Structural Mean Models For Nonlinear Outcomes

Why a Structural Mean Model?

For a binary outcome, an ordinary outcome-regression model describes the outcome mean across observed treatment and covariate values. A structural mean model instead describes how a treatment changes a counterfactual mean. This distinction matters when treatment is endogenous: the observed treated and untreated groups may differ in their untreated outcome risks, so their observed outcome contrast need not represent a treatment effect.

The model's parameter is not automatically identified just because it is written in a convenient form. Instrumental-variable (IV) assumptions constrain the relationship between the instrument, treatment, and potential outcomes. Identification also requires a correctly specified structural model and enough independent information in the resulting restrictions. A valid instrument may not identify every parameter of a flexible model.

Definition: Let $A\in\{0,1\}$ denote treatment, $Y\in\{0,1\}$ a binary outcome, $X$ measured covariates, and $Y^a$ the outcome that would be observed under treatment level $a$. A structural mean model (SMM) specifies a treatment contrast between conditional counterfactual means. One general form is $$ g\{E(Y^a\mid A,X)\}-g\{E(Y^0\mid A,X)\} =\gamma(a,X;\psi), $$ where $g$ is a specified link, $\gamma(0,X;\psi)=0$, and $\psi$ is the structural parameter. For binary treatment, a common parameterization is $\gamma(a,X;\psi)=a\,\psi^{T}f(X)$, with a specified covariate function $f(X)$. The model concerns counterfactual means within actual-treatment and covariate strata; it is not, by itself, a model for the observed conditional mean $E(Y\mid A,X)$.

Consistency states that the observed outcome equals the potential outcome corresponding to the treatment received: $Y=Y^A$. Thus, among people with $A=1$, $E(Y\mid A=1,X)=E(Y^1\mid A=1,X)$; among those with $A=0$, $E(Y\mid A=0,X)=E(Y^0\mid A=0,X)$. The other counterfactual mean in each treatment group is not directly observed. The SMM links those observed and unobserved means according to its chosen scale and structural form.

The SMM is distinct from an outcome-regression model such as $E(Y\mid A,X)=m(A,X;\beta)$. The latter models observed conditional means and may be useful for prediction or adjustment. Its coefficient for $A$, however, is not automatically the SMM parameter: it need not describe the specified contrast between counterfactual means, particularly when treatment is associated with potential outcomes. The introductory material on structural and reduced-form models for nonlinear outcomes distinguishes these observed-data and causal targets.

Reading the Model Without Confusing Its Target

A link $g$ transforms a mean to the scale on which the structural contrast is specified. For a binary outcome, each conditional mean is a risk between zero and one, but the model need not express its treatment effect as a risk difference. The link and the function $\gamma$ define the scale and structure. This tutorial leaves the choice among particular effect scales to the later material devoted to them.

For example, when $A=1$, the SMM relates the observed treated-group mean to the unobserved mean that the same treated group would have had under $A=0$. It does not assert that the untreated group's observed mean is that counterfactual mean. Nor does it assert that the treatment effect is identical for every person: a model with $\psi^{T}f(X)$ specifies a conditional mean contrast, not individual potential outcomes.

Worked Example: Translating a Nonlinear Structural Contrast

Suppose, hypothetically, that at a fixed covariate value the specified link is $g(p)=\log\{p/(1-p)\}$, and the SMM says the treated-versus-untreated contrast on that transformed-mean scale is $\psi=0.50$. In the observed treated group, the outcome risk is $E(Y\mid A=1,X)=0.60$. What counterfactual untreated risk among that same group is implied by the model?

By consistency, the observed treated risk is $E(Y^1\mid A=1,X)=0.60$. The structural model gives $$ g\{E(Y^1\mid A=1,X)\}-g\{E(Y^0\mid A=1,X)\}=0.50. $$

Since $\log\{0.60/(1-0.60)\}=\log(1.5)\approx0.4055$, the transformed counterfactual risk is $$ g\{E(Y^0\mid A=1,X)\}=0.4055-0.50=-0.0945. $$ Applying the inverse link gives $$ E(Y^0\mid A=1,X)=\frac{\exp(-0.0945)}{1+\exp(-0.0945)}\approx0.4764. $$

Under the stipulated model, the implied untreated risk among the treated is about 0.4764. This is a model implication, not an estimate established by the observed treated risk alone: the value of $\psi$ must be identified from suitable assumptions and data. The observed risk among untreated people would not replace this counterfactual risk without an additional assumption justifying that equality.

