Ventilation Decisions Under Changing Critical Illness Severity

Why Ventilation Decisions Need a Longitudinal Comparison

In critical illness, ventilation is not a one-time baseline choice. Clinicians make decisions repeatedly as oxygenation, organ function, and overall severity change. Those same measurements predict mortality and are affected by prior treatment. For example, ventilation today may influence tomorrow’s organ function, which then helps determine tomorrow’s ventilation decision. A comparison that simply adjusts for the latest severity in an ordinary outcome regression can therefore mishandle the evolving relationship between treatment and prognosis.

A marginal structural model (MSM) analysis instead asks a policy-level question: what would the mortality risk be if eligible patients followed one specified daily ventilation strategy rather than another? The strategies must be stated precisely, and the observational data must contain enough treatment variation to support them. The estimates are causal only under explicit assumptions; weighting does not make an unsupported comparison reliable.

Specify the Strategies and the Target Risk

Let $i$ index patients and $t=0,\ldots,K-1$ index daily treatment decisions. Let $A_{it}=1$ indicate invasive ventilation during decision interval $t$, and $A_{it}=0$ indicate no invasive ventilation. Let $L_{it}$ collect severity and organ-function measurements recorded before that day’s decision. Write $H_{it}=(L_{i0},A_{i0},\ldots,A_{i,t-1},L_{it})$ for the observed history available when $A_{it}$ is chosen. Let $Y_i=1$ indicate death by the prespecified horizon and $Y_i=0$ otherwise. Complete outcome follow-up through that horizon is assumed here.

A dynamic strategy $g$ specifies the action $g_t(H_{it})$ for each decision day as a function of the history available then. Let $Y_i^g$ be the potential mortality outcome if patient $i$ followed strategy $g$. The marginal mortality risk under $g$ is $R(g)=P(Y^g=1)$; the risk difference comparing strategies $g_1$ and $g_0$ is $R(g_1)-R(g_0)$.

Definition: For each day $t$, let $p_t(a\mid H_t)=P(A_t=a\mid H_t)$ be the observed probability of receiving action $a$ given the measured history just before that decision. The inverse-probability weight for strategy $g$ is $$ W^g=\prod_{t=0}^{K-1} \frac{I\{A_t=g_t(H_t)\}} {p_t\{g_t(H_t)\mid H_t\}}, $$ where $I\{\cdot\}$ equals 1 when the patient’s action matches the strategy and 0 otherwise. The weight is zero for a patient who departs from the strategy and, for an adherent patient, is the inverse of the probability of the observed treatment path.

A clinical strategy should describe a plausible decision rule, not an instruction to ventilate every patient regardless of clinical need. It should also specify how prior ventilation affects later actions. For instance, a strategy that initiates ventilation at a threshold and then continues it needs an explicit continuation rule; otherwise $g_t$ is not fully defined.

Worked Example: Turning Severity Criteria Into Daily Strategies

Consider two illustrative rules based on the pre-decision ratio of arterial oxygen pressure to inspired oxygen concentration, denoted $Q_t$. The early rule initiates ventilation when $Q_t\leq150$; the rescue rule initiates when $Q_t\leq100$. For this example, assume both rules continue ventilation once it has been initiated.

A patient is not ventilated on day 0, when $Q_0=180$. On day 1, before the next decision, $Q_1=140$. Under the early rule the day-1 action is 1, while under the rescue rule it is 0. If instead $Q_1=90$, both rules call for ventilation on day 1. These rules illustrate different policies, not clinical recommendations.

The day-1 value of $Q_1$ can be affected by day-0 treatment and can also predict mortality. It is therefore a time-varying confounder for the day-1 decision, with treatment-confounder feedback across days. A policy comparison must account for that history without simply conditioning away treatment effects through an ordinary regression adjustment.

