Control Functions For Ordinal Outcomes

Why an Ordinal Outcome Needs Its Own Control-Function Model

An ordinal outcome has ordered categories, such as mild, moderate, and severe, but the distances between categories are not measured on a meaningful numerical scale. Treating category labels as ordinary numbers in a linear outcome model therefore imposes an interpretation the data do not supply. An ordered probit instead represents categories as intervals of an underlying continuous latent response.

When treatment \(T\) is endogenous, an ordered probit that simply includes \(T\) need not have a causal interpretation. A control-function approach models the source of treatment variation and includes a function of the first-stage disturbance in the ordinal outcome model. This tutorial develops that construction for a continuous treatment with a linear first stage. The resulting category probabilities depend on treatment, covariates, and the control function; a coefficient in the latent index is not itself a category-probability difference.

Model and Assumptions

Let \(X\) be a vector of observed covariates and \(Z\) a vector of instruments, with \(Z\) excluded from the outcome equation. For a continuous treatment, specify the linear first stage

Here \(V\) is the first-stage disturbance. The latent ordinal outcome model is

where \(\kappa_0=-\infty\), \(\kappa_J=\infty\), and the finite thresholds satisfy \(\kappa_1<\cdots<\kappa_{J-1}\). The treatment coefficient \(\tau\) describes a shift in the latent response, not directly a change in the probability of any particular category.

A sufficient control-function structure for the derivation below is that, conditional on \(X,Z\), \(V\) and \(\varepsilon\) are jointly normal with mean zero, \(\operatorname{Var}(V)=\sigma_V^2>0\), and \(\operatorname{Var}(\varepsilon)=\sigma_\varepsilon^2\). Write

Under joint normality, this decomposition gives \(U\) independent of \(V\); the structural model also requires that the remaining outcome disturbance is independent of the instruments and covariates as specified. Endogeneity corresponds to dependence between \(V\) and \(\varepsilon\), equivalently \(\lambda\ne0\) in this setup. The instruments must be relevant, so \(Z\) contributes variation to \(T\) conditional on \(X\), and valid: they are independent of unobserved outcome determinants conditional on \(X\) and have no direct effect on \(Y^*\). These assumptions, plus adequate support for the treatment comparisons, are substantive requirements, not consequences of fitting the model.

Conditional Probabilities and Threshold Interpretation

Substituting the error decomposition into the latent outcome equation gives \(Y^*=X'\beta+\tau T+\lambda V+U\). Conditional on \(X,T,V\), the remaining error is normal with variance \(\sigma_U^2\). Thus the cumulative probability through category \(j\) is

The division by \(\sigma_U\) matters: the latent error after conditioning on \(V\) need not have variance one. In an ordered probit, location and scale are not separately identified from the observed categories. A common estimation convention fixes the scale and reports normalized thresholds and coefficients, each divided by \(\sigma_U\). Under that convention, the displayed formula is written without the denominator, but all its parameters must be normalized consistently. Do not combine unnormalized parameters with a standard-normal CDF.

Category probabilities are differences between successive cumulative probabilities. Set \(F_0=0\), \(F_J=1\), and for finite thresholds let \(F_j=P(Y\leq j\mid X,T,V)\). Then

A positive \(\tau\), for example, shifts the latent response upward when treatment increases. It lowers the cumulative probabilities \(P(Y\leq j)\) at fixed \(X,V\), but the effect on an individual middle-category probability can be positive or negative: that probability is the difference between two cumulative probabilities. Consequently, interpretation should use predicted category probabilities or contrasts on those probabilities, not just the sign or size of \(\tau\).

From Fitted Thresholds to Category Contrasts

The following example uses normalized parameters, so \(\sigma_U\) is absorbed into the thresholds and coefficients. It illustrates how to compute probabilities while holding the control-function value fixed. Suppose there are three categories and, at a specified covariate value, the normalized thresholds are \(\tilde\kappa_1=-0.2\) and \(\tilde\kappa_2=1.1\), with \(\tilde\tau=0.7\) and \(\tilde\lambda=1\). The cumulative-probit arguments are \(\tilde\kappa_j-X'\tilde\beta-\tilde\tau T-\tilde\lambda V\); take \(X'\tilde\beta=0\) for this illustration.

What the Contrast Does—and Does Not—Mean

The calculation above changes \(T\) while holding each \(V\) value fixed, then averages over a chosen distribution of \(V\). Under a correctly specified structural model and the required IV assumptions, such a contrast can represent a model-based counterfactual comparison. The averaging distribution must be stated: averaging over an empirical residual distribution, a target population, or a small illustrative set of residual values answers different questions. The two equally weighted values in the example are only a transparent arithmetic illustration, not a population estimate.

A fitted coefficient \(\tilde\tau\) is not itself the average change in the probability of a category. Probability contrasts depend on thresholds, covariates, treatment values, and the control-function value. For a continuous treatment, the contrast between two doses also depends on which doses are compared. If a causal interpretation is intended, the model must support the relevant counterfactual treatment values; extrapolating well beyond observed support is not justified merely because the probit formula returns a number.

The control-function coefficient is also not a standalone test of endogeneity in every specification. Its meaning depends on the assumed first-stage and error structure. A residual-inclusion model is a particular parametric correction, not a general guarantee that all sources of confounding have been removed.

Estimation, Diagnostics, and Common Pitfalls

Diagnostics cannot establish instrument validity, but they can reveal problems. Inspect first-stage strength and the distribution of fitted residuals; examine whether treatment and residual values used in predictions are represented in the data; and check whether model-predicted category frequencies are grossly inconsistent with observed frequencies across relevant covariate and treatment groups. The ordered-probit threshold structure assumes a common latent-index shift across cumulative cutpoints. Poor fit may indicate that this structure, the linear first stage, the normality assumptions, or the control-function form is inadequate.

A common mistake is to use the raw residual from a linear regression as a valid control function for any endogenous treatment. The correction derived here relies on a linear first stage for a continuous treatment and the specified error relationship. For a binary or otherwise nonlinear first stage, the raw residual is generally not the appropriate control function; do not transfer this formula mechanically. A further mistake is to calculate category probabilities using rounded cumulative probabilities and then subtract them. Compute differences and averages using full precision, then round the final results, as in the worked example.

Compared with a standard ordered probit that omits \(\widehat V\), residual inclusion explicitly models dependence between treatment and the latent outcome disturbance, but requires stronger structural assumptions. A linear two-stage substitution that replaces \(T\) with its fitted value is not equivalent: nonlinear outcome probabilities do not generally preserve the required relationship through predictor substitution. A joint latent-variable model can encode dependence directly, but it also requires distributional and functional-form assumptions. No approach removes the need to justify instrument relevance, exclusion, and the outcome model.