Binomial mixed model
Verification
Scope
Coefficients, inference, fitted values, random effects, variance components, and likelihoods of a binomial logit model with one random intercept, at the Laplace approximation and 9, 25, and 61 adaptive quadrature points. Log likelihood also against an independent SciPy evaluation of the same approximation.
References
- Reference
- R lme4 glmer(family = binomial), bobyqa, tolPwrss 1e-12
- Versions
- R 4.6.1, lme4 2.0.6, PHP 8.5.10, python 3.14.7, numpy 2.5.4, scipy 1.18.1, statsmodels 0.15.0
Data
- Inputs
- Seven data sets, including lme4's cbpp, rare events, grouped outcomes with offsets, large group variation, and a zero-variance boundary.
Method
Each case is fitted by lme4 and phit, and every quantity is compared with lme4. The same approximation is also evaluated independently with SciPy root finding and Gauss–Hermite quadrature. Each case's status is recomputed from the saved values.
- Check
- Comparison with a reference implementation
- Recorded
- 2026-11-01, PHP 8.5.10, macOS 27.0.1 (arm64), commit 066e2e1
Tolerances
|x − r| ≤ 2e-5 + 2e-4 × |r|
Applies to: Estimates
Coefficients, random effects, variance components, fitted values
|x − r| ≤ 2e-5 + 0.002 × |r|
Applies to: Inference
Covariance, standard errors, z, p, intervals
|x − r| ≤ 1e-6 + 1e-8 × |r|
Applies to: Likelihood
Log likelihood, deviance, AIC, BIC
|x − r| ≤ 2e-8
Applies to: Independent likelihood
SciPy evaluation of the same approximation
x: package value; r: reference value. Absolute error: |x − r|. Max errors are over the values a row compares.
Results
588 comparisons in 28 cases: 588 pass, 0 fail, 0 skipped.
By case
| Case | Comparisons | Largest abs. error | Status |
|---|---|---|---|
| Binary outcomes, Laplace | 21 | 2.1e-5 | pass |
| Binary outcomes, 9 quadrature points | 21 | 4.2e-6 | pass |
| Binary outcomes, 25 quadrature points | 21 | 2.2e-6 | pass |
| Binary outcomes, 61 quadrature points | 21 | 2.6e-6 | pass |
| No fixed intercept, Laplace | 21 | 4.1e-6 | pass |
| No fixed intercept, 9 quadrature points | 21 | 4.5e-7 | pass |
| No fixed intercept, 25 quadrature points | 21 | 9.9e-7 | pass |
| No fixed intercept, 61 quadrature points | 21 | 6.7e-7 | pass |
| Grouped outcomes with offsets, Laplace | 21 | 3.5e-5 | pass |
| Grouped outcomes with offsets, 9 quadrature points | 21 | 4.1e-6 | pass |
| Grouped outcomes with offsets, 25 quadrature points | 21 | 3.6e-6 | pass |
| Grouped outcomes with offsets, 61 quadrature points | 21 | 8.7e-7 | pass |
| Rare events, Laplace | 21 | 2.9e-4 | pass |
| Rare events, 9 quadrature points | 21 | 7.1e-5 | pass |
| Rare events, 25 quadrature points | 21 | 2.0e-5 | pass |
| Rare events, 61 quadrature points | 21 | 1.5e-5 | pass |
| Large group variation, Laplace | 21 | 4.1e-5 | pass |
| Large group variation, 9 quadrature points | 21 | 6.5e-7 | pass |
| Large group variation, 25 quadrature points | 21 | 5.5e-7 | pass |
| Large group variation, 61 quadrature points | 21 | 8.9e-7 | pass |
| Zero group variance (boundary), Laplace | 21 | 2.9e-8 | pass |
| Zero group variance (boundary), 9 quadrature points | 21 | 1.7e-8 | pass |
| Zero group variance (boundary), 25 quadrature points | 21 | 1.7e-8 | pass |
| Zero group variance (boundary), 61 quadrature points | 21 | 1.7e-8 | pass |
| cbpp (lme4), Laplace | 21 | 5.6e-6 | pass |
| cbpp (lme4), 9 quadrature points | 21 | 1.0e-6 | pass |
| cbpp (lme4), 25 quadrature points | 21 | 1.2e-6 | pass |
| cbpp (lme4), 61 quadrature points | 21 | 8.6e-7 | pass |
All 588 comparisons
| Case | Quantity | Criterion | Values | Package value | Reference value | Max abs. error | Status |
|---|---|---|---|---|---|---|---|
| Binary outcomes, Laplace | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 3 | 3.5e-7 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 9 | 3.3e-6 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 3.0e-6 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 2.1e-5 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 1.2e-7 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 6 | 6.3e-6 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 7.1e-8 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 7.3e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 5.9e-7 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 7.3e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 3.6e-7 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 1.3e-7 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.024953238 | 1.02495296 | 2.8e-7 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.05052914 | 1.050528571 | 5.7e-7 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -106.5160823 | -106.5160823 | 1.0e-8 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 213.0321645 | 213.0321646 | 2.1e-8 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 221.0321645 | 221.0321646 | 2.1e-8 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 233.9135878 | 233.9135879 | 2.1e-8 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 4 | 4 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 181 | 181 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -106.5160823 | -106.5160823 | 4.3e-14 | pass | |
| Binary outcomes, 9 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 3 | 4.9e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 9 | 6.3e-7 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 5.8e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 4.2e-6 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 3.6e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 6 | 1.2e-6 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.5e-8 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.6e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.2e-7 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.6e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 6.0e-8 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 2.1e-8 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.036172174 | 1.036172234 | 6.0e-8 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.073652774 | 1.073652898 | 1.2e-7 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -106.4237305 | -106.4237305 | 2.1e-13 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 212.847461 | 212.847461 | 4.3e-13 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 220.847461 | 220.847461 | 4.3e-13 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 233.7288843 | 233.7288843 | 4.3e-13 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 4 | 4 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 181 | 181 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -106.4237305 | -106.4237305 | 0 | pass | |