Instrumental-Variable Restrictions and Identification

Let $Z$ denote an instrument, and let $A^z$ denote the treatment that would occur under instrument value $z$. In the standard potential-outcome formulation, the following assumptions are central:

Consistency and no interference: observed treatment and outcome agree with their corresponding potential values, and one person's instrument or treatment does not change another person's outcome.
Instrument independence: conditional on $X$, $Z$ is independent of the relevant potential outcomes and treatment assignments, commonly written $Z\perp(Y^0,Y^1,A^0,A^1)\mid X$.
Exclusion: the instrument affects the outcome only through treatment, so potential outcomes do not change directly with $Z$.
Relevance and positivity: at covariate values used for inference, the instrument changes the probability of treatment and has adequate support. A weak or nearly constant first stage provides little information.
Structural-model adequacy: the chosen link, covariate function, and structural restrictions correctly describe the target contrasts being modeled.

These assumptions make the instrument's variation useful; they do not alone guarantee identification of an arbitrary SMM parameter. The model and the IV restrictions together imply restrictions on the observed distribution of $(Y,A,Z,X)$. The parameter is identified when those restrictions determine a unique value of $\psi$ (or a unique target function of it). With multiple parameters, the restrictions must provide enough independent information, often described by a rank condition.

A key caution is that instrument independence before treatment does not imply independence after conditioning on treatment. If $Z$ affects $A$, then selecting people with $A=1$ can associate $Z$ with other causes of treatment and outcome. Consequently, one cannot generally compare $E(Y\mid A=1,Z,X)$ across instrument values and treat the difference as a randomized comparison within the treated group. Identification arguments must use restrictions that respect the treatment-selection process and the particular SMM.

Worked Example: Checking an Instrument's Reduced-Form Information

Consider hypothetical data with a binary instrument. The treatment rate is 0.25 when $Z=0$ and 0.55 when $Z=1$. The outcome risk is 0.20 when $Z=0$ and 0.32 when $Z=1$. Suppose the instrument was randomized, exclusion is credible, and treatment is monotone in the instrument (no one would become less likely to receive treatment when assigned the higher instrument value).

The first-stage difference is $$ P(A=1\mid Z=1)-P(A=1\mid Z=0)=0.55-0.25=0.30. $$ The reduced-form outcome difference is $$ E(Y\mid Z=1)-E(Y\mid Z=0)=0.32-0.20=0.12. $$

Under random assignment of $Z$, exclusion, consistency, and monotonicity, the Wald ratio is $0.12/0.30=0.40$. For a binary outcome, this is the average risk difference among compliers: those whose treatment is changed by the instrument. The calculation confirms that the instrument has a nonzero first stage and illustrates what these assumptions can identify in this setting.

It does not establish that a general SMM parameter equals 0.40. That conclusion would require the SMM's target to match this complier contrast, or further restrictions linking the model parameter to it. Nor does the calculation verify exclusion or independence; those are substantive design and scientific assumptions, not consequences of the observed differences.

Why Rank and Model Restrictions Matter

The identification logic can be stated in terms of model-implied moment restrictions. Suppose a specified SMM and valid IV assumptions imply two population restrictions $m_1(\psi)=c_1$ and $m_2(\psi)=c_2$, where the $m_j$ are determined by the model and the $c_j$ by the observed-data distribution. The values of the restrictions are not universal SMM formulas: they depend on the particular model, instrument, covariates, and assumptions. To identify two parameters, the restrictions must distinguish their values.

Locally, a standard check is whether the Jacobian matrix of the restrictions with respect to the parameters has full column rank at the solution. If it does not, small changes in different parameter directions cannot be distinguished using those restrictions. Full rank is a local condition; it does not rule out multiple separated solutions elsewhere.

