From a Clinical Strategy to a Rule
A strategy such as “treat patients with the new drug” does not yet describe what treatment is to be delivered. It leaves open when treatment begins, what dose is used, whether it continues throughout follow-up, and how clinicians respond to changes in a patient’s condition. Different answers to those questions can produce different treatment histories, even when the strategy has the same short label.
An operational strategy resolves these choices in advance. It states what action is called for at each decision time, using information available at that time. This makes the strategy interpretable as a clinical plan and makes it possible to check whether a recorded treatment history followed that plan. The purpose here is to define the plan, not to specify how participants are assigned to it or how its causal effect is estimated.
The treatment action \(A_t\) must be defined at a useful level of detail. It might be a binary indicator of treatment, a dose, a named regimen, or a combined action such as “continue 10 mg daily.” If the strategy distinguishes starting treatment from continuing it, the action set must represent those possibilities. The history \(H_t\) must contain the information the rule uses, recorded before the action is taken; a rule cannot use a future measurement to decide a current action.
A complete specification names the decision times and states the action at every feasible history. Depending on the clinical question, the specification may cover:
- Initiation: whether treatment begins, and at which decision time or under what observed condition.
- Timing: the schedule of decisions and, when relevant, the allowed interval between a trigger and the action.
- Duration: how long treatment continues and when it ends.
- Dose and regimen: the dose, route, frequency, and any planned titration.
- Permitted changes: the conditions for escalation, reduction, temporary holds, switching, rescue treatment, or permanent discontinuation.
These are parts of one rule, not optional details that can safely be supplied after the strategy has been named. A strategy may intentionally allow some choices, but it must say what those choices are and under what conditions they are allowed. For example, “dose may be reduced for toxicity” is incomplete unless toxicity is defined in observable terms and the allowed reduction is stated.
Checking Whether a Rule Is Complete
A practical check is to take each decision time in turn and ask: for every history that could arise by then, does the rule identify exactly one action? If the rule leaves a clinician to decide what “appropriate” means, the action is not determined. If it gives incompatible instructions for the same history, the rule is not unique. A fully specified deterministic strategy is both total on its feasible histories and single-valued: it returns an action for each such history, and never returns two different actions for the same history.
Proof. By definition, a deterministic rule \(g_t\) is a function from feasible histories to actions. A function must be defined at every element of its domain and must assign only one output to each input. Thus, if the strategy is operationally complete, every feasible history has exactly one specified action. Conversely, if every feasible history has exactly one specified action, these assignments define a function \(g_t\) on the full feasible-history domain. Applying this argument at each decision time gives the sequence \(g=(g_0,\ldots,g_{K-1})\). Therefore the strategy is operationally complete exactly when the stated condition holds. \(\square\)
Worked Example: Turning “Start Treatment Promptly” Into a Rule
Suppose decisions occur at baseline and at weeks 2 and 4. The action is \(A_t=1\) if a participant takes the study drug during that interval and \(A_t=0\) otherwise. A clinician’s note says, “Start treatment promptly.” This does not specify an action at any particular time: “promptly” could mean baseline, within several days, or at the next visit.
A precise alternative is: “Begin 10 mg orally once daily at baseline; at weeks 2 and 4, continue 10 mg daily if treatment has begun and no stopping criterion has occurred.” This rule specifies an initiation time, dose, frequency, and continuation plan. If the action is recorded as \(A_t\), the prescribed sequence in the absence of stopping criteria is \(A_0=1\), \(A_1=1\), and \(A_2=1\).
If treatment may be delayed, that possibility must be made explicit. For instance, a rule could state that a named laboratory test is measured at baseline, treatment begins at week 2 only if the result is below a stated threshold, and otherwise treatment is not started. The threshold, measurement time, and actions for both possible results are necessary parts of the rule. Merely adding “depending on the test” would still leave the action undetermined.
Worked Rules for Dose Changes and Duration
A dynamic strategy uses current history to choose an action. The rule can depend on symptoms or measurements, but the decision condition must be stated in terms of information available before that decision. It also needs a specified action on each branch, including boundary cases such as a dose already at its maximum.
Worked Example: A Blood-Pressure-Based Dose Rule
At monthly visits \(t=0,1,2\), a medication can be prescribed at 0, 5, 10, or 15 mg daily. Let \(D_t\) be the dose just before visit \(t\), and let \(S_t\) be the systolic blood pressure measured before the dose decision. Use the following rule: if \(S_t<110\), prescribe 0 mg; if \(110\leq S_t<140\), keep the current dose; if \(S_t\geq140\), increase the dose by 5 mg, up to a maximum of 15 mg. A participant with \(D_t=0\) who has \(S_t\geq140\) starts at 5 mg.
For a participant with \(D_0=0\), suppose the recorded pre-decision pressures are \(S_0=146\), \(S_1=134\), and \(S_2=151\) mmHg. At visit 0, \(146\geq140\), so \(D_1=5\) mg. At visit 1, \(110\leq134<140\), so the dose is unchanged and \(D_2=5\) mg. At visit 2, \(151\geq140\), so the dose increases to 10 mg. The resulting prescribed doses are 5, 5, and 10 mg for the three intervals.
The cap is part of the rule: if a participant is already at 15 mg and has \(S_t\geq140\), the dose remains 15 mg. Without that clause, the rule would request an unavailable 20 mg dose at this history and would not define a feasible action.
Duration can also be specified as part of the action rule. A treatment course that ends at a fixed time differs from a strategy that continues until a condition is met. If there are permitted exceptions, the rule should state their trigger and what happens afterward; otherwise, “continue for 12 weeks” and “continue for 12 weeks unless needed” do not identify the same plan.
