Smoking Cessation Medication and Relapse

Why Repeated Medication Use Creates a Feedback Problem

A person’s decision to use cessation medication can change as their quit attempt unfolds. Someone who has smoked since the last visit, or who reports severe withdrawal, may be more likely to start or continue medication. Those same experiences also predict whether the person will be abstinent later. If medication at one visit can itself affect smoking status or withdrawal at the next visit, then those predictors are partly consequences of earlier treatment. This is time-varying confounding affected by prior treatment.

Consider a two-decision study. At baseline, $L_0$ records smoking history and withdrawal symptoms; $A_0$ indicates medication use during the first interval. At the next visit, $L_1$ records smoking status and withdrawal symptoms, and $A_1$ indicates medication use during the next interval. Let $Y=1$ mean abstinent at the end of follow-up and $Y=0$ otherwise. The target could be the probability of abstinence if everyone followed a specified medication strategy, such as medication in both intervals, compared with no medication in either interval.

Ordinary adjustment for $L_1$ in a conventional outcome regression can block part of the effect of $A_0$, because $L_1$ may lie on the pathway from earlier medication to later abstinence. Omitting $L_1$, however, leaves the association between $A_1$ and abstinence confounded. Marginal structural models address this feedback by weighting observations according to their treatment histories.

Definition: For a medication strategy $\bar a=(a_0,a_1)$, let $Y^{a_0,a_1}$ denote the abstinence outcome that would be observed if medication use were set to $a_0$ in the first interval and $a_1$ in the second. The marginal abstinence probability under that strategy is $P(Y^{a_0,a_1}=1)$, where the probability averages over the study population’s baseline characteristics.

Assumptions and Treatment Weights

The causal interpretation requires a well-defined intervention: for example, the same specified medication regimen and adherence rule for everyone assigned $A_t=1$. It also requires consistency (the observed outcome equals the potential outcome for the treatment history actually received), no interference between participants, and complete outcome ascertainment for the analysis described here. In addition, treatment must be sequentially exchangeable: conditional on measured history, medication use at each decision is independent of the potential outcomes under the strategies of interest. For two decisions, the relevant histories are $L_0$ before $A_0$, and $(L_0,A_0,L_1)$ before $A_1$. Finally, each treatment choice required by a strategy must have positive probability in every history that occurs in the target population.

Write $p_0(A_0\mid L_0)=P(A_0\mid L_0)$ and $p_1(A_1\mid A_0,L_0,L_1)=P(A_1\mid A_0,L_0,L_1)$ for the treatment probabilities given the observed histories. A stabilized weight uses less detailed treatment probabilities in its numerator and history-conditional probabilities in its denominator:

SW = \frac{P(A_0)}{P(A_0\mid L_0)} \;\frac{P(A_1\mid A_0)}{P(A_1\mid A_0,L_0,L_1)}.

These probabilities are evaluated at each person’s observed treatment values. Thus, for a person who used medication at both decisions, the numerators and denominators are the probabilities of $A_0=1$ and $A_1=1$. The denominators account for why that person’s treatment history was likely given smoking and withdrawal history; the numerators stabilize the weights by retaining dependence on the prior treatment history. In practice, these probabilities are estimated from treatment models, and uncertainty in their estimation should be reflected in inference.

Worked Example: A Medication Weight After a Lapse

Suppose a participant used medication at the first decision, then reported smoking since the last visit and severe withdrawal. The fitted treatment models give $P(A_0=1)=0.40$, $P(A_0=1\mid L_0)=0.20$, $P(A_1=1\mid A_0=1)=0.60$, and $P(A_1=1\mid A_0=1,L_0,L_1)=0.30$. Calculate the stabilized weight.

The first factor is $0.40/0.20=2$. The second factor is $0.60/0.30=2$. Therefore,

SW = \frac{0.40}{0.20}\frac{0.60}{0.30}=2\times2=4.

The observed history made this treatment sequence less likely than the numerator model alone would suggest, so the participant represents more than one person in the weighted analysis. The weight is not a personal causal effect, and it does not mean this individual’s abstinence probability is four times larger.

