Structural Nested Mean Models And Causal Blips

Why Separate Treatment Effects by Time?

With treatment measured repeatedly, a comparison of two complete treatment strategies can combine several effects. For example, a contrast between treatment at both visits and no treatment at either visit may reflect the effect of the first treatment, the effect of the second treatment, and how the first treatment changes the effect of the second. A structural nested mean model describes treatment effects one time at a time, conditional on the history available before each treatment decision.

The central quantity is a causal blip: a conditional mean contrast for changing treatment at one time from a reference value to another, with prior history held fixed and later treatment specified by a reference continuation. The word “blip” emphasizes a local change in treatment history. It is not, by itself, the contrast between two treatment strategies averaged over a population.

Histories, Reference Treatment, and Blips

Suppose treatment is assigned at times $t=1,\ldots,K$. Write $A_t$ for treatment at time $t$, and $L_t$ for covariates measured before treatment at that time. The history immediately before treatment $A_t$ is $H_t=(\bar A_{t-1},\bar L_t)$, where $\bar A_{t-1}=(A_1,\ldots,A_{t-1})$ and $\bar L_t=(L_1,\ldots,L_t)$. At the first time, $H_1$ consists of baseline covariates $L_1$. For clarity, consider binary treatment with 0 as the reference value.

Let $Y^{\bar a}$ denote the outcome that would occur under the treatment history $\bar a=(a_1,\ldots,a_K)$. To define the blip at time $t$, compare histories that agree through time $t-1$, differ at time $t$, and then follow the reference treatment value 0. The history before time $t$ is held fixed at the history being considered.

Definition: A causal blip at time $t$, for a pre-treatment history $h_t$, is the conditional mean contrast between setting $A_t=1$ and setting $A_t=0$, while holding the prior treatment history in $h_t$ fixed and setting later treatments to 0: $$ \gamma_t(h_t) = E\!\left[ Y^{(\bar A_{t-1},1,\underline{0})} - Y^{(\bar A_{t-1},0,\underline{0})} \mid H_t=h_t \right]. $$ Here $\underline{0}$ means reference treatment at every later time. A structural nested mean model specifies this contrast as a function of history and parameters, such as $\gamma_t(h_t;\psi)=g_t(h_t;\psi)$.

The contrast is on the mean outcome scale: if $Y$ is measured in days, the blip is measured in days. It is conditional because it is defined at a particular pre-treatment history $h_t$, not averaged over all histories in a population. It is nested because the contrast at time $t$ is defined within a specified prior treatment history and reference continuation.

The model’s functional form is a substantive choice. For instance, a constant blip assumes the treatment contrast is the same at every history, while a model including covariates or prior treatments allows effect variation across histories. These forms describe different causal effect patterns; neither is implied merely by calling a model “structural nested.”

Worked Examples: Interpreting and Calculating Blips

Worked Example: A History-Specific Blip

Consider a hypothetical two-time model for a continuous outcome measured in days. Suppose its time-specific blips are $$ \gamma_1(L_1)=2+0.5L_1, \qquad \gamma_2(A_1,L_1,L_2)=3+A_1+0.2L_2. $$ These are stipulated model values, not estimates from a real study. At the first time, the blip compares treatment with no treatment, with later treatment set to 0. At the second time, it compares the two current treatment values at the specified prior history.

For a person with $L_1=4$, the first-time blip is $$ \gamma_1(4)=2+0.5(4)=2+2=4\text{ days}. $$ Under this model, the conditional mean outcome under treatment at time 1 is 4 days higher than under no treatment at time 1, among the specified history, when later treatment follows the reference continuation.

At time 2, suppose $A_1=1$ and $L_2=5$. Then $$ \gamma_2(1,4,5)=3+1+0.2(5)=3+1+1=5\text{ days}. $$ This is the time-2 blip at that particular history. It is not automatically the time-2 effect for people with $A_1=0$, because the model includes $A_1$.

Worked Example: A Regime Contrast Combines Blips

Continue with a hypothetical two-time model, now specified for a fixed covariate history with $L_1=4$ and $L_2=5$. Assume these covariates do not change under the treatment interventions in this example. Let the conditional mean outcome under each treatment history be $$ E[Y^{(a_1,a_2)}]=50+(2+0.5L_1)a_1+(3+a_1+0.2L_2)a_2. $$ The outcome is in days. The first treatment blip is 4 days, and the second-time blip is $3+a_1+1=4+a_1$ days.

Under no treatment at either time, $$ E[Y^{(0,0)}]=50+4(0)+(4+0)(0)=50\text{ days}. $$ Under treatment at time 1 only, $$ E[Y^{(1,0)}]=50+4(1)+(4+1)(0)=54\text{ days}. $$ Under treatment at time 2 only, $$ E[Y^{(0,1)}]=50+4(0)+(4+0)(1)=54\text{ days}. $$ Under treatment at both times, $$ E[Y^{(1,1)}]=50+4(1)+(4+1)(1)=59\text{ days}. $$ Thus the contrast between the both-treatment and neither-treatment regimes is $59-50=9$ days. It comprises the 4-day first-time blip and the 5-day second-time blip evaluated at prior treatment $A_1=1$. The 4-day second-time blip when $A_1=0$ would instead correspond to comparing $(0,1)$ with $(0,0)$.