| Binary outcomes, 25 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 3 | 1.7e-7 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 9 | 3.8e-7 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 3.5e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 2.2e-6 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 1.4e-7 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 6 | 8.5e-7 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 3.8e-8 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 4.3e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.8e-7 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 4.3e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 1.1e-7 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 1.7e-8 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.036173553 | 1.036173614 | 6.2e-8 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.073655632 | 1.073655759 | 1.3e-7 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -106.4237285 | -106.4237285 | 5.7e-14 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 212.8474571 | 212.8474571 | 1.1e-13 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 220.8474571 | 220.8474571 | 1.1e-13 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 233.7288804 | 233.7288804 | 1.1e-13 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 4 | 4 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 181 | 181 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -106.4237285 | -106.4237285 | 0 | pass | |
| Binary outcomes, 61 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 3 | 5.1e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 9 | 4.1e-7 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 3.7e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 2.6e-6 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 1.3e-7 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 6 | 7.8e-7 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 2.6e-8 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.9e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.2e-7 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.9e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 1.6e-8 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 4.0e-9 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.036173545 | 1.036173551 | 6.0e-9 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.073655614 | 1.073655627 | 1.2e-8 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -106.4237285 | -106.4237285 | 1.4e-14 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 212.8474571 | 212.8474571 | 2.8e-14 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 220.8474571 | 220.8474571 | 2.8e-14 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 233.7288804 | 233.7288804 | 2.8e-14 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 4 | 4 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 181 | 181 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -106.4237285 | -106.4237285 | 1.4e-14 | pass | |
| No fixed intercept, Laplace | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 2 | 3.0e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 4.0e-8 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 2.3e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 4.1e-6 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 8.4e-7 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 4.8e-7 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 5.1e-8 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 5.4e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 3.7e-7 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 5.4e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 4.1e-7 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 9.2e-8 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.822756605 | 1.822756234 | 3.7e-7 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3.32244164 | 3.32244029 | 1.4e-6 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -113.822073 | -113.822073 | 8.3e-9 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 227.6441459 | 227.644146 | 1.7e-8 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 233.6441459 | 233.644146 | 1.7e-8 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 243.3052134 | 243.3052134 | 1.7e-8 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3 | 3 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 182 | 182 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -113.822073 | -113.822073 | 5.4e-13 | pass | |
| No fixed intercept, 9 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 2 | 8.7e-9 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 4.9e-9 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 2.8e-8 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 4.5e-7 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 8.9e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 6.4e-8 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.3e-8 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.1e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 9.5e-8 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.1e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 1.0e-7 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 3.3e-8 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.843266299 | 1.843266179 | 1.2e-7 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3.397630647 | 3.397630206 | 4.4e-7 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -113.6330606 | -113.6330606 | 7.3e-12 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 227.2661213 | 227.2661213 | 1.5e-11 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 233.2661213 | 233.2661213 | 1.5e-11 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 242.9271887 | 242.9271887 | 1.5e-11 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3 | 3 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 182 | 182 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -113.6330606 | -113.6330606 | 0 | pass | |
| No fixed intercept, 25 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 2 | 2.7e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 3.8e-10 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 2.2e-9 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 3.5e-7 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 7.0e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 3.1e-8 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 4.0e-8 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 2.9e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 2.5e-7 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 2.9e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 2.8e-7 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 6.8e-8 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.843342873 | 1.843342605 | 2.7e-7 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3.397912949 | 3.397911959 | 9.9e-7 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -113.6329667 | -113.6329667 | 7.1e-14 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 227.2659333 | 227.2659333 | 1.4e-13 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 233.2659333 | 233.2659333 | 1.4e-13 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 242.9270008 | 242.9270008 | 1.4e-13 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3 | 3 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 182 | 182 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -113.6329667 | -113.6329667 | 0 | pass | |
