Choosing Encouragement Intensity Under Uptake and Cost Constraints

Introduction

A stronger encouragement can reach more people, but it can also require more staff time, money, or contact. Choosing an intensity therefore involves more than asking which arm produced the highest uptake. The relevant decision is which feasible offer best serves the policy objective, given the outcomes it produces and the resources it uses.

The key distinction is between the effect of offering an encouragement intensity and the effect of receiving treatment. A randomized comparison of encouragement arms can estimate the former, along with each arm’s uptake and implementation cost. Higher uptake alone does not establish that treatment caused an outcome improvement.

What Is Compared Across Intensities?

Suppose people are randomly assigned to one of several encouragement arms, indexed by \(z\). Let \(z=0\) denote a reference arm, such as usual information, and let larger values represent more intensive offers. Let \(D\) indicate whether a person takes up the treatment, and let \(Y\) be an outcome measured on a defined scale. For each person, \(D(z)\) is the uptake that would occur under assignment to arm \(z\), and \(Y(z)\) is the outcome that would occur under that assignment. The latter includes all consequences of the offer, whether or not the person takes up treatment.

Let \(C(z)\) denote the implementation cost associated with assigning one person to arm \(z\), measured on a consistent accounting basis. The mean cost per assigned person is \(E[C(z)]\). It is important to specify the denominator: a per-person-assigned cost includes people who do not take up treatment, while a cost per treated person does not. For comparing offers as policies, cost per person assigned is often the more direct quantity.

A valid randomized comparison requires that assignment to the encouragement arms is genuinely random, that the observed outcome and uptake for each person correspond to their assigned arm, and that one person’s assignment does not change another person’s outcome or uptake. Under these conditions, arm-specific sample means estimate the corresponding mean potential outcomes. Randomization identifies the assignment effects even if some people do not take up treatment and even if the encouragement itself can affect outcomes by pathways other than treatment receipt.

To see why, random assignment makes \(Z\) independent of the potential outcomes, so \(E[Y(z)\mid Z=z]=E[Y(z)]\) and \(E[Y(0)\mid Z=0]=E[Y(0)]\). Consistency means that the observed outcome is the potential outcome under the assigned arm: \(Y=Y(Z)\). Thus the observed difference in arm means equals the assignment effect. The same reasoning applies to uptake and, when costs are well-defined under assignment, to costs.

Feasibility Before Preference

A decision rule should state constraints before selecting an arm. For example, a program may require uptake of at least a specified level and limit mean implementation cost per person assigned. An arm is feasible only if it satisfies every required constraint. Among feasible arms, the decision-maker can then apply a stated objective, such as choosing the largest assignment effect on the outcome.

This separates two questions. Feasibility asks whether an intensity meets operational requirements. Preference asks which feasible intensity is best according to the policy objective. A high-uptake arm can fail a cost ceiling; a low-cost arm can fail a minimum-uptake requirement. Neither should be declared best based on one metric alone.

Comparing Feasible Options With a Valued Outcome

Sometimes more than one arm meets all constraints. A decision-maker may then specify how much an outcome improvement is worth relative to cost. This requires an explicit value and compatible units; it cannot be inferred from uptake rates. For example, if one outcome point is assigned a value of $10 per person, a simple incremental net benefit relative to the reference arm is that value multiplied by the assignment effect, minus incremental cost per person assigned.

Here \(v\) is the decision-maker’s stated monetary value per outcome point. This calculation is meaningful only if the outcome scale, time horizon, and cost accounting are appropriate to the decision. It is a decision criterion for encouragement policies, not an estimate of the treatment effect.

What Uptake Can and Cannot Tell You

Uptake is an important implementation outcome. It tells a program how many people start treatment under each offer and helps assess reach, staffing, and capacity. But a change in uptake is a change in an intermediate response to assignment, not an outcome effect. People induced to take up treatment may differ from those who would take it up under either offer, and the encouragement could also affect the measured outcome through other pathways.

A treatment-effect interpretation requires additional causal assumptions beyond random assignment to the encouragement. In particular, translating an assignment contrast into a treatment-receipt effect generally depends on assumptions about whether the encouragement affects the outcome only through treatment and about how uptake responds to assignment. The assignment-effect comparison presented here does not require that interpretation. If the decision concerns which offer to implement, the randomized assignment effect is often directly relevant precisely because it includes all consequences of implementing that offer.

One can also report the cost per additional person taking up treatment, calculated as the incremental cost divided by the incremental uptake proportion. For the high-versus-low comparison above, this is \(\$12/0.25=\$48\) per additional person taking up treatment. This is an implementation ratio, not a cost per person whose outcome improved, and not a treatment-effect estimate. It can be unstable when the uptake difference is small, so it should be reported alongside the separate cost and uptake differences.

Uncertainty and Practical Selection

Arm means, uptake proportions, and costs are estimated from data, so a decision based on their point estimates can be uncertain. A result just meeting a minimum-uptake threshold may not reliably meet it in a new implementation. Likewise, apparent differences in mean outcomes across arms may be imprecise. Decision-makers should examine uncertainty in the quantities that determine feasibility and preference, and should specify in advance how uncertainty affects the rule—for example, whether feasibility is judged by a point estimate or by a conservative confidence bound.

The randomized design supports causal interpretation of arm comparisons only if assignment was implemented as planned and outcome measurement and follow-up are sufficiently complete for the target comparison. Differential missing outcomes, deviations in assignment, or spillovers between participants can undermine a simple comparison of observed arm means. Cost estimates also need a common perspective and time horizon: comparing direct delivery costs in one arm with total program costs in another would not be a coherent ranking.

Common Mistake

Wrong reasoning: “The high-intensity arm increased uptake by 25 percentage points compared with the low-intensity arm, so treatment caused a 25-percentage-point improvement in health.”

This confuses an uptake difference with an outcome effect. Uptake is whether people began treatment; health is a separate outcome. The correction is to report each randomized contrast in its own units: here, high versus low intensity increased uptake by 25 percentage points, increased the mean health score by 1.5 points, and increased implementation cost by $12 per person assigned. Those figures describe the effect of assigning the higher-intensity offer, not the effect of receiving treatment.