Why the Effect Scale Matters
A binary outcome has only two possible values, but its treatment effect can be expressed in several valid ways. A change from a 20% risk to a 40% risk is a 20-percentage-point increase, a doubling of risk, and an increase in odds by a factor of about 2.67. These statements describe the same pair of risks, but they are not interchangeable parameters. Each corresponds to a different structural mean model and answers a different scientific question.
In an instrumental-variable (IV) analysis, the scale also determines how the structural model represents the treatment effect and how that effect is estimated. The chosen scale does not repair an invalid instrument, and choosing a familiar link function does not make its coefficient automatically meaningful as a causal effect. The model, its assumptions, and the scale-specific target must be considered together.
Three Structural Mean Scales
For a concise comparison, let \(m_a(A,Z,L)=E[Y(a)\mid A,Z,L]\), the conditional mean of the potential outcome under treatment level \(a\). Since \(Y\) is binary, each \(m_a\) is a risk. A simple homogeneous structural specification makes the treatment contrast constant across the conditioning variables; more flexible models can let the parameter depend on \(L\). In either case, the scale defines what “constant effect” means.
Additive: \(m_1-m_0=\psi\). The parameter \(\psi\) is a risk difference, measured in probability points.
Multiplicative: \(m_1/m_0=\exp(\psi)\). The parameter \(\exp(\psi)\) is a risk ratio, and \(\psi\) is the log risk ratio.
Logistic: \(\operatorname{logit}(m_1)-\operatorname{logit}(m_0)=\psi\), where \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). The parameter \(\exp(\psi)\) is an odds ratio, and \(\psi\) is the log odds ratio.
These equations describe structural treatment contrasts, not ordinary regression coefficients by default. The structural baseline risk \(m_0\) can vary with \(A,Z,L\); the model constrains the treatment contrast on its designated scale. With effect modification, for example, an additive model could specify \(m_1-m_0=\psi(L)\) rather than one common \(\psi\).
The scales impose different patterns. A constant additive effect means the same absolute risk change at different baseline risks. A constant risk ratio means the same proportional change. A constant odds ratio means the same ratio of odds. Because risks are bounded between zero and one, these patterns cannot be assumed equivalent across strata.
Worked Example: Three Effects From the Same Pair of Risks
Suppose, for a specified target population and covariate stratum, the structural model implies a no-treatment risk of \(m_0=0.20\) and a treatment risk of \(m_1=0.40\). Calculate each scale-specific contrast.
Additive effect: \[ m_1-m_0=0.40-0.20=0.20. \] The treatment risk is 20 percentage points higher.
Risk ratio: \[ \frac{m_1}{m_0}=\frac{0.40}{0.20}=2.00. \] The treatment risk is twice the no-treatment risk.
Odds ratio: The treatment odds are \(0.40/(1-0.40)=0.6667\), and the no-treatment odds are \(0.20/(1-0.20)=0.2500\). Therefore \[ \frac{0.40/0.60}{0.20/0.80}=\frac{0.6667}{0.2500}=2.667. \] The corresponding logistic structural parameter is \(\psi=\log(2.667)\approx0.9808\) log-odds units. These are three descriptions of the same two risks, not three estimates of one shared parameter.
What IV Identification Requires
Choosing a scale is only one part of the analysis. To use \(Z\) to identify a structural treatment effect, the instrument must satisfy assumptions appropriate to the design and target. In a standard conditional IV formulation, these include:
- Relevance: conditional on \(L\), the distribution of treatment \(A\) changes with \(Z\). Without a treatment contrast induced by the instrument, the data do not contain the needed IV information.
- Instrument independence: conditional on \(L\), \(Z\) is independent of the potential outcomes and potential treatment assignments relevant to the model. This is a design or substantive assumption, not something established by a nonsignificant test.
- Exclusion: \(Z\) affects \(Y\) only through \(A\), conditional on \(L\). A direct effect of the instrument violates this condition.
- Consistency: the observed outcome equals the potential outcome under the treatment actually received: \(Y=Y(A)\). Treatment versions must be sufficiently well defined for this equality to be meaningful.
- Structural-model validity and identification: the chosen scale and its restrictions must adequately represent the structural contrast, and the resulting IV moment conditions must identify the parameter. In particular, instrument strength and the available variation must support estimation.
G-estimation uses the structural model to remove, or “blip down,” the modeled treatment effect and then uses the instrument restriction to estimate the parameter. The operation is scale-specific: the additive, multiplicative, and logistic models do not use the same treatment-effect transformation. For the logistic model, for example, the model works with a contrast in conditional logits; simply subtracting a constant from each observed binary outcome is not a valid logistic blip-down operation. The earlier material on G-estimation for binary structural mean models develops the estimating procedure; here the key point is that changing the scale changes the structural restriction being estimated.
How Scale Changes the Structural Restriction
A short calculation shows why the models can lead to different substantive conclusions across covariate strata. Consider two strata with the following hypothetical potential-outcome risks:
| Stratum | No-treatment risk \(m_0\) | Treatment risk \(m_1\) |
|---|---|---|
| \(L=0\) | 0.10 | 0.20 |
| \(L=1\) | 0.40 | 0.50 |
Worked Example: Which Scale Is Constant Across Strata?
For \(L=0\), the additive effect is \(0.20-0.10=0.10\), the risk ratio is \(0.20/0.10=2.00\), and the odds ratio is \((0.20/0.80)/(0.10/0.90)=2.25\).