Identification Assumptions and the Weighting Result

The interpretation of $R(g)$ requires the following assumptions. First, consistency: for a patient whose observed treatment history follows $g$, the observed outcome equals $Y^g$. This also requires that the treatment strategy be sufficiently well-defined to avoid treating clinically different versions of ventilation as identical. Second, sequential exchangeability: at each decision time, conditional on the measured history $H_t$, the treatment choice is independent of the counterfactual future outcomes and covariates under the strategy being considered. This rules out unmeasured common causes of treatment decisions and those counterfactuals, conditional on the recorded history. Third, sequential positivity: for histories that can arise under $g$, the probability $p_t\{g_t(H_t)\mid H_t\}$ is positive. Finally, assume no interference between patients and independent sampling from the target population.

Theorem: Under consistency, sequential exchangeability, sequential positivity, and the stated sampling assumptions, the marginal mortality risk under strategy $g$ is identified by $$ R(g)=E[W^gY]. $$ Also, $E[W^g]=1$. Thus, if the treatment probabilities are known, a sample estimate is the ratio $\widehat R(g)=\sum_{i=1}^n W_i^gY_i/\sum_{i=1}^n W_i^g$.

Proof. For clarity, suppose the covariates have densities; the same argument uses sums for discrete variables. Under strategy $g$, the joint density for the baseline covariates, subsequent covariate history, and outcome factors into the baseline density, the conditional covariate densities generated after the strategy’s earlier actions, and the outcome distribution at the end. By consistency and sequential exchangeability, these covariate and outcome distributions agree with the corresponding observed-data distributions among histories that follow the strategy.

In the observed data, the density for a history that follows $g$ includes the additional factor $\prod_t p_t\{g_t(H_t)\mid H_t\}$, the probability of making each required treatment choice. Multiplying by $W^g$ cancels that factor: the indicators restrict the history to one following $g$, and the inverse probabilities remove the observed treatment-choice factors. The remaining density is exactly the density of covariate and outcome histories under $g$. Integrating $Y$ against it gives $E[W^gY]=E[Y^g]=R(g)$. Replacing $Y$ by 1 in the same calculation gives $E[W^g]=1$. The sample ratio estimates the risk because the corresponding population numerator is $R(g)$ and denominator is 1. $\square$

Worked Example: Calculating Strategy-Adherence Weights

Suppose a two-day strategy requires action 1 on day 0 and action 0 on day 1. For four patients, the observed probabilities of their strategy-required actions at each decision are as follows. Patient 3 takes the wrong action on day 0, so that patient’s strategy weight is zero.

Patient	Required-action probability, day 0	Required-action probability, day 1	Observed adherence	Outcome $Y$
1	0.50	0.40	Yes	1
2	0.50	0.80	Yes	0
3	0.50	0.50	No, day 0	1
4	0.50	0.50	Yes	0

For patient 1, $W^g=1/(0.50\times0.40)=5$. For patient 2, $W^g=1/(0.50\times0.80)=2.5$. Patient 3 has $W^g=0$, irrespective of the later probability, because the day-0 action did not match. Patient 4 has $W^g=1/(0.50\times0.50)=4$.

The weighted numerator for mortality is $5(1)+2.5(0)+0(1)+4(0)=5$. The sum of weights is $5+2.5+0+4=11.5$, so the normalized weighted risk is $5/11.5\approx0.435$. This tiny sample is only an arithmetic illustration: it is not precise evidence about a clinical population.

From Weighted Risks to a Policy Contrast

The estimate of interest is a difference between marginal risks, not a comparison of outcomes among patients who happened to receive ventilation on a particular day. Estimate each strategy’s risk using its own adherence weights, then subtract the estimates. Each weight set represents a different reweighted population; the comparison is meaningful under the identification assumptions, not because the weighted patients are literally the same people.

Worked Example: Comparing Two Weighted Mortality Risks

Suppose an analysis of a larger cohort produces the following weighted sums for two strategies. Under the early strategy, the sum of weights is 8 and the weighted sum of deaths is 3. Under the rescue strategy, the corresponding values are 6 and 2.

The estimated early-strategy mortality risk is $3/8=0.375$. The estimated rescue-strategy risk is $2/6=0.333$ to three decimal places. The estimated risk difference, early minus rescue, is $0.375-0.333=0.042$, or about 4.2 percentage points.

