Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
At the full frame J ≈ 80 schools (20 with a computer lab, 60 without), with a median grades 2-4 cohort of m ≈ 13 pupils per school and a mean cohort size of approximately 25 pupils per school (≈2,000 pupils assessed total; the gap between median and mean reflects larger cohorts concentrated at the 12 secondary schools relative to the 68 basic schools) and a baseline oral-English covariate, the main-effect minimum detectable effect (MDE) is ≈ 0.24 SD at the headline ICC = 0.25 (≈ 0.27 SD at a conservative ICC ≈ 0.39), under the efficient inverse-variance-weighted (precision-weighted / GLS) estimator, which is the pre-registered primary specification. A naive unweighted OLS with cluster-robust standard errors is materially weaker (≈ 0.30-0.32 SD), because per-school cohort sizes vary widely; this is why the primary specification is precision-weighted rather than unweighted.
The covariate R² is ≈ 0.51 alone, ≈ 0.66 combined with stratification, and ≈ 0.62 after a cross-administrator reliability discount. The headline 0.24 SD MDE uses R² ≈ 0.62 (the reliability-discounted figure); R² ≈ 0.66 is reported for comparison as the pre-discount value. These design parameters (covariate R², the ICCs, and the reliability discount lambda) were re-estimated on the grades 2-4 band the trial actually assesses, which lowered the combined R² from 0.85 (the prior, broader-band estimate) to 0.66, moving the headline MDE from ≈ 0.19 to ≈ 0.24 SD. They will be re-confirmed, not re-derived, once baseline data are collected; no outcome data inform these parameters prior to baseline.
The pre-specified 0.20 SD target is not cleared, and we do not claim it is: 0.20 SD is retained as the policy-relevant threshold the design is built around, and ≈ 0.24 SD is the honest minimum detectable effect on the conservative grades 2-4 proxy.
The binding exposure constraint is consent-gated adoption rather than pupil attrition: the intent-to-treat MDE stays near 0.24-0.27 SD across plausible school-level loss (0-20%), but the Bloom-adjusted treatment-on-the-treated effect among adopters must be at least 0.36, 0.46, or 0.63 SD at adoption rates of 70%, 55%, or 40% respectively. Adequate power therefore rests on adoption being reasonably high, a bar the baseline pilot will establish.
Within-stratum and interaction tests are powered only to sign a direction, not to size an effect: within-stratum MDE ≈ 0.29 SD in the non-lab stratum (J = 60) to ≈ 0.41 SD in the lab stratum (J = 20); the lab × phone interaction MDE is ≈ 0.47 SD.