Logistic Structural Mean Models For Binary Outcomes

Why Model the Effect on the Odds Scale?

For a binary outcome, a treatment effect can be expressed as a difference in risks, a ratio of risks, or a ratio of odds. A logistic structural mean model (logistic SMM) specifies the treatment effect on the odds scale while allowing the untreated counterfactual risk to vary across people and covariates. This matters when treatment is endogenous: the observed association between treatment and outcome need not equal the causal effect, and a conventional logistic regression does not by itself resolve that problem.

Let $A\in\{0,1\}$ denote treatment, $Y\in\{0,1\}$ the outcome, $Z$ an instrumental variable, and $L$ measured covariates. Write $Y^a$ for the outcome that would be observed if treatment were set to $a$. The observed outcome satisfies $Y=Y^A$ under consistency. The logistic SMM describes how the conditional mean of $Y^a$ changes with $a$, relative to the no-treatment counterfactual.

Definition: A logistic structural mean model for a binary treatment and binary outcome specifies $$ \operatorname{logit} E[Y^a\mid A,Z,L] - \operatorname{logit} E[Y^0\mid A,Z,L] = \psi a,\qquad a\in\{0,1\}, $$ where $\operatorname{logit}(p)=\log\{p/(1-p)\}$. The parameter $\psi$ is the structural log odds ratio, and $\exp(\psi)$ is the structural odds ratio. The baseline risk $E[Y^0\mid A,Z,L]$ may vary with the observed treatment, instrument, and covariates.

The conditioning variables describe subgroups in the structural model; they do not turn $A$ into a randomized exposure. In particular, the baseline counterfactual risk is allowed to differ between people who did and did not receive treatment. The model's restriction is that the treatment changes the log odds by the same amount $\psi$ across these strata. This is a homogeneous effect assumption on the odds scale, not a claim that the risk difference is constant.

Counterfactual Risks and the Structural Odds Ratio

For $a=0$, the model compares $Y^0$ with itself, so the equation is automatically satisfied. For $a=1$, it says that the odds of the treated counterfactual outcome equal the odds of the untreated counterfactual outcome multiplied by $\exp(\psi)$. If $q_0=E[Y^0\mid A,Z,L]$, then

E[Y^1\mid A,Z,L] = \frac{\exp(\psi)q_0}{1-q_0+\exp(\psi)q_0}.

This follows by writing the untreated odds as $q_0/(1-q_0)$, multiplying by $\exp(\psi)$, and converting the resulting odds back to a probability. The formula is useful because an odds ratio does not itself state how many percentage points the risk changes. The risk difference depends on the baseline risk $q_0$, even when $\psi$ is fixed.

Worked Example: Translating a Structural Odds Ratio Into Risks

Suppose a correctly specified logistic SMM has $\psi=\log(1.8)$. For a particular covariate and instrument stratum, suppose the untreated counterfactual risk is $q_0=0.20$. Calculate the treated counterfactual risk and the risk difference.

The structural odds ratio is $\exp(\psi)=1.8$. The untreated odds are $0.20/(1-0.20)=0.25$, so the treated odds are $1.8(0.25)=0.45$. Converting to a risk gives

q_1=\frac{0.45}{1+0.45}=0.3103.

Thus the treated counterfactual risk is about 31.0%, compared with an untreated risk of 20.0%. The risk difference is $0.3103-0.2000=0.1103$, or 11.0 percentage points. These are counterfactual risks for the specified stratum; the calculation does not establish that the assumptions needed to identify them hold.

The same structural odds ratio can correspond to different risk differences. For example, with the same odds ratio of 1.8 but $q_0=0.50$, the treated risk is $0.45/(1+0.45)=0.3103$ only if the odds are 0.45; here the untreated odds are actually 1, so treated odds are 1.8 and the treated risk is $1.8/2.8=0.6429$. The risk difference is 14.3 percentage points. A structural odds ratio is therefore not a universal percentage-point effect.

Structural Effects Are Not Ordinary Logistic Coefficients

A conventional logistic regression models the observed conditional outcome probability, for example

\operatorname{logit}P(Y=1\mid A,Z,L) = \alpha+\beta A+\gamma Z+\delta^\top L.

