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R/multilevel.r2.R
In misty: Miscellaneous Functions 'T. Yanagida'
Defines functions
multilevel.r2
Documented in
multilevel.r2
#' R-Squared Measures for Multilevel and Linear Mixed Effects Models
#'
#' This function computes R-squared measures by Raudenbush and Bryk (2002),
#' Snijders and Bosker (1994), Nakagawa and Schielzeth (2013) as extended by
#' Johnson (2014), and Rights and Sterba (2019) for multilevel and linear mixed
#' effects models estimated by using the \code{lmer()} function in the package
#' \pkg{lme4} or \code{lme()} function in the package \pkg{nlme}.
#'
#' @param model a fitted model of class \code{"lmerMod"} from the \pkg{lme4}
#' package or \code{"lme"} from the \pkg{nlme} package.
#' @param print a character vector indicating which R-squared measures to be
#' printed on the console, i.e., \code{RB} for measures from
#' Raudenbush and Bryk (2002), \code{SB} for measures from Snijders
#' and Bosker (1994), \code{NS} for measures from Nakagawa and
#' Schielzeth (2013) as extended by Johnson (2014), and \code{RS}
#' for measures from Rights and Sterba (2019). The default setting
#' is \code{print = "RS"}.
#' @param digits an integer value indicating the number of decimal places to be used.
#' @param plot logical: if \code{TRUE}, bar chart showing the decomposition of
#' scaled total, within-cluster, and between-cluster outcome variance
#' into five (total), three (within-cluster), and two (between-cluster)
#' proportions is drawn. Note that the \pkg{ggplot2} package is required
#' to draw the bar chart.
#' @param gray logical: if \code{TRUE}, graphical parameter to draw the bar chart
#' in gray scale.
#' @param start a numeric value between 0 and 1, graphical parameter to specify
#' the gray value at the low end of the palette.
#' @param end a numeric value between 0 and 1, graphical parameter to specify
#' the gray value at the high end of the palette.
#' @param color a character vector, graphical parameter indicating the color of
#' bars in the bar chart in the following order: Fixed slopes (Within),
#' Fixed slopes (Between), Slope variation (Within), Intercept variation
#' (Between), and Residual (Within). By default, colors from the
#' colorblind-friendly palettes are used
#' @param check logical: if \code{TRUE}, argument specification is checked.
#' @param output logical: if \code{TRUE}, output is shown on the console.
#'
#' @details
#' A number of R-squared measures for multilevel and linear mixed effects models have
#' been developed in the methodological literature (see Rights & Sterba, 2018).
#' Based on these measures, following measures were implemented in the current function:
#' \describe{
#' \item{\strong{Raudenbush and Bryk (2002)}}{R-squared measures by Raudenbush
#' and Bryk (2002) are based on the proportional reduction of unexplained variance
#' when predictors are added. More specifically, variance estimates from the
#' baseline/null model (i.e., \eqn{\sigma^2_{e|b}} and \eqn{\sigma^2_{u0|b}})
#' and variance estimates from the model including predictors (i.e., \eqn{\sigma^2_{e|m}}
#' and \eqn{\sigma^2_{u0|m}}) are used to compute the proportional reduction in
#' variance between baseline/null model and the complete model by:
#'
#' \deqn{R^2_1(RB) = \frac{\sigma^2_{e|b} - \sigma^2_{e|m}}{\sigma^2_{e|b}}}
#'
#' for the proportional reduction at level-1 (within-cluster) and by:
#'
#' \deqn{R^2_2(RB) = \frac{\sigma^2_{u0|b} - \sigma^2_{u0|m}}{\sigma^2_{u0|b}}}
#'
#' for the proportional reduction at level-2 (between-cluster), where \eqn{|b}
#' and \eqn{|m} represent the baseline and full models, respectively (Hox et al.,
#' 2018; Roberts et al., 2010).
#'
#' A major disadvantage of these measures is that adding predictors can increases
#' rather than decreases some of the variance components and it is even possible
#' to obtain negative values for \eqn{R^2} with these formulas (Snijders & Bosker,
#' 2012). According to Snijders and Bosker (1994) this can occur because the
#' between-group variance is a function of both level-1 and level-2 variance:
#'
#' \deqn{var(\bar{Y}_j) = \sigma^2_{u0} + \frac{\sigma^2_e}{n_j}}
#'
#' Hence, adding a predictor (e.g., cluster-mean centered predictor) that explains
#' proportion of the within-group variance will decrease the estimate of \eqn{\sigma^2_e}
#' and increase the estimate \eqn{\sigma^2_{u0}} if this predictor does not explain
#' a proportion of the between-group variance to balance out the decrease in
#' \eqn{\sigma^2_e} (LaHuis et al., 2014). Negative estimates for \eqn{R^2} can
#' also simply occur due to chance fluctuation in sample estimates from the two
#' models.
#'
#' Another disadvantage of these measures is that \eqn{R^2_2(RB)} for the explained
#' variance at level-2 has been shown to perform poorly in simulation studies even
#' with \eqn{j = 200} clusters with group cluster size of \eqn{n_j = 50} (LaHuis
#' et al., 2014; Rights & Sterba, 2019).
#'
#' Moreover, when there is missing data in the level-1 predictors, it is possible
#' that sample sizes for the baseline and complete models differ.
#'
#' Finally, it should be noted that R-squared measures by Raudenbush and Bryk (2002)
#' are appropriate for random intercept models, but not for random intercept and
#' slope models. For random slope models, Snijders and Bosker (2012) suggested to
#' re-estimate the model as random intercept models with the same predictors while
#' omitting the random slopes to compute the R-squared measures. However, the
#' simulation study by LaHuis (2014) suggested that the R-squared measures showed
#' an acceptable performance when there was little slope variance, but did not
#' perform well in the presence of higher levels of slope variance.}
#'
#' \item{\strong{Snijders and Bosker (1994)}}{R-squared measures by Snijders and
#' Bosker (1994) are based on the proportional reduction of mean squared prediction
#' error and is computed using the formula:
#'
#' \deqn{R^2_1(SB) = \frac{\hat{\sigma}^2_{e|m} + \hat{\sigma}^2_{u0|m}}{\hat{\sigma}^2_{e|b} + \hat{\sigma}^2_{u0|b}}}
#'
#' for computing the proportional reduction of error at level-1 representing
#' the total amount of explained variance and using the formula:
#'
#' \deqn{R^2_2(SB) = \frac{\hat{\sigma}^2_{e|m} / n_j + \hat{\sigma}^2_{u0|m}}{\hat{\sigma}^2_{e|b} / n_j + \hat{\sigma}^2_{u0|b}}}
#'
#' for computing the proportional reduction of error at level-2 by dividing the
#' \eqn{\hat{\sigma}^2_e} by the group cluster size \eqn{n_j} or by the average
#' cluster size for unbalanced data (Roberts et al., 2010). Note that the function
#' uses the harmonic mean of the group sizes as recommended by Snijders and Bosker
#' (1994). The population values of \eqn{R^2} based on these measures cannot be
#' negative because the interplay of level-1 and level-2 variance components is
#' considered. However, sample estimates of \eqn{R^2} can be negative either due
#' to chance fluctuation when sample sizes are small or due to model misspecification
#' (Snijders and Bosker, 2012).
#'
#' When there is missing data in the level-1 predictors, it is possible that sample
#' sizes for the baseline and complete models differ.
#'
#' Similar to the R-squared measures by Raudenbush and Bryk (2002), the measures
#' by Snijders and Bosker (1994) are appropriate for random intercept models, but
#' not for random intercept and slope models. Accordingly, for random slope models,
#' Snijders and Bosker (2012) suggested to re-estimate the model as random intercept
#' models with the same predictors while omitting the random slopes to compute the
#' R-squared measures. The simulation study by LaHuis et al. (2014) revealed that
#' the R-squared measures showed an acceptable performance, but it should be noted
#' that \eqn{R^2_2(SB)} the explained variance at level-2 was not investigated in
#' their study.}
#'
#' \item{\strong{Nakagawa and Schielzeth (2013)}}{R-squared measures by Nakagawa
#' and Schielzeth (2013) are based on partitioning model-implied variance from a
#' single fitted model and uses the variance of predicted values of \eqn{var(\hat{Y}_{ij})}
#' to form both the outcome variance in the denominator and the explained variance
#' in the numerator of the formulas:
#'
#' \deqn{R^2_m(NS) = \frac{var(\hat{Y}_{ij})}{var(\hat{Y}_{ij}) + \sigma^2_{u0} + \sigma^2_{e}}}
#'
#' for marginal total \eqn{R^2_m(NS)} and:
#'
#' \deqn{R^2_c(NS) = \frac{var(\hat{Y}_{ij}) + \sigma^2_{u0}}{var(\hat{Y}_{ij}) + \sigma^2_{u0} + \sigma^2_{e}}}
#'
#' for conditional total \eqn{R^2_c(NS)}. In the former formula \eqn{R^2} predicted
#' scores are marginalized across random effects to indicate the variance explained
#' by fixed effects and in the latter formula \eqn{R^2} predicted scores are conditioned
#' on random effects to indicate the variance explained by fixed and random effects
#' (Rights and Sterba, 2019).
#'
#' The advantage of these measures is that they can never become negative and
#' that they can also be extended to generalized linear mixed effects models (GLMM)
#' when outcome variables are not continuous (e.g., binary outcome variables).
#' Note that currently the function does not provide \eqn{R^2} measures for GLMMs,
#' but these measures can be obtained using the \code{r.squaredGLMM()} function in
#' the \pkg{MuMIn} package.
#'
#' A disadvantage is that these measures do not allow random slopes and are restricted
#' to the simplest random effect structure (i.e., random intercept model). In other
#' words, these measures do not fully reflect the structure of the fitted model when
#' using random intercept and slope models. However, Johnson (2014) extended these
#' measures to allow random slope by taking into account the contribution of random
#' slopes, intercept-slope covariances, and the covariance matrix of random slope
#' to the variance in \eqn{Y_{ij}}. As a result, R-squared measures by Nakagawa
#' and Schielzeth (2013) as extended by Johnson (2014) can be used for both random
#' intercept, and random intercept and slope models.
#'
#' The major criticism of the R-squared measures by Nakagawa and Schielzeth (2013)
#' as extended by Johnson (2014) is that these measures do not decompose outcome
#' variance into each of total, within-cluster, and between-cluster variance which
#' precludes from computing level-specific \eqn{R^2} measures. In addition, these
#' measures do not distinguish variance attributable to level-1 versus level-2
#' predictors via fixed effects, and they also do not distinguish between random
#' intercept and random slope variation (Rights and Sterba, 2019).}
#'
#' \item{\strong{Rights and Sterba (2019)}}{R-squared measures by Rights and Sterba
#' (2019) provide an integrative framework of R-squared measures for multilevel
#' and linear mixed effects models with random intercepts and/or slopes. Their
#' measures are also based on partitioning model implied variance from a single
#' fitted model, but they provide a full partitioning of the total outcome variance
#' to one of five specific sources:
#'
#' \itemize{
#'
#' \item variance attributable to level-1 predictors via fixed slopes (shorthand:
#' variance attributable to \code{f1})
#' \item variance attributable to level-2 predictors via fixed slopes (shorthand:
#' variance attributable to \code{f2})
#' \item variance attributable to level-1 predictors via random slope variation/
#' covariation (shorthand: variance attributable to \code{v})
#' \item variance attributable to cluster-specific outcome means via random
#' intercept variation (shorthand: variance attributable to \code{m})
#' \item variance attributable to level-1 residuals
#' }
#' \eqn{R^2} measures are based on the outcome variance of interest (total,
#' within-cluster, or between-cluster) in the denominator, and the source contributing
#' to explained variance in the numerator:
#' \describe{
#'
#' \item{\strong{Total \eqn{R^2} measures}}{incorporate both within-cluster
#' and between cluster variance in the denominator and quantify variance
#' explained in an omnibus sense:
#' \itemize{
#' \item{\eqn{R^{2(f_1)}_t}}: Proportion of total outcome variance explained
#' by level-1 predictors via fixed slopes.