Worked Example: A Rank Check for Two Structural Parameters

For illustration, suppose a correctly specified model and its valid IV restrictions produce the following two population equations for $\psi_1$ and $\psi_2$: $$ \psi_1+2\psi_2=0.90,\qquad 2\psi_1+5\psi_2=1.80. $$ These equations are hypothetical model-implied restrictions, not a universal formula for binary-outcome SMMs.

Twice the first equation is $2\psi_1+4\psi_2=1.80$. Subtracting it from the second equation gives $\psi_2=0$, and substituting into the first gives $\psi_1=0.90$. The coefficient matrix has determinant $$ \begin{vmatrix}1&2\\2&5\end{vmatrix}=1(5)-2(2)=1\ne0, $$ so the two equations are linearly independent and have a unique solution.

If the second equation instead were $2\psi_1+4\psi_2=1.80$, it would duplicate twice the first equation. Every pair satisfying $\psi_1+2\psi_2=0.90$ would fit; the two parameters would not be separately identified. More data can reduce sampling uncertainty, but cannot repair this population-level lack of independent restrictions.

Diagnostics, Alternatives, and Limitations

Before interpreting an SMM estimate, check both the instrument and the model. Examine treatment rates across instrument levels, conditional on relevant $X$, to detect a weak or absent first stage. Consider whether the instrument could affect the outcome through a pathway other than treatment, and whether its assignment could be associated with potential outcomes after accounting for $X$. These checks cannot prove independence or exclusion, but they can reveal serious incompatibilities. Also assess whether the fitted structural restrictions yield a unique and stable parameter solution; near rank deficiency can produce large uncertainty and sensitivity to small changes in the data.

An outcome-regression approach models observed $E(Y\mid A,X)$, and can support causal interpretation under adequate adjustment for measured confounding and correct model assumptions. An IV approach instead uses instrument variation and requires valid instrument assumptions; it may address unmeasured treatment-outcome confounding but can target a population or parameter different from the outcome-regression target. Neither approach is assumption-free. A structural mean model makes its causal contrast explicit, but misspecifying its link or structural form can undermine interpretation even when the instrument is valid.

A common mistake is to equate the observed mean in the untreated group with the counterfactual untreated mean among treated people. For instance, the claim “the untreated outcome risk is 0.25, so that is the treated group's untreated risk” is unjustified unless an assumption establishes that equality. Treatment selection can make the two groups' potential outcomes differ. Consistency identifies $E(Y^0\mid A=0,X)$ from untreated observations; it does not identify $E(Y^0\mid A=1,X)$. An SMM specifies how counterfactual means relate, while valid IV restrictions and sufficient model information are needed to identify its parameter.

Finally, SMM parameters are scale-dependent and their interpretation depends on the population and conditioning variables in the model. IV evidence may identify only a restricted or local contrast, and nonlinear models do not inherit the simple substitution properties of linear regression. Report the instrument assumptions, the structural form and target, and the identification conditions alongside any estimate.

Check Your Understanding

Answer each question, then compare with the brief solutions.

Under consistency, which counterfactual mean is observed among people with $A=1$, and which remains unobserved?
Why does $Z\perp(Y^0,Y^1,A^0,A^1)\mid X$ not imply that $Z$ is independent of outcomes after conditioning on $A$?
In the reduced-form example, what does the Wald ratio 0.40 identify under the stated assumptions, and what does it not automatically identify?
What does a rank-deficient set of IV restrictions imply about two structural parameters?
Name two reasons a valid instrument might still fail to identify a specified SMM parameter.

Solutions. (1) $E(Y^1\mid A=1,X)=E(Y\mid A=1,X)$ is observed; $E(Y^0\mid A=1,X)$ is counterfactual. (2) Conditioning on treatment, which the instrument affects, can create selection associations among its causes. (3) Under the assumptions in that example, it identifies a complier average risk difference, not automatically a general SMM parameter. (4) The restrictions cannot distinguish all parameter values; unique identification fails. (5) The model may lack enough independent restrictions, or its parameterization may be misspecified; a weak first stage can also make identification practically uninformative.