Treatment-Policy Versus Hypothetical Effects

Two Different Questions About Treatment

Suppose a treatment is studied for its effect on disease progression, but some patients die before progression can occur. One question is: what is the risk of progression if patients receive each treatment and death occurs as it ordinarily would? A second asks: what would the progression risk be if death could somehow be prevented? These are different causal questions, and their answers need not agree.

The first is a treatment-policy contrast: it compares treatment strategies while allowing post-treatment events, including competing events, to follow their natural course under each strategy. The second is a hypothetical contrast: it compares strategies in a setting where the competing event is prevented. The introductory material on choosing an estimand motivates why the target question must be specified first; here the focus is on calculating and interpreting this particular pair of risks.

Definition: Let \(A=a\) denote a treatment strategy, let \(Y\) be the event of interest, let \(D\) be a competing event, and let \(\tau\) be the end of follow-up. The treatment-policy risk, \(R_a^{TP}(\tau)\), is the probability of experiencing \(Y\) by \(\tau\) before \(D\), if the population followed strategy \(a\) and competing events occurred naturally. The hypothetical prevention risk, \(R_a^{H}(\tau)\), is the probability of experiencing \(Y\) by \(\tau\) if the population followed strategy \(a\) and an intervention prevented \(D\) throughout follow-up.

Both are risks of the event of interest, but their post-treatment worlds differ. Their causal risk differences are \(R_1^{TP}(\tau)-R_0^{TP}(\tau)\) and \(R_1^{H}(\tau)-R_0^{H}(\tau)\), respectively. A negative risk difference means the event is less frequent under treatment than under control for the specified estimand. Neither contrast is automatically the effect of treatment on a biological process isolated from competing events.

Calculating Risks in Discrete Time

For a simple calculation, divide follow-up into intervals. Among individuals event-free at the start of interval \(k\), let \(q_{Yk}^a\) be the probability of \(Y\) during that interval under strategy \(a\), and \(q_{Dk}^a\) the probability of \(D\). These are conditional probabilities in the event-free risk set. The two events are mutually exclusive within an interval, so \(q_{Yk}^a+q_{Dk}^a\leq 1\). The probability of remaining event-free through interval \(k\), given that the individual was event-free at its start, is \(1-q_{Yk}^a-q_{Dk}^a\).

Under the natural course, the event-of-interest risk accumulates only among those who have experienced neither event earlier. For two intervals, for example, the treatment-policy risk is \(q_{Y1}^a+(1-q_{Y1}^a-q_{D1}^a)q_{Y2}^a\). Under hypothetical prevention, the competing-event probability is set to zero, while the event-of-interest probability is assumed to remain \(q_{Yk}^a\) among those still event-free. With that assumption, the hypothetical risk over \(K\) intervals is \(1-\prod_{k=1}^{K}(1-q_{Yk}^a)\).

R_a^{TP}(\tau) = \sum_{k=1}^{K} \left[ \prod_{j=1}^{k-1} (1-q_{Yj}^a-q_{Dj}^a) \right]q_{Yk}^a, \qquad R_a^{H}(\tau) = 1-\prod_{k=1}^{K}(1-q_{Yk}^a).

The empty product for \(k=1\) is 1. These equations are a discrete-time illustration, not a universal identification formula: in real data, the quantities may vary with measured history, and estimating the hypothetical risk requires additional assumptions and appropriate adjustment.

Assumptions for a causal interpretation: The treatment strategies must be sufficiently well-defined for consistency to apply; treatment exchangeability (from randomization or adequate control of confounding), positivity, and suitable handling of censoring are needed for treatment risks. A hypothetical prevention risk also requires a well-defined prevention intervention, support for the relevant treatment and event histories, and an assumption that allows the event-of-interest process under prevention to be learned from the available data. In the simple calculation above, that additional assumption is that preventing \(D\) does not itself alter the conditional probability of \(Y\) in an interval, given the modeled history. The dismissible component conditions and weighting for hypothetical competing-event elimination are relevant earlier material on these assumptions and estimation strategies.

Worked Risks When Competing Events Occur Naturally

Worked Example: A One-Interval Treatment-Policy Risk

Consider a hypothetical randomized study with one year of follow-up. Among people event-free at baseline, 18% in the control group experience the event of interest, 12% experience the competing event, and the rest experience neither. In the treatment group the corresponding probabilities are 14%, 20%, and 66%. Assume complete follow-up, valid randomization, and well-defined treatment strategies.