#' \item{\eqn{R^{2(f_2)}_t}}: Proportion of total outcome variance explained
#' by level-2 predictors via fixed slopes.
#' \item{\eqn{R^{2(f)}_t}}: Proportion of total outcome variance explained
#' by all predictors via fixed slopes.
#' \item{\eqn{R^{2(v)}_t}}: Proportion of total outcome variance explained
#' by level-1 predictors via random slope variation/covariation.
#' \item{\eqn{R^{2(m)}_t}}: Proportion of total outcome variance explained
#' by cluster-specific outcome means via random intercept variation.
#' \item{\eqn{R^{2(fv)}_t}}: Proportion of total outcome variance explained
#' by predictors via fixed slopes and random slope variation/covariation.
#' \item{\eqn{R^{2(fvm)}_t}}: Proportion of total outcome variance explained
#' by predictors via fixed slopes and random slope variation/covariation
#' and by cluster-specific outcome means via random intercept variation.
#' }
#' }
#' \item{\strong{Within-Cluster \eqn{R^2} measures}}{incorporate only within-cluster
#' variance in the denominator and indicate
#' the degree to which within-cluster variance can be explained by a given model:
#' \itemize{
#' \item{\eqn{R^{2(f_1)}_w}}: Proportion of within-cluster outcome variance
#' explained by level-1 predictors via fixed slopes.
#' \item{\eqn{R^{2(v)}_w}}: Proportion of within-cluster outcome variance
#' explained by level-1 predictors via random slope variation/covariation.
#' \item{\eqn{R^{2(f_1v)}_w}}: Proportion of within-cluster outcome variance
#' explained by level-1 predictors via fixed slopes and random slope
#' variation/covariation.
#' }
#' }
#' \item{\strong{Between-Cluster \eqn{R^2} measures}}{incorporate only between-cluster
#' variance in the denominator and indicate the degree to which between-cluster
#' variance can be explained by a given model:
#' \itemize{
#' \item{\eqn{R^{2(f_2)}_b}}: Proportion of between-cluster outcome variance
#' explained by level-2 predictors via fixed slopes.
#' \item{\eqn{R^{2(m)}_b}}: Proportion of between-cluster outcome variance
#' explained by cluster-specific outcome means via random intercept variation.
#' }
#' }
#' }
#' The decomposition of the total outcome variance can be visualized in a bar
#' chart by specifying \code{plot = TRUE}. The first column of the bar chart
#' decomposes scaled total variance into five distinct proportions (i.e.,
#' \eqn{R^{2(f_1)}_t}, \eqn{R^{2(f_2)}_t}, \eqn{R^{2(f)}_t}, \eqn{R^{2(v)}_t},
#' \eqn{R^{2(m)}_t}, \eqn{R^{2(fv)}_t}, and \eqn{R^{2(fvm)}_t}), the second
#' column decomposes scaled within-cluster variance into three distinct proportions
#' (i.e., \eqn{R^{2(f_1)}_w}, \eqn{R^{2(v)}_w}, and \eqn{R^{2(f_1v)}_w}), and
#' the third column decomposes scaled between-cluster variance into two distinct
#' proportions (i.e., \eqn{R^{2(f_2)}_b}, \eqn{R^{2(m)}_b}).
#'
#' Note that the function assumes that all level-1 predictors are centered within
#' cluster (i.e., group-mean or cluster-mean centering) as has been widely recommended
#' (e.g., Enders & Tofighi, D., 2007; Rights et al., 2019). In fact, it does not
#' matter whether a lower-level predictor is merely a control variable, or is
#' quantitative or categorical (Yaremych et al., 2021), cluster-mean centering
#' should always be used for lower-level predictors to obtain an orthogonal
#' between-within partitioning of a lower-level predictor's variance that directly
#' parallels what happens to a level-1 outcome (Hoffman & Walters, 2022). In the
#' absence of cluster-mean-centering, however, the function provides total \eqn{R^2}
#' measures, but does not provide any within-cluster or between-cluster \eqn{R^2}
#' measures.}
#' }
#' By default, the function only computes R-squared measures by Rights and Sterba
#' (2019) because the other R-squared measures reflect the same population quantity
#' provided by Rights and Sterba (2019). That is, R-squared measures \eqn{R^2_1(RB)}
#' and \eqn{R^2_2(RB)} by Raudenbush and Bryk (2002) are equivalent to \eqn{R^{2(f_1v)}_w}
#' and \eqn{R^{2(f_2)}_b}, R-squared measures \eqn{R^2_1(SB)} and \eqn{R^2_2(SB)}
#' are equivalent to \eqn{R^{2(f)}_t} and \eqn{R^{2(f_2)}_b}, and R-squared measures
#' \eqn{R^2_m(NS)} and \eqn{R^2_c(NS)} by Nakagawa and Schielzeth (2013) as extended
#' by Johnson (2014) are equivalent to \eqn{R^{2(f)}_t} and \eqn{R^{2(fvm)}_t}
#' (see Rights and Sterba, Table 3).
#'
#' Note that none of these measures provide an \eqn{R^2} for the random slope
#' variance explained by cross-level interactions, a quantity that is frequently
#' of interest (Hoffman & Walters, 2022).
#'
#' @author
#' Simon Grund, Alexander Robitzsch, Oliver Luedtk, Mairead Shaw, Jason D. Rights,
#' Sonya K. Sterba, Jessica K. Flake, and Takuya Yanagida
#'
#' @seealso
#' \code{\link{multilevel.r2.manual}}, \code{\link{multilevel.cor}},
#' \code{\link{multilevel.descript}}, \code{\link{multilevel.icc}},
#' \code{\link{multilevel.indirect}}
#'
#' @references
#' Enders, C. K., & Tofighi, D. (2007). Centering predictor variables in
#' cross-sectional multilevel models: A new look at an old issue.
#' \emph{Psychological Methods, 12}, 121-138. https://doi.org/10.1037/1082-989X.12.2.121
#'
#' Hoffmann, L., & Walter, W. R. (2022). Catching up on multilevel modeling.
#' \emph{Annual Review of Psychology, 73}, 629-658. https://doi.org/10.1146/annurev-psych-020821-103525
#'
#' Hox, J., Moerbeek, M., & van de Schoot, R. (2018). \emph{Multilevel Analysis:
#' Techniques and Applications} (3rd ed.) Routledge.
#'
#' Johnson, P. C. D. (2014). Extension of Nakagawa & Schielzeth’s R2 GLMM to random
#' slopes models. \emph{Methods in Ecology and Evolution, 5}(9), 944-946.
#' https://doi.org/10.1111/2041-210X.12225
#'
#' LaHuis, D. M., Hartman, M. J., Hakoyama, S., & Clark, P. C. (2014). Explained
#' variance measures for multilevel models. \emph{Organizational Research Methods, 17},
#' 433-451. https://doi.org/10.1177/1094428114541701
#'
#' Nakagawa, S., & Schielzeth, H. (2013). A general and simple method for obtaining
#' R2 from generalized linear mixed-effects models. \emph{Methods in Ecology and Evolution, 4}(2),
#' 133-142. https://doi.org/10.1111/j.2041-210x.2012.00261.x
#'
#' Raudenbush, S. W., & Bryk, A. S., (2002). \emph{Hierarchical linear models: Applications
#' and data analysis methods}. Sage.
#'
#' Rights, J. D., Preacher, K. J., & Cole, D. A. (2020). The danger of conflating
#' level‐specific effects of control variables when primary interest lies in level‐2
#' effects. \emph{British Journal of Mathematical and Statistical Psychology, 73}(Suppl 1),
#' 194-211. https://doi.org/10.1111/bmsp.12194
#'
#' Rights, J. D., & Cole, D. A. (2018). Effect size measures for multilevel models
#' in clinical child and adolescent research: New r-squared methods and recommendations.
#' \emph{Journal of Clinical Child and Adolescent Psychology, 47}, 863-873.
#' https://doi.org/10.1080/15374416.2018.1528550
#'
#' Rights, J. D., & Sterba, S. K. (2019). Quantifying explained variance in multilevel
#' models: An integrative framework for defining R-squared measures. \emph{Psychological Methods, 24},
#' 309-338. https://doi.org/10.1037/met0000184
#'
#' Roberts, K. J., Monaco, J. P., Stovall, H., & Foster, V. (2011). Explained variance
#' in multilevel models (pp. 219-230). In J. J. Hox & J. K. Roberts (Eds.), \emph{Handbook
#' of Advanced Multilevel Analysis}. Routledge.
#'
#' Snijders, T. A. B., & Bosker, R. (1994). Modeled variance in two-level models.
#' \emph{Sociological methods and research, 22}, 342-363. https://doi.org/10.1177/0049124194022003004
#'
#' Snijders, T. A. B., & Bosker, R. J. (2012). \emph{Multilevel analysis: An introduction
#' to basic and advanced multilevel modeling} (2nd ed.). Sage.
#'
#' Yaremych, H. E., Preacher, K. J., & Hedeker, D. (2021). Centering categorical
#' predictors in multilevel models: Best practices and interpretation. \emph{Psychological
#' Methods}. Advance online publication. https://doi.org/10.1037/met0000434
#'
#' @return
#' Returns an object of class \code{misty.object}, which is a list with following
#' entries:
#' \tabular{ll}{
#' \code{call} \tab function call \cr
#' \code{type} \tab type of analysis \cr
#' \code{model} \tab model specified in \code{model} \cr
#' \code{plot} \tab ggplot2 object for plotting the results \cr
#' \code{args} \tab specification of function arguments \cr
#' \code{result} \tab list with result tables \cr
#' }
#'
#' @note
#' This function is based on the \code{multilevelR2()} function from the \pkg{mitml}
#' package by Simon Grund, Alexander Robitzsch and Oliver Luedtke (2021), and a
#' copy of the function \code{r2mlm} in the \pkg{r2mlm} package by Mairead Shaw,
#' Jason Rights, Sonya Sterba, and Jessica Flake.
