Mixed models for repeated measurements

Use a mixed model when observations share a participant, site or other grouping unit. lmer() fits a continuous Gaussian response with ML or REML. glmer() fits binomial or Poisson responses with a Laplace approximation.

The snippets below assume a pandas data frame data with score, time, treatment and participant columns. Set treatment to a categorical column to make its levels and ordering explicit. For a complete runnable dataset, see the quickstart and mixed-model example.

Fit a random intercept or slope

from rparity import lmer

model = lmer("score ~ treatment + time + (1 | participant)", data)
print(model.summary())
print(model.fixef())
print(model.VarCorr())

slopes = lmer("score ~ treatment + time + (time | participant)", data)

(1 | participant) allows each participant a different intercept. (time | participant) adds a participant-specific time slope and estimates its correlation with the intercept. Use (time || participant) for independent intercept and slope components. REML is the default; set reml=False for maximum likelihood.

Predict for existing or new participants

conditional = model.predict(data)
population = model.predict(data, re_form="NA")
new_predictions = model.predict(newdata, allow_new_levels=True)

Here newdata is a data frame containing the predictors in the fitted formula. Default predictions include the fitted random effects for known participants. re_form="NA" sets those effects to zero. With allow_new_levels=True, a new participant contributes a zero random effect.

Test effects and compare models

from rparity import Anova, anova, emmeans, pairs

print(Anova(model, type=2))
means = emmeans(model, "treatment", lmer_df="satterthwaite")
print(pairs(means, adjust="tukey").summary())

reduced = lmer("score ~ time + (1 | participant)", data)
print(anova(reduced, model))

Anova() tests fixed-effect terms. anova(reduced, model) compares likelihoods and refits REML fits using ML by default. Use models fitted to the same observations when comparing their likelihoods. For Type III main-effect tests involving interactions, fit with contrasts="sum".

Binary and count responses

If data instead contains a binary success column or a nonnegative integer count column, choose the corresponding family:

from rparity import glmer

binary = glmer("success ~ time + (1 | participant)", data,
               family="binomial")
counts = glmer("count ~ time + (1 | participant)", data,
               family="poisson")

Use glmmTMB() for negative-binomial counts, zero inflation or modeled dispersion. Supported glmer() links and further formula examples appear in the lme4 migration guide. Review singular-fit and convergence warnings before interpreting uncertainty; see numerical limitations.