Choosing Blip Functions And Time Structures

Why Blip-Function Choices Matter

A structural nested mean model describes how a treatment changes an outcome through causal contrasts called blips. Choosing a blip function means deciding how those contrasts vary with treatment amount, treatment time, and the history available before treatment. That choice is substantive: it determines which causal patterns the model can represent and which it rules out.

A compact model can be useful when its restrictions reflect the target scientific question. But a simple equation is not automatically a good choice. A constant treatment coefficient, for example, says that the treatment blip is the same at every modeled occasion and history unless the model includes terms that allow it to vary. The aim is not to make the model as flexible as possible; it is to choose a structure that represents the intended contrast without imposing unsupported restrictions.

This definition makes the reference value and outcome scale explicit. For binary treatment, 0 is commonly the untreated reference and the contrast of interest is treatment versus no treatment. For a continuous dose, the blip at dose \(a_t\) is relative to dose 0. A different reference can be chosen, but the function must then be defined relative to it. In an additive mean model, the blip is measured in outcome units.

Functional-form choice is separate from identification. A causal interpretation still requires assumptions such as consistency, sequential exchangeability given the measured history, and positivity for the treatment levels being compared. Even when those conditions hold, a misspecified blip function may fail to represent the causal contrasts of interest. Conversely, a well-chosen function does not repair a failure of identification assumptions.

Time Structures: One Effect or Several?

When treatment can occur at multiple times, the model must say how the blip varies across those times. A constant effect structure uses one common parameter. A time-specific structure permits separate blips at each occasion. A trend structure lies between those choices: it allows effects to vary, but only according to a specified pattern.

The common-effect form says that a unit of treatment at each modeled time has the same blip, on the chosen scale and conditional on the included history. The time-specific form makes no such equality restriction across visits. The trend form assumes a particular, smooth change in the blip across the chosen time index. Equally spaced visits may justify using the visit number as a linear index; irregularly spaced visits may call for actual elapsed time instead.

Choosing the Treatment-Response Shape

Treatment need not be binary. For a continuous dose, a linear blip assumes that each additional unit of dose changes the causal contrast by the same amount throughout the modeled range. A nonlinear blip can express curvature, thresholds, or diminishing changes. The treatment scale and its units must be stated, because coefficients depend on them.

For any dose-response form with a zero-dose reference, the blip should generally satisfy \(\gamma_t(h_t,0;\psi)=0\). A model with an intercept that violates this condition would assign a nonzero contrast to the reference dose unless it were redefined accordingly. This is a useful structural check before interpreting coefficients.

Matching the Time Index to Treatment Timing

“Time-varying” can refer to different scientific structures. A calendar-time model lets the blip depend on the visit or elapsed study time. A duration model lets it depend on how long the person has been exposed. These are not interchangeable when people start treatment at different times. The index should follow the causal contrast: ask whether the intended change is by study occasion, by time since initiation, or by another pre-treatment feature of the history.

For example, a model indexed only by visit number can assign different blips to visits 1 and 3 even for people with different treatment start dates. A duration-indexed model instead assigns blips according to the time elapsed since initiation. If that duration is determined from treatment history, it must be defined using information available before the treatment whose blip is being modeled.

A Practical Specification Process

A disciplined choice starts with the contrast, not the formula. In particular, state the outcome scale, reference treatment, time index, and the kinds of variation the target requires. Then decide which restrictions are defensible: a common treatment effect, a smooth trend, distinct visit effects, or a nonlinear dose curve. If effect modification is relevant, include it only in a form tied to the modifier and contrast of interest; the same logic applies to time and dose.

These choices matter because G-estimation uses the specified blip model in its estimating conditions. Under the identification assumptions, a correctly specified blip function can support the intended causal interpretation. If the function is misspecified, the fitted parameters need not equal the causal contrasts the analyst intended to describe. A flexible model is not automatically preferable, either: it may require more information than the data provide and may estimate the target less precisely. The central task is to balance scientific alignment and a defensible level of restriction.

Common Mistake

A common mistake is to call any model containing a time term “time-varying enough.” For example, a linear time trend does allow blips to change, but it also requires the changes to follow a straight line in the chosen time index. It cannot generally reproduce arbitrary visit-specific effects, and it may use the wrong index if the scientific target is time since treatment initiation rather than calendar time.

The correction is to interpret the restrictions explicitly. Compare the implied blips at relevant treatment times and levels, as in the worked examples. Then choose a shared effect, trend, or separate effects according to the target contrast and scientific rationale—not merely because one form is convenient to write down.