multilevel.icc: Intraclass Correlation Coefficient, ICC(1) and ICC(2)

Description

This function computes the intraclass correlation coefficient ICC(1), i.e., proportion of the total variance explained by the grouping structure, and ICC(2), i.e., reliability of aggregated variables in a two-level and three-level model.

Usage

multilevel.icc(..., data = NULL, cluster, type = c("1a", "1b", "2"),
               method = c("aov", "lme4", "nlme"), REML = TRUE,
               as.na = NULL, check = TRUE)

Value

Returns a numeric vector or matrix with intraclass correlation coefficient(s). In a three level model, the label L2 is used for ICCs at Level 2 and L3 for ICCs at Level 3.

Arguments

...: a numeric vector, matrix, or data frame. Alternatively, an expression indicating the variable names in data. Note that the operators ., +, -, ~, :, ::, and ! can also be used to select variables, see 'Details' in the df.subset function.
data: a data frame when specifying one or more variables in the argument .... Note that the argument is NULL when specifying a numeric vector, matrix, or data frame for the argument ....
cluster: a character string indicating the name of the cluster variable in ... or data for two-level data, a character vector indicating the names of the cluster variables in ... for three-level data, or a vector or data frame representing the nested grouping structure (i.e., group or cluster variables). Alternatively, a character string or character vector indicating the variable name(s) of the cluster variable(s) in data. Note that the cluster variable at Level 3 come first in a three-level model, i.e., cluster = c("level3", "level2").
type: a character string indicating the type of intraclass correlation coefficient, i.e., type = "1a" (default) for ICC(1) and type = "2" for ICC(2) when specifying a two-level model (i.e., one cluster variable), and type = "1a" (default) for ICC(1) representing the proportion of variance at Level 2 and Level 3, type = "1b" representing an estimate of the expected correlation between two randomly chosen elements in the same group, and type = "2" for ICC(2) when specifying a three-level model (i.e., two cluster variables). See 'Details' for the formula used in this function.
method: a character string indicating the method used to estimate intraclass correlation coefficients, i.e., method = "aov" ICC estimated using the aov function, method = "lme4" (default) ICC estimated using the lmer function in the lme4 package, method = "nlme" ICC estimated using the lme function in the nlme package. Note that if the lme4 or nlme package is needed when estimating ICCs in a three-level model.
REML: logical: if TRUE (default), restricted maximum likelihood is used to estimate the null model when using the lmer function in the lme4 package or the lme function in the nlme package.
as.na: a numeric vector indicating user-defined missing values, i.e. these values are converted to NA before conducting the analysis. Note that as.na() function is only applied to x but not to cluster.
check: logical: if TRUE (default), argument specification is checked.

Author

Takuya Yanagida takuya.yanagida@univie.ac.at

Details

Two-Level Model

In a two-level model, the intraclass correlation coefficients are computed in the random intercept-only model:

$$Y_{ij} = \gamma_{00} + u_{0j} + r_{ij}$$

where the variance in $Y$ is decomposed into two independent components: $\sigma^2_{u_{0}}$, which represents the variance at Level 2, and $\sigma^2_{r}$, which represents the variance at Level 1 (Hox et al., 2018). These two variances sum up to the total variance and are referred to as variance components. The intraclass correlation coefficient, ICC(1) $\rho$ requested by type = "1a" represents the proportion of the total variance explained by the grouping structure and is defined by the equation

$$\rho = \frac{\sigma^2_{u_{0}}}{\sigma^2_{u_{0}} + \sigma^2_{r}}$$

The intraclass correlation coefficient, ICC(2) $\lambda_j$ requested by type = "2" represents the reliability of aggregated variables and is defined by the equation

$$\lambda_j = \frac{\sigma^2_{u_{0}}}{\sigma^2_{u_{0}} + \frac{\sigma^2_{r}}{n_j}} = \frac{n_j\rho}{1 + (n_j - 1)\rho}$$

where $n_j$ is the average group size (Snijders & Bosker, 2012).

Three-Level Model

In a three-level model, the intraclass correlation coefficients are computed in the random intercept-only model:

$$Y_{ijk} = \gamma_{000} + v_{0k} + u_{0jk} + r_{ijk}$$

where the variance in $Y$ is decomposed into three independent components: $\sigma^2_{v_{0}}$, which represents the variance at Level 3, $\sigma^2_{u_{0}}$, which represents the variance at Level 2, and $\sigma^2_{r}$, which represents the variance at Level 1 (Hox et al., 2018). There are two ways to compute the intraclass correlation coefficient in a three-level model. The first method requested by type = "1a" represents the proportion of variance at Level 2 and Level 3 and should be used if we are interested in a decomposition of the variance across levels. The intraclass correlation coefficient, ICC(1) $\rho_{L2}$ at Level 2 is defined as:

$$\rho_{L2} = \frac{\sigma^2_{u_{0}}}{\sigma^2_{v_{0}} + \sigma^2_{u_{0}} + \sigma^2_{r}}$$