Why Weighting Identifies Strategy-Specific Means

The inverse-probability identity makes the role of the treatment models explicit. For a fixed strategy $(a_0,a_1)$, define its unstabilized regime weight as the indicator of receiving that exact treatment history divided by its conditional probability:

W^{a_0,a_1} = \frac{I(A_0=a_0,A_1=a_1)} {P(A_0=a_0\mid L_0)\, P(A_1=a_1\mid A_0=a_0,L_0,L_1)}.

Theorem: Under consistency, sequential exchangeability at both treatment decisions, and positivity for $(a_0,a_1)$, $$ E[W^{a_0,a_1}Y]=E[Y^{a_0,a_1}] \quad\text{and}\quad E[W^{a_0,a_1}]=1. $$ Thus the ratio of these expectations is the marginal mean outcome under the strategy.

Proof. Let $g_t$ denote the true treatment probability in the denominator at decision $t$. Conditional on the history $(L_0,A_0,L_1)$, sequential exchangeability at the second decision and consistency imply

E\!\left[ \frac{I(A_1=a_1)Y}{g_1} \middle| L_0,A_0=a_0,L_1 \right] = E[Y^{a_0,a_1}\mid L_0,A_0=a_0,L_1].

Indeed, the conditional probability of $A_1=a_1$ is $g_1$, so division by $g_1$ cancels the treatment-selection probability; among those receiving $a_1$, consistency identifies the observed outcome with the corresponding potential outcome. Now condition on $L_0$ and average over $A_0$. Exchangeability at the first decision identifies the conditional mean under $A_0=a_0$ with the mean of the potential outcome under that first decision. Multiplication by $I(A_0=a_0)/g_0$ cancels the first treatment-selection probability in the same way. Averaging over $L_0$ gives $E[W^{a_0,a_1}Y]=E[Y^{a_0,a_1}]$. Repeating these steps with $Y$ replaced by the constant 1 gives $E[W^{a_0,a_1}]=1$. Positivity ensures that both denominators are nonzero for histories with positive probability. $\square$

In a sample, the corresponding strategy-specific estimate is a weighted mean among participants who followed that strategy. With known treatment probabilities, independent participants, and positive probabilities bounded away from zero, the weights are bounded; the law of large numbers then makes the sample numerator and denominator converge to their expectations. Their ratio converges to $E[Y^{a_0,a_1}]$, since the denominator converges to 1. With estimated probabilities, consistent treatment models and appropriate regularity conditions are additionally needed for this argument to apply.

Worked Example: Estimating Two Abstinence Probabilities

For a small illustrative sample, suppose the fitted stabilized weights and outcomes among participants following each of two strategies are as follows. These values illustrate the weighted-mean calculation; the small sample is not intended to support precise inference.

Strategy followed	Participant	Stabilized weight	Abstinent $Y$
Medication in both intervals	1	2.0	1
Medication in both intervals	2	1.0	0
No medication in either interval	3	1.0	1
No medication in either interval	4	3.0	0

For medication in both intervals, the weighted outcome total is $2.0(1)+1.0(0)=2.0$, and the total weight is $2.0+1.0=3.0$. The estimated marginal abstinence probability is therefore $2.0/3.0\approx0.667$.

For no medication in either interval, the weighted outcome total is $1.0(1)+3.0(0)=1.0$, and the total weight is $1.0+3.0=4.0$. The estimated probability is $1.0/4.0=0.25$. The estimated risk difference, medication in both intervals minus neither, is $0.667-0.25=0.417$, or about 42 percentage points. This contrast is descriptive of the illustrative weighted sample; its causal interpretation depends on the assumptions above and its uncertainty depends on the actual study size and analysis.

Using a Marginal Structural Model

Rather than estimate each strategy-specific mean by a separate weighted average, one can fit a weighted marginal structural model (MSM) for the outcome. For example, a saturated model for two binary treatment decisions represents the four strategy-specific means:

E[Y^{a_0,a_1}] = \beta_0+\beta_1a_0+\beta_2a_1+\beta_3a_0a_1.