When this regression is correctly specified, $\exp(\beta)$ is a conditional association between observed treatment and observed outcome, adjusted for the included covariates. It is not automatically the structural odds ratio $\exp(\psi)$. If treatment is associated with unmeasured causes of the outcome, the observed conditional association can differ from the causal effect. Further, logistic odds ratios are non-collapsible: conditional and marginal odds ratios can differ even without confounding.

Worked Example: Interpreting an Observed Logistic Coefficient

In hypothetical observational data, a fitted logistic regression gives $\hat\beta=\log(3)$ for treatment after adjustment for measured covariates. The model's estimated conditional odds ratio is

\exp(\hat\beta)=\exp\{\log(3)\}=3.

The correct direct interpretation is that, at the same included covariate values, the fitted model associates treatment with three times the observed outcome odds. It is not justified to say that treatment causally triples the odds. Unmeasured confounding, model misspecification, or differences between conditional and marginal quantities may matter. By contrast, a logistic SMM's $\exp(\psi)$ is defined as a counterfactual treatment odds ratio under the structural model. An instrument can help identify that structural parameter only if the required assumptions and model restrictions are credible; the numerical coefficient from an ordinary logistic regression is not a substitute.

Instrumental-Variable Assumptions and the Blipped-Down Outcome

An instrumental-variable interpretation requires assumptions about how $Z$ relates to treatment and potential outcomes. The following are stated conditional on $L$, allowing measured covariates to account for relevant differences.

Consistency and no interference: each person's observed outcome equals their potential outcome under their received treatment, and one person's treatment does not alter another person's outcome.
Instrument relevance: the distribution of $A$ changes with $Z$ for at least some covariate strata. Without such a change, the instrument supplies no treatment variation for identifying the effect.
Conditional instrument independence: $Z$ is independent of the potential outcomes and potential treatments given $L$. In particular, $Z$ must not carry information about $Y^0$ after conditioning on $L$.
Exclusion restriction: the instrument affects the outcome only through treatment. In potential-outcome notation, changing $Z$ while holding treatment fixed does not change $Y$.
Positivity: the relevant treatment and instrument values occur with positive probability in the covariate strata used for the analysis.
Structural-model correctness: the logistic SMM's odds-scale blip is adequately represented by the specified $\psi$, including its assumption of a common log odds effect if no effect modifiers are included.

A useful justification for the model's identifying restriction is to remove, or “blip down,” the modeled treatment effect from the observed risk. Define $p(A,Z,L)=P(Y=1\mid A,Z,L)$. Since $Y=Y^A$, the SMM implies

\operatorname{logit}E[Y^0\mid A,Z,L] = \operatorname{logit}p(A,Z,L)-\psi A.

Applying the inverse logit gives a transformed risk

H(\psi)= \operatorname{expit}\{\operatorname{logit}p(A,Z,L)-\psi A\}, \qquad \operatorname{expit}(x)=\frac{\exp(x)}{1+\exp(x)}.

Under the correct structural model, $H(\psi)=E[Y^0\mid A,Z,L]$. Averaging over treatment within instrument and covariate strata yields $E[H(\psi)\mid Z,L]=E[Y^0\mid Z,L]$. Conditional instrument independence then gives the restriction

E[H(\psi)\mid Z,L]=E[Y^0\mid L].

Thus, after removing the treatment effect on the log-odds scale, the transformed outcome risk should not vary with the instrument within $L$ strata. This is a justification for an instrumental-variable restriction, not a guarantee: exclusion, independence, relevance, and correct model specification must still be defensible. How to estimate $\psi$ from such restrictions is a separate estimation problem.

Worked Example: Checking the Blipped-Down Risk

Consider hypothetical data in a single covariate stratum. The instrument has two values. When $Z=0$, 20% receive treatment; when $Z=1$, 60% receive treatment. The observed outcome risks are 0.100 among untreated people and 0.1818 among treated people in both instrument groups. Suppose the structural odds ratio is 2, so $\psi=\log(2)$. Calculate the transformed risk for each treatment group and compare the instrument groups.