This contrast is not a conditional effect among patients with a particular severity score. It compares the estimated population risks under the two specified daily policies. The calculation alone does not establish that the contrast is precise or unbiased: the assumptions, treatment support, and uncertainty still require assessment.

Large-Sample Justification and Practical Diagnostics

Result: If patients are independently sampled, the treatment probabilities are known, and the weights are bounded, then $\widehat R(g)\xrightarrow{p}R(g)$.

Proof. Bounded weights ensure that both $W^g$ and $W^gY$ have finite expectations. The law of large numbers gives $n^{-1}\sum_i W_i^gY_i\xrightarrow{p}E[W^gY]=R(g)$ and $n^{-1}\sum_i W_i^g\xrightarrow{p}E[W^g]=1$. Since the limiting denominator is nonzero, the continuous mapping theorem for ratios gives $\widehat R(g)\xrightarrow{p}R(g)$. $\square$

In practice, the probabilities $p_t(a\mid H_t)$ are estimated rather than known, so this result does not by itself guarantee valid uncertainty estimates or protect against model error. Before interpreting a weighted comparison, check whether the analysis is going wrong in ways visible in the data:

Verify that each severity and organ-function measurement was recorded before the corresponding ventilation decision. Reversing that order can condition on a consequence of treatment.
Inspect whether patients with histories relevant to a strategy actually received the strategy-required action. Sparse or absent support makes a policy risk depend heavily on extrapolation.
Check the fitted treatment probabilities across days and histories. Probabilities close to zero imply large inverse factors; a small number of observations may then dominate the estimate.
Compare measured histories across treatment choices after weighting, particularly severity and organ function at each decision. Persistent imbalance suggests that the treatment model or measured history may be inadequate.
Confirm that the rules, horizon, and outcome are defined consistently. Changing the initiation threshold or the follow-up horizon changes the estimand.

Why Ordinary Adjustment Can Mislead

A regression that predicts mortality from ventilation and the latest severity score may appear to control confounding. But if prior ventilation changes current severity, and current severity affects both today’s ventilation and mortality, adjusting for that score can block part of the earlier treatment effect while also being necessary to address confounding of the current decision. This is treatment-confounder feedback. The MSM approach uses treatment weights to create a pseudo-population in which measured treatment histories are not associated with the recorded decision history, under the assumptions above.

The longitudinal g-formula is a main alternative: it models the covariate and outcome distributions and then predicts outcomes under specified strategies. It targets the same policy risks under suitable assumptions, but relies on correct specification of those models rather than treatment-probability models. A randomized trial of the strategies would address treatment assignment by design, although a trial may be infeasible or unethical. Neither weighting nor g-computation fixes unmeasured confounding, an ill-defined policy, or absent treatment support. The introductory material on marginal structural models provides the broader framework; here the central issue is applying it to daily critical-care decisions.

Common pitfall: It is tempting to compare mortality among patients ventilated today with mortality among patients not ventilated today, then “adjust for severity.” That comparison can confuse treatment effects with the reasons clinicians chose ventilation, and ordinary adjustment can mishandle severity affected by earlier treatment. Define strategies over the full decision history and use a longitudinal identification method; a single-day contrast does not answer the policy question.

Check Your Understanding

Answer each question, then check the concise solutions below.

What is the outcome whose marginal risk is estimated under a ventilation strategy?
Why must the severity measure used for a day’s treatment probability be recorded before that day’s decision?
A patient follows a strategy on two days, with required-action probabilities 0.25 and 0.80. What is the adherence weight?
State one reason a large inverse-probability weight may signal a problem for a policy comparison.
What does sequential exchangeability rule out, conditional on the measured history?

Solutions. (1) $Y^g$, the indicator of death by the prespecified horizon under strategy $g$; the target is $P(Y^g=1)$. (2) A post-decision measurement may be affected by that day’s treatment, so using it as if it were pre-treatment can distort the decision model. (3) The weight is $1/(0.25\times0.80)=5$. (4) It can indicate that the required action was very unlikely for that history, so the estimate may rely heavily on a few patients and weak treatment support. (5) It rules out unmeasured common causes of the treatment decision and the relevant counterfactual future, after conditioning on the recorded history.