Why the Link Determines the Effect Scale
A marginal structural model (MSM) describes a population-level causal quantity, such as the mean outcome that would occur if everyone followed one treatment strategy rather than another. Weighting can support estimation of that quantity under appropriate causal assumptions, but the model’s link function determines how treatment contrasts are represented. Choosing a link is therefore not just a computational choice: it sets the scale on which the model expresses effects.
For a binary outcome measured at a fixed follow-up time, the marginal mean under strategy \(g\) is the counterfactual risk \(p(g)=E[Y^g]\), where \(Y^g\) is the outcome that would be observed under that strategy. For two strategies, \(g=0\) and \(g=1\), the three common effect scales are the risk difference \(p(1)-p(0)\), risk ratio \(p(1)/p(0)\), and odds ratio comparing \(p(1)/(1-p(1))\) with \(p(0)/(1-p(0))\). These are distinct summaries of the same pair of risks.
The causal interpretation requires more than a particular link. In a longitudinal treatment setting, consistency, sequential exchangeability given the measured histories, and positivity are needed for the observed data and weighting procedure to identify the relevant counterfactual means. Any required censoring adjustment must also be adequate. The chosen MSM must then represent the target marginal means sufficiently well. These assumptions justify interpreting the fitted marginal contrasts causally; the link alone does not.
Three Links, Three Interpretations
Write a simple MSM for a binary strategy indicator \(g\), with \(\beta_0\) as the intercept and \(\beta_1\) as its coefficient. The interpretation of \(\beta_1\) depends on the link.
These interpretations follow by substituting \(g=0\) and \(g=1\). Under the identity link, \(p(0)=\beta_0\) and \(p(1)=\beta_0+\beta_1\), so subtraction gives \(\beta_1\). Under the log link, the two risks are \(\exp(\beta_0)\) and \(\exp(\beta_0+\beta_1)\); their ratio is \(\exp(\beta_1)\). Under the logit link, the difference between the two log odds is \(\beta_1\), so exponentiating yields the odds ratio.
Thus coefficients from different links cannot be compared as though they were expressed in the same units. An identity-link coefficient is a difference in probability, often reported in percentage points. A log-link coefficient is a log risk ratio, and a logit-link coefficient is a log odds ratio; exponentiating either gives the corresponding ratio. A ratio is unitless, but its value is not interchangeable with a probability difference.
Worked Example: Expressing Two Strategy Risks on Three Scales
Suppose a hypothetical weighted MSM estimates a 20% risk under a reference strategy and a 35% risk under an alternative strategy at a fixed follow-up time. Assume the causal identification conditions are credible and the model correctly represents these two strategy-specific risks. Calculate each contrast.
Risk difference. The identity-link coefficient for the alternative-strategy indicator is \(0.35-0.20=0.15\). The estimated risk is 15 percentage points higher under the alternative strategy.
Risk ratio. The log-link coefficient is the log of the risk ratio: \(\beta_1=\log(0.35/0.20)=\log(1.75)=0.5596\). Exponentiating gives \(\exp(\beta_1)=1.75\): the estimated risk under the alternative strategy is 1.75 times the reference-strategy risk.
Odds ratio. The reference odds are \(0.20/0.80=0.25\), and the alternative-strategy odds are \(0.35/0.65=0.5385\). Therefore, \(\exp(\beta_1)=0.5385/0.25=2.154\), and \(\beta_1=\log(2.154)=0.7673\). The estimated odds are about 2.15 times as high under the alternative strategy.
All three calculations describe the same pair of estimated risks, but answer different questions. In particular, an odds ratio of 2.154 does not mean the risk is 2.154 times as high; the risk ratio here is 1.75.
What Each Choice Assumes and Constrains
A link determines the form of the model over treatment strategies. If a model includes only a binary indicator for two strategies, it can represent their two risks on any of these scales, provided the resulting probabilities are valid. With several treatment histories, a model usually imposes a structure across histories. For example, a linear predictor with additive history terms asserts a particular pattern of contrasts on the selected link scale. Adding terms or interactions changes that structure; it does not make the link scales equivalent.
| Link | Effect scale for binary strategy contrast | Main probability constraint |
|---|---|---|
| Identity | Risk difference | Every fitted risk must be between 0 and 1 |
| Log | Risk ratio | Every fitted risk must be positive and no greater than 1 |
| Logit | Odds ratio | Fitted risks are between 0 and 1 by construction |
The identity link is attractive when an absolute change in risk is the scientific target, such as additional cases per 100 people. Its linear form is easy to interpret, but the model does not automatically keep predictions within the probability range. The log link directly represents relative risk, often useful when proportional changes are meaningful. However, exponentiating a linear predictor guarantees positive values, not values at most 1. The logit link always maps finite linear predictors to probabilities strictly between 0 and 1, but its coefficient is an odds ratio rather than a risk ratio or risk difference.
For models that include a quantitative exposure or multiple history terms, the coefficient generally describes a change associated with a specified change in that model term, on the chosen link scale. It is not automatically the contrast between any two complete strategies. To report a strategy contrast, obtain the model-implied marginal risks for each strategy and then calculate their difference, ratio, or odds ratio as appropriate. This also makes clear which strategies are being compared.