#'
#' @export
#'
#' @examples
#' \dontrun{
#' # Load misty, lme4, nlme, and ggplot2 package
#' library(misty)
#' library(lme4)
#' library(nlme)
#' library(ggplot2)
#'
#' # Load data set "Demo.twolevel" in the lavaan package
#' data("Demo.twolevel", package = "lavaan")
#'
#' #---------------------------
#'
#' # Cluster mean centering, center() from the misty package
#' Demo.twolevel$x2.c <- center(Demo.twolevel$x2, type = "CWC",
#' cluster = Demo.twolevel$cluster)
#'
#' # Compute group means, cluster.scores() from the misty package
#' Demo.twolevel$x2.b <- cluster.scores(Demo.twolevel$x2,
#' cluster = Demo.twolevel$cluster)
#'
#' # Estimate multilevel model using the lme4 package
#' mod1a <- lmer(y1 ~ x2.c + x2.b + w1 + (1 + x2.c | cluster), data = Demo.twolevel,
#' REML = FALSE, control = lmerControl(optimizer = "bobyqa"))
#'
#' #---------------------------
#'
#' # R-squared measures according to Rights and Sterba (2019)
#' multilevel.r2(mod1a)
#'
#' # Estimate multilevel model using the nlme package
#' mod1b <- lme(y1 ~ x2.c + x2.b + w1, random = ~ 1 + x2.c | cluster, data = Demo.twolevel,
#' method = "ML")
#'
#' # R-squared measures according to Rights and Sterba (2019)
#' multilevel.r2(mod1b)
#'
#' #-----------------------------------------
#'
#' # Bar chart showing the decomposition of scaled total, within-cluster,
#' # and between-cluster outcome variance
#' multilevel.r2(mod1a, plot = TRUE)
#'
#' # Bar chart in gray scale
#' multilevel.r2(mod1a, plot = TRUE, gray = TRUE)
#'
#' # Save bar chart, ggsave() from the ggplot2 package
#' ggsave("Proportion_of_Variance.png", dpi = 600, width = 5.5, height = 5.5)
#'
#' #-----------------------------------------
#'
#' # Estimate multilevel model without random slopes
#' # Note. R-squared measures by Raudenbush and Bryk (2002), and Snijders and
#' # Bosker (2012) should be computed based on the random intercept model
#' mod2 <- lmer(y1 ~ x2.c + x2.b + w1 + (1 | cluster), data = Demo.twolevel,
#' REML = FALSE, control = lmerControl(optimizer = "bobyqa"))
#'
#' # Print all available R-squared measures
#' multilevel.r2(mod2, print = "all")
#'
#' #-----------------------------------------
#'
#' # Draw bar chart manually
#' mod1a.r2 <- multilevel.r2(mod1a, output = FALSE)
#'
#' # Prepare data frame for ggplot()
#' df <- data.frame(var = factor(rep(c("Total", "Within", "Between"), each = 5),
#' level = c("Total", "Within", "Between")),
#' part = factor(c("Fixed Slopes (Within)", "Fixed Slopes (Between)",
#' "Slope Variation (Within)", "Intercept Variation (Between)",
#' "Residual (Within)"),
#' level = c("Residual (Within)", "Intercept Variation (Between)",
#' "Slope Variation (Within)", "Fixed Slopes (Between)",
#' "Fixed Slopes (Within)")),
#' y = as.vector(mod1a.r2$result$rs$decomp))
#'
#' # Draw bar chart in line with the default setting of multilevel.r2()
#' ggplot(df, aes(x = var, y = y, fill = part)) +
#' theme_bw() +
#' geom_bar(stat = "identity") +
#' scale_fill_manual(values = c("#E69F00", "#009E73", "#CC79A7", "#0072B2", "#D55E00")) +
#' scale_y_continuous(name = "Proportion of Variance", breaks = seq(0, 1, by = 0.1)) +
#' theme(axis.title.x = element_blank(),
#' axis.ticks.x = element_blank(),
#' legend.title = element_blank(),
#' legend.position = "bottom",
#' legend.box.margin = margin(-10, 6, 6, 6)) +
#' guides(fill = guide_legend(nrow = 2, reverse = TRUE))
#' }
multilevel.r2 <- function(model, print = c("all", "RB", "SB", "NS", "RS"), digits = 3,
plot = FALSE, gray = FALSE, start = 0.15, end = 0.85,
color = c("#D55E00", "#0072B2", "#CC79A7", "#009E73", "#E69F00"),
check = TRUE, output = TRUE) {
#_____________________________________________________________________________
#
# Initial Check --------------------------------------------------------------
# Check if input 'model' is missing or null
if (isTRUE(missing(model) || is.null(model))) { stop("Please specify a fitted model of class \"lmerMod\" or \"lme\" for the argument 'model'.", call. = FALSE) }
# Check method
if (isTRUE(inherits(model, "merMod"))) {
method <- "lme4"
} else if (isTRUE(inherits(model, "lme"))) {
method <- "nlme"
} else {
stop("Calculation of multilevel R-squared measures not supported for models of class \"", class(model), "\".", call. = TRUE)
}
#_____________________________________________________________________________
#
# Input Check ----------------------------------------------------------------
# Check input 'check'
if (isTRUE(!is.logical(check))) { stop("Please specify TRUE or FALSE for the argument 'check'.", call. = FALSE) }
if (isTRUE(check)) {
# ggplot2 package
if (isTRUE(!nzchar(system.file(package = "ggplot2")))) { warning("Package \"ggplot2\" is needed for drawing a bar chart, please install the package.", call. = FALSE) }
# Check input 'print'
if (isTRUE(!all(print %in% c("all", "RB", "SB", "NS", "RS")))) { stop("Character strings in the argument 'print' do not all match with \"all\", \"RB\", \"SB\", \"NS\", or \"RS\".", call. = FALSE) }
# Check input 'digits'
if (isTRUE(digits %% 1L != 0L || digits < 0L)) { stop("Specify a positive integer value for the argument 'digits'.", call. = FALSE) }
# Check input 'plot'
if (isTRUE(!is.logical(plot))) { stop("Please specify TRUE or FALSE for the argument 'plot'", call. = FALSE) }
# Check input 'gray'
if (isTRUE(!is.logical(gray))) { stop("Please specify TRUE or FALSE for the argument 'gray'", call. = FALSE) }
# Check input 'start'
if (isTRUE(start < 0L || start > 1L)) { stop("Please specify a numeric value between 0 and 1 for the argument 'start'", call. = FALSE) }
# Check input 'end'
if (isTRUE(end < 0L || end > 1L)) { stop("Please specify a numeric value between 0 and 1 for the argument 'end'", call. = FALSE) }
# Check input 'output'
if (isTRUE(!is.logical(output))) { stop("Please specify TRUE or FALSE for the argument 'output'", call. = FALSE) }
}
#_____________________________________________________________________________
#
# Arguments ------------------------------------------------------------------
if (isTRUE(all(c("all", "RB", "SB", "NS", "RS") %in% print))) { print <- "RS" }
if (isTRUE(length(print) == 1L && print == "all")) { print <- c("RB", "SB", "NS", "RS") }
if (isTRUE(plot) & isTRUE(!"RS" %in% print)) { warning("Bar char is only available when \"RS\" is specified in the argument 'print'.", call. = FALSE) }
#_____________________________________________________________________________
#
# Main Function --------------------------------------------------------------
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## RB, SB, or NS R-squared measures ####
if (isTRUE(any(c("RB", "SB", "NS") %in% print))) {
# Check if refit is necessary
refit <- any(c("RB", "SB") %in% print)
#...................
### lme4 ####
switch(method, lme4 = {
# Model terms
trm <- terms(model)
if (isTRUE(!as.logical(attr(trm, "intercept")))) stop("Model must contain intercept.", call. = FALSE)
# Outcome variable
yvr <- as.character(attr(trm, "variables")[-1L])[attr(trm, "response")]
# Cluster variable
flist <- lme4::getME(model, "flist")
cvr <- names(flist)
if (isTRUE(length(cvr) > 1L)) stop("Calculation of R-squared only support for models with a single cluster variable.", call. = FALSE)
# Cluster size harmonic mean
n.j <- 1 / mean(1 / table(flist[1]))
# Check for random slopes
random <- length(lme4::getME(model, "theta")) > 1L
if (isTRUE(refit)) {
# Fit null model
model0 <- update(model, formula(paste0(yvr, "~ 1 + (1 |", cvr, ")")))
# Variance components under Null
vc0 <- lme4::VarCorr(model0)
s0 <- attr(vc0, "sc")^2L
t0.0 <- vc0[[cvr]][1L, 1L]
}
# Alternative model components
beta <- lme4::fixef(model)[-1L]
X <- lme4::getME(model, "X")[, -1L, drop = FALSE]
Z <- lme4::getME(model, "mmList")[[1L]][, -1L, drop = FALSE]
muZ <- colMeans(Z)
vZ <- cov(Z)
# Predicted and total variance
vc1 <- lme4::VarCorr(model)
t0.1 <- vc1[[cvr]][1L, 1L]
t10.1 <- vc1[[cvr]][1L, -1L]
t11.1 <- vc1[[cvr]][-1L, -1L, drop = FALSE]
s1 <- attr(vc1, "sc")^2L
#...................
### nlme ####
}, nlme = {
# Model terms
trm <- terms(model)
if(isTRUE(!as.logical(attr(trm, "intercept")))) stop("Model must contain intercept.", call. = FALSE)
# Outcome variable
yvr <- as.character(attr(trm, "variables")[-1L])[attr(trm, "response")]
# Cluster variable
cvr <- attr(nlme::getGroups(model), "label")
if(isTRUE(length(nlme::getGroupsFormula(model, asList = TRUE)) > 1L)) stop("Calculation of R-squared only support for models with a single cluster variable.", call. = FALSE)
# Cluster size harmonic mean
n.j <- 1L / mean(1L / table(nlme::getGroups(model)))
# Check for random slopes
random <- ncol(nlme::ranef(model)) > 1L
if (isTRUE(refit)) {
# Fit Null model
ffml0 <- formula(paste0(yvr, "~ 1"))
rfml0 <- formula(paste0("~ 1 |", cvr, ""))
if (is.null(nlme::getData(model))) stop("No data sets found in 'lme' fit. See '?testModels' for an example.", call. = FALSE)
model0 <- update(model, fixed = ffml0, random = rfml0, data = model$data)
# Variance components under Null
s0 <- model0$sigma^2L
t0.0 <- nlme::getVarCov(model0)[1L, 1L]
}
# Alternative model components
beta <- nlme::fixef(model)[-1L]
fe <- model$terms
X <- model.matrix(fe, nlme::getData(model))[, -1L, drop = FALSE]
re <- attr(model$modelStruct$reStruct[[1L]], "formula")
Z <- model.matrix(re, nlme::getData(model))[, -1L, drop = FALSE]
muZ <- colMeans(Z)
vZ <- cov(Z)
# Predicted and total variance
vc1 <- nlme::getVarCov(model)
t0.1 <- vc1[1L, 1L]
t10.1 <- vc1[1L, -1L]
t11.1 <- vc1[-1L, -1L, drop = FALSE]
s1 <- model$sigma^2L
})
#...................
### Random slope model ####
if (isTRUE(random)) {
if (isTRUE(all(c("RB", "SB") %in% print))) {
warning("R-squared measures by Raudenbush and Bryk (2002), and Snijders and Bosker (1994) should be computed based on the random intercept model.", call. = FALSE)
} else if (isTRUE("RB" %in% print)) {
warning("R-squared measure by Raudenbush and Bryk should be computed based on the random intercept model.", call. = FALSE)
} else if (isTRUE("SB" %in% print)) {
warning("R-squared measure by Snijders and Bosker (1994) should be computed based on the random intercept model.", call. = FALSE)
}
}
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Calculate R2 ####
# Raudenbush and Bryk (2002)
if (isTRUE("RB" %in% print)) {
# Within-Cluster R2
rb1 <- 1L - s1 / s0
# Between-Cluster R2
rb2 <- 1L - t0.1 / t0.0
} else {
rb1 <- rb2 <- NA
}
# Snijders and Bosker (1994)
if (isTRUE("SB" %in% print)) {
# Within-Cluster R2
sb1 <- 1L - (s1 + t0.1) / (s0 + t0.0)
# Between-Cluster R2
sb2 <- 1L - (s1 / n.j + t0.1) / (s0 / n.j + t0.0)
} else {
sb1 <- sb2 <- NA
}
# Nakagawa and Schielzeth (2013); Johnson (2014)
if (isTRUE("NS" %in% print)) {
vyhat <- var(X %*% beta)
vy <- vyhat + t0.1 + 2L*(muZ %*% t10.1) + muZ %*% t11.1 %*% muZ + sum(diag(t11.1 %*% vZ)) + s1
# Marginal R2
marg <- as.vector(vyhat / vy)
# Conditional R2
cond <- as.vector((vyhat + t0.1 + 2L*(muZ %*% t10.1) + muZ %*% t11.1 %*% muZ + sum(diag(t11.1 %*% vZ))) / vy)
} else {
marg <- cond <- NA
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Rights and Sterba (2019) R-squared measures ####
if (isTRUE("RS" %in% print)) {
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Internal functions from the r2mlm package ####
#### r2mlm() Function ####
r2mlm <- function(model) {
temp_formula <- formula(model)
grepl_array <- grepl("I(", temp_formula, fixed = TRUE)
for (bool in grepl_array) {
if (isTRUE(bool)) {
stop("Error: r2mlm does not allow for models fit using the I() function; user must thus manually include any desired transformed predictor variables such as x^2 or x^3 as separate columns in dataset.")
}
}
# call appropriate r2mlm helper function
if (isTRUE(typeof(model) == "list")) {
r2mlm_nlme(model)
} else if (isTRUE(typeof(model) == "S4")) {
r2mlm_lmer(model)
} else {
stop("You must input a model generated using either the lme4 or nlme package.")
