What this report shows

MCPower generates a dataset from a formula, then fits it back to recover the coefficients. This report checks the fitting half: given one fixed dataset, does MCPower’s own solver return the same numbers a trusted, independent R fit returns?

For each formula we take the exact bytes MCPower saved (the same design matrix and outcome, byte-for-byte), fit them two ways —

• R (the reference): stats::lm through the origin on the design matrix, the textbook least-squares solution.
• MCPower: the engine’s own load_data() path, which runs the same solver the power simulation uses.

— and compare the fitted coefficients, the test statistics, and the decision threshold. Because both fit the identical bytes, sampling noise cancels: any difference is a solver discrepancy, and the allowed differences are tiny.

How the check works

1. Provenance. Re-generate the dataset from the committed seed and confirm its content hash matches the saved file — proof the bytes haven’t drifted.
2. B (R) vs C (MCPower) on those bytes: coefficients, the marginal t statistics, and the critical value.
3. C vs A (a readable sanity overlay): MCPower’s recovered coefficients next to the formula’s true values.

The thresholds

These are the B↔C gates from tolerances.R. OLS is closed-form, so it gets the tight estimate_rel_ols band — set just above the measured worst agreement (≈1.2e-12) so a real solver regression would trip it. The iterative GLM/MLE fits use the looser estimate_rel_iter band in their own reports; the per-formula tables below show OLS’s actual margin.

All OLS formulas pass: MCPower’s solver matches R to machine precision on every saved dataset, and every file reproduces from its seed.

png 2

Each point is one fitted quantity on one saved dataset; the red line is the gate. Every β and statistic agrees with R orders of magnitude inside tolerance — exact (bit-identical) matches pile up at the left floor.

y = 0.25·x1 + noise

R formula y ~ x1 · ordinary least squares (continuous outcome) · n=400, seed=2137. File reproduces from seed: yes.

y = 0.40·x1 + noise

R formula y ~ x1 · ordinary least squares (continuous outcome) · n=400, seed=2138. File reproduces from seed: yes.

y = 0.25·x1 + 0.10·x2 + noise

R formula y ~ x1 + x2 · ordinary least squares (continuous outcome) · n=400, seed=2137. File reproduces from seed: yes.

y = 0.40·x1 + 0.25·x2 + noise

R formula y ~ x1 + x2 · ordinary least squares (continuous outcome) · n=400, seed=2138. File reproduces from seed: yes.

y = 0.25·x1 + 0.00·x2 + noise

R formula y ~ x1 + x2 · ordinary least squares (continuous outcome) · n=400, seed=2137. File reproduces from seed: yes.

y = 0.25·x1 + 0.10·x2 + noise

R formula y ~ x1 + x2 · ordinary least squares (continuous outcome) · n=400, seed=2137. File reproduces from seed: yes.

y = 0.40·x1 + 0.25·x2 + noise

R formula y ~ x1 + x2 · ordinary least squares (continuous outcome) · n=400, seed=2138. File reproduces from seed: yes.

y = 0.25·x1 + 0.10·x2 + -0.20·x1:x2 + noise

R formula y ~ x1*x2 · ordinary least squares (continuous outcome) · n=600, seed=2137. File reproduces from seed: yes.

y = 0.40·x1 + 0.25·x2 + 0.15·x1:x2 + noise

R formula y ~ x1*x2 · ordinary least squares (continuous outcome) · n=600, seed=2138. File reproduces from seed: yes.

y = 0.25·x1 + 0.50·g[2] + 0.80·g[3] + noise

R formula y ~ x1 + g · ordinary least squares (continuous outcome) · n=600, seed=2137. File reproduces from seed: yes.

y = 0.40·x1 + 0.20·g[2] + 0.50·g[3] + noise

R formula y ~ x1 + g · ordinary least squares (continuous outcome) · n=600, seed=2138. File reproduces from seed: yes.

y = 0.30·x1 + 0.40·g[2] + 0.60·g[3] + 0.20·x1:g[2] + 0.30·x1:g[3] + noise

R formula y ~ x1*g · ordinary least squares (continuous outcome) · n=800, seed=2137. File reproduces from seed: yes.

y = 0.40·x1 + 0.50·g[2] + 0.80·g[3] + 0.25·x1:g[2] + 0.40·x1:g[3] + noise

R formula y ~ x1*g · ordinary least squares (continuous outcome) · n=800, seed=2138. File reproduces from seed: yes.

y = 0.50·g1[2] + 0.40·g2[2] + 0.30·g1[2]:g2[2] + noise

R formula y ~ g1*g2 · ordinary least squares (continuous outcome) · n=800, seed=2137. File reproduces from seed: yes.

y = 0.20·g1[2] + 0.80·g2[2] + 0.50·g1[2]:g2[2] + noise

R formula y ~ g1*g2 · ordinary least squares (continuous outcome) · n=800, seed=2138. File reproduces from seed: yes.

Reproduce: Rscript mcpower/validation/data_generation.r then rmarkdown::render("mcpower/validation/validation_OLS_solving.rmd", output_dir = "mcpower/web/documentation/validation").