Since there is just one interval, the treatment-policy risk of the event of interest is its observed probability in each group: \(R_0^{TP}=0.18\) and \(R_1^{TP}=0.14\). The risk difference is \(0.14-0.18=-0.04\), or 4 percentage points lower under treatment. The risk ratio is \(0.14/0.18=0.778\). In this hypothetical population, the treatment-policy risk is 14% rather than 18% over one year; the risk ratio compares those risks, not the probability that treatment benefits an individual.

The probabilities are coherent: control has \(0.18+0.12=0.30\) total event probability, and treatment has \(0.14+0.20=0.34\), both no greater than 1. The comparison is treatment-policy because it allows the competing event to occur in each group.

Why the Two Effects Can Differ

The treatment can affect the event of interest and the competing event in different directions. If treatment increases the competing event, fewer treated individuals remain at risk of the event of interest. A treatment-policy risk can therefore be low even when the treatment does not reduce, or may increase, the event-of-interest transition probability. The hypothetical risk removes this pathway by asking what would happen under prevention of the competing event, but that answer depends on the prevention intervention and its assumptions.

Worked Example: A Reversal Between Estimands

Use two hypothetical intervals with no covariates. For control, in each interval among those still event-free, let \(q_Y^0=0.20\) and \(q_D^0=0.10\). For treatment, let \(q_Y^1=0.25\) and \(q_D^1=0.50\). Assume these probabilities are constant across the two intervals. For the hypothetical calculation, additionally assume preventing \(D\) leaves each interval’s \(q_Y^a\) unchanged. The probabilities are valid: their sums are 0.30 for control and 0.75 for treatment.

For control, the treatment-policy risk is the first-interval event probability plus the second-interval probability among those who avoided both events in interval 1:

R_0^{TP} = 0.20+(1-0.20-0.10)(0.20) = 0.20+(0.70)(0.20) = 0.34.

For treatment, the corresponding calculation is:

R_1^{TP} = 0.25+(1-0.25-0.50)(0.25) = 0.25+(0.25)(0.25) = 0.3125.

Thus the treatment-policy risk difference is \(0.3125-0.34=-0.0275\), or 2.75 percentage points lower under treatment. This lower risk does not establish that treatment improves the event-of-interest process: treatment has a much higher competing-event probability in this example.

Under hypothetical prevention, the risk is calculated with the competing event removed from both intervals. Therefore:

R_0^{H}=1-(1-0.20)^2=1-0.64=0.36, \qquad R_1^{H}=1-(1-0.25)^2=1-0.5625=0.4375.

The hypothetical risk difference is \(0.4375-0.36=0.0775\), or 7.75 percentage points higher under treatment. The directions of the two contrasts differ. The reversal follows from the numbers and the prevention assumption; it is not a mathematical guarantee that treatment-policy and hypothetical effects will disagree.

Deriving the Prevention Formula

The hypothetical formula follows by tracking the probability of avoiding \(Y\) when \(D\) is prevented. At the start of follow-up everyone is free of \(Y\). Under prevention, a person who has avoided \(Y\) through interval \(k-1\) remains in the risk set for interval \(k\), because \(D\) cannot remove that person from it. The probability of avoiding \(Y\) in interval \(k\), conditional on being event-free at its start, is \(1-q_{Yk}^a\). Multiplying these conditional probabilities gives the probability of avoiding \(Y\) throughout follow-up; subtracting from 1 gives the risk.

P(\text{no }Y\text{ by }\tau\mid a,\ D\text{ prevented}) = \prod_{k=1}^{K}(1-q_{Yk}^a), \qquad R_a^{H}(\tau) = 1- P(\text{no }Y\text{ by }\tau\mid a,\ D\text{ prevented}).

This derivation depends on the conditional probabilities being appropriate for the prevention world. If people who would experience \(D\) differ in their subsequent \(Y\) probabilities, simply reusing observed probabilities among those still event-free may not identify the hypothetical risk. Conditioning on remaining event-free can select a different group over time, which is why the prevention assumption and any required adjustment matter.

Worked Example: Standardizing Across Baseline Strata

Consider another hypothetical two-interval setting with two baseline strata, low-risk (weight 0.70 in the target population) and high-risk (weight 0.30). In each stratum, the conditional probabilities are constant across intervals. In the low-risk stratum, control has \((q_Y,q_D)=(0.10,0.05)\) and treatment has \((0.08,0.30)\). In the high-risk stratum, control has \((0.40,0.10)\) and treatment has \((0.35,0.40)\). Assume treatment exchangeability within strata, positivity, no censoring, and, for the hypothetical risks, that preventing \(D\) leaves \(q_Y\) unchanged within each stratum.

