Random-effect intervals

A mixed model estimates each school's random effect as a conditional mode, the most likely value of the effect given the data and the estimated variance components. Phit\fit_lmer() also returns the conditional variance of each mode, the quantity that lme4 reports with ranef(condVar = TRUE). Those variances give each school an interval.

This example fits the school scores data with random intercepts, then with random study-hours slopes. phiz's random_effects_plot() draws the caterpillar plot from the fit alone, using the reported variances. The intervals treat the fixed effects and variance components as known, as lme4's do.

Random effects Random intercepts Correlated random slopes Independent random slopes

Interval

$fit = DataFrame::fromQuery(StudentScore::query()) |> lmer('score ~ study_hours + (1 | school)');

$fit['random'][0]['effects'];           // conditional modes, one row per school
$fit['random'][0]['effects_cond_var'];  // their conditional covariance, one matrix per school

random_effects_plot($fit, level: 0.95);

$fit = phplyr(DataFrame::fromQuery(StudentScore::query()))->lmer('score ~ study_hours + (1 | school)');

$fit['random'][0]['effects'];           // conditional modes, one row per school
$fit['random'][0]['effects_cond_var'];  // their conditional covariance, one matrix per school

random_effects_plot($fit, level: 0.95);

Caterpillar plot

Each point is a school's conditional mode, its estimated deviation from the fixed effects. Each bar is a 95% interval of ±z standard deviations, where the standard deviation is the square root of the conditional variance. Schools are sorted by their intercept.

Variance components

Term	Variance	Std. dev.
school: (Intercept)	20.8666	4.5680
Residual	29.0940	5.3939

REML fit on 250 students in 12 schools. AIC = 1,591.3.

Conditional modes and standard deviations

School	Students	(Intercept)	Cond. SD	Closed form
school-01	18	0.8926	1.2248	1.2248
school-02	24	1.7890	1.0704	1.0704
school-03	15	-2.5419	1.3322	1.3322
school-04	22	-5.7326	1.1152	1.1152
school-05	20	0.5818	1.1661	1.1661
school-06	27	2.8431	1.0122	1.0122
school-07	19	9.9943	1.1944	1.1944
school-08	21	-5.5026	1.1398	1.1398
school-09	16	2.4371	1.2933	1.2933
school-10	25	0.2354	1.0499	1.0499
school-11	23	0.1785	1.0921	1.0921
school-12	20	-5.1747	1.1661	1.1661

With one random intercept, the conditional variance of a school's effect is τ²σ² / (σ² + nτ²), where τ² is the school variance, σ² is the residual variance, and n is the number of students. The closed-form column computes it from the variance components and matches the reported value. Schools with more students have smaller conditional standard deviations.