For untreated people, $A=0$, so the transformed risk is unchanged: $H=0.100$. For treated people, the observed odds are $0.1818/(1-0.1818)$, approximately $0.2222$. Blipping down divides these odds by 2:

H=\frac{0.2222/2}{1+0.2222/2} =\frac{0.1111}{1.1111} \approx 0.100.

Consequently, the mean transformed risk is $0.100$ for $Z=0$, and it is also $0.100$ for $Z=1$, regardless of the different treatment proportions. This agreement is consistent with the instrument restriction in this stratum. It does not prove instrument independence or exclusion; observed agreement can occur by chance or because violations offset one another.

Diagnostics, Comparisons, and Limitations

A practical assessment asks whether the assumptions and the structural form are plausible, rather than treating a fitted value of $\psi$ as self-validating. Inspect whether the instrument meaningfully shifts treatment within covariate strata; sparse or nearly deterministic treatment patterns undermine positivity and can make estimates unstable. Consider whether $Z$ could affect the outcome through a pathway other than treatment, and whether common causes of $Z$ and the outcome remain after conditioning on $L$. These are design and subject-matter questions; a regression fit cannot establish their absence.

The structural model itself also has diagnostic pressure points. A common $\psi$ assumes no variation in the log odds treatment effect across included strata. If clinically plausible effect modifiers exist, an unmodified model may be inadequate. Extreme risks or sparse binary-outcome cells can also make log odds estimates unstable. The blipped-down restriction can be examined across instrument levels and covariate strata, but failure to see a discrepancy is not evidence that every assumption holds, especially with limited data.

The logistic SMM differs from other approaches in both scale and target. An additive structural mean model specifies an effect on the risk-difference scale, while a multiplicative structural mean model specifies a risk ratio; these scales can imply different effect patterns. A conventional logistic regression targets an observed conditional association unless causal identification assumptions justify more. Control-function models instead represent endogeneity through a specified outcome model and a function of first-stage information. These approaches are not interchangeable merely because they all involve nonlinear outcomes or instruments. The appropriate choice depends on the causal estimand, assumptions, and plausibility of the model.

The logistic SMM has important limitations. Its odds ratio can be difficult to communicate clinically, and converting it to a risk difference requires a baseline counterfactual risk. Its homogeneous effect restriction may be unrealistic. Instrumental-variable identification may be weak when the instrument barely changes treatment, and it depends on assumptions that are often not empirically testable. Finally, a structural odds ratio should not be mistaken for a population-average risk difference or for the effect among a particular subgroup unless that estimand has been separately defined and identified.

Common pitfall: “The treatment coefficient in my logistic regression is an odds ratio, so it estimates the logistic SMM parameter.” This is wrong because the conventional coefficient describes the observed conditional association, while the SMM parameter describes a counterfactual contrast. The two may differ under endogeneity, and even unconfounded conditional and marginal odds ratios need not agree. An instrumental-variable interpretation requires its own assumptions and structural model.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

If $\psi=\log(1.5)$, what is the structural odds ratio? What does it compare?
For untreated counterfactual risk $q_0=0.10$ and structural odds ratio 2, calculate the treated counterfactual risk.
Why is an ordinary logistic regression treatment coefficient not automatically the logistic SMM parameter?
What does the blipped-down restriction say about $H(\psi)$ and the instrument, conditional on $L$?
Name two assumptions, other than structural-model correctness, needed for an instrumental-variable interpretation.

Solutions. (1) $\exp\{\log(1.5)\}=1.5$; it is the modeled ratio of the treated to untreated counterfactual odds, conditional on the structural model's strata. (2) Untreated odds are $0.10/0.90=0.1111$; treated odds are $2(0.1111)=0.2222$, so treated risk is $0.2222/(1+0.2222)=0.1818$, about 18.2%. (3) The regression coefficient describes observed conditional association; endogeneity and non-collapsibility can separate it from the structural counterfactual parameter. (4) Under the correct model and instrument assumptions, $E[H(\psi)\mid Z,L]=E[Y^0\mid L]$, so this mean does not vary with $Z$ within $L$. (5) Examples include instrument relevance, conditional instrument independence, and the exclusion restriction; consistency and positivity are also required.