Why Longitudinal Positivity Problems Accumulate

Why One Visit Is Not the Whole Positivity Problem

At a single visit, a treatment probability can look comfortably away from zero. But a longitudinal analysis assigns weight to a person’s sequence of treatment decisions, not to each decision in isolation. The probability associated with a sequence is built from conditional probabilities at successive visits. Even moderately sized factors can produce a small product, and the inverse of that product can be large.

The key distinction is between checking whether each treatment decision is possible at a given history and assessing how much probability supports a particular sequence across follow-up. This tutorial derives that multiplication and shows its consequences for inverse-probability weights. It does not imply that every small path probability is a structural violation: small probabilities can arise through accumulation even when no single visit has a probability near zero.

From Visit-Specific Probabilities to a Path Probability

Suppose treatment is assigned at visits \(t=1,\ldots,T\). Let \(A_t\) denote treatment at visit \(t\), and let \(H_t\) denote the observed history available immediately before that treatment decision, including prior treatments and measured covariates. For a specified treatment sequence \(\bar a=(a_1,\ldots,a_T)\), define

at the history encountered along that sequence. Each \(q_t\) is a conditional probability at one decision point. The product of these conditional probabilities is the treatment-assignment likelihood contribution for the specified sequence along its successive histories. It is not, in general, the marginal probability of that sequence conditional only on baseline covariates: histories can evolve over time, and a marginal probability may require averaging over those histories.

This product follows directly from the chain rule. Each factor conditions on the history available at its decision, so later factors are evaluated after the earlier treatment choices and the resulting observed histories. The inverse weight becomes large when the product is small, even if no individual factor is exceptionally small.

Proof. Since each \(q_t\geq\epsilon>0\), multiplying the \(T\) inequalities gives \(Q_T=\prod_{t=1}^{T}q_t\geq\prod_{t=1}^{T}\epsilon=\epsilon^T\). Taking reciprocals reverses the inequality between positive quantities, so \(W_T=1/Q_T\leq\epsilon^{-T}\). This bound is for each \(T\). If \(0<\epsilon<1\), then \(\epsilon^T\) tends to zero and \(\epsilon^{-T}\) grows exponentially as \(T\) increases; a uniform per-visit lower bound below one does not stop cumulative probabilities from shrinking. Only the special bound \(\epsilon=1\) makes this particular lower bound remain one. \(\square\)

Worked Calculations

Why the Product Is a Useful Diagnostic Quantity

The product makes cumulative scarcity explicit. Taking logarithms turns multiplication into addition:

Because \(0<q_t\leq1\), each term \(-\log(q_t)\) is nonnegative. Every visit with \(q_t<1\) adds to the log weight, and a visit with a smaller probability adds more. This explains why a sequence of individually unremarkable contributions can yield a large total weight. It also motivates examining weight growth across follow-up and log-scale diagnostics, alongside the cumulative probabilities themselves.

The multiplication has an important conditional interpretation. It describes the treatment-assignment contribution along the realized sequence of histories. It should not be read as asserting that every person has the same path probability, or that the product is the population-marginal probability of a path after averaging over covariate histories. Across people, the \(q_t\) values can differ because their histories differ; their resulting products and weights can therefore differ as well.

Assumptions, Comparisons, and Failure Modes

The calculations require that the treatment probabilities refer to the specified treatment choices at the relevant histories, and that the factors used in the product are positive. If a required \(q_t\) is zero, then the sequence has zero treatment-assignment probability at that history and its inverse weight is undefined. A positive but very small factor instead produces a finite, potentially very large weight. These are different situations: structural impossibility is not the same as practical scarcity.

A per-visit diagnostic asks whether treatment probabilities are small at individual decision points. A cumulative diagnostic asks what happens after those probabilities are multiplied along a sequence. Neither replaces the other. A lower bound \(q_t\geq\epsilon\) is informative for a fixed follow-up length, but when \(\epsilon<1\) its implied worst-case weight bound \(\epsilon^{-T}\) can itself increase rapidly with \(T\). For example, with \(\epsilon=0.8\), the bound at eight visits is \(0.8^{-8}\approx5.960\), while at twelve visits it is \(0.8^{-12}\approx14.552\).

Stabilized weights are one alternative to using the unstabilized inverse product: their numerator can reduce the size of the ratio, as the third example showed. But comparing those approaches does not change the denominator’s role. Nor does a large weight by itself identify why it arose: it can reflect a long sequence of moderately unlikely decisions, one particularly small conditional probability, or both. Earlier material on reading marginal weight distributions, tail quantiles, log-scale diagnostics, and tracing weight growth provides complementary ways to inspect the resulting weights.

A practical warning sign is a pattern in which cumulative probabilities decline with follow-up while the largest weights or upper tail increase. Examine the factors and the products at their corresponding histories, and distinguish a single difficult visit from gradual accumulation. Remember that empirical diagnostics describe the fitted probabilities available in the data; they do not establish that those probabilities are correct or that all relevant histories are represented.

Common mistake: “Every visit-specific treatment probability is at least 0.5, so the longitudinal weight cannot become very large.” This treats a per-visit bound as though it were a bound on the entire product. The correction is to multiply across visits: if \(q_t=0.5\) for ten visits, then \(Q_{10}=0.5^{10}=0.0009765625\) and \(W_{10}=1{,}024\). The per-visit probabilities are all positive and identical, but their product is small.

The product calculation is a diagnostic explanation, not a guarantee that a weighting analysis is adequate. It does not by itself establish the validity of the treatment model, resolve structural support failures, or determine how a particular analysis should be altered. Its value is narrower and essential: it shows how the probability of a longitudinal sequence can deteriorate through follow-up even when each decision looks reasonably supported in isolation.