Randomization Method
The randomization is conducted at the household level rather than the child level, to preserve household integrity: all eligible children in a selected household are admitted together. This rule reflects both the way INAIPI operates (families enrol siblings jointly) and the way we
will analyse outcomes (intra-household dynamics are part of the theory of change).
Household-level randomization combined with age-group capacity caps creates a non-trivial allocation problem: a uniform draw at the household level can violate age-group capacity in any single age cell. To respect both capacity and household integrity while pre-serving well-defined ex-ante admission probabilities, we use a scarcity-ordered stratified lottery with household integrity (hereafter, the bucket-lottery protocol). The protocol proceeds within each center as follows:
(i) Eligible age cells are defined at one-year resolution: g0–1, g1–2, g2–3, g3–4, g4–5. Per-cell capacities are set ex ante based on the center’s expected age distribution. Two indicative vectors were used in simulations, both summing to 195 seats per center: a conservative allocation (23, 28, 44, 50, 50) and a benevolent allocation (30, 36, 39, 45, 45) (scenarios S-A and S-C, see attached documents for more information).
(ii) Each household is assigned to a bucket corresponding to the age cell in which it is hardest to accommodate (its scarcest occupied cell). Households with children in multiple cells are bucketed by the scarcest occupied cell; households with children in a single cell are bucketed by that cell.
(iii) Buckets are processed in descending order of demand-to-capacity ratio (scarcest age cell first). Within each bucket, a uniform random ordering of households is drawn. Households are admitted in order until the bucket’s target age cell is filled (the treatment tranche), accounting for any spillover of siblings into other cells from earlier buckets; all remaining households constitute the control tranche. The same ordering is retained for the substitution rule described below.
(iv) A residual fill-up step admits additional households (drawn uniformly from eligible buckets) if any cell remains under-filled after all primary buckets are processed. Ex ante simulations show this step does not activate under realistic scenarios (zero activations in 4,000 replications), but it is pre-specified for completeness.
This protocol delivers (a) household integrity by construction, (b) known ex-ante admission probabilities per bucket, and (c) capacity filled by design. Analysis of treatment effects proceeds with center × bucket fixed effects, which fully absorb the differential admission
probabilities across strata.
Substitution for declined slots. A share of admitted households is expected to decline the offered slot (central assumption: 10%; see the compliance discussion in Section 5.3). Declined slots must be filled for operational reasons. To do so without compromising the design,
the same uniform random ordering of households drawn within each bucket in step (iii) is used to define a substitution order : when an admitted household declines, its slot is offered to the next control-tranche household in that pre-drawn order, subject to the same age-cell
capacity checks as steps (iii)–(iv). The household that accepts the substituted slot is retired from the analysis (it counts neither as treatment nor as control), because its enrolment depended on the realised pattern of declines rather than on the initial assignment. This rule
preserves one-sided noncompliance: only households promoted through the substitution order can transition from the control tranche to enrolment, and those households are excluded from the confirmatory comparison, so no household that remains in the control tranche for
analysis can become enrolled. The ITT/TOT framework therefore applies unchanged. The cost is a modest reduction in the size of the confirmatory control group, in expectation equal to the realised number of declined slots.
Two sources of heterogeneity in our sample motivated this design. First, based on existing survey data, we expect around 45% of households to be single-mother households (no male partner present), and the other 55% to be households with a male partner. This implies that for only around half of the sampled households will we have data on the male partner and on intra-household dynamics. Second, INAIPI attends children from 0 to 5; treatment relevance and outcome instruments differ across this age span, motivating the age-cell structure above. Stratification on mothers’ employment and marital status at baseline is not necessary because the realised variation in these dimensions (around 40% of women are employed; roughly one third are single mothers) is sufficient to support heterogeneity analyses without ex ante balancing.