First calculate the stratum-specific treatment-policy risks. In the low-risk stratum they are \(0.10+(0.85)(0.10)=0.185\) for control and \(0.08+(0.62)(0.08)=0.1296\) for treatment. In the high-risk stratum they are \(0.40+(0.50)(0.40)=0.60\) for control and \(0.35+(0.25)(0.35)=0.4375\) for treatment. Standardize using the same target-population weights in both treatment groups:

R_0^{TP}=0.70(0.185)+0.30(0.60)=0.3095, \qquad R_1^{TP}=0.70(0.1296)+0.30(0.4375)=0.22197.

The standardized treatment-policy risk difference is \(0.22197-0.3095=-0.08753\), or 8.753 percentage points lower under treatment. For hypothetical prevention, the stratum-specific risks are \(1-(0.90)^2=0.19\) and \(1-(0.92)^2=0.1536\) in the low-risk stratum, and \(1-(0.60)^2=0.64\) and \(1-(0.65)^2=0.5775\) in the high-risk stratum, for control and treatment respectively. Standardizing gives \(R_0^{H}=0.70(0.19)+0.30(0.64)=0.325\) and \(R_1^{H}=0.70(0.1536)+0.30(0.5775)=0.28077\). The hypothetical risk difference is \(0.28077-0.325=-0.04423\), or 4.423 percentage points lower under treatment. Both contrasts use the same target-population stratum weights, but they answer different questions.

Interpretation, Diagnostics, and Alternatives

Treatment-policy and hypothetical effects should be reported with the event, competing event, treatment strategies, time horizon, population, and effect scale stated. Risk differences are absolute contrasts in percentage points; risk ratios are relative contrasts. A risk ratio of 0.8 means the risk under one strategy is 0.8 times the risk under the comparator for that estimand and time horizon. It does not mean an 80% probability that treatment works.

The treatment-policy risk is commonly represented by the cumulative incidence of the event of interest when competing events occur naturally. A hypothetical risk instead targets a prevention world and cannot generally be obtained by merely censoring people at the competing event and applying a standard survival calculation. That approach treats competing events as if their subsequent event-of-interest experience could be recovered under censoring assumptions; it does not, by itself, establish the effect of an intervention that prevents them. Cause-specific event probabilities can contribute to a hypothetical calculation only when the assumptions connecting them to the prevention world are justified.

Several checks help reveal a faulty analysis. Verify that event probabilities are conditional on the correct risk set and that mutually exclusive probabilities sum to no more than 1. Check that cumulative risks stay between 0 and 1 and do not decrease over time. For adjusted analyses, inspect whether treatment or event histories needed for the contrast are represented in the data; sparse histories can make estimates unstable. Compare treatment-policy and hypothetical estimates as distinct targets, not as competing estimates of one uniquely correct effect.

The hypothetical estimand has important limitations. “Preventing death” or another competing event may not describe a feasible or unique intervention: different prevention methods could also change the event of interest. The prevention assumption may be scientifically implausible or unsupported by observed data, and positivity can fail when some histories have no chance of remaining free of the competing event. A precise numerical estimate does not remove these problems. The treatment-policy estimand avoids the need to posit prevention, but it includes pathways through the competing event and may not isolate the effect on the event-of-interest process.

Common pitfall: “Death is censoring, so I can censor deaths and interpret the resulting survival risk as the risk if nobody died.” This is incorrect without assumptions that identify the hypothetical prevention world. The treatment-policy risk counts the event of interest before death under the natural course; a hypothetical risk requires a specified prevention intervention and assumptions that link its outcome to observed data.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In one interval, control has event-of-interest probability 0.12 and competing-event probability 0.08; treatment has probabilities 0.10 and 0.25. Calculate both treatment-policy risks and their risk difference.
For two intervals, \(q_Y=0.30\) in each interval and the competing event is hypothetically prevented. Calculate the risk of the event of interest.
In a treatment-policy analysis, why does a higher competing-event probability potentially lower the observed risk of the event of interest?
Why is censoring competing events and using a standard survival risk not automatically an estimate of hypothetical prevention risk?

Solutions. (1) The risks are 0.12 and 0.10, so the treatment-minus-control risk difference is \(0.10-0.12=-0.02\), or 2 percentage points lower under treatment. (2) The risk is \(1-(1-0.30)^2=1-0.49=0.51\), or 51%. (3) Individuals experiencing the competing event leave the event-free risk set and cannot subsequently experience the event of interest first; therefore fewer remain at risk for it. (4) That calculation requires assumptions linking the censoring process and the event-of-interest process to the prevention world; censoring alone does not define or identify an intervention that prevents the competing event.