The ICC(1) $\rho_{L3}$ at Level 3 is defined as:

$$\rho_{L3} = \frac{\sigma^2_{v_{0}}}{\sigma^2_{v_{0}} + \sigma^2_{u_{0}} + \sigma^2_{r}}$$

The second method requested by type = "1b" represents the expected correlation between two randomly chosen elements in the same group. The intraclass correlation coefficient, ICC(1) $\rho_{L2}$ at Level 2 is defined as:

$$\rho_{L2} = \frac{\sigma^2_{v_{0}} + \sigma^2_{u_{0}}}{\sigma^2_{v_{0}} + \sigma^2_{u_{0}} + \sigma^2_{r}}$$

The ICC(1) $\rho_L3$ at Level 3 is defined as:

$$\rho_{L3} = \frac{\sigma^2_{v_{0}}}{\sigma^2_{v_{0}} + \sigma^2_{u_{0}} + \sigma^2_{r}}$$

Note that both formula are correct, but express different aspects of the data, which happen to coincide when there are only two levels (Hox et al., 2018).

The intraclass correlation coefficients, ICC(2) requested by type = "2" represent the reliability of aggregated variables at Level 2 and Level 3. The ICC(2) $\lambda_j$ at Level 2 is defined as:

$$\lambda_j = \frac{\sigma^2_{u_{0}}}{\sigma^2_{u_{0}} + \frac{\sigma^2_{r}}{n_j}}$$

The ICC(2) $\lambda_k$ at Level 3 is defined as:

$$\lambda_k = \frac{\sigma^2_{v_{0}}}{\frac{{\sigma^2_{v_{0}} + \sigma^2_{u_{0}}}}{n_{j}} + \frac{\sigma^2_{r}}{n_k \cdot n_j}}$$

where $n_j$ is the average group size at Level 2 and $n_j$ is the average group size at Level 3 (Hox et al., 2018).

References

Hox, J., Moerbeek, M., & van de Schoot, R. (2018). Multilevel analysis: Techniques and applications (3rd. ed.). Routledge.

Snijders, T. A. B., & Bosker, R. J. (2012). Multilevel analysis: An introduction to basic and advanced multilevel modeling (2nd ed.). Sage Publishers.

Examples

Run this code

# Load data set "Demo.twolevel" in the lavaan package
data("Demo.twolevel", package = "lavaan")

#----------------------------------------------------------------------------
# Two-Level Models

#..........
# Cluster variable specification

# Example 1a: Cluster variable 'cluster' in '...'
multilevel.icc(Demo.twolevel[, c("y1", "cluster")], cluster = "cluster")

# Example 1b: Cluster variable 'cluster' not in '...'
multilevel.icc(Demo.twolevel$y1, cluster = Demo.twolevel$cluster)

# Example 1c: Alternative specification using the 'data' argument
multilevel.icc(y1, data = Demo.twolevel, cluster = "cluster")

#..........

# Example 2: ICC(1) for 'y1'
multilevel.icc(Demo.twolevel$y1, cluster = Demo.twolevel$cluster)

# Example 3: ICC(2)
multilevel.icc(Demo.twolevel$y1, cluster = Demo.twolevel$cluster, type = 2)

# Example 4: ICC(1)
# use lme() function in the lme4 package to estimate ICC
multilevel.icc(Demo.twolevel$y1, cluster = Demo.twolevel$cluster, method = "nlme")

# Example 5a: ICC(1) for 'y1', 'y2', and 'y3'
multilevel.icc(Demo.twolevel[, c("y1", "y2", "y3")], cluster = Demo.twolevel$cluster)

# Example 5b: Alternative specification using the 'data' argument
multilevel.icc(y1:y3, data = Demo.twolevel, cluster = "cluster")

#----------------------------------------------------------------------------
# Three-Level Models

# Create arbitrary three-level data
Demo.threelevel <- data.frame(Demo.twolevel, cluster2 = Demo.twolevel$cluster,
                                             cluster3 = rep(1:10, each = 250))

#..........
# Cluster variable specification

# Example 6a: Cluster variables 'cluster' in '...'
multilevel.icc(Demo.threelevel[, c("y1", "cluster3", "cluster2")],
               cluster = c("cluster3", "cluster2"))

# Example 6b: Cluster variables 'cluster' not in '...'
multilevel.icc(Demo.threelevel$y1, cluster = Demo.threelevel[, c("cluster3", "cluster2")])

# Example 6c: Alternative specification using the 'data' argument
multilevel.icc(y1, data = Demo.threelevel, cluster = c("cluster3", "cluster2"))

#..........

# Example 7a: ICC(1), proportion of variance at Level 2 and Level 3
multilevel.icc(y1, data = Demo.threelevel, cluster = c("cluster3", "cluster2"))

# Example 7b: ICC(1), expected correlation between two randomly chosen elements
# in the same group
multilevel.icc(y1, data = Demo.threelevel, cluster = c("cluster3", "cluster2"),
               type = "1b")

# Example 7c: ICC(2)
multilevel.icc(y1, data = Demo.threelevel, cluster = c("cluster3", "cluster2"),
type = "2")

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