derivatives. Because the probability model is nonlinear, the AME generally differs from the derivative evaluated at the average covariates.

Definition: For \(n\) observations, the sample AME for category \(j\) and covariate \(x_r\) is \(n^{-1}\sum_{i=1}^n \partial p_j(x_i)/\partial x_r\). It describes the average of the observation-specific model derivatives in the sample; it is not necessarily the derivative at the sample mean.

Worked Example: Averaging Effects Across Two Observations

In a hypothetical three-category ordered-logit model, let the thresholds be \(-1\) and \(1\), with \(\beta_r=1\). Suppose two observations have \(\eta=0\) and \(\eta=1\). Their category-specific derivatives can be calculated from the logistic density \(f(z)=F(z)[1-F(z)]\).

For the first observation, the shifted thresholds are \(-1\) and \(1\), and \(f(-1)=f(1)=0.196612\). The three derivatives are \(-0.196612\), \(0.196612-0.196612=0\), and \(0.196612\).

For the second observation, the shifted thresholds are \(-2\) and \(0\), and \(f(-2)=0.104994\), \(f(0)=0.250000\). Its derivatives are \(-0.104994\), \(0.104994-0.250000=-0.145006\), and \(0.250000\). Averaging each category’s two derivatives gives the AMEs:

\left( \frac{-0.196612-0.104994}{2}, \frac{0-0.145006}{2}, \frac{0.196612+0.250000}{2} \right) = (-0.1508,-0.0725,0.2233).

The AMEs sum to zero, subject to rounding. They summarize the average local change in category probability per one-unit increase in \(x_r\) across these two observations. They are not the same as the derivatives for either observation alone.

Common Mistakes and Interpretation

Reading the latent-index coefficient as a probability effect. The coefficient \(\beta_r\) shifts the index; the category probability effect also depends on the thresholds, the density, and the covariate values.
Assuming every category moves in the same direction. Category probabilities sum to one, so an increase in some probabilities must be balanced by decreases elsewhere. Interior-category effects depend on two density values.
Using a derivative for a discrete change without qualification. A derivative is local. For an indicator changing from zero to one, calculate the two probabilities and take their difference.
Forgetting to hold other covariates fixed. A stated discrete change or derivative isolates the change in \(x_r\); changing other covariates at the same time answers a different question.
Confusing the AME with the effect at the mean. In a nonlinear probability model, averaging observation-specific derivatives generally does not equal evaluating the derivative at average covariate values.

Key takeaway: Ordered-response marginal effects are category-specific. Derivatives use the density at adjacent shifted thresholds; finite changes compare probabilities at two specified covariate values. Across all categories, the effects sum to zero because the probabilities sum to one.

Check Your Understanding

Answer each question using the category-probability formulas and the covariate values specified.

For an interior category, which two density values determine the continuous marginal effect, and why can its sign differ from the sign of \(\beta_r\)?
Why must the continuous marginal effects across all categories sum to zero?
When is a finite probability difference more appropriate than a derivative?
In an ordered logit, a positive index shift raises an interior-category probability. Does that contradict probability accounting? Explain.
Why can the sample AME differ from the marginal effect evaluated at the sample mean covariates?