}
}
#### r2mlm_lmer() Function ####
r2mlm_lmer <- function(model) {
# Visible global function defnition
is <- fixef <- getME <- NULL
# Step 1) check if model has_intercept
if (isTRUE(attr(terms(model), which = "intercept") == 1L)) {
has_intercept <- TRUE
} else {
has_intercept <- FALSE
}
# Step 2) Pull all variable names from the formula
all_variables <- all.vars(formula(model))
cluster_variable <- all_variables[length(all_variables)]
# Step 3a) Pull and prepare data
data <- prepare_data(model, "lme4", cluster_variable)
# Step 3b) Determine whether data is appropriate format
# a) Pull all variables except for cluster
outcome_and_predictors <- all_variables[1L:(length(all_variables) - 1L)]
# b) If any of those variables is non-numeric, then throw an error
for (variable in outcome_and_predictors) {
if (isTRUE(!is(data[[variable]], "integer") && !is(data[[variable]], "numeric"))) {
stop("Your data must be numeric. Only the cluster variable can be a factor.")
}
}
# Step 4a) Define variables you'll be sorting
if (isTRUE(length(outcome_and_predictors) == 1L)) {
predictors <- get_interaction_vars(model)
} else {
predictors <- append(outcome_and_predictors[2L:length(outcome_and_predictors)], get_interaction_vars(model))
}
# Step 4b) Create and fill vectors
l1_vars <- sort_variables(data, predictors, cluster_variable)$l1_vars
l2_vars <- sort_variables(data, predictors, cluster_variable)$l2_vars
# Step 5) Pull variable names for L1 predictors with random slopes into a variable called random_slope_vars
random_slope_vars <- get_random_slope_vars(model, has_intercept, "lme4")
# Step 6) Determine value of centeredwithincluster
if (is.null(l1_vars)) {
centeredwithincluster <- TRUE
} else {
centeredwithincluster <- get_cwc(l1_vars, cluster_variable, data)
}
# 7a) within_covs (l1 variables)
within <- get_covs(l1_vars, data)
# 7b) pull column numbers for between_covs (l2 variables)
between <- get_covs(l2_vars, data)
# 7c) pull column numbers for random_covs (l1 variables with random slopes)
random <- get_covs(random_slope_vars, data)
# 8a) gamma_w, fixed slopes for L1 variables (from l1_vars list)
gammaw <- c()
i <- 1L
for (variable in l1_vars) {
gammaw[i] <- fixef(model)[variable]
i <- i + 1L
}
# 8b) gamma_b, intercept value if hasintercept = TRUE, and fixed slopes for L2 variables (from between list)
gammab <- c()
if (isTRUE(has_intercept)) {
gammab[1L] <- fixef(model)[1L]
i <- 2L
} else {
i <- 1L
}
for (var in l2_vars) {
gammab[i] <- fixef(model)[var]
i = i + 1L
}
# Step 9) Tau matrix, results from VarCorr(model)
vcov <- VarCorr(model)
tau <- as.matrix(Matrix::bdiag(vcov))
# Step 10) Sigma^2 value, Rij
sigma2 <- getME(model, "sigma")^2L
# Step 11) Input everything into r2mlm_manual
r2mlm_manual(as.data.frame(data), within_covs = within, between_covs = between, random_covs = random,
gamma_w = gammaw, gamma_b = gammab, Tau = tau, sigma2 = sigma2, has_intercept = has_intercept, clustermeancentered = centeredwithincluster)
}
#### r2mlm_nlme() Function ####
r2mlm_nlme <- function(model) {
# Visible global function defnition
is <- getME <- NULL
# Step 1) check if model has_intercept
if (isTRUE(attr(terms(model), which = "intercept") == 1L)) {
has_intercept = TRUE
} else {
has_intercept = FALSE
}
# Step 2) Pull all variable names from the formula
all_variables <- all.vars(formula(model))
cluster_variable <- attr(nlme::getGroups(model), which = "label")
# Add the grouping var to list of all variables, and calculate formula length (for later use, to iterate)
all_variables[length(all_variables) + 1L] <- cluster_variable
formula_length <- length(all_variables) # this returns the number of elements in the all_vars list TODO remove this
# Step 3a) Pull and prepare data
data <- prepare_data(model, "nlme", cluster_variable)
# Step 3b) Determine whether data is appropriate format. Only the cluster variable can be a factor, for now
outcome_and_predictors <- all_variables[1L:length(all_variables) - 1L]
for (variable in outcome_and_predictors) {
if (isTRUE(!is(data[[variable]], "integer") && !is(data[[variable]], "numeric"))) {
stop("Your data must be numeric. Only the cluster variable can be a factor.")
}
}
# Step 4a) Define variables you'll be sorting
if (isTRUE(length(outcome_and_predictors) == 1L)) {
predictors <- get_interaction_vars(model)
} else {
predictors <- append(outcome_and_predictors[2:length(outcome_and_predictors)], get_interaction_vars(model))
}
# Step 4b) Create and fill vectors
l1_vars <- sort_variables(data, predictors, cluster_variable)$l1_vars
l2_vars <- sort_variables(data, predictors, cluster_variable)$l2_vars
# Step 5) pull variable names for L1 predictors with random slopes into a variable called random_slope_vars
random_slope_vars <- get_random_slope_vars(model, has_intercept, "nlme")
# Step 6) determine value of centeredwithincluster
if (isTRUE(is.null(l1_vars))) {
centeredwithincluster <- TRUE
} else {
centeredwithincluster <- get_cwc(l1_vars, cluster_variable, data)
}
# 7a) within_covs (l1 variables)
within <- get_covs(l1_vars, data)
# 7b) Pull column numbers for between_covs (l2 variables)
between <- get_covs(l2_vars, data)
# 7c) Pull column numbers for random_covs (l1 variables with random slopes)
random <- get_covs(random_slope_vars, data)
# 8a) gamma_w, fixed slopes for L1 variables (from l1_vars list)
gammaw <- c()
i = 1L
for (variable in l1_vars) {
gammaw[i] <- nlme::fixef(model)[variable]
i = i + 1L
}
# 8b) gamma_b, intercept value if hasintercept = TRUE, and fixed slopes for L2 variables (from between list)
gammab <- c()
if (isTRUE(has_intercept)) {
gammab[1L] <- nlme::fixef(model)[1L]
i = 2L
} else {
i = 1L
}
for (variable in l2_vars) {
gammab[i] <- nlme::fixef(model)[variable]
i = i + 1L
}
# Step 9) Tau matrix
tau <- nlme::getVarCov(model)
# Step 10) sigma^2 value, Rij
sigma2 <- model$sigma^2
# Step 11) Input everything into r2mlm_manual
r2mlm_manual(as.data.frame(data), within_covs = within, between_covs = between, random_covs = random,
gamma_w = gammaw, gamma_b = gammab, Tau = tau, sigma2 = sigma2, has_intercept = has_intercept, clustermeancentered = centeredwithincluster)
}
#### r2mlm_manual() Function ####
r2mlm_manual <- function(data, within_covs, between_covs, random_covs,
gamma_w, gamma_b, Tau, sigma2, has_intercept = TRUE, clustermeancentered = TRUE) {
if (isTRUE(has_intercept)) {
if (isTRUE(length(gamma_b) > 1L)) gamma <- c(1L, gamma_w, gamma_b[2:length(gamma_b)])
if (isTRUE(length(gamma_b) == 1L)) gamma <- c(1L, gamma_w)
if (isTRUE(is.null(within_covs))) gamma_w <- 0L
}
if (isTRUE(!has_intercept)) {
gamma <- c(gamma_w, gamma_b)
if (isTRUE(is.null(within_covs))) gamma_w <- 0L
if (isTRUE(is.null(between_covs))) gamma_b <- 0L
}
if (isTRUE(is.null(gamma))) gamma <- 0L
# Compute phi
phi <- var(cbind(1L, data[, c(within_covs)], data[, c(between_covs)]), na.rm = TRUE)
if (isTRUE(!has_intercept)) phi <- var(cbind(data[, c(within_covs)], data[, c(between_covs)]), na.rm = TRUE)
if (isTRUE(is.null(within_covs) && is.null(within_covs) && !has_intercept)) phi <- 0L
phi_w <- var(data[, within_covs], na.rm = TRUE)
if (isTRUE(is.null(within_covs))) phi_w <- 0L
phi_b <- var(cbind(1L, data[, between_covs]), na.rm = TRUE)
if (isTRUE(is.null(between_covs))) phi_b <- 0L
# Compute psi and kappa
var_randomcovs <- var(cbind(1, data[, c(random_covs)]), na.rm = TRUE)
if (isTRUE(length(Tau) > 1L)) psi <- matrix(c(diag(Tau)), ncol = 1L)
if (isTRUE(length(Tau) == 1L)) psi <- Tau
if (isTRUE(length(Tau) > 1L)) kappa <- matrix(c(Tau[lower.tri(Tau) == TRUE]), ncol = 1)
if (isTRUE(length(Tau) == 1L)) kappa <- 0L
v <- matrix(c(diag(var_randomcovs)), ncol = 1L)
r <- matrix(c(var_randomcovs[lower.tri(var_randomcovs) == TRUE]), ncol = 1L)
if (isTRUE(is.null(random_covs))) {
v <- 0L
r <- 0L
m <- matrix(1L, ncol = 1L)
}
if (isTRUE(length(random_covs) > 0L)) m <- matrix(c(colMeans(cbind(1, data[, c(random_covs)]), na.rm = TRUE)), ncol = 1L)
# Total variance
totalvar_notdecomp <- t(v)%*%psi + 2L*(t(r)%*%kappa) + t(gamma)%*%phi%*%gamma + t(m)%*%Tau%*%m + sigma2
totalwithinvar <- (t(gamma_w)%*%phi_w%*%gamma_w) + (t(v)%*%psi + 2L*(t(r)%*%kappa)) + sigma2
totalbetweenvar <- (t(gamma_b)%*%phi_b%*%gamma_b) + Tau[1L]
totalvar <- totalwithinvar + totalbetweenvar
# Total decomp
decomp_fixed_notdecomp <- (t(gamma)%*%phi%*%gamma) / totalvar_notdecomp
decomp_varslopes_notdecomp <- (t(v)%*%psi + 2L*(t(r)%*%kappa)) / totalvar_notdecomp
decomp_varmeans_notdecomp <- (t(m)%*%Tau%*%m) / totalvar_notdecomp
decomp_sigma_notdecomp <- sigma2/totalvar_notdecomp
decomp_fixed_within <- (t(gamma_w)%*%phi_w%*%gamma_w) / totalvar
decomp_fixed_between <- (t(gamma_b)%*%phi_b%*%gamma_b) / totalvar
decomp_fixed <- decomp_fixed_within + decomp_fixed_between
decomp_varslopes <- (t(v)%*%psi + 2L*(t(r)%*%kappa)) / totalvar
decomp_varmeans <- (t(m)%*%Tau%*%m) / totalvar
decomp_sigma <- sigma2 / totalvar
# Within decomp
decomp_fixed_within_w <- (t(gamma_w)%*%phi_w%*%gamma_w) / totalwithinvar
decomp_varslopes_w <- (t(v)%*%psi + 2L*(t(r)%*%kappa)) / totalwithinvar
decomp_sigma_w <- sigma2 / totalwithinvar
# Between decomp
decomp_fixed_between_b <- (t(gamma_b)%*%phi_b%*%gamma_b) / totalbetweenvar
decomp_varmeans_b <- Tau[1L] / totalbetweenvar
# New measures
if (isTRUE(clustermeancentered)) {
R2_f <- decomp_fixed
R2_f1 <- decomp_fixed_within
R2_f2 <- decomp_fixed_between
R2_fv <- decomp_fixed + decomp_varslopes
R2_fvm <- decomp_fixed + decomp_varslopes + decomp_varmeans
R2_v <- decomp_varslopes
R2_m <- decomp_varmeans
R2_f_w <- decomp_fixed_within_w
R2_f_b <- decomp_fixed_between_b
R2_fv_w <- decomp_fixed_within_w + decomp_varslopes_w
R2_v_w <- decomp_varslopes_w
R2_m_b <- decomp_varmeans_b
}
if (isTRUE(!clustermeancentered)) {
R2_f <- decomp_fixed_notdecomp
R2_fv <- decomp_fixed_notdecomp + decomp_varslopes_notdecomp
R2_fvm <- decomp_fixed_notdecomp + decomp_varslopes_notdecomp + decomp_varmeans_notdecomp
R2_v <- decomp_varslopes_notdecomp
R2_m <- decomp_varmeans_notdecomp
}
if (isTRUE(clustermeancentered)) {
decomp_table <- matrix(c(decomp_fixed_within, decomp_fixed_between, decomp_varslopes, decomp_varmeans, decomp_sigma,
