# Between-within approximation for SEs, CIs and p-values

Source:`R/ci_betwithin.R`

, `R/dof_betwithin.R`

, `R/p_value_betwithin.R`

`p_value_betwithin.Rd`

Approximation of degrees of freedom based on a "between-within" heuristic.

## Usage

```
ci_betwithin(model, ci = 0.95, ...)
dof_betwithin(model)
p_value_betwithin(model, dof = NULL, ...)
```

## 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
*Between-within* denominator degrees of freedom approximation is
recommended in particular for (generalized) linear mixed models with repeated
measurements (longitudinal design). `dof_betwithin()`

implements a heuristic
based on the between-within approach. **Note** that this implementation
does not return exactly the same results as shown in *Li and Redden 2015*,
but similar.

### Degrees of Freedom for Longitudinal Designs (Repeated Measures)

In particular for repeated measure designs (longitudinal data analysis),
the *between-within* heuristic is likely to be more accurate than simply
using the residual or infinite degrees of freedom, because `dof_betwithin()`

returns different degrees of freedom for within-cluster and between-cluster
effects.

## 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