| No fixed intercept, 61 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 2 | 1.8e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 3.2e-9 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 1.9e-8 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 1.5e-7 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 3.0e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 5.4e-8 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 2.7e-8 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 2.2e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.7e-7 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 2.2e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 2.0e-7 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 4.7e-8 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.843342841 | 1.843342659 | 1.8e-7 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3.397912829 | 3.397912158 | 6.7e-7 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -113.6329667 | -113.6329667 | 7.1e-14 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 227.2659333 | 227.2659333 | 1.4e-13 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 233.2659333 | 233.2659333 | 1.4e-13 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 242.9270008 | 242.9270008 | 1.4e-13 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3 | 3 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 182 | 182 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -113.6329667 | -113.6329667 | 2.8e-14 | pass | |
| Grouped outcomes with offsets, Laplace | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 3 | 1.6e-6 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 9 | 2.3e-6 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 2.7e-6 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 3.5e-5 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 2.0e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 6 | 6.8e-6 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 3.9e-7 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 3.3e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 3.7e-7 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 3.3e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 1.6e-6 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 4.6e-8 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.428117459 | 1.428118288 | 8.3e-7 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 2.039519478 | 2.039521844 | 2.4e-6 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -350.4316056 | -350.4316056 | 2.0e-11 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 700.8632113 | 700.8632113 | 4.1e-11 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 708.8632113 | 708.8632113 | 4.1e-11 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 721.7446346 | 721.7446346 | 4.1e-11 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 4 | 4 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 181 | 181 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -350.4316056 | -350.4316056 | 0 | pass | |
| Grouped outcomes with offsets, 9 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 3 | 1.2e-6 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 9 | 1.0e-7 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 1.2e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 4.1e-6 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 2.3e-9 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 6 | 1.4e-6 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 2.8e-7 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.2e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.6e-7 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.2e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 1.1e-6 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 1.3e-8 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.428641935 | 1.428642038 | 1.0e-7 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 2.041017777 | 2.041018072 | 2.9e-7 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -350.4100063 | -350.4100063 | 1.0e-12 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 700.8200126 | 700.8200126 | 2.0e-12 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 708.8200126 | 708.8200126 | 2.0e-12 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 721.7014359 | 721.7014359 | 2.0e-12 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 4 | 4 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 181 | 181 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -350.4100063 | -350.4100063 | 0 | pass | |
| Grouped outcomes with offsets, 25 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 3 | 6.9e-7 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 9 | 3.8e-7 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 4.4e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 3.6e-6 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 1.9e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 6 | 1.6e-6 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.7e-7 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.1e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.2e-7 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 1.1e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 6.9e-7 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 1.5e-8 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.428642945 | 1.428642704 | 2.4e-7 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 2.041020665 | 2.041019977 | 6.9e-7 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -350.4100013 | -350.4100013 | 1.8e-12 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 700.8200026 | 700.8200026 | 3.6e-12 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 708.8200026 | 708.8200026 | 3.6e-12 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 721.7014259 | 721.7014259 | 3.6e-12 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 4 | 4 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 181 | 181 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -350.4100013 | -350.4100013 | 5.7e-14 | pass | |