decomp_fixed_within_w, "NA", decomp_varslopes_w, "NA", decomp_sigma_w,
"NA", decomp_fixed_between_b, "NA", decomp_varmeans_b, "NA"), ncol = 3L)
decomp_table <- suppressWarnings(apply(decomp_table, 2, as.numeric))
rownames(decomp_table) <- c("fixed, within", "fixed, between", "slope variation", "mean variation", "sigma2")
colnames(decomp_table) <- c("total", "within", "between")
R2_table <- matrix(c(R2_f1, R2_f2, R2_v, R2_m, R2_f, R2_fv, R2_fvm,
R2_f_w, "NA", R2_v_w, "NA", "NA", R2_fv_w, "NA",
"NA", R2_f_b, "NA", R2_m_b, "NA", "NA", "NA"), ncol = 3L)
R2_table <- suppressWarnings(apply(R2_table, 2, as.numeric)) # make values numeric, not character
rownames(R2_table) <- c("f1", "f2", "v", "m", "f", "fv", "fvm")
colnames(R2_table) <- c("total", "within", "between")
}
if (isTRUE(!clustermeancentered)) {
decomp_table <- matrix(c(decomp_fixed_notdecomp, decomp_varslopes_notdecomp, decomp_varmeans_notdecomp, decomp_sigma_notdecomp), ncol = 1L)
decomp_table <- suppressWarnings(apply(decomp_table, 2L, as.numeric))
rownames(decomp_table) <- c("fixed", "slope variation", "mean variation", "sigma2")
colnames(decomp_table) <- c("total")
R2_table <- matrix(c(R2_f, R2_v, R2_m, R2_fv, R2_fvm), ncol = 1L)
R2_table <- suppressWarnings(apply(R2_table, 2L, as.numeric))
rownames(R2_table) <- c("f", "v", "m", "fv", "fvm")
colnames(R2_table) <- c("total")
}
Output <- list(noquote(decomp_table), noquote(R2_table))
names(Output) <- c("Decompositions", "R2s")
return(Output)
}
#### prepare_data() Function ####
prepare_data <- function(model, calling_function, cluster_variable, second_model = NULL) {
# Step 1a: Pull dataframe associated with model
if (isTRUE(calling_function == "lme4")) {
data <- model@frame
} else {
data <- model[["data"]]
}
# Step 2a) pull interaction terms into list
interaction_vars <- get_interaction_vars(model)
if (isTRUE(!is.null(second_model))) {
interaction_vars_2 <- get_interaction_vars(second_model)
interaction_vars <- unique(append(interaction_vars, interaction_vars_2))
}
# Step 2b) split interaction terms into halves, multiply halves to create new columns in dataframe
data <- add_interaction_vars_to_data(data, interaction_vars)
return(data)
}
#### add_interaction_vars_to_data() Function ####
add_interaction_vars_to_data <- function(data, interaction_vars) {
for (whole in interaction_vars) {
half1 <- unlist(strsplit(whole, ":"))[1L]
half2 <- unlist(strsplit(whole, ":"))[2L]
newcol <- data[[half1]] * data[[half2]]
data <- within(data, assign(whole, newcol))
}
return(data)
}
#### get_covs() Function ####
get_covs <- function(variable_list, data) {
cov_list <- c()
i <- 1L
for (variable in variable_list) {
tmp <- match(variable, names(data))
cov_list[i] <- tmp
i <- i + 1L
}
return(cov_list)
}
#### get_random_slope_vars() Function ####
get_random_slope_vars <- function(model, has_intercept, calling_function) {
# Visible global function defnition
ranef <- NULL
if (isTRUE(calling_function == "lme4")) {
temp_cov_list <- ranef(model)[[1L]]
} else if (calling_function == "nlme") {
temp_cov_list <- nlme::ranef(model)
}
if (isTRUE(has_intercept == 1L)) {
running_count <- 2L
} else {
running_count <- 1L
}
random_slope_vars <- c()
x <- 1L
while (running_count <= length(temp_cov_list)) {
random_slope_vars[x] <- names(temp_cov_list[running_count])
x <- x + 1L
running_count <- running_count + 1L
}
return(random_slope_vars)
}
#### get_cwc() Function ####
get_cwc <- function(l1_vars, cluster_variable, data) {
for (variable in l1_vars) {
t <- tapply(data[, variable], data[, cluster_variable], sum, na.rm = TRUE)
temp_tracker <- 0L
# sum all of the sums
for (i in t) { temp_tracker <- temp_tracker + i }
if (abs(temp_tracker) < 0.0000001) {
centeredwithincluster <- TRUE
} else {
centeredwithincluster <- FALSE
break
}
}
return(centeredwithincluster)
}
#### get_interaction_vars() Function ####
get_interaction_vars <- function(model) {
interaction_vars <- c()
x <- 1L
for (term in attr(terms(model), "term.labels")) {
if (isTRUE(grepl(":", term))) {
interaction_vars[x] <- term
x <- x + 1L
}
}
return(interaction_vars)
}
#### sort_variables() Function ####
sort_variables <- function(data, predictors, cluster_variable) {
l1_vars <- c()
l2_vars <- c()
l1_counter <- 1L
l2_counter <- 1L
for (variable in predictors) {
t <- tapply(data[, variable], data[, cluster_variable], var, na.rm = TRUE)
counter <- 1L
while (counter <= length(t)) {
if (is.na(t[[counter]])) { t[[counter]] <- 0L }
counter <- counter + 1L
}
variance_tracker <- 0L
for (i in t) { variance_tracker <- variance_tracker + i }
if (isTRUE(variance_tracker == 0L)) {
l2_vars[l2_counter] <- variable
l2_counter <- l2_counter + 1L
} else {
l1_vars[l1_counter] <- variable
l1_counter <- l1_counter + 1L
}
}
return(list("l1_vars" = l1_vars, "l2_vars" = l2_vars))
}
#...................
### R-squared measures ####
r2mlm.out <- r2mlm(model)
rs <- suppressWarnings(list(decomp = matrix(apply(r2mlm.out$Decomposition, 2L, as.numeric),
ncol = ncol(r2mlm.out$Decomposition),
dimnames = list(rownames(r2mlm.out$Decompositions),
colnames(r2mlm.out$Decompositions))),
r2 = matrix(apply(r2mlm.out$R2s, 2L, as.numeric), ncol = ncol(r2mlm.out$R2s),
dimnames = list(rownames(r2mlm.out$R2s), colnames(r2mlm.out$R2s)))))
#...................
### Plot ####
part <- NULL
# Predictors are not cluster-mean-centered
if (isTRUE(ncol(rs$decomp) == 1L)) {
df <- data.frame(var = factor(rep("Total", times = 4L)),
part = factor(c("Fixed Slopes", "Slope Variation", "Intercept Variation", "Residual"),
levels = c("Residual", "Intercept Variation", "Slope Variation", "Fixed Slopes")),
y = as.vector(rs$decomp))
# Predictors are cluster-mean-centered
} else {
df <- data.frame(var = factor(rep(c("Total", "Within", "Between"), each = 5L),
levels = c("Total", "Within", "Between")),
part = factor(c("Fixed Slopes (Within)", "Fixed Slopes (Between)","Slope Variation (Within)", "Intercept Variation (Between)", "Residual (Within)"),
levels = c("Residual (Within)", "Intercept Variation (Between)", "Slope Variation (Within)", "Fixed Slopes (Between)", "Fixed Slopes (Within)")),
y = as.vector(rs$decomp))
}
p <- ggplot2::ggplot(df, ggplot2::aes(x = var, y = y, fill = part)) +
ggplot2::geom_bar(stat = "identity") +
ggplot2::scale_y_continuous(name = "Proportion of Variance",
breaks = seq(0L, 1L, by = 0.1)) +
ggplot2::theme_bw() +
ggplot2::theme(axis.title.x = ggplot2::element_blank(),
axis.ticks.x = ggplot2::element_blank(),
legend.title = ggplot2::element_blank(),
legend.position = "bottom",
legend.box.margin = ggplot2::margin(-10L, 6L, 6L, 6L)) +
ggplot2::guides(fill = ggplot2::guide_legend(nrow = 2L, reverse = TRUE))
# Gray color scales
if (isTRUE(gray)) {
p <- p + ggplot2::scale_fill_grey(start = end, end = start)
} else {
p <- p + ggplot2::scale_fill_manual(values = rev(color))
}
# Print plot
if (isTRUE(plot)) { suppressWarnings(print(p)) }
} else {
p <- NULL
rs <- list(decomp = matrix(NA, ncol = 3L, nrow = 5L,
dimnames = list(c("fixed, within", "fixed, between", "slope variation", "mean variation", "sigma2"),
c("total", "within", "between"))),
r2 = matrix(NA, ncol = 3L, nrow = 7L,
dimnames = list(c("f1", "f2", "v", "m", "f", "fv", "fvm"),
c("total", "within", "between"))))
}
#_____________________________________________________________________________
#
# Return object --------------------------------------------------------------
object <- list(call = match.call(),
type = "multilevel.r2",
model = model,
plot = p,
args = list(print = print, digits = digits, plot = plot, gray = gray,
start = start, end = end, color = color, check = check,
output = output),
result = list(rb = data.frame(rb1 = rb1, rb2 = rb2),
sb = data.frame(sb1 = sb1, sb2 = sb2),
ns = data.frame(marg = marg, cond = cond),
rs = list(decomp = rs$decomp,
total = data.frame(f1 = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "f1", "total"], NA),
f2 = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "f2", "total"], NA),
f = rs$r2[row.names(rs$r2) == "f", "total"],
v = rs$r2[row.names(rs$r2) == "v", "total"],
m = rs$r2[row.names(rs$r2) == "m", "total"],
fv = rs$r2[row.names(rs$r2) == "fv", "total"],
fvm = rs$r2[row.names(rs$r2) == "fvm", "total"]),
within = data.frame(f1 = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "f1", "within"], NA),
v = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "v", "within"], NA),
f1v = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "fv", "within"], NA)),
between = data.frame(f2 = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "f2", "between"], NA),
m = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "m", "between"], NA)))))
class(object) <- "misty.object"
#_____________________________________________________________________________
#
# Output ---------------------------------------------------------------------
if (isTRUE(output)) { print(object, check = FALSE) }
return(invisible(object))
}
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Defines functions multilevel.r2
Documented in multilevel.r2
#' R-Squared Measures for Multilevel and Linear Mixed Effects Models
#'
#' This function computes R-squared measures by Raudenbush and Bryk (2002),
#' Snijders and Bosker (1994), Nakagawa and Schielzeth (2013) as extended by
#' Johnson (2014), and Rights and Sterba (2019) for multilevel and linear mixed
#' effects models estimated by using the \code{lmer()} function in the package
#' \pkg{lme4} or \code{lme()} function in the package \pkg{nlme}.
#'
#' @param model a fitted model of class \code{"lmerMod"} from the \pkg{lme4}
#' package or \code{"lme"} from the \pkg{nlme} package.
#' @param print a character vector indicating which R-squared measures to be
#' printed on the console, i.e., \code{RB} for measures from
#' Raudenbush and Bryk (2002), \code{SB} for measures from Snijders
#' and Bosker (1994), \code{NS} for measures from Nakagawa and
#' Schielzeth (2013) as extended by Johnson (2014), and \code{RS}
#' for measures from Rights and Sterba (2019). The default setting
#' is \code{print = "RS"}.