Worked Example: A Coefficient of 1.5 Does Not Have One Meaning
Consider three hypothetical MSMs, each with a reference-strategy risk of 0.10. The treatment coefficient is chosen so that the implied treatment contrast is 0.10 for the identity-link model and 1.5 on the ratio scale for each other model. These are alternative models, not three estimates required to agree.
Identity link. With \(\beta_0=0.10\) and \(\beta_1=0.10\), the alternative risk is \(0.10+0.10=0.20\). The coefficient is an increase of 10 percentage points.
Log link. A risk ratio of 1.5 gives an alternative risk of \(0.10(1.5)=0.15\). The log-link coefficient is \(\log(1.5)=0.4055\), not 1.5; its exponent is the risk ratio.
Logit link. An odds ratio of 1.5 applied to a reference risk of 0.10 gives alternative odds \(1.5[0.10/(1-0.10)]=1.5(0.1111)=0.1667\). Converting odds to risk gives \(0.1667/(1+0.1667)=0.1429\). The logit coefficient is \(\log(1.5)=0.4055\), while the resulting risk difference is \(0.1429-0.10=0.0429\), or 4.29 percentage points.
The same numerical ratio, 1.5, implies a 50% higher risk in the log-link model but 50% higher odds in the logit-link model. At this reference risk, those correspond to different alternative risks: 0.15 and 0.1429. The identity model’s coefficient has yet another interpretation and unit.
Choosing a Scale for the Scientific Question
A practical choice begins with the effect that needs to be communicated. If decisions depend on excess cases per 1,000 people, an absolute risk difference may be most directly relevant. If the scientific question concerns proportional changes in risk, a risk ratio is a direct target. If the analysis uses a logistic model for a binary outcome or a discrete-time event process, odds ratios may be convenient, but they must be described as odds contrasts.
The scale can affect both interpretation and model behavior. Identity and log links may require checking that all relevant fitted risks remain in the allowable range. The logit link avoids invalid fitted probabilities, but a stable logistic fit does not establish that the odds ratio is the most informative effect measure. In every case, estimates depend on both the weighting assumptions and the specified MSM; a convenient link does not correct a misspecified model or unsupported treatment histories.
For a discrete-time outcome, pooled logistic MSMs commonly model the marginal event hazard in each interval among those still at risk. A treatment coefficient in such a model is an odds ratio for the interval-specific hazard, conditional on remaining event-free up to that interval. It is not automatically an odds ratio for cumulative risk by the end of follow-up. Cumulative risk depends on hazards across intervals, so it must be obtained from those hazards rather than read directly from a single interval coefficient.
Worked Example: Interval Hazard Odds Are Not Cumulative-Risk Odds
Suppose, hypothetically, that under one strategy the event hazards in two successive intervals are 0.10 and 0.20, and under another they are 0.20 and 0.30. Assume these are the relevant marginal hazards among people still event-free at the start of each interval. Calculate the cumulative risks through the second interval.
For the first strategy, the probability of remaining event-free through both intervals is \((1-0.10)(1-0.20)=0.90(0.80)=0.72\). Its cumulative risk is therefore \(1-0.72=0.28\).
For the second strategy, the probability of remaining event-free is \((1-0.20)(1-0.30)=0.80(0.70)=0.56\). Its cumulative risk is \(1-0.56=0.44\).
The cumulative risk difference is \(0.44-0.28=0.16\), and the cumulative risk ratio is \(0.44/0.28=1.571\). The cumulative odds ratio is \([0.44/(1-0.44)]/[0.28/(1-0.28)]=(0.7857)/(0.3889)=2.020\). These cumulative contrasts are calculated from the pair of risks; they are not obtained by treating an interval-specific logistic coefficient as a cumulative-risk odds ratio.
Common Mistakes and a Practical Selection Process
A frequent error is to exponentiate a coefficient and call the result a risk ratio without checking the link. Exponentiation of a log-link treatment coefficient gives a risk ratio; exponentiation of a logit-link coefficient gives an odds ratio. Another mistake is to present a link-scale coefficient as though it were the only scientifically meaningful contrast. When absolute consequences matter, a model fitted with another link can still be used to derive model-implied risks and their difference, as long as those predictions are appropriate.
Specify the treatment strategies, outcome, follow-up time, and whether the primary summary is an absolute risk difference, risk ratio, or odds ratio.
Use the identity, log, or logit link when its corresponding contrast is the intended model coefficient and the model form is suitable.
Confirm the model-implied risks are valid, state the contrast and units, and distinguish risks from odds and interval hazards from cumulative risks.
If the link is selected for computational convenience rather than because its coefficient is the target, report the strategy-specific risks and calculate the desired contrast explicitly. The central discipline is to name the scale before interpreting a coefficient.
Check Your Understanding
Answer each question by identifying the effect scale and interpreting it in context.
- A log-link MSM has a treatment-strategy coefficient of 0.30. What is the estimated risk ratio, and what calculation gives it?
- A logit-link MSM reports an exponentiated coefficient of 1.8. What does this compare, and what does it not directly say about risks?
- Why must fitted risks from an identity-link or log-link MSM be checked, even though a logit link maps predictions into the probability range?
- Two strategies have estimated risks 0.12 and 0.18. Calculate their risk difference and risk ratio. Which link’s treatment coefficient directly corresponds to each contrast in the simple binary-strategy model?
- Why is an interval-specific odds ratio from a pooled logistic MSM not automatically the odds ratio for cumulative risk at the end of follow-up?