| Grouped outcomes with offsets, 61 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 3 | 1.7e-7 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 9 | 9.3e-8 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 1.1e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 8.7e-7 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 3 | 1.7e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 6 | 3.9e-7 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 4.1e-8 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 4.2e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 3.2e-8 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 185 | 4.2e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 1.6e-7 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 1.5e-9 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.428642845 | 1.428642839 | 6.4e-9 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 2.041020379 | 2.041020361 | 1.8e-8 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -350.4100013 | -350.4100013 | 1.1e-13 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 700.8200026 | 700.8200026 | 2.3e-13 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 708.8200026 | 708.8200026 | 2.3e-13 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 721.7014259 | 721.7014259 | 2.3e-13 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 4 | 4 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 181 | 181 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -350.4100013 | -350.4100013 | 1.1e-13 | pass | |
| Rare events, Laplace | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 2 | 1.2e-6 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 2.9e-4 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 3.7e-5 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 2.7e-5 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 4.3e-7 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 7.4e-5 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 2.3e-9 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 3.6e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 1.7e-6 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 3.6e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 16 | 1.2e-6 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 16 | 2.0e-5 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 7.298560319 | 7.298561664 | 1.3e-6 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 53.26898273 | 53.26900236 | 2.0e-5 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -14.30921541 | -14.30921541 | 4.9e-9 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 28.61843081 | 28.61843082 | 9.7e-9 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 34.61843081 | 34.61843082 | 9.7e-9 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 42.31147539 | 42.3114754 | 9.7e-9 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3 | 3 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 93 | 93 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -14.30921541 | -14.30921541 | 1.4e-12 | pass | |
| Rare events, 9 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 2 | 1.3e-7 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 7.1e-5 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 1.1e-5 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 8.5e-6 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 3.8e-7 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 2.2e-5 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 4.1e-10 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 7.4e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 2.6e-7 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 7.4e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 16 | 8.8e-8 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 16 | 1.0e-7 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3.499107409 | 3.499107415 | 6.1e-9 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 12.24375266 | 12.2437527 | 4.3e-8 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -15.66374786 | -15.66374786 | 1.7e-12 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 31.32749572 | 31.32749572 | 3.3e-12 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 37.32749572 | 37.32749572 | 3.3e-12 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 45.0205403 | 45.0205403 | 3.3e-12 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3 | 3 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 93 | 93 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -15.66374786 | -15.66374786 | 1.6e-13 | pass | |
| Rare events, 25 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 2 | 2.9e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 2.0e-5 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 3.3e-6 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 2.6e-6 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 1.1e-7 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 6.5e-6 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 8.3e-10 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 1.6e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 3.9e-8 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 1.6e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 16 | 2.2e-8 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 16 | 2.2e-8 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3.356056643 | 3.35605665 | 6.9e-9 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 11.26311619 | 11.26311623 | 4.6e-8 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -15.70761449 | -15.70761449 | 2.3e-14 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 31.41522899 | 31.41522899 | 4.6e-14 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 37.41522899 | 37.41522899 | 4.3e-14 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 45.10827356 | 45.10827356 | 5.0e-14 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3 | 3 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 93 | 93 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -15.70761449 | -15.70761449 | 7.1e-15 | pass | |