#' @param digits an integer value indicating the number of decimal places to be used.
#' @param plot logical: if \code{TRUE}, bar chart showing the decomposition of
#' scaled total, within-cluster, and between-cluster outcome variance
#' into five (total), three (within-cluster), and two (between-cluster)
#' proportions is drawn. Note that the \pkg{ggplot2} package is required
#' to draw the bar chart.
#' @param gray logical: if \code{TRUE}, graphical parameter to draw the bar chart
#' in gray scale.
#' @param start a numeric value between 0 and 1, graphical parameter to specify
#' the gray value at the low end of the palette.
#' @param end a numeric value between 0 and 1, graphical parameter to specify
#' the gray value at the high end of the palette.
#' @param color a character vector, graphical parameter indicating the color of
#' bars in the bar chart in the following order: Fixed slopes (Within),
#' Fixed slopes (Between), Slope variation (Within), Intercept variation
#' (Between), and Residual (Within). By default, colors from the
#' colorblind-friendly palettes are used
#' @param check logical: if \code{TRUE}, argument specification is checked.
#' @param output logical: if \code{TRUE}, output is shown on the console.
#'
#' @details
#' A number of R-squared measures for multilevel and linear mixed effects models have
#' been developed in the methodological literature (see Rights & Sterba, 2018).
#' Based on these measures, following measures were implemented in the current function:
#' \describe{
#' \item{\strong{Raudenbush and Bryk (2002)}}{R-squared measures by Raudenbush
#' and Bryk (2002) are based on the proportional reduction of unexplained variance
#' when predictors are added. More specifically, variance estimates from the
#' baseline/null model (i.e., \eqn{\sigma^2_{e|b}} and \eqn{\sigma^2_{u0|b}})
#' and variance estimates from the model including predictors (i.e., \eqn{\sigma^2_{e|m}}
#' and \eqn{\sigma^2_{u0|m}}) are used to compute the proportional reduction in
#' variance between baseline/null model and the complete model by:
#'
#' \deqn{R^2_1(RB) = \frac{\sigma^2_{e|b} - \sigma^2_{e|m}}{\sigma^2_{e|b}}}
#'
#' for the proportional reduction at level-1 (within-cluster) and by:
#'
#' \deqn{R^2_2(RB) = \frac{\sigma^2_{u0|b} - \sigma^2_{u0|m}}{\sigma^2_{u0|b}}}
#'
#' for the proportional reduction at level-2 (between-cluster), where \eqn{|b}
#' and \eqn{|m} represent the baseline and full models, respectively (Hox et al.,
#' 2018; Roberts et al., 2010).
#'
#' A major disadvantage of these measures is that adding predictors can increases
#' rather than decreases some of the variance components and it is even possible
#' to obtain negative values for \eqn{R^2} with these formulas (Snijders & Bosker,
#' 2012). According to Snijders and Bosker (1994) this can occur because the
#' between-group variance is a function of both level-1 and level-2 variance:
#'
#' \deqn{var(\bar{Y}_j) = \sigma^2_{u0} + \frac{\sigma^2_e}{n_j}}
#'
#' Hence, adding a predictor (e.g., cluster-mean centered predictor) that explains
#' proportion of the within-group variance will decrease the estimate of \eqn{\sigma^2_e}
#' and increase the estimate \eqn{\sigma^2_{u0}} if this predictor does not explain
#' a proportion of the between-group variance to balance out the decrease in
#' \eqn{\sigma^2_e} (LaHuis et al., 2014). Negative estimates for \eqn{R^2} can
#' also simply occur due to chance fluctuation in sample estimates from the two
#' models.
#'
#' Another disadvantage of these measures is that \eqn{R^2_2(RB)} for the explained
#' variance at level-2 has been shown to perform poorly in simulation studies even
#' with \eqn{j = 200} clusters with group cluster size of \eqn{n_j = 50} (LaHuis
#' et al., 2014; Rights & Sterba, 2019).
#'
#' Moreover, when there is missing data in the level-1 predictors, it is possible
#' that sample sizes for the baseline and complete models differ.
#'
#' Finally, it should be noted that R-squared measures by Raudenbush and Bryk (2002)
#' are appropriate for random intercept models, but not for random intercept and
#' slope models. For random slope models, Snijders and Bosker (2012) suggested to
#' re-estimate the model as random intercept models with the same predictors while
#' omitting the random slopes to compute the R-squared measures. However, the
#' simulation study by LaHuis (2014) suggested that the R-squared measures showed
#' an acceptable performance when there was little slope variance, but did not
#' perform well in the presence of higher levels of slope variance.}
#'
#' \item{\strong{Snijders and Bosker (1994)}}{R-squared measures by Snijders and
#' Bosker (1994) are based on the proportional reduction of mean squared prediction
#' error and is computed using the formula:
#'
#' \deqn{R^2_1(SB) = \frac{\hat{\sigma}^2_{e|m} + \hat{\sigma}^2_{u0|m}}{\hat{\sigma}^2_{e|b} + \hat{\sigma}^2_{u0|b}}}
#'
#' for computing the proportional reduction of error at level-1 representing
#' the total amount of explained variance and using the formula:
#'
#' \deqn{R^2_2(SB) = \frac{\hat{\sigma}^2_{e|m} / n_j + \hat{\sigma}^2_{u0|m}}{\hat{\sigma}^2_{e|b} / n_j + \hat{\sigma}^2_{u0|b}}}
#'
#' for computing the proportional reduction of error at level-2 by dividing the
#' \eqn{\hat{\sigma}^2_e} by the group cluster size \eqn{n_j} or by the average
#' cluster size for unbalanced data (Roberts et al., 2010). Note that the function
#' uses the harmonic mean of the group sizes as recommended by Snijders and Bosker
#' (1994). The population values of \eqn{R^2} based on these measures cannot be
#' negative because the interplay of level-1 and level-2 variance components is
#' considered. However, sample estimates of \eqn{R^2} can be negative either due
#' to chance fluctuation when sample sizes are small or due to model misspecification
#' (Snijders and Bosker, 2012).
#'
#' When there is missing data in the level-1 predictors, it is possible that sample
#' sizes for the baseline and complete models differ.
#'
#' Similar to the R-squared measures by Raudenbush and Bryk (2002), the measures
#' by Snijders and Bosker (1994) are appropriate for random intercept models, but
#' not for random intercept and slope models. Accordingly, for random slope models,
#' Snijders and Bosker (2012) suggested to re-estimate the model as random intercept
#' models with the same predictors while omitting the random slopes to compute the
#' R-squared measures. The simulation study by LaHuis et al. (2014) revealed that
#' the R-squared measures showed an acceptable performance, but it should be noted
#' that \eqn{R^2_2(SB)} the explained variance at level-2 was not investigated in
#' their study.}
#'
#' \item{\strong{Nakagawa and Schielzeth (2013)}}{R-squared measures by Nakagawa
#' and Schielzeth (2013) are based on partitioning model-implied variance from a
#' single fitted model and uses the variance of predicted values of \eqn{var(\hat{Y}_{ij})}
#' to form both the outcome variance in the denominator and the explained variance
#' in the numerator of the formulas:
#'
#' \deqn{R^2_m(NS) = \frac{var(\hat{Y}_{ij})}{var(\hat{Y}_{ij}) + \sigma^2_{u0} + \sigma^2_{e}}}
#'
#' for marginal total \eqn{R^2_m(NS)} and:
#'
#' \deqn{R^2_c(NS) = \frac{var(\hat{Y}_{ij}) + \sigma^2_{u0}}{var(\hat{Y}_{ij}) + \sigma^2_{u0} + \sigma^2_{e}}}
#'
#' for conditional total \eqn{R^2_c(NS)}. In the former formula \eqn{R^2} predicted
#' scores are marginalized across random effects to indicate the variance explained
#' by fixed effects and in the latter formula \eqn{R^2} predicted scores are conditioned
#' on random effects to indicate the variance explained by fixed and random effects
#' (Rights and Sterba, 2019).
#'
#' The advantage of these measures is that they can never become negative and
#' that they can also be extended to generalized linear mixed effects models (GLMM)
#' when outcome variables are not continuous (e.g., binary outcome variables).
#' Note that currently the function does not provide \eqn{R^2} measures for GLMMs,
#' but these measures can be obtained using the \code{r.squaredGLMM()} function in
#' the \pkg{MuMIn} package.
#'
#' A disadvantage is that these measures do not allow random slopes and are restricted
#' to the simplest random effect structure (i.e., random intercept model). In other
#' words, these measures do not fully reflect the structure of the fitted model when
#' using random intercept and slope models. However, Johnson (2014) extended these
#' measures to allow random slope by taking into account the contribution of random
#' slopes, intercept-slope covariances, and the covariance matrix of random slope
#' to the variance in \eqn{Y_{ij}}. As a result, R-squared measures by Nakagawa
#' and Schielzeth (2013) as extended by Johnson (2014) can be used for both random
#' intercept, and random intercept and slope models.
#'
#' The major criticism of the R-squared measures by Nakagawa and Schielzeth (2013)
#' as extended by Johnson (2014) is that these measures do not decompose outcome
#' variance into each of total, within-cluster, and between-cluster variance which
#' precludes from computing level-specific \eqn{R^2} measures. In addition, these
#' measures do not distinguish variance attributable to level-1 versus level-2
#' predictors via fixed effects, and they also do not distinguish between random
#' intercept and random slope variation (Rights and Sterba, 2019).}
#'
#' \item{\strong{Rights and Sterba (2019)}}{R-squared measures by Rights and Sterba
#' (2019) provide an integrative framework of R-squared measures for multilevel
#' and linear mixed effects models with random intercepts and/or slopes. Their
#' measures are also based on partitioning model implied variance from a single
#' fitted model, but they provide a full partitioning of the total outcome variance
#' to one of five specific sources:
#'
#' \itemize{
#'
#' \item variance attributable to level-1 predictors via fixed slopes (shorthand:
#' variance attributable to \code{f1})
#' \item variance attributable to level-2 predictors via fixed slopes (shorthand:
#' variance attributable to \code{f2})
#' \item variance attributable to level-1 predictors via random slope variation/
#' covariation (shorthand: variance attributable to \code{v})
#' \item variance attributable to cluster-specific outcome means via random
#' intercept variation (shorthand: variance attributable to \code{m})
#' \item variance attributable to level-1 residuals
#' }
#' \eqn{R^2} measures are based on the outcome variance of interest (total,
#' within-cluster, or between-cluster) in the denominator, and the source contributing
#' to explained variance in the numerator:
#' \describe{
#'
#' \item{\strong{Total \eqn{R^2} measures}}{incorporate both within-cluster
#' and between cluster variance in the denominator and quantify variance
#' explained in an omnibus sense:
#' \itemize{
#' \item{\eqn{R^{2(f_1)}_t}}: Proportion of total outcome variance explained
#' by level-1 predictors via fixed slopes.
#' \item{\eqn{R^{2(f_2)}_t}}: Proportion of total outcome variance explained
#' by level-2 predictors via fixed slopes.
#' \item{\eqn{R^{2(f)}_t}}: Proportion of total outcome variance explained
#' by all predictors via fixed slopes.
#' \item{\eqn{R^{2(v)}_t}}: Proportion of total outcome variance explained
#' by level-1 predictors via random slope variation/covariation.
#' \item{\eqn{R^{2(m)}_t}}: Proportion of total outcome variance explained
#' by cluster-specific outcome means via random intercept variation.
#' \item{\eqn{R^{2(fv)}_t}}: Proportion of total outcome variance explained
#' by predictors via fixed slopes and random slope variation/covariation.
#' \item{\eqn{R^{2(fvm)}_t}}: Proportion of total outcome variance explained
#' by predictors via fixed slopes and random slope variation/covariation
#' and by cluster-specific outcome means via random intercept variation.
#' }
#' }
#' \item{\strong{Within-Cluster \eqn{R^2} measures}}{incorporate only within-cluster
#' variance in the denominator and indicate
#' the degree to which within-cluster variance can be explained by a given model:
#' \itemize{
#' \item{\eqn{R^{2(f_1)}_w}}: Proportion of within-cluster outcome variance
#' explained by level-1 predictors via fixed slopes.
