Random Assignment and Dynamic Policy Contrasts

Conceptual map

1. II-03.01ITT and factorial contrasts
2. II-03.02Sequential randomization
3. II-03.03Policy-loop experiments
4. II-03.04Contrasts across time windows
5. II-03.05Selection and nonresponse

1. An assignment mechanism defines the causal contrast

A dynamic policy contrast is indexed by assignment unit, treatment history, availability, interference set, and outcome horizon. Known assignment probabilities identify only contrasts supported by that design.

For one unit in a \(2\times2\) factorial experiment, let \(A,B\in\{0,1\}\) be independently assigned with probabilities \(\pi _{A},\pi _{B}\in(0,1)\). Let \(Y(a,b)\) be potential outcomes and assume consistency and no interference across the declared units. The main effect of \(A\), averaged over the design distribution of \(B\), is

2. Main and interaction effects use different contrasts

Let \(Y(a,b)=1+2a+3b+4ab\), with both assignments fair. The two effects of \(A\) conditional on \(B\) are two and six, so \(\tau _{A}=4\). The factorial interaction is

The equality of these two numerical values is incidental. One averages two treatment contrasts; the other is their difference.

3. Histories and availability enter sequential weights

At decision time \(t\), let \(H_{t}\) be recorded history and \(I_{t}\in\{0,1\}\) indicate availability. Assignment probability \(\pi _{t}(H_{t})\) is defined among available units. A proximal excursion contrast compares assignment one and zero at time \(t\) among a declared distribution of available histories, while later assignments follow a specified reference rule.

Inverse-probability terms contain \(I_{t}[A_{t}/\pi _{t}-(1-A_{t})/(1-\pi _{t})]\). Positivity is needed only on histories assigned positive weight by the estimand. An unavailable decision point contributes no randomized contrast at that time.

4. Cluster paths use the cluster as assignment unit

If an entire organization receives a policy path or loop, the randomization unit is the organization. Outcomes within it can remain dependent, and standard errors must respect the assignment clusters. A terminal probe randomized after the path identifies a probe contrast conditional on the assigned path under its own probability. Observational improvement in predicting the terminal state is a different quantity.

At assignment probabilities zero or one, the corresponding inverse weight is undefined and that contrast leaves the design support. Attrition and nonresponse add observation rules; inverse assignment weights alone do not correct them. Cross-cluster interference requires an exposure mapping or a broader assignment model.

5. Implementation, exercises, and sources

Version the randomization table before outcomes are observed. Store unit, cluster, history, availability, assignment probability, treatment realization, response window, and missingness reason. Check probability bounds and weight tails, reproduce design-based means, and use variance estimators matched to the randomization level.

Exercises

1. Enumerate all four assignments in Section 2 and verify (2)–(3).
2. Write the regime weight for a two-stage design with history-dependent probabilities.
3. Construct a cluster path experiment with a terminal probe and list the separately identified contrasts.

1. Guido W. Imbens and Donald B. Rubin, Causal Inference for Statistics, Social, and Biomedical Sciences, chapters on randomized experiments.Potential outcomes and assignment mechanisms.
2. James M. Robins (1986), “A New Approach to Causal Inference in Mortality Studies with a Sustained Exposure Period—Application to Control of the Healthy Worker Survivor Effect,” Mathematical Modelling 7, 1393–1512.Sequential treatment histories.
3. Susan A. Murphy (2003), “Optimal Dynamic Treatment Regimes,” Journal of the Royal Statistical Society B 65, 331–355.Dynamic treatment rules and sequential randomization.