| Rare events, 61 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 2 | 1.0e-7 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 1.5e-5 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 2.4e-6 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 1.9e-6 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 8.3e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 4.8e-6 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 1.8e-9 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 5.8e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 1.7e-7 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 96 | 5.8e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 16 | 9.6e-8 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 16 | 4.2e-7 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3.356286954 | 3.356287017 | 6.4e-8 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 11.26466212 | 11.26466254 | 4.3e-7 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -15.70790636 | -15.70790636 | 1.6e-14 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 31.41581273 | 31.41581273 | 3.2e-14 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 37.41581273 | 37.41581273 | 2.8e-14 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 45.1088573 | 45.1088573 | 3.6e-14 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3 | 3 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 93 | 93 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -15.70790636 | -15.70790636 | 2.0e-14 | pass | |
| Large group variation, Laplace | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 2 | 1.5e-6 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 9.2e-6 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 1.0e-5 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 4.1e-5 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 4.4e-6 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 2.1e-5 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 3.3e-7 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 3.0e-7 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 1.7e-6 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 3.0e-7 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 14 | 7.4e-7 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 14 | 7.8e-7 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.771034545 | 1.771033881 | 6.6e-7 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3.13656336 | 3.136561009 | 2.4e-6 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -60.96051285 | -60.96051304 | 1.9e-7 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 121.9210257 | 121.9210261 | 3.8e-7 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 127.9210257 | 127.9210261 | 3.8e-7 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 136.0765223 | 136.0765227 | 3.8e-7 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3 | 3 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 109 | 109 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -60.96051285 | -60.96051285 | 8.3e-13 | pass | |
| Large group variation, 9 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 2 | 5.4e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 3.4e-7 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 3.1e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 4.7e-7 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 2.9e-7 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 6.5e-7 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 2.5e-8 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 1.3e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 6.5e-8 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 1.3e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 14 | 4.6e-8 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 14 | 5.1e-7 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.83057019 | 1.830570191 | 1.5e-9 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3.350987219 | 3.350987225 | 5.6e-9 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -60.65815085 | -60.65815085 | 8.2e-10 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 121.3163017 | 121.3163017 | 1.6e-9 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 127.3163017 | 127.3163017 | 1.6e-9 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 135.4717983 | 135.4717983 | 1.6e-9 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3 | 3 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 109 | 109 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -60.65815085 | -60.65815085 | 0 | pass | |
| Large group variation, 25 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 2 | 1.5e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 2.6e-7 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 2.3e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 3.5e-7 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 1.7e-7 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 4.6e-7 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 3.7e-9 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 3.2e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 4.0e-8 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 3.2e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 14 | 3.3e-8 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 14 | 5.5e-7 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.830523929 | 1.830523894 | 3.5e-8 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3.350817854 | 3.350817727 | 1.3e-7 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -60.65818087 | -60.65818087 | 1.1e-13 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 121.3163617 | 121.3163617 | 2.1e-13 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 127.3163617 | 127.3163617 | 2.1e-13 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 135.4718584 | 135.4718584 | 2.3e-13 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3 | 3 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 109 | 109 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -60.65818087 | -60.65818087 | 0 | pass | |
| Large group variation, 61 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 2 | 3.3e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 3.5e-7 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 3.1e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 8.9e-7 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 2.6e-7 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 6.2e-7 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 1.1e-8 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 7.6e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 3.9e-8 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 112 | 7.6e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 14 | 2.2e-8 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 14 | 5.1e-7 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 1.830523917 | 1.830523928 | 1.1e-8 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3.35081781 | 3.35081785 | 4.0e-8 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -60.65818088 | -60.65818088 | 0 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 121.3163618 | 121.3163618 | 0 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 127.3163618 | 127.3163618 | 0 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 135.4718584 | 135.4718584 | 0 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 3 | 3 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 109 | 109 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -60.65818088 | -60.65818088 | 1.4e-14 | pass | |