Worked Example: Fixed Duration With an Explicit Hold Rule
Consider a 6-week course of 20 mg daily, with decisions at weeks 0, 2, 4, and 6. Let \(L_t\) be a laboratory value measured before the decision, with the upper limit of normal equal to 40 units. Define the rule as follows: begin 20 mg daily at week 0; at weeks 2 and 4, continue 20 mg if \(L_t\leq120\), and otherwise hold treatment until the next scheduled decision; at week 6, stop treatment regardless of the laboratory value. A held treatment is restarted at 20 mg if the next measured value is at most 120.
Suppose the values at weeks 2 and 4 are 100 and 130. At week 0, treatment begins at 20 mg. At week 2, \(100\leq120\), so 20 mg continues. At week 4, \(130>120\), so treatment is held for the next interval. At week 6, the rule says to stop, irrespective of the value then measured. Thus the plan has a fixed end time, a defined hold trigger, and a defined terminal action.
The rule does not prescribe what to do between scheduled decisions if symptoms occur. If such a response is intended, it must be added as another decision condition and action. Until then, the strategy is specifically a scheduled-decision plan, not an all-purpose clinical management protocol.
Verifying an Observed Treatment History
Once a strategy is written, its agreement with a recorded history can be checked one decision at a time. Let \(A_t\) be the observed action and \(g_t(H_t)\) the action required by the rule. Define the indicator \(I_t=1\) when they agree and \(I_t=0\) otherwise. The product of these indicators records whether every decision in the observed sequence agrees with the rule.
Proof. Each factor is either 0 or 1. If every observed action agrees with the rule, every factor equals 1, and their product is 1. If at least one action disagrees, its factor is 0, so the product is 0. These are the only possibilities, proving the equivalence. \(\square\)
This check requires that the recorded history contain the rule’s inputs and that treatment actions be measured at the decision times and in the units the rule specifies. If dose is recorded only as “treated” or “untreated,” a rule distinguishing 5 mg from 10 mg cannot be checked from that record. A missing measurement used by the rule can likewise make the required action unclassifiable; it should not be silently treated as a normal value.
Worked Example: Classifying Decisions Under a Rule
Use the blood-pressure rule above. A participant has a dose of 5 mg before a visit, a measured pressure of 105 mmHg, and an observed next dose of 0 mg. Since \(105<110\), the rule prescribes 0 mg; therefore \(I_t=1\) at this visit.
At the next visit, suppose the current dose is 0 mg and the pressure is 145 mmHg, but the observed next dose is 10 mg. Since \(145\geq140\), the rule calls for an increase of 5 mg from the current dose, which means 5 mg, not 10 mg. Thus \(I_t=0\) at this visit.
For these two decisions, \(C=1\times0=0\). The observed sequence therefore does not agree with the strategy at every recorded decision. This is a classification against the stated rule; it does not, by itself, explain why the action differed.
Common Pitfalls and Practical Checks
The most common error is to treat a treatment label as though it were a complete strategy. “Standard care,” “early treatment,” or “dose as needed” can conceal multiple possible schedules and actions. A second error is to specify a response for common histories but omit edge cases: the maximum dose, a treatment hold followed by a low measurement, simultaneous triggers, or the end of the planned course. A third is to use information that is not available at the time of the decision.
A useful diagnostic is to write a decision table or flow rule and challenge it with histories at each boundary. Ask what happens when a measurement equals the threshold, when the participant is already at the minimum or maximum dose, when a stopping condition and an escalation condition occur together, and when a required measurement is missing. If any answer depends on an unstated clinical judgment, the strategy still needs clarification.
Compared with a static strategy that assigns the same action regardless of evolving history, a dynamic rule can adapt treatment to observed history. The added flexibility has a cost: every branch, threshold, measurement time, and permitted change must be defined. A protocol that deliberately allows clinician discretion may be clinically realistic, but it is not equivalent to a single deterministic rule unless the allowed discretion is itself precisely characterized. Strategies can be compared only after their action rules are clear enough to identify where they agree and differ.
Operational detail does not guarantee that the strategy is clinically appropriate, that the relevant variables were measured well, or that the strategy can be followed in practice. It does ensure that the intended actions are explicit. The key test is reproducibility: two people given the same pre-decision history should identify the same prescribed action from the written rule.
Check Your Understanding
For each question, identify the decision rule or the missing specification. Solutions follow.
- A plan says, “Start the drug early and continue as tolerated.” Name two details needed to make this an operational rule.
- A rule says to increase a dose by 5 mg whenever a marker is at least 50, but does not state a maximum dose. Why might this rule be incomplete?
- At a decision time, the rule prescribes 10 mg and the observed dose is 10 mg. What is \(I_t\)? If all other decisions also agree, what is \(C\)?
- A strategy says, “If the pre-decision measurement exceeds 8, hold treatment; otherwise continue.” What action does it prescribe when the measurement equals 8?
- Explain why a rule that uses a measurement taken after the treatment decision is not an operational pre-decision rule.
Solutions. (1) It must state when “early” means treatment begins, and define “as tolerated” using an observable condition and a specified action; it should also give the dose and planned duration. (2) Repeated qualifying measurements could call for doses beyond those available or intended, so the maximum and action at the cap must be stated. (3) \(I_t=1\); if every other decision agrees, all factors equal 1 and \(C=1\). (4) The condition is “exceeds 8,” so a value equal to 8 falls in the otherwise branch and treatment continues. (5) The measurement was not yet available when the action had to be chosen, so the rule cannot determine that decision from information available at that time.