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Static and Dynamic Treatment Regimes

Represent a longitudinal treatment strategy as either a fixed schedule or a rule that assigns treatment using the history available at each decision time.

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What You'll Learn

  • Define a sustained static treatment regime as a fixed treatment schedule over decision times.
  • Distinguish a fixed schedule from a dynamic regime that responds to observed history.
  • Write the pre-decision history and the treatment rule at each time point.
  • Apply static and dynamic rules to hypothetical patient histories.
  • Recognize why a treatment strategy is not the same as an observed treatment trajectory.

Why Specify a Treatment Regime?

In longitudinal studies, treatment may be started, stopped, or changed at several decision times. A question such as “What is the effect of treatment?” is therefore incomplete unless it describes which treatment is assigned at each time. A strategy might prescribe treatment at every visit, prescribe no treatment, or change the prescription when a patient’s recorded condition crosses a threshold.

A treatment regime gives that prescription precisely. It describes how treatment is to be assigned over a specified sequence of decision times. A regime can prescribe the same treatment regardless of a person’s evolving history, or it can use that history to select the treatment at each decision. The distinction is central in longitudinal causal inference: the same treatment at the first visit may lead to different later prescriptions under different strategies.

Decision Times and Histories

Use decision times \(t=0,1,\ldots,K\). Let \(A_t\) denote the treatment assigned at time \(t\), and let \(L_t\) denote covariates observed before that treatment decision. The covariates may include disease severity, laboratory measurements, or other information relevant to treatment. At time 0, \(L_0\) is the baseline information. The pre-decision history at time \(t\) is

$$ H_t=(L_0,A_0,L_1,A_1,\ldots,L_{t-1},A_{t-1},L_t). $$

Thus \(H_0=L_0\). For \(t>0\), the history contains the earlier treatments and covariates, as well as the covariates measured just before the current treatment decision. It does not contain \(A_t\), which has not yet been assigned. This ordering makes clear what information a treatment rule is allowed to use at each decision time.

For simplicity, suppose treatment at each time takes values in a set \(\mathcal{A}_t\), such as \(\{0,1\}\) for no treatment or treatment. A deterministic regime specifies one permitted treatment value at every decision time for every history to which its rule applies.

Definition: A sustained static treatment regime is a fixed schedule \(\bar a=(a_0,a_1,\ldots,a_K)\), where \(a_t\in\mathcal{A}_t\). Under this strategy, treatment at time \(t\) is prescribed to be \(a_t\), regardless of the person’s history at that time. The schedule is sustained because it specifies treatment across all decision times, not just the initial decision.

“Static” means history-independent; it does not require the treatment value to be identical at every time. For example, a schedule prescribing treatment at visits 0 and 1 and no treatment at visit 2 is static, even though its values change over time. The special schedule \((1,1,\ldots,1)\) prescribes treatment at every time, while \((0,0,\ldots,0)\) prescribes no treatment at any time.

Definition: A dynamic treatment regime is a sequence of rules \(\bar d=(d_0,d_1,\ldots,d_K)\), where each \(d_t\) maps the pre-decision history \(H_t\) to an allowed treatment value: \(d_t:H_t\to\mathcal{A}_t\). Under the regime, the prescribed treatment at time \(t\) is \(d_t(H_t)\).

A dynamic rule may depend on baseline characteristics, current covariates, and previously observed treatments. Its key feature is that it uses only information available by the decision time. For instance, a rule can use a laboratory value measured immediately before a visit, but it cannot use a measurement that will be taken after the treatment decision. A sequence of rules is needed because the relevant history can change from one decision time to the next.

Worked Example: A Fixed Three-Visit Schedule

Worked Example: A Fixed Three-Visit Schedule

Suppose a hypothetical clinic makes treatment decisions at visits \(t=0,1,2\). At each visit, \(A_t=1\) means the patient receives a medication and \(A_t=0\) means the patient does not. Consider the schedule \(\bar a=(1,1,0)\).

The static prescription is

$$ A_0=a_0=1,\qquad A_1=a_1=1,\qquad A_2=a_2=0. $$

The patient receives medication at visits 0 and 1, then does not receive it at visit 2. The rule does not change if the patient’s recorded severity is low or high, or if the patient received a different treatment in the past. Those observations may occur, but this particular regime does not use them to determine treatment.

By contrast, \((1,1,1)\) would prescribe medication at every visit, and \((0,0,0)\) would prescribe none. Each is also a sustained static regime. The important point is that each schedule specifies the treatment at all three decision times.

Worked Example: A Rule Based on Current Severity

Worked Example: A Rule Based on Current Severity

Consider a hypothetical three-visit program with binary treatment \(A_t\). At each visit, a severity score \(L_t\) is recorded before the treatment decision. Define a dynamic regime that starts medication at visit 0, then gives medication at visits 1 and 2 only when the current severity score is at least 6.

