nested_fully-crossed_cross-classified_models.Rmd

---
title: "Nested and Crossed Random Effects Models"
author: "Daniel Lüdecke"
date: "26 3 2019"
output: html_document
---

```{r setup, include=FALSE,echo=FALSE}
library(knitr)
knitr::opts_chunk$set(
  echo = TRUE,
  collapse = TRUE, 
  warning = FALSE,
  comment = "#>", 
  dev = "png"
)
```

# Nested and Crossed Random Effects Models

Say we have a model with a dependent variable **DV**, independent variable **IV** and groups as random effects (**Cluster**, **Subject**). The **IV** varies across **Cluster** and **Subject**.

Is this a nested, fully crossed or cross-classified design?

## Nested design

The key distinction is whether each **Subject** receives a completely different **Cluster** set. If this is the case the design is _nested_, which simply means: _not crossed_.

```{r eval=FALSE}
lmer(DV ~ IV + (1 + IV | Cluster / Subject), data = ...)
```

which expands to...

```{r eval=FALSE}
lmer(DV ~ IV + (1 + IV | Cluster ) + (1 + IV | Cluster:Subject), data = ...)
```


## Fully-crossed or cross-classified models

If each **Subject** receives the same **Cluster** (i.e. subjects appear in all clusters), it is a _fully crossed_ random factors design. If there is some mixture it is _cross-classified_. The appropriate model notation for a crossed design would be:

```{r eval=FALSE}
lmer(DV ~ IV + (1 + IV | Cluster) + (1 + IV | Subject), data = ...)
```

## Easily check if group factors are nested or crossed

You can use the [**sjmisc**-package](https://strengejacke.github.io/sjmisc/) to check whether group factors are (fully) crossed, nested or cross-classified.

`is_cross_classified()` returns `TRUE`, so a cross-classified design would be appropriate for this random effects structure.

```{r}
# data with cross-classified distribution of "cluster" and "subject"
data <- data.frame(
  cluster = rep(1:5, each = 3),
  subject = c(1,2,3, 1, 2, 4, 1, 2, 3, 1, 2, 5, 1, 2, 4)
)

# the table output indicates that data is not nested, but also not fully crossed
table(data)

# check nesting / crossing of group factors
library(sjmisc)
is_nested(data$cluster, data$subject)
is_crossed(data$cluster, data$subject)
is_cross_classified(data$cluster, data$subject)
```

# References

* Related post: https://www.researchgate.net/post/Multilevel_modelling_in_R
* See also: https://stats.stackexchange.com/a/228814/54740