| Zero group variance (boundary), Laplace | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 8.2e-9 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 1.4e-17 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 0 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 2.9e-8 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 2.3e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 8.2e-9 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 2.1e-9 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 2.1e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 8.2e-9 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 2.1e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 0 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 0 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0 | 0 | 0 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0 | 0 | 0 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -33.27106467 | -33.27106467 | 7.1e-15 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 66.54212933 | 66.54212933 | 1.4e-14 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 70.54212933 | 70.54212933 | 1.4e-14 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 74.28453136 | 74.28453136 | 1.4e-14 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 2 | 2 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 46 | 46 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -33.27106467 | -33.27106467 | 0 | pass | |
| Zero group variance (boundary), 9 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 4.8e-9 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 1.4e-17 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 0 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 1.7e-8 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 1.3e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 4.8e-9 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 1.2e-9 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 1.2e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 4.8e-9 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 1.2e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 0 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 0 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0 | 0 | 0 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0 | 0 | 0 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -33.27106467 | -33.27106467 | 0 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 66.54212933 | 66.54212933 | 0 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 70.54212933 | 70.54212933 | 0 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 74.28453136 | 74.28453136 | 0 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 2 | 2 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 46 | 46 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -33.27106467 | -33.27106467 | 0 | pass | |
| Zero group variance (boundary), 25 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 4.8e-9 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 1.4e-17 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 5.6e-17 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 1.7e-8 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 1.3e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 4.8e-9 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 1.2e-9 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 1.2e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 4.8e-9 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 1.2e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 0 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 0 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0 | 0 | 0 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0 | 0 | 0 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -33.27106467 | -33.27106467 | 0 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 66.54212933 | 66.54212933 | 0 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 70.54212933 | 70.54212933 | 0 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 74.28453136 | 74.28453136 | 0 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 2 | 2 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 46 | 46 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -33.27106467 | -33.27106467 | 7.1e-15 | pass | |
| Zero group variance (boundary), 61 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 4.8e-9 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 1.4e-17 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 5.6e-17 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 1.7e-8 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 1 | 1.3e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 2 | 4.8e-9 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 1.2e-9 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 1.2e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 4.8e-9 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 48 | 1.2e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 0 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 12 | 0 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0 | 0 | 0 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0 | 0 | 0 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -33.27106467 | -33.27106467 | 0 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 66.54212933 | 66.54212933 | 0 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 70.54212933 | 70.54212933 | 0 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 74.28453136 | 74.28453136 | 0 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 2 | 2 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 46 | 46 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -33.27106467 | -33.27106467 | 0 | pass | |
