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)
dof_ml1(model)
p_value_ml1(model, dof = NULL)
se_ml1(model)
```

## Arguments

model |
A mixed model. |

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

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 statitics (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`

), standard errors (`se_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

## See also

`dof_ml1()`

and `se_ml1()`

are small helper-functions
to calculate approximated degrees of freedom and standard errors of model
parameters, based on the "m-l-1" heuristic.

## 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
# }
```