For \(L=1\), the additive effect is \(0.50-0.40=0.10\), the risk ratio is \(0.50/0.40=1.25\), and the odds ratio is \((0.50/0.50)/(0.40/0.60)=1.50\).
The additive contrast is constant at 0.10, but the risk ratio and odds ratio differ across strata. A constant-effect additive model is compatible with these risks; a model imposing one common risk ratio or odds ratio is not. This comparison concerns the model’s structural restriction, not which model should be selected from these two rows alone.
The same reasoning can be used in reverse: a constant multiplicative or logistic effect can imply different risk differences at different baseline risks. A scale-specific model is not a claim that every other contrast is zero or irrelevant; it is a restriction on which contrast is treated as common or modeled systematically.
Worked Example: A Constant Odds Ratio Does Not Mean a Constant Risk Difference
Suppose the structural odds ratio is 2.00 in each of two strata. To convert a baseline risk \(p_0\) and odds ratio \(\theta\) to the corresponding treatment risk, start with \[ \frac{p_1}{1-p_1}=\theta\frac{p_0}{1-p_0}. \] Solving for \(p_1\) gives \[ p_1=\frac{\theta p_0}{1-p_0+\theta p_0}. \]
In a stratum with \(p_0=0.10\), the treatment risk is \[ p_1=\frac{2(0.10)}{1-0.10+2(0.10)}=\frac{0.20}{1.10}=0.1818. \] The risk difference is \(0.1818-0.10=0.0818\).
In a stratum with \(p_0=0.50\), the treatment risk is \[ p_1=\frac{2(0.50)}{1-0.50+2(0.50)}=\frac{1.00}{1.50}=0.6667. \] The risk difference is \(0.6667-0.50=0.1667\). Thus the same odds ratio of 2.00 corresponds to risk differences of about 8.18 and 16.67 percentage points at these two baseline risks.
Selecting and Evaluating a Scale
The best scale is not determined solely by which model returns a convenient coefficient. Begin with the substantive question. If the decision concerns absolute excess cases or a risk reduction in percentage points, the additive scale may be most directly relevant. If the question concerns proportional change in risk, the multiplicative scale may be natural. If odds are the scientific or modeling target, the logistic scale may be appropriate. The outcome’s prevalence, the intended target population, and the way the effect will be communicated also matter.
Then examine whether the structural restriction is plausible. Relevant checks include whether predicted risks remain between zero and one; whether the chosen model is compatible with observed patterns across covariate strata; whether the instrument produces a useful treatment contrast; and whether estimates are highly sensitive to modeling choices. These diagnostics can reveal problems, but they do not prove instrument independence or exclusion. Weak relevance can produce unstable estimates and wide uncertainty on any scale.
Model comparisons require care. A numerical coefficient of 0.20 on the risk-difference scale is not directly comparable to a coefficient of 0.20 in log-risk or log-odds units. Nor does a better-fitting outcome model alone establish that its structural causal assumptions are correct. Whenever models are compared, describe the scale and units, and distinguish assumptions about the instrument from assumptions about the treatment-effect model.
| Scale | Parameter interpretation | Useful question | Important limitation |
|---|---|---|---|
| Additive | Risk difference | How many percentage points does risk change? | A common difference can imply different relative changes at different baseline risks. |
| Multiplicative | Risk ratio | By what factor does risk change? | Implied risks must remain in the interval from zero to one. |
| Logistic | Odds ratio | By what factor do the odds change? | Odds ratios are not risk ratios; their risk implications depend on baseline risk. |
Common Pitfalls and Limitations
- Calling an odds ratio a risk ratio. They are different contrasts. The distinction can be substantial when risks are not small.
- Assuming one scale’s constant effect is constant on another scale. The calculations above show that constancy is scale-specific.
- Reading the structural coefficient as a population-wide effect without checking its target. The conditioning set, structural restrictions, and IV identification determine what population or conditional contrast the parameter represents.
- Treating a fitted model or diagnostic as proof of IV validity. Relevance can be investigated empirically, but independence and exclusion require design knowledge and substantive justification.
- Ignoring feasibility and instability. A proposed multiplicative or other nonlinear model can imply impossible risks, and weak instruments can make estimates unstable even when the scale is scientifically appealing.
No effect scale is universally best. Different scales can be useful for different decisions, and reporting more than one can clarify how the result translates across interpretations. But the estimates should not be presented as interchangeable: each parameter answers a distinct question under its own structural restriction. The subsequent step of translating nonlinear IV estimates into marginal effects or counterfactual risks requires additional care beyond simply reading off a model coefficient.
Check Your Understanding
Answer each question, then check the brief solutions below.
- A structural model gives \(m_0=0.30\) and \(m_1=0.45\). What are the risk difference and risk ratio?
- What does a constant logistic structural parameter mean: a constant risk difference, risk ratio, or odds ratio?
- For an odds ratio of 1.50 and baseline risk \(p_0=0.20\), calculate the implied treatment risk using \(p_1=\theta p_0/(1-p_0+\theta p_0)\).
- Name two IV assumptions that are not established merely by finding a statistically significant association between \(Z\) and \(A\).
Solutions. (1) The risk difference is \(0.45-0.30=0.15\), or 15 percentage points; the risk ratio is \(0.45/0.30=1.50\). (2) It means a constant log-odds difference, equivalently a constant odds ratio, under the specified model. (3) \(p_1=1.50(0.20)/(0.80+1.50(0.20))=0.30/1.10=0.2727\), approximately 27.27%. (4) Instrument independence and exclusion require substantive justification; a treatment association addresses relevance, not those assumptions.