| cbpp (lme4), Laplace | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 4 | 4.7e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 16 | 5.5e-7 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 6.5e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 5.6e-6 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 1.6e-8 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 8 | 1.3e-6 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 3.0e-9 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 1.1e-8 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 7.6e-8 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 1.1e-8 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 15 | 4.1e-8 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 15 | 1.7e-8 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0.6422614614 | 0.6422614173 | 4.4e-8 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0.4124997849 | 0.4124997282 | 5.7e-8 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -92.02628186 | -92.02628187 | 6.7e-9 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 184.0525637 | 184.0525637 | 1.3e-8 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 194.0525637 | 194.0525637 | 1.3e-8 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 204.1793222 | 204.1793222 | 1.3e-8 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 5 | 5 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 51 | 51 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -92.02628186 | -92.02628186 | 1.2e-12 | pass | |
| cbpp (lme4), 9 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 4 | 7.6e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 16 | 8.6e-8 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 1.0e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 1.0e-6 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 2.7e-9 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 8 | 2.7e-7 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 3.1e-9 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 8.6e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 8.3e-8 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 8.6e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 15 | 1.5e-8 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 15 | 5.2e-9 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0.6475177228 | 0.6475177086 | 1.4e-8 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0.4192792013 | 0.419279183 | 1.8e-8 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -91.9833718 | -91.9833718 | 8.7e-13 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 183.9667436 | 183.9667436 | 1.7e-12 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 193.9667436 | 193.9667436 | 1.7e-12 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 204.093502 | 204.093502 | 1.8e-12 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 5 | 5 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 51 | 51 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -91.9833718 | -91.9833718 | 7.4e-13 | pass | |
| cbpp (lme4), 25 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 4 | 6.1e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 16 | 1.2e-7 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 1.4e-7 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 1.2e-6 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 2.4e-9 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 8 | 2.8e-7 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 5.1e-9 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 9.4e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 7.6e-8 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 9.4e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 15 | 2.4e-8 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 15 | 7.9e-9 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0.647518456 | 0.6475184778 | 2.2e-8 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0.4192801508 | 0.4192801791 | 2.8e-8 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -91.98336904 | -91.98336904 | 6.8e-13 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 183.9667381 | 183.9667381 | 1.4e-12 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 193.9667381 | 193.9667381 | 1.4e-12 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 204.0934965 | 204.0934965 | 1.4e-12 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 5 | 5 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 51 | 51 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -91.98336904 | -91.98336904 | 7.2e-13 | pass | |
| cbpp (lme4), 61 quadrature points | Coefficients | |x − r| ≤ 2e-5 + 2e-4 × |r| | 4 | 5.6e-8 | pass | ||
| Covariance matrix | |x − r| ≤ 2e-5 + 0.002 × |r| | 16 | 5.8e-8 | pass | |||
| Standard errors | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 8.7e-8 | pass | |||
| z statistics | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 8.6e-7 | pass | |||
| p-values | |x − r| ≤ 2e-5 + 0.002 × |r| | 4 | 3.7e-9 | pass | |||
| Confidence intervals | |x − r| ≤ 2e-5 + 0.002 × |r| | 8 | 1.9e-7 | pass | |||
| Fitted values (fixed effects only) | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 7.1e-9 | pass | |||
| Fitted values | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 7.9e-9 | pass | |||
| Linear predictor | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 5.9e-8 | pass | |||
| Residuals | |x − r| ≤ 2e-5 + 2e-4 × |r| | 56 | 7.9e-9 | pass | |||
| Random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 15 | 3.5e-8 | pass | |||
| Conditional variances of the random effects | |x − r| ≤ 2e-5 + 2e-4 × |r| | 15 | 4.6e-10 | pass | |||
| Random-intercept standard deviation | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0.6475184777 | 0.6475184732 | 4.5e-9 | pass | |
| Random-intercept variance | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 0.419280179 | 0.4192801731 | 5.9e-9 | pass | |
| Log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | -91.98336904 | -91.98336904 | 8.0e-13 | pass | |
| −2 log likelihood | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 183.9667381 | 183.9667381 | 1.6e-12 | pass | |
| AIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 193.9667381 | 193.9667381 | 1.6e-12 | pass | |
| BIC | |x − r| ≤ 1e-6 + 1e-8 × |r| | 1 | 204.0934965 | 204.0934965 | 1.6e-12 | pass | |
| Number of parameters | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 5 | 5 | 0 | pass | |
| Residual degrees of freedom | |x − r| ≤ 2e-5 + 2e-4 × |r| | 1 | 51 | 51 | 0 | pass | |
| Log likelihood vs independent SciPy evaluation | |x − r| ≤ 2e-8 | 1 | -91.98336904 | -91.98336904 | 7.7e-13 | pass |
Limitations
Exclusions
- statsmodels' Bayesian binomial mixed estimator has a different objective and is not compared.
Known differences
- lme4's default tolPwrss (1e-7) leaves its conditional modes slightly short of convergence; the references use tolPwrss 1e-12, where lme4 and phit agree.
- Agreement with lme4 is not integration accuracy: integration error is measured separately with QUADPACK and reported in the source results.
Reproduction
From the repository root
cargo build -p phit --release --locked
.context/notebooks-venv/bin/python docs/verifications/compare_logistic_mixed.py- Raw record
- site/verifications/logistic-mixed.json
- Full results
- docs/verifications/logistic-mixed-results.json
- Input data
- tests/fixtures/logistic-mixed-r.json