"m-l-1" approximation for SEs, CIs and p-values

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

Usage

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

dof_ml1(model)

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

Arguments

model: A mixed model.
ci: Confidence Interval (CI) level. Default to 0.95 (95%).
...: Additional arguments passed down to the underlying functions. E.g., arguments like vcov or vcov_args can be used to compute confidence intervals using a specific variance-covariance matrix for the standard errors.
dof: Degrees of Freedom.

Value

A data frame.

Details

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.

References

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

Examples

# \donttest{
if (require("lme4")) {
  model <- lmer(Petal.Length ~ Sepal.Length + (1 | Species), data = iris)
  p_value_ml1(model)
}
#>      Parameter          p
#> 1  (Intercept) 0.96504927
#> 2 Sepal.Length 0.04534945
# }