#' \item{\eqn{R^{2(v)}_w}}: Proportion of within-cluster outcome variance
#' explained by level-1 predictors via random slope variation/covariation.
#' \item{\eqn{R^{2(f_1v)}_w}}: Proportion of within-cluster outcome variance
#' explained by level-1 predictors via fixed slopes and random slope
#' variation/covariation.
#' }
#' }
#' \item{\strong{Between-Cluster \eqn{R^2} measures}}{incorporate only between-cluster
#' variance in the denominator and indicate the degree to which between-cluster
#' variance can be explained by a given model:
#' \itemize{
#' \item{\eqn{R^{2(f_2)}_b}}: Proportion of between-cluster outcome variance
#' explained by level-2 predictors via fixed slopes.
#' \item{\eqn{R^{2(m)}_b}}: Proportion of between-cluster outcome variance
#' explained by cluster-specific outcome means via random intercept variation.
#' }
#' }
#' }
#' The decomposition of the total outcome variance can be visualized in a bar
#' chart by specifying \code{plot = TRUE}. The first column of the bar chart
#' decomposes scaled total variance into five distinct proportions (i.e.,
#' \eqn{R^{2(f_1)}_t}, \eqn{R^{2(f_2)}_t}, \eqn{R^{2(f)}_t}, \eqn{R^{2(v)}_t},
#' \eqn{R^{2(m)}_t}, \eqn{R^{2(fv)}_t}, and \eqn{R^{2(fvm)}_t}), the second
#' column decomposes scaled within-cluster variance into three distinct proportions
#' (i.e., \eqn{R^{2(f_1)}_w}, \eqn{R^{2(v)}_w}, and \eqn{R^{2(f_1v)}_w}), and
#' the third column decomposes scaled between-cluster variance into two distinct
#' proportions (i.e., \eqn{R^{2(f_2)}_b}, \eqn{R^{2(m)}_b}).
#'
#' Note that the function assumes that all level-1 predictors are centered within
#' cluster (i.e., group-mean or cluster-mean centering) as has been widely recommended
#' (e.g., Enders & Tofighi, D., 2007; Rights et al., 2019). In fact, it does not
#' matter whether a lower-level predictor is merely a control variable, or is
#' quantitative or categorical (Yaremych et al., 2021), cluster-mean centering
#' should always be used for lower-level predictors to obtain an orthogonal
#' between-within partitioning of a lower-level predictor's variance that directly
#' parallels what happens to a level-1 outcome (Hoffman & Walters, 2022). In the
#' absence of cluster-mean-centering, however, the function provides total \eqn{R^2}
#' measures, but does not provide any within-cluster or between-cluster \eqn{R^2}
#' measures.}
#' }
#' By default, the function only computes R-squared measures by Rights and Sterba
#' (2019) because the other R-squared measures reflect the same population quantity
#' provided by Rights and Sterba (2019). That is, R-squared measures \eqn{R^2_1(RB)}
#' and \eqn{R^2_2(RB)} by Raudenbush and Bryk (2002) are equivalent to \eqn{R^{2(f_1v)}_w}
#' and \eqn{R^{2(f_2)}_b}, R-squared measures \eqn{R^2_1(SB)} and \eqn{R^2_2(SB)}
#' are equivalent to \eqn{R^{2(f)}_t} and \eqn{R^{2(f_2)}_b}, and R-squared measures
#' \eqn{R^2_m(NS)} and \eqn{R^2_c(NS)} by Nakagawa and Schielzeth (2013) as extended
#' by Johnson (2014) are equivalent to \eqn{R^{2(f)}_t} and \eqn{R^{2(fvm)}_t}
#' (see Rights and Sterba, Table 3).
#'
#' Note that none of these measures provide an \eqn{R^2} for the random slope
#' variance explained by cross-level interactions, a quantity that is frequently
#' of interest (Hoffman & Walters, 2022).
#'
#' @author
#' Simon Grund, Alexander Robitzsch, Oliver Luedtk, Mairead Shaw, Jason D. Rights,
#' Sonya K. Sterba, Jessica K. Flake, and Takuya Yanagida
#'
#' @seealso
#' \code{\link{multilevel.r2.manual}}, \code{\link{multilevel.cor}},
#' \code{\link{multilevel.descript}}, \code{\link{multilevel.icc}},
#' \code{\link{multilevel.indirect}}
#'
#' @references
#' Enders, C. K., & Tofighi, D. (2007). Centering predictor variables in
#' cross-sectional multilevel models: A new look at an old issue.
#' \emph{Psychological Methods, 12}, 121-138. https://doi.org/10.1037/1082-989X.12.2.121
#'
#' Hoffmann, L., & Walter, W. R. (2022). Catching up on multilevel modeling.
#' \emph{Annual Review of Psychology, 73}, 629-658. https://doi.org/10.1146/annurev-psych-020821-103525
#'
#' Hox, J., Moerbeek, M., & van de Schoot, R. (2018). \emph{Multilevel Analysis:
#' Techniques and Applications} (3rd ed.) Routledge.
#'
#' Johnson, P. C. D. (2014). Extension of Nakagawa & Schielzeth’s R2 GLMM to random
#' slopes models. \emph{Methods in Ecology and Evolution, 5}(9), 944-946.
#' https://doi.org/10.1111/2041-210X.12225
#'
#' LaHuis, D. M., Hartman, M. J., Hakoyama, S., & Clark, P. C. (2014). Explained
#' variance measures for multilevel models. \emph{Organizational Research Methods, 17},
#' 433-451. https://doi.org/10.1177/1094428114541701
#'
#' Nakagawa, S., & Schielzeth, H. (2013). A general and simple method for obtaining
#' R2 from generalized linear mixed-effects models. \emph{Methods in Ecology and Evolution, 4}(2),
#' 133-142. https://doi.org/10.1111/j.2041-210x.2012.00261.x
#'
#' Raudenbush, S. W., & Bryk, A. S., (2002). \emph{Hierarchical linear models: Applications
#' and data analysis methods}. Sage.
#'
#' Rights, J. D., Preacher, K. J., & Cole, D. A. (2020). The danger of conflating
#' level‐specific effects of control variables when primary interest lies in level‐2
#' effects. \emph{British Journal of Mathematical and Statistical Psychology, 73}(Suppl 1),
#' 194-211. https://doi.org/10.1111/bmsp.12194
#'
#' Rights, J. D., & Cole, D. A. (2018). Effect size measures for multilevel models
#' in clinical child and adolescent research: New r-squared methods and recommendations.
#' \emph{Journal of Clinical Child and Adolescent Psychology, 47}, 863-873.
#' https://doi.org/10.1080/15374416.2018.1528550
#'
#' Rights, J. D., & Sterba, S. K. (2019). Quantifying explained variance in multilevel
#' models: An integrative framework for defining R-squared measures. \emph{Psychological Methods, 24},
#' 309-338. https://doi.org/10.1037/met0000184
#'
#' Roberts, K. J., Monaco, J. P., Stovall, H., & Foster, V. (2011). Explained variance
#' in multilevel models (pp. 219-230). In J. J. Hox & J. K. Roberts (Eds.), \emph{Handbook
#' of Advanced Multilevel Analysis}. Routledge.
#'
#' Snijders, T. A. B., & Bosker, R. (1994). Modeled variance in two-level models.
#' \emph{Sociological methods and research, 22}, 342-363. https://doi.org/10.1177/0049124194022003004
#'
#' Snijders, T. A. B., & Bosker, R. J. (2012). \emph{Multilevel analysis: An introduction
#' to basic and advanced multilevel modeling} (2nd ed.). Sage.
#'
#' Yaremych, H. E., Preacher, K. J., & Hedeker, D. (2021). Centering categorical
#' predictors in multilevel models: Best practices and interpretation. \emph{Psychological
#' Methods}. Advance online publication. https://doi.org/10.1037/met0000434
#'
#' @return
#' Returns an object of class \code{misty.object}, which is a list with following
#' entries:
#' \tabular{ll}{
#' \code{call} \tab function call \cr
#' \code{type} \tab type of analysis \cr
#' \code{model} \tab model specified in \code{model} \cr
#' \code{plot} \tab ggplot2 object for plotting the results \cr
#' \code{args} \tab specification of function arguments \cr
#' \code{result} \tab list with result tables \cr
#' }
#'
#' @note
#' This function is based on the \code{multilevelR2()} function from the \pkg{mitml}
#' package by Simon Grund, Alexander Robitzsch and Oliver Luedtke (2021), and a
#' copy of the function \code{r2mlm} in the \pkg{r2mlm} package by Mairead Shaw,
#' Jason Rights, Sonya Sterba, and Jessica Flake.
