# A07 prospective statistical analysis specification

Version 1.0 · no local trial data exist in this package

## Design, estimands and gates

Candidate design: 24 district blocks, two eligible schools per district, twelve consenting teachers per school recruited and measured before assignment, 48 schools and 576 planned teachers. An independent custodian assigns one school per pair to a protected-time charter after the common AI orientation. Recruitment, paid time agreements, funding, independent ethics determination, instrument pilot, locked analysis plan and public registration precede allocation. IDs, dates and permissions remain null until obtained.

Primary estimand: equal-school average assignment effect on change in teachers' mean weekday job-related minutes outside the agreed paid timetable, baseline to week 12. Negative favors the charter. Teacher means use the two assigned baseline and three assigned endline teaching weekdays; the finite scheduled-day target includes missing assigned days, never only self-selected completed days. Schools receive equal weight, then district pairs equal weight. Staff departure or tool discontinuation does not remove a baseline cohort member from follow-up.

Primary estimator after the declared missing-data procedure: for pair b, D_b = change_intervention_school − change_control_school. tau_hat = mean(D_b), SE = SD(D_b)/sqrt(B); 95% t interval with B−1 degrees of freedom. Main two-sided alpha .05. Report raw arm means, dispersion, all denominators and an individual-weighted secondary estimand. Baseline-adjusted school ANCOVA is a sensitivity analysis, not a data-dependent replacement chosen for a smaller p value. Actual trial inference requires a finalized executable analysis implementation; the included Python tool executes only source auditing and artificial planning simulations.

A complete-outcome Fisher sharp-null check uses all 2^24 possible pair sign flips if computationally feasible, or 99,999 reproducible independent flips plus the observed arrangement, p=(1+exceedances)/(R+1); comparison statistic absolute paired mean. The null asserts no effect on any school's recorded outcome. It is not the same hypothesis as the average-effect null. Do not apply complete-case permutation inference to an incomplete cohort as if missingness were randomized.

Key quality guardrail: equal-school difference in 0–20 blinded quality-score change. Candidate noninferiority margin −0.5 points, lower one-sided 95% confidence limit exceeding −0.5; margin and task equivalence require external educator/pilot justification before allocation. The pilot is excluded from the main efficacy cohort. A failed superiority test for quality is not evidence of noninferiority. A 'sustainable dividend' conclusion requires the primary time benefit, the independently justified quality guardrail, and no unresolved material safety issue. Report every component even if this conjunction fails. The published simulation does not calculate joint power for this conjunction.

## Missing data

Record scheduled and observed diary days, entire-day nonresponse, partial unknown episodes and reasons by arm and school. An incomplete diary is not padded with zero work. The primary target remains the baseline consenting cohort. Before allocation, implement and validate a multilevel multiple-imputation workflow with 50 imputations of bounded 0–1,440 minute daily endpoints, predictors assignment, baseline weekday measure, weekday, grade assignment, contracted fraction, caregiving indicator, school, district pair, and assignment-by-baseline term. Use a bounded predictive mean matching donor procedure within a prespecified hierarchy, retaining school dependence via random effects or an appropriate multilevel imputation model. If sparse domains lack compatible donors, the planned alternative and diagnostics must be specified before use; do not silently switch algorithms after seeing treatment differences.

Pool the equal-pair estimate and variance using Rubin's rules with a finite-df adjustment. Inspect imputed versus observed distributions, boundary pileup, donor diversity, convergence and between-imputation variance. Report the model, implementation/version and deviations. This specification is a requirement to implement before study activation, not a claim that imputation has already been coded or run. If the required pilot and implementation cannot support credible imputation, redesign before allocation.

Sensitivity: shift imputed unobserved intervention-day work by +15,+30,+60 minutes relative to MAR, bounded at 1,440; repeat shifts in control and symmetric unfavorable directions; show tipping points. Compute worst-case bounds using 0 and 1,440 for all missing assigned days. Missing whole schools and pair failures receive an explicit case audit; no postallocation replacement school. An incomplete follow-up may make the result inconclusive.

## Secondary outcomes and equity

Secondary families: total work and checking burden; realized uninterrupted blocks and paid-time allocation; exploratory wellbeing/control; process interviews. Apply Holm adjustment within each numerical family and label all subgroup effects exploratory. Report baseline contracted fraction, experience and caregiving burden only with consent and disclosure safeguards. Interaction estimates have confidence intervals and no unsupported ranking of small demographic groups. Outcome raters and the final primary analyst remain blinded until scoring and data cleaning rules are locked.

## ATUS secondary analysis

Join unique respondent TUCASEID to many activity episodes (TUCASEID,TUACTIVITY_N); sum TUACTDUR24, not untruncated TUACTDUR. Require exactly 1,440 minutes per diary. Select TELFS=1 or 2 and TEIO1OCD in 2310/2320/2330 using 2018 Census codes; exclude ambiguous preschool/kindergarten code 2300, postsecondary and assistants from the selected definition. Weekdays TUDIARYDAY=2–6. Main-job work is exact TRCODE=050101. Compute ratio sum(TUFINLWGT*minutes)/sum(TUFINLWGT) with zeros retained for nonworking diary days. Conditional working-day mean uses the positive-minute denominator and answers a different question. Do not divide weights by 365 for a ratio mean. No replicate SE, hypothesis test, annual teacher workload total, AI effect or district-level linkage is supplied. Published means are descriptive researcher computations with small-cell caveats, not official BLS estimates.

## Artificial design sensitivity

For each B in 16,24,32 and SD of paired school changes in 30,45,60 minutes, generate 10,000 independent B-vectors from a Gaussian distribution under null mean zero and alternative −20. Test each with a paired two-sided t statistic using B−1 df. Seed 20261015. Report empirical rejection, power, MCSE=sqrt[p(1−p)/10000], and approximate MDE=(t_.975,B−1+z_.80)*SD/sqrt(B). This MDE is a normal-approximation planning summary, not an exact noncentral-t solution. The SD assumptions incorporate a hypothetical aggregate school design; they are not ATUS-derived ICCs, teacher variance estimates or empirical local parameters. No missingness, unequal cluster-size penalty, contamination or quality-guardrail power is included.

## Mandatory pre-enrollment activation gate

The 48-school/24-pair configuration is provisional, not an adequately powered final sample. Before main-cohort recruitment, an external measurement-only pilot of at least twelve school pairs, excluded from efficacy, estimates paired-change SDs for time and quality and diary/form reliability. Use each SD's upper 80% confidence bound (Gaussian pilot approximation: SD*sqrt((P−1)/chi2.ppf(.20,P−1)); inspect nonnormality and use a prespecified robust alternative if required). Independently justify the quality margin and establish task equivalence. Evaluate candidate recruitment sizes 24–120 pairs, allowing loss of 10% of pairs or a larger pilot-supported rate. With finalized estimator/missingness workflow, run 50,000 replicates and select the smallest funded size whose lower 95% Monte Carlo power bound reaches .90 for each of the time and quality components under the declared −20minute time effect and zero quality deterioration. Two .90 marginal powers imply joint power at least .80 by union bound under that model; do not multiply powers assuming independence. Require independent statistical review, registration and design freeze before main recruitment/allocation. If no affordable candidate passes, no confirmatory launch: redesign or explicitly recast as feasibility. These gate calculations have not been executed and the current research tool does not claim to implement them.