Worked Example: Averaging Blips Over Baseline Histories

Suppose a hypothetical population has two baseline groups. Three quarters have $L_1=0$, and one quarter have $L_1=4$. A fitted first-time blip model is $\gamma_1(L_1)=1+0.75L_1$ days. Assume the target population has exactly these group proportions and that the comparison uses the same reference continuation for everyone.

For the group with $L_1=0$, the blip is $1+0.75(0)=1$ day. For the group with $L_1=4$, it is $1+0.75(4)=4$ days. The population-average of these conditional blips is $$ 0.75(1)+0.25(4)=0.75+1=1.75\text{ days}. $$ This arithmetic illustrates how a mean over history-specific effects can be formed when the relevant history distribution is specified. It does not make either conditional blip equal to the marginal mean. In more complex longitudinal settings, the distribution of histories being averaged can depend on the treatment regime, so a marginal regime contrast requires the appropriate target-population averaging, not just a coefficient read from the blip model.

From Conditional Blips to Marginal Contrasts

A marginal contrast compares mean outcomes under two complete treatment histories or strategies in a target population. It averages over the covariate and history distribution relevant to that population. A blip, by contrast, is conditional on a history and changes treatment at one time while specifying a reference continuation. The two quantities answer different questions.

A simple algebraic decomposition helps show why. For two treatment times, the contrast between histories $(1,1)$ and $(0,0)$ can be split by inserting the intermediate history $(1,0)$:

Y^{(1,1)}-Y^{(0,0)} = \left[Y^{(1,1)}-Y^{(1,0)}\right] + \left[Y^{(1,0)}-Y^{(0,0)}\right].

The second bracket changes treatment at time 1 while setting time-2 treatment to the reference value. The first bracket changes treatment at time 2, conditional on the prior treatment history $A_1=1$. Taking expectations preserves the equality, provided all terms refer to the same target population and covariate history. The regime contrast therefore uses blips evaluated along a particular sequence of histories; it is not generally one blip evaluated at an arbitrary history.

In a longitudinal setting, histories may include covariates affected by earlier treatment. The covariate distribution under one regime can differ from the distribution under another. Consequently, averaging conditional blips over a single fixed distribution may not recover a desired marginal regime contrast. One must specify the target contrast and the histories over which it is averaged.

The structural nested mean model is a model for the sequence of conditional causal contrasts. It does not, on its own, guarantee that those contrasts are identified from observed data. Causal interpretation requires assumptions appropriate to the study, including consistency, no interference between units, adequate positivity for the treatment choices being compared, and sequential exchangeability given the measured pre-treatment history. The model’s conditional mean form must also be adequate for the blips being represented. These are distinct requirements: a convenient blip model cannot repair a failure of causal identification assumptions.

Why the Distinction Matters

Marginal structural models and structural nested mean models organize causal effects differently. A marginal structural model directly represents a marginal mean outcome or contrast across treatment regimes. A structural nested mean model represents conditional treatment changes at each time, indexed by the history before treatment. The chosen model should match the scientific question: a population-level comparison of strategies or a history-specific treatment effect.

Keeping the estimands separate also prevents misleading interpretations. A positive time-specific blip does not imply that every complete regime containing that treatment has the same positive marginal contrast. Later effects may depend on earlier treatment, and the distribution of covariate histories may differ across regimes. Conversely, a marginal contrast can conceal substantial differences in blips across histories.

Key takeaway: A structural nested mean model specifies conditional mean causal blips for treatment changes at particular times and histories. A marginal contrast compares complete treatment strategies after averaging over a target population. Connecting them requires evaluating the relevant blips along the regime comparison and using the appropriate history distribution.

Common Mistakes

Treating a blip as a regime effect. A time-$t$ blip is not automatically the effect of an entire treatment strategy; it changes treatment at one time under a specified prior history and continuation.
Ignoring the conditioning history. If a blip function includes prior treatment or covariates, its value depends on those values. Reporting it without stating the history obscures the estimand.
Adding blips without checking where they are evaluated. A regime contrast may involve a later-time blip evaluated under earlier treatment, not under the reference treatment. The sequence of histories matters.
Equating a model with identification. A specified conditional mean model is not enough to establish that observed data identify its causal blips; the relevant causal assumptions and data support must also be considered.

Check Your Understanding

Use the definitions and examples above to answer these questions.

In a two-time study, what treatment values are compared by the time-1 blip when 0 is the reference treatment?
Why does a time-2 blip potentially differ between histories with $A_1=0$ and $A_1=1$?
In the fixed-history example, how does the 9-day contrast between $(1,1)$ and $(0,0)$ decompose into time-specific contrasts?
When can an average of conditional blips be interpreted as a population-average effect, and why should the averaging distribution be specified?
What is the difference between specifying a structural nested mean model and establishing causal identification of its blips?