Cross-Subject Change: General vs Subject-Specific

When a school's Reading scores rise, do Writing and Math rise too? This page decomposes year-over-year school-level changes into a general cohort factor and subject-specific residuals.

Key findings

Section 1: Scatter plots of cross-subject changes

Each dot is one school's year-over-year change (pp) in one subject plotted against another. The strong positive correlation confirms that most of the movement at a school moves in lockstep across subjects.

Section 2: Variance decomposition — General vs subject-specific

Principal component analysis on the joint distribution of (ΔReading, ΔWriting, ΔMath) extracts the first component that maximally captures shared variance. Call this the general cohort factor (PC1). The remaining two components capture subject-specific divergence.

The general factor accounts for ~75% of G3 and ~67% of G6 cross-subject change variance. The two subject-specific components share the remaining ~26–33% roughly equally, meaning no single subject systematically diverges from the others — subject-specific variation is scattered.

Why is G6 slightly lower? G6 Math–literacy coupling (r ≈ 0.43) is weaker than in G3 (r ≈ 0.60), likely reflecting greater specialisation in Grade 6 instruction and cohort-level Math trajectory differences.

Section 3: Year-to-year consistency

The general-factor structure is consistent across both year pairs (2022–23 → 2023–24 and 2023–24 → 2024–25).

Both year pairs show very similar correlations — the cross-subject structure is consistent across 2022–23 through 2024–25 with no meaningful shift between the two transitions.

Section 4: Does school size change the structure?

If the general factor were merely a statistical artifact of correlated noise, small schools (high noise per subject) should show lower cross-subject correlations than large schools (low noise). They don't.

The SD of changes halves from small (n < 20) to large (n ≥ 50) schools — confirming that small schools are noisy. Yet the cross-subject correlations barely budge. This means the noise is correlated across subjects, not independent — see Section 5.

Section 5: Distribution of general factor vs subject-specific residuals

The general factor score (PC1) summarises the school's overall direction that year. The subject-specific residuals show how much each subject deviated from that shared trend.

The subject-specific residuals are centred on zero and symmetrically spread — consistent with genuine subject-specific effects plus independent noise, not systematic bias in any subject.

Section 7: Correlation matrices — full summary

Interpretation and implications

The general cohort factor is real and large. ~75% (G3) and ~67% (G6) of school-level year-to-year change variance is shared across all three subjects. A school that improves in Reading almost certainly improved in Writing and Math too. These figures come from a clean sample: schools with complete, non-suppressed data across three consecutive years and without anomalous Math-vs-Reading registration gaps.

Two mechanisms contribute:

Real cohort/school effects — changes in staffing, leadership, demographics, or the cohort's prior preparation affect all subjects. The general factor cannot be fully explained away by participation alone.

Practical implications for interpreting school scores:

• A single-subject YoY change is mostly a general signal, not a subject-specific one. Before attributing a Math improvement to a numeracy programme, check whether Reading and Writing improved by the same amount.
• Subject-specific changes (the residuals) tend to be ≤ 5–10 pp even for large schools, and are mostly noise at the school level. Sustained multi-year divergence in one subject is more meaningful.
• The G6 Math independence (r ≈ 0.43 vs 0.66 for R–W) suggests that interventions targeting Grade 6 numeracy have a realistic chance of producing a detectable subject-specific signal — unlike Grade 3 where the three subjects move together more tightly (r ≈ 0.60 for literacy–Math).
• Small schools should not be compared on subject-specific changes at all — the SD is high for n < 20, dwarfing any plausible intervention effect. Even the general factor is unreliable at this scale.