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Approximation of degrees of freedom based on a "m-l-1" heuristic as suggested by Elff et al. (2019).


ci_ml1(model, ci = 0.95, ...)


p_value_ml1(model, dof = NULL, ...)



A mixed model.


Confidence Interval (CI) level. Default to 0.95 (95%).


Additional arguments


Degrees of Freedom.


A data frame.


Small Sample Cluster corrected Degrees of Freedom

Inferential statistics (like p-values, confidence intervals and standard errors) may be biased in mixed models when the number of clusters is small (even if the sample size of level-1 units is high). In such cases it is recommended to approximate a more accurate number of degrees of freedom for such inferential statistics (see Li and Redden 2015). The m-l-1 heuristic is such an approach that uses a t-distribution with fewer degrees of freedom (dof_ml1()) to calculate p-values (p_value_ml1()) and confidence intervals (ci(method = "ml1")).

Degrees of Freedom for Longitudinal Designs (Repeated Measures)

In particular for repeated measure designs (longitudinal data analysis), the m-l-1 heuristic is likely to be more accurate than simply using the residual or infinite degrees of freedom, because dof_ml1() returns different degrees of freedom for within-cluster and between-cluster effects.

Limitations of the "m-l-1" Heuristic

Note that the "m-l-1" heuristic is not applicable (or at least less accurate) for complex multilevel designs, e.g. with cross-classified clusters. In such cases, more accurate approaches like the Kenward-Roger approximation (dof_kenward()) is recommended. However, the "m-l-1" heuristic also applies to generalized mixed models, while approaches like Kenward-Roger or Satterthwaite are limited to linear mixed models only.


  • Elff, M.; Heisig, J.P.; Schaeffer, M.; Shikano, S. (2019). Multilevel Analysis with Few Clusters: Improving Likelihood-based Methods to Provide Unbiased Estimates and Accurate Inference, British Journal of Political Science.

  • Li, P., Redden, D. T. (2015). Comparing denominator degrees of freedom approximations for the generalized linear mixed model in analyzing binary outcome in small sample cluster-randomized trials. BMC Medical Research Methodology, 15(1), 38. doi:10.1186/s12874-015-0026-x

See also

dof_ml1() is a small helper-function to calculate approximated degrees of freedom of model parameters, based on the "m-l-1" heuristic.


# \donttest{
if (require("lme4")) {
  model <- lmer(Petal.Length ~ Sepal.Length + (1 | Species), data = iris)
#>      Parameter          p
#> 1  (Intercept) 0.96504927
#> 2 Sepal.Length 0.04534945
# }