Write the rules as

$$ d_0(H_0)=1,\qquad d_1(H_1)=\mathbb{1}(L_1\geq 6),\qquad d_2(H_2)=\mathbb{1}(L_2\geq 6), $$

where \(\mathbb{1}(C)=1\) when condition \(C\) is true and equals 0 otherwise. Suppose one hypothetical patient has severity scores \(L_0=4\), \(L_1=7\), and \(L_2=5\). At visit 0, the rule gives \(A_0=d_0(H_0)=1\). At visit 1, \(7\geq6\), so \(A_1=d_1(H_1)=1\). At visit 2, \(5\geq6\) is false, so \(A_2=d_2(H_2)=0\). The prescribed sequence for this patient’s observed covariate history is therefore \((1,1,0)\).

A second hypothetical patient with \(L_1=3\) and \(L_2=8\) would receive \(A_0=1\), then \(A_1=0\) because \(3<6\), and \(A_2=1\) because \(8\geq6\). The regime is the same for both patients; their treatment sequences differ because the rule responds to their histories.

How Static and Dynamic Regimes Relate

A static schedule can be written in the same rule-based form as a dynamic regime. For a fixed schedule \(\bar a\), define \(d_t(h)=a_t\) for every possible history \(h\) at time \(t\). This rule is formally a function of history, but it returns the same treatment for every history. In that sense, static regimes are a special case of dynamic regimes: they are dynamic-rule sequences with no actual dependence on history.

The reverse is not generally true. A dynamic rule that prescribes treatment according to current severity cannot usually be represented by one fixed schedule that applies to every patient. A fixed schedule can match the dynamic rule along a particular history, but a different history may produce a different treatment sequence under the rule.

Worked Example: A Rule That Uses Past Treatment

Suppose a hypothetical program makes two decisions, at \(t=0\) and \(t=1\). Treatment is binary. At the first decision, it assigns treatment if baseline severity \(L_0\) is at least 5. At the second decision, it continues treatment only if the patient received treatment at time 0 and the current severity \(L_1\) is at least 4.

The rule is

$$ d_0(H_0)=\mathbb{1}(L_0\geq5),\qquad d_1(H_1)=\mathbb{1}(A_0=1\ \text{and}\ L_1\geq4). $$

For a patient with \(L_0=6\) and \(L_1=5\), the first decision is \(A_0=\mathbb{1}(6\geq5)=1\). At the second decision, both conditions hold: \(A_0=1\) and \(5\geq4\). Therefore \(A_1=1\), giving the sequence \((1,1)\).

For a patient with \(L_0=6\) and \(L_1=3\), the first decision is still \(A_0=1\), but the second condition fails because \(3<4\). Thus \(A_1=0\), giving \((1,0)\). For a patient with \(L_0=3\), the first decision is \(A_0=0\); regardless of the later severity, the rule’s requirement \(A_0=1\) is false, so \(A_1=0\). This example shows why the rule must state precisely which parts of the prior history it uses.

Why the Rule Must Match the Decision Process

A regime is not a description of what happened to one patient. It is a prescription that can be applied to eligible histories at every decision time. An observed sequence such as \((1,0,1)\) records treatments received; by itself, it does not tell us whether the sequence arose from a fixed schedule, a severity-based rule, clinician judgment, or chance.

To specify a regime clearly, state the decision times, the available treatment options, and the information used by each rule. In particular, say whether the rule uses current covariates, earlier treatment, or both. If the rule refers to a threshold, state the threshold and what happens at equality. This precision distinguishes strategies that may look similar at baseline but prescribe different later treatment.

The timing of the inputs matters. If \(L_t\) is used by \(d_t\), it must be measured before \(A_t\) is assigned. If an outcome or covariate is observed only after that decision, it cannot be an input to that decision’s rule. A clinically sensible description can still be an invalid formal rule if it relies on information that is not yet available.

Key takeaway: A sustained static regime assigns a fixed treatment at every decision time, without regard to history. A dynamic regime assigns treatment through a sequence of rules, each using the history available immediately before that decision. Both describe treatment strategies; neither is merely an observed treatment sequence.

Common Mistakes

  • Calling only an “always treat” or “never treat” plan static. A fixed schedule can vary across time, such as treat at the first two visits and stop at the third. Static means history-independent, not constant over time.
  • Calling an observed sequence a regime. The sequence \((1,0,1)\) says what happened for one person. A regime must say how treatment is assigned across decision times and, for a dynamic regime, how histories determine those assignments.
  • Using information from after the decision. A rule for \(A_t\) cannot use a covariate measured after \(A_t\). Define the pre-decision history first, then write the rule using that history.
  • Leaving a threshold ambiguous. “Treat when severity is high” is not a complete rule. Specify the score and the cutoff, including whether equality leads to treatment.

Check Your Understanding

For each question, identify the treatment rule and the information it uses.

  1. With binary treatment at three visits, is \((1,0,1)\) a static regime? Explain what makes it static.
  2. Write a dynamic rule for two visits that assigns treatment at time 0 when \(L_0\geq 3\), and at time 1 when \(L_1\geq 5\).
  3. A rule assigns \(A_1=1\) if \(A_0=1\) and \(L_1<4\). For \(A_0=1\) and \(L_1=3\), what does it prescribe? For \(A_0=0\) and \(L_1=3\)?
  4. Why is a treatment sequence observed in one patient not, by itself, enough to identify the regime that generated it?
  5. A proposed rule for treatment at time \(t\) uses a laboratory value measured after the treatment decision. What is the timing problem?

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