A weighted regression of $Y$ on $A_0$, $A_1$, and their interaction uses the stabilized weights. Because $Y$ is binary, an identity-link model can express probabilities directly when fitted values remain between 0 and 1; alternatively, a logistic MSM models log odds and requires converting fitted values back to probabilities. The model form determines which contrasts it can represent. A saturated model avoids imposing a no-interaction restriction for two binary decisions. The model’s predicted probabilities are marginal over the measured histories in the weighted pseudo-population, rather than conditional on $L_1$ as in a conventional outcome model that adjusts for the post-baseline covariate.

Worked Example: Interpreting the Interaction Term

Suppose an identity-link MSM has $\beta_0=0.25$, $\beta_1=0.10$, $\beta_2=0.08$, and $\beta_3=0.237$. For neither interval on medication, the fitted probability is $0.25$. For medication in both intervals, it is

$$ 0.25+0.10+0.08+0.237=0.667. $$

The interaction term allows the joint strategy’s fitted probability to differ from what would be obtained by adding the two separate treatment coefficients alone. Here, without the interaction, the prediction for medication in both intervals would be $0.25+0.10+0.08=0.43$, not $0.667$. The probabilities in this example lie between 0 and 1, as required for their interpretation as risks.

Diagnostics, Alternatives, and Limitations

A weighted analysis depends on the treatment models adequately representing the histories that predict medication use. Inspect the estimated treatment probabilities and weights, including their range and whether a few observations dominate weighted summaries. Check whether each strategy has observed support across relevant smoking and withdrawal histories. A fitted treatment probability near zero signals a practical positivity problem: the required comparison relies on very few observations and may be unstable. Also check weighted covariate balance across treatment groups at each decision, especially for the measured histories used in the treatment models. These checks can reveal problems; they cannot establish that unmeasured confounding is absent.

The main alternative is the parametric g-formula, which models the evolving covariate and outcome distributions and then predicts outcomes under specified treatment strategies. It handles treatment-confounder feedback without constructing inverse-probability weights, but depends on correct models for those distributions. A conventional regression that simply adjusts for all measured variables is not generally an equivalent solution: adjustment for $L_1$ can remove part of the effect of $A_0$, while leaving it out may confound the $A_1$ comparison. The weighted MSM instead uses treatment weights to create a pseudo-population in which measured treatment predictors are balanced while modeling marginal outcomes by treatment history.

The analysis is limited by unmeasured confounding, poorly defined interventions, treatment-model misspecification, and inadequate positivity. The binary outcome makes abstinence probabilities easy to interpret, but does not remove those concerns. If follow-up is incomplete, the analysis also needs to address selection due to loss to follow-up; treating missing outcomes as if they were observed does not meet the complete-ascertainment assumption. A causal conclusion should therefore be reported alongside the intervention definition, assumptions, weight diagnostics, and uncertainty—not as a consequence of weighting alone.

Common pitfall: A tempting argument is, “Smoking status and withdrawal predict later medication, so I will adjust for both in one ordinary outcome regression.” That ignores that $L_1$ may be affected by $A_0$; conditioning on it can block part of the earlier medication effect and does not by itself resolve the feedback problem. The correction is to use a method suited to time-varying confounders affected by prior treatment, such as a weighted MSM, while stating and assessing its assumptions.

Check Your Understanding

Answer each question, then compare with the brief solutions.

Why can $L_1$, the smoking-status and withdrawal history before the second medication decision, be both a confounder for $A_1$ and a consequence of $A_0$?
For a participant with $A_0=0,A_1=1$, suppose $P(A_0=0)=0.60$, $P(A_0=0\mid L_0)=0.30$, $P(A_1=1\mid A_0=0)=0.50$, and $P(A_1=1\mid A_0=0,L_0,L_1)=0.25$. Calculate the stabilized weight.
In a weighted group, abstinent outcomes have total weight 15 and the total weight of all participants is 40. What is the weighted abstinence probability?
Name two assumptions needed to interpret the weighted contrast causally, and give one diagnostic relevant to the treatment weights.

Solutions. (1) Earlier medication may change subsequent smoking or withdrawal, and those histories can predict both later medication and final abstinence. (2) The weight is $(0.60/0.30)(0.50/0.25)=2\times2=4$. (3) The estimate is $15/40=0.375$. (4) Examples include sequential exchangeability and positivity; a relevant diagnostic is examining treatment probabilities or weight distributions and checking weighted balance in measured treatment histories.