Mixed Model Basics

Upon Completing This Topic

You will be able to link fixed effects and random effects to the level of study design. You will be able to avoid arbitrarily assigning grouping variables as random effects and to interpret replication, clustering structures, and convergence warnings as part of model diagnostics.

Why Include Hierarchical Structure in the Model

Observations from the same subject, plate, or site may share common variation. Treating all replications as independent samples can underestimate uncertainty, while summarizing with simple averages may lose time or cluster information. A mixed model is one approach that separates fixed effects of interest from group-level variation.

Fixed and Random Effects

Fixed effects are the effects of conditions or covariates that the researcher has predefined for interpretation. Random effects are a structure that allows for different intercepts or slopes per group, treating the observed groups as coming from a larger set of groups. This distinction is not determined solely by the data type of the variable but depends on the study design and the target of inference.

Minimal Structure of MixedLM

import statsmodels.formula.api as smf

model = smf.mixedlm(
    "outcome ~ exposure + time",
    data=data,
    groups=data["subject_id"],
)
result = model.fit()
print(result.fe_params)

subject_id is the group key for repeated observations, and the meanings of exposure and time should be defined in the data contract. Even if fitting results are returned, review convergence warnings, random effect structures, residuals, and missingness.

Structure Verification Over Model Selection

Do not apply mixed models to all replicated data. Verify whether there are sufficient groups, what population the groups represent, whether random slopes are necessary for the research question, and the relationship between measurement time points and missingness. Convergence failure is not a matter of simply hiding warnings but a signal to investigate issues with the model, data, or scale.

Random intercepts allow for different intercepts per group, while random slopes allow the relationship with covariates or time to vary by group. More complex random effect structures are not always better. You must verify whether the data contains information to distinguish that variation and whether the model fits stably.

If convergence warnings occur, do not repeatedly call the optimizer and select only the results where the warning disappeared. Check input scales, the number of observations per group, whether variance estimates are on the boundary, and whether the model structure is overly complex. Record all attempts and warnings.

Returning to Research Questions for Interpretation

Fixed effects represent the average relationship conditional on the specified group structure, while random effects express variation between groups. Do not automatically interpret these as causal effects or biological mechanisms. Report grouping levels, time, missingness, and extrapolation together.

Common Failures

• Do not use subject ID as a covariate with numerical magnitude.
• Do not include all categorical variables as random effects.
• Do not hide convergence warnings and finalize coefficients.
• Do not use mixed models as a panacea without verifying the repeated measures structure.

Key Takeaways

• Mixed models are one way to express hierarchical and repeated structures.
• The distinction between fixed and random effects depends on study design and the target of inference.
• Convergence and diagnostics are part of the results.

Next Topic

In the next section, we will address longitudinal changes and dropout/missingness in conjunction with time structure.

References

• statsmodels API: https://www.statsmodels.org/stable/api.html
• Bioconductor OSTA experimental design: https://bioconductor.org/books/release/OSTA/pages/bkg-exp-design.html

The hierarchical schema and code in this section were written independently by BioStatPy.