For R / lme4 users

If you use lme4::lmer() in R, interlace is the closest Python equivalent for models with crossed random intercepts. This page maps lme4 concepts and syntax to interlace, notes where the two differ, and points to the reference literature they share.

Formula syntax

lme4 encodes random effects inside the model formula using (term | group) notation. interlace separates fixed-effect formula from grouping factors for simple random intercepts, and accepts lme4-style (term | group) notation via the random parameter for random slopes:

lme4 (R)	interlace (Python)	Notes
y ~ x + (1|g)	formula="y ~ x", groups="g"	Single random intercept
y ~ x + (1|g1) + (1|g2)	formula="y ~ x", groups=["g1", "g2"]	Crossed random intercepts
y ~ x + (x|g)	formula="y ~ x", random=["(1 + x | g)"]	Correlated random intercept + slope (v0.2.1+)
y ~ x + (x||g)	formula="y ~ x", random=["(1 + x || g)"]	Independent (uncorrelated) parameterisation (v0.2.1+)
y ~ x + (1|g1/g2)	formula="y ~ x", random=["(1 | g1/g2)"]	Nested designs (v0.2.4+); expands to (1|g1) + (1|g1:g2) automatically

groups vs random: use groups= for random intercepts only (shorter syntax); use random= when you need random slopes or want to mix intercept-only and slope terms across grouping factors. See Quickstart for a side-by-side example.

Side-by-side examples

Crossed random intercepts

The following example fits the same model in both languages, using a synthetic dataset of reading times with subject and item as crossed grouping factors — the canonical setup in psycholinguistics and item response theory.

R (lme4)

library(lme4)

fm <- lmer(
  rt ~ condition + (1 | subject) + (1 | item),
  data   = df,
  REML   = TRUE
)

summary(fm)
fixef(fm)           # fixed-effect coefficients
ranef(fm)           # conditional modes (BLUPs)
VarCorr(fm)         # variance components

Python (interlace)

import interlace

result = interlace.fit(
    "rt ~ condition",
    data   = df,
    groups = ["subject", "item"],
    method = "REML",          # default
)

print(result.fe_params)           # fixed-effect coefficients
print(result.random_effects)      # BLUPs, keyed by grouping factor
print(result.variance_components) # sigma^2 per grouping factor + residual

Random slopes (v0.2.1+)

When subjects differ not just in their baseline RT but also in how much condition affects them, add a by-subject random slope:

R (lme4)

fm_slopes <- lmer(
  rt ~ condition + (1 + condition | subject) + (1 | item),
  data = df,
  REML = TRUE
)

ranef(fm_slopes)$subject  # DataFrame: columns (Intercept) + condition

Python (interlace)

result_slopes = interlace.fit(
    "rt ~ condition",
    data   = df,
    random = [
        "(1 + condition | subject)",  # correlated intercept + slope
        "(1 | item)",                 # intercept only
    ],
)

# random_effects["subject"] is now a DataFrame, one column per term
print(result_slopes.random_effects["subject"])
#             (Intercept)  condition
# subject_01       -12.3       0.42
# subject_02         8.7      -0.31
# ...

# Full random-effect covariance matrix
print(result_slopes.varcov)

For the independent (uncorrelated) parameterisation — equivalent to lme4’s || — use "(1 + condition || subject)". See the Random Slopes Guide for a full walkthrough including interpretation and when to use each parameterisation.

Comparing output

result.fe_params is a pandas.Series with the same ordering as fixef(fm). result.random_effects is a dict of {group_col: pandas.Series}, equivalent to ranef(fm). Variance components match as.data.frame(VarCorr(fm))$vcov (the variance, not the standard deviation).

Estimation

Both lme4 and interlace use profiled REML by default, with the same Lambda-theta parameterisation described in Bates et al. (2015). The sparse Cholesky factor is the core computational primitive in both implementations — lme4 uses the Eigen C++ library; interlace uses scipy.sparse.linalg.

Fixed-effect estimates agree to within 1e-4 (absolute) and variance components to within 5% (relative) on standard benchmarks.

Diagnostics

lme4 delegates post-fit diagnostics to companion packages. interlace bundles an equivalent suite directly, inspired by the R package HLMdiag (Loy & Hofmann, 2014):

HLMdiag / lme4 (R)	interlace (Python)
hlm_resid(fm)	interlace.hlm_resid(result)
leverage(fm)	interlace.leverage(result)
hlm_influence(fm)	interlace.hlm_influence(result)
cooks.distance(fm)	interlace.cooks_distance(result)
mdffits(fm)	interlace.mdffits(result)
dotplot_diag(...)	interlace.dotplot_diag(result, ...)

See the Diagnostics notebook for full diagnostic workflows, and Influence diagnostics for the function reference.

References

Primary citation for lme4:

Douglas Bates, Martin Mächler, Ben Bolker, Steve Walker (2015). Fitting Linear Mixed-Effects Models Using lme4. Journal of Statistical Software, 67(1), 1–48. doi:10.18637/jss.v067.i01

Crossed random effects in item-response models:

Harold Doran, Douglas Bates, Paul Bliese, Maritza Dowling (2007). Estimating the Multilevel Rasch Model: With the lme4 Package. Journal of Statistical Software, 20(2), 1–18. doi:10.18637/jss.v020.i02

HLMdiag diagnostics:

Adam Loy, Heike Hofmann (2014). HLMdiag: A Suite of Diagnostics for Hierarchical Linear Models in R. Journal of Statistical Software, 56(5), 1–28. doi:10.18637/jss.v056.i05