# Model specification

Two-level hierarchical linear models, students (i) within schools (j),
estimated separately for each system and pooled with system fixed effects.

## Unconditional model

```
ICTEFFIC_ij = beta_0j + e_ij
beta_0j     = gamma_00 + u_0j

e_ij ~ N(0, sigma^2);  u_0j ~ N(0, tau_00)
ICC: rho = tau_00 / (tau_00 + sigma^2)
```

## Conditional model (level 1)

```
ICTEFFIC_ij = beta_0j + beta_1j ESCS_ij + beta_2j Girl_ij + beta_3j Immig_ij
            + beta_4j ICTQUAL_ij + beta_5j ICTHOME_ij + beta_6j ICTUSE_ij
            + e_ij
```

`ICTUSE` denotes the block ICTWKDY, ICTWKEND, ICTOUT. Continuous level-1
predictors are group-mean centered (Enders & Tofighi, 2007). Age and relative
grade are included as controls.

## School level

```
beta_0j = gamma_00 + gamma_01 meanESCS_j + gamma_02 RATCMP1_j
        + gamma_03 EDUSHORT_j + gamma_04 Private_j + gamma_05 Rural_j + u_0j
beta_1j = gamma_10 + gamma_11 meanESCS_j + u_1j     (cross-level interaction)
beta_2j = gamma_20 + u_2j
beta_qj = gamma_q0,  q = 3, ..., 6
```

School-level continuous predictors are grand-mean centered within system.

## Estimation

- REML via `statsmodels` MixedLM (Python).
- Country-specific models: random intercept + random ESCS slope.
- Pooled models (M1 student predictors; M2 + school predictors; M3 +
  cross-level interactions `meanESCS x ESCS`, `meanESCS x Girl`): random
  intercept, system fixed effects.
- Effect sizes: proportional reduction in variance (PRV) relative to the
  pooled unconditional model.
- Descriptive statistics use final student weights (W_FSTUWT) rescaled within
  systems.

## Provisional status and planned robustness

Current estimates are complete-case and unweighted at the model stage.
Planned: survey-weighted multilevel estimation (WeMix, R), standard errors by
Fay-adjusted balanced repeated replication (80 replicate weights), and
multiple imputation by chained equations combined under Rubin's rules.
Plausible-value methodology is not required because the outcome is a
questionnaire index, not a cognitive score.

## Key references

- Enders & Tofighi (2007), *Psychological Methods*, 12(2), 121–138.
- Raudenbush & Bryk (2002), *Hierarchical Linear Models* (2nd ed.).
- Rutkowski et al. (2010), *Educational Researcher*, 39(2), 142–151.
- OECD (2024), *PISA 2022 Technical Report*.