#'
#' @export
#'
#' @examples
#' \dontrun{
#' # Load misty, lme4, nlme, and ggplot2 package
#' library(misty)
#' library(lme4)
#' library(nlme)
#' library(ggplot2)
#'
#' # Load data set "Demo.twolevel" in the lavaan package
#' data("Demo.twolevel", package = "lavaan")
#'
#' #---------------------------
#'
#' # Cluster mean centering, center() from the misty package
#' Demo.twolevel$x2.c <- center(Demo.twolevel$x2, type = "CWC",
#' cluster = Demo.twolevel$cluster)
#'
#' # Compute group means, cluster.scores() from the misty package
#' Demo.twolevel$x2.b <- cluster.scores(Demo.twolevel$x2,
#' cluster = Demo.twolevel$cluster)
#'
#' # Estimate multilevel model using the lme4 package
#' mod1a <- lmer(y1 ~ x2.c + x2.b + w1 + (1 + x2.c | cluster), data = Demo.twolevel,
#' REML = FALSE, control = lmerControl(optimizer = "bobyqa"))
#'
#' #---------------------------
#'
#' # R-squared measures according to Rights and Sterba (2019)
#' multilevel.r2(mod1a)
#'
#' # Estimate multilevel model using the nlme package
#' mod1b <- lme(y1 ~ x2.c + x2.b + w1, random = ~ 1 + x2.c | cluster, data = Demo.twolevel,
#' method = "ML")
#'
#' # R-squared measures according to Rights and Sterba (2019)
#' multilevel.r2(mod1b)
#'
#' #-----------------------------------------
#'
#' # Bar chart showing the decomposition of scaled total, within-cluster,
#' # and between-cluster outcome variance
#' multilevel.r2(mod1a, plot = TRUE)
#'
#' # Bar chart in gray scale
#' multilevel.r2(mod1a, plot = TRUE, gray = TRUE)
#'
#' # Save bar chart, ggsave() from the ggplot2 package
#' ggsave("Proportion_of_Variance.png", dpi = 600, width = 5.5, height = 5.5)
#'
#' #-----------------------------------------
#'
#' # Estimate multilevel model without random slopes
#' # Note. R-squared measures by Raudenbush and Bryk (2002), and Snijders and
#' # Bosker (2012) should be computed based on the random intercept model
#' mod2 <- lmer(y1 ~ x2.c + x2.b + w1 + (1 | cluster), data = Demo.twolevel,
#' REML = FALSE, control = lmerControl(optimizer = "bobyqa"))
#'
#' # Print all available R-squared measures
#' multilevel.r2(mod2, print = "all")
#'
#' #-----------------------------------------
#'
#' # Draw bar chart manually
#' mod1a.r2 <- multilevel.r2(mod1a, output = FALSE)
#'
#' # Prepare data frame for ggplot()
#' df <- data.frame(var = factor(rep(c("Total", "Within", "Between"), each = 5),
#' level = c("Total", "Within", "Between")),
#' part = factor(c("Fixed Slopes (Within)", "Fixed Slopes (Between)",
#' "Slope Variation (Within)", "Intercept Variation (Between)",
#' "Residual (Within)"),
#' level = c("Residual (Within)", "Intercept Variation (Between)",
#' "Slope Variation (Within)", "Fixed Slopes (Between)",
#' "Fixed Slopes (Within)")),
#' y = as.vector(mod1a.r2$result$rs$decomp))
#'
#' # Draw bar chart in line with the default setting of multilevel.r2()
#' ggplot(df, aes(x = var, y = y, fill = part)) +
#' theme_bw() +
#' geom_bar(stat = "identity") +
#' scale_fill_manual(values = c("#E69F00", "#009E73", "#CC79A7", "#0072B2", "#D55E00")) +
#' scale_y_continuous(name = "Proportion of Variance", breaks = seq(0, 1, by = 0.1)) +
#' theme(axis.title.x = element_blank(),
#' axis.ticks.x = element_blank(),
#' legend.title = element_blank(),
#' legend.position = "bottom",
#' legend.box.margin = margin(-10, 6, 6, 6)) +
#' guides(fill = guide_legend(nrow = 2, reverse = TRUE))
#' }
multilevel.r2 <- function(model, print = c("all", "RB", "SB", "NS", "RS"), digits = 3,
plot = FALSE, gray = FALSE, start = 0.15, end = 0.85,
color = c("#D55E00", "#0072B2", "#CC79A7", "#009E73", "#E69F00"),
check = TRUE, output = TRUE) {
#_____________________________________________________________________________
#
# Initial Check --------------------------------------------------------------
# Check if input 'model' is missing or null
if (isTRUE(missing(model) || is.null(model))) { stop("Please specify a fitted model of class \"lmerMod\" or \"lme\" for the argument 'model'.", call. = FALSE) }
# Check method
if (isTRUE(inherits(model, "merMod"))) {
method <- "lme4"
} else if (isTRUE(inherits(model, "lme"))) {
method <- "nlme"
} else {
stop("Calculation of multilevel R-squared measures not supported for models of class \"", class(model), "\".", call. = TRUE)
}
#_____________________________________________________________________________
#
# Input Check ----------------------------------------------------------------
# Check input 'check'
if (isTRUE(!is.logical(check))) { stop("Please specify TRUE or FALSE for the argument 'check'.", call. = FALSE) }
if (isTRUE(check)) {
# ggplot2 package
if (isTRUE(!nzchar(system.file(package = "ggplot2")))) { warning("Package \"ggplot2\" is needed for drawing a bar chart, please install the package.", call. = FALSE) }
# Check input 'print'
if (isTRUE(!all(print %in% c("all", "RB", "SB", "NS", "RS")))) { stop("Character strings in the argument 'print' do not all match with \"all\", \"RB\", \"SB\", \"NS\", or \"RS\".", call. = FALSE) }
# Check input 'digits'
if (isTRUE(digits %% 1L != 0L || digits < 0L)) { stop("Specify a positive integer value for the argument 'digits'.", call. = FALSE) }
# Check input 'plot'
if (isTRUE(!is.logical(plot))) { stop("Please specify TRUE or FALSE for the argument 'plot'", call. = FALSE) }
# Check input 'gray'
if (isTRUE(!is.logical(gray))) { stop("Please specify TRUE or FALSE for the argument 'gray'", call. = FALSE) }
# Check input 'start'
if (isTRUE(start < 0L || start > 1L)) { stop("Please specify a numeric value between 0 and 1 for the argument 'start'", call. = FALSE) }
# Check input 'end'
if (isTRUE(end < 0L || end > 1L)) { stop("Please specify a numeric value between 0 and 1 for the argument 'end'", call. = FALSE) }
# Check input 'output'
if (isTRUE(!is.logical(output))) { stop("Please specify TRUE or FALSE for the argument 'output'", call. = FALSE) }
}
#_____________________________________________________________________________
#
# Arguments ------------------------------------------------------------------
if (isTRUE(all(c("all", "RB", "SB", "NS", "RS") %in% print))) { print <- "RS" }
if (isTRUE(length(print) == 1L && print == "all")) { print <- c("RB", "SB", "NS", "RS") }
if (isTRUE(plot) & isTRUE(!"RS" %in% print)) { warning("Bar char is only available when \"RS\" is specified in the argument 'print'.", call. = FALSE) }
#_____________________________________________________________________________
#
# Main Function --------------------------------------------------------------
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## RB, SB, or NS R-squared measures ####
if (isTRUE(any(c("RB", "SB", "NS") %in% print))) {
# Check if refit is necessary
refit <- any(c("RB", "SB") %in% print)
#...................
### lme4 ####
switch(method, lme4 = {
# Model terms
trm <- terms(model)
if (isTRUE(!as.logical(attr(trm, "intercept")))) stop("Model must contain intercept.", call. = FALSE)
# Outcome variable
yvr <- as.character(attr(trm, "variables")[-1L])[attr(trm, "response")]
# Cluster variable
flist <- lme4::getME(model, "flist")
cvr <- names(flist)
if (isTRUE(length(cvr) > 1L)) stop("Calculation of R-squared only support for models with a single cluster variable.", call. = FALSE)
# Cluster size harmonic mean
n.j <- 1 / mean(1 / table(flist[1]))
# Check for random slopes
random <- length(lme4::getME(model, "theta")) > 1L
if (isTRUE(refit)) {
# Fit null model
model0 <- update(model, formula(paste0(yvr, "~ 1 + (1 |", cvr, ")")))
# Variance components under Null
vc0 <- lme4::VarCorr(model0)
s0 <- attr(vc0, "sc")^2L
t0.0 <- vc0[[cvr]][1L, 1L]
}
# Alternative model components
beta <- lme4::fixef(model)[-1L]
X <- lme4::getME(model, "X")[, -1L, drop = FALSE]
Z <- lme4::getME(model, "mmList")[[1L]][, -1L, drop = FALSE]
muZ <- colMeans(Z)
vZ <- cov(Z)
# Predicted and total variance
vc1 <- lme4::VarCorr(model)
t0.1 <- vc1[[cvr]][1L, 1L]
t10.1 <- vc1[[cvr]][1L, -1L]
t11.1 <- vc1[[cvr]][-1L, -1L, drop = FALSE]
s1 <- attr(vc1, "sc")^2L
#...................
### nlme ####
}, nlme = {
# Model terms
trm <- terms(model)
if(isTRUE(!as.logical(attr(trm, "intercept")))) stop("Model must contain intercept.", call. = FALSE)
# Outcome variable
yvr <- as.character(attr(trm, "variables")[-1L])[attr(trm, "response")]
# Cluster variable
cvr <- attr(nlme::getGroups(model), "label")
if(isTRUE(length(nlme::getGroupsFormula(model, asList = TRUE)) > 1L)) stop("Calculation of R-squared only support for models with a single cluster variable.", call. = FALSE)
# Cluster size harmonic mean
n.j <- 1L / mean(1L / table(nlme::getGroups(model)))
# Check for random slopes
random <- ncol(nlme::ranef(model)) > 1L
if (isTRUE(refit)) {
# Fit Null model
ffml0 <- formula(paste0(yvr, "~ 1"))
rfml0 <- formula(paste0("~ 1 |", cvr, ""))
if (is.null(nlme::getData(model))) stop("No data sets found in 'lme' fit. See '?testModels' for an example.", call. = FALSE)
model0 <- update(model, fixed = ffml0, random = rfml0, data = model$data)
# Variance components under Null
s0 <- model0$sigma^2L
t0.0 <- nlme::getVarCov(model0)[1L, 1L]
}
# Alternative model components
beta <- nlme::fixef(model)[-1L]
fe <- model$terms
X <- model.matrix(fe, nlme::getData(model))[, -1L, drop = FALSE]
re <- attr(model$modelStruct$reStruct[[1L]], "formula")
Z <- model.matrix(re, nlme::getData(model))[, -1L, drop = FALSE]
muZ <- colMeans(Z)
vZ <- cov(Z)
# Predicted and total variance
vc1 <- nlme::getVarCov(model)
t0.1 <- vc1[1L, 1L]
t10.1 <- vc1[1L, -1L]
t11.1 <- vc1[-1L, -1L, drop = FALSE]
s1 <- model$sigma^2L
})
#...................
### Random slope model ####
if (isTRUE(random)) {
if (isTRUE(all(c("RB", "SB") %in% print))) {
warning("R-squared measures by Raudenbush and Bryk (2002), and Snijders and Bosker (1994) should be computed based on the random intercept model.", call. = FALSE)
} else if (isTRUE("RB" %in% print)) {
warning("R-squared measure by Raudenbush and Bryk should be computed based on the random intercept model.", call. = FALSE)
} else if (isTRUE("SB" %in% print)) {
warning("R-squared measure by Snijders and Bosker (1994) should be computed based on the random intercept model.", call. = FALSE)
}
}
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Calculate R2 ####
# Raudenbush and Bryk (2002)
if (isTRUE("RB" %in% print)) {
# Within-Cluster R2
rb1 <- 1L - s1 / s0
# Between-Cluster R2
rb2 <- 1L - t0.1 / t0.0
} else {
rb1 <- rb2 <- NA
}
# Snijders and Bosker (1994)
if (isTRUE("SB" %in% print)) {
# Within-Cluster R2
sb1 <- 1L - (s1 + t0.1) / (s0 + t0.0)
# Between-Cluster R2
sb2 <- 1L - (s1 / n.j + t0.1) / (s0 / n.j + t0.0)
} else {
sb1 <- sb2 <- NA
}
# Nakagawa and Schielzeth (2013); Johnson (2014)
if (isTRUE("NS" %in% print)) {
vyhat <- var(X %*% beta)
vy <- vyhat + t0.1 + 2L*(muZ %*% t10.1) + muZ %*% t11.1 %*% muZ + sum(diag(t11.1 %*% vZ)) + s1
# Marginal R2
marg <- as.vector(vyhat / vy)
# Conditional R2
cond <- as.vector((vyhat + t0.1 + 2L*(muZ %*% t10.1) + muZ %*% t11.1 %*% muZ + sum(diag(t11.1 %*% vZ))) / vy)
} else {
marg <- cond <- NA
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Rights and Sterba (2019) R-squared measures ####
if (isTRUE("RS" %in% print)) {
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Internal functions from the r2mlm package ####
#### r2mlm() Function ####
r2mlm <- function(model) {
temp_formula <- formula(model)
grepl_array <- grepl("I(", temp_formula, fixed = TRUE)
for (bool in grepl_array) {
if (isTRUE(bool)) {
stop("Error: r2mlm does not allow for models fit using the I() function; user must thus manually include any desired transformed predictor variables such as x^2 or x^3 as separate columns in dataset.")
}
}
# call appropriate r2mlm helper function
if (isTRUE(typeof(model) == "list")) {
r2mlm_nlme(model)
} else if (isTRUE(typeof(model) == "S4")) {
r2mlm_lmer(model)
} else {
stop("You must input a model generated using either the lme4 or nlme package.")
}
}
#### r2mlm_lmer() Function ####
r2mlm_lmer <- function(model) {
# Visible global function defnition
is <- fixef <- getME <- NULL
# Step 1) check if model has_intercept
if (isTRUE(attr(terms(model), which = "intercept") == 1L)) {
has_intercept <- TRUE
} else {
has_intercept <- FALSE
}
# Step 2) Pull all variable names from the formula
all_variables <- all.vars(formula(model))
cluster_variable <- all_variables[length(all_variables)]
# Step 3a) Pull and prepare data
data <- prepare_data(model, "lme4", cluster_variable)
# Step 3b) Determine whether data is appropriate format
# a) Pull all variables except for cluster
outcome_and_predictors <- all_variables[1L:(length(all_variables) - 1L)]
# b) If any of those variables is non-numeric, then throw an error
for (variable in outcome_and_predictors) {
if (isTRUE(!is(data[[variable]], "integer") && !is(data[[variable]], "numeric"))) {
stop("Your data must be numeric. Only the cluster variable can be a factor.")
}
}
# Step 4a) Define variables you'll be sorting
if (isTRUE(length(outcome_and_predictors) == 1L)) {
predictors <- get_interaction_vars(model)
} else {
predictors <- append(outcome_and_predictors[2L:length(outcome_and_predictors)], get_interaction_vars(model))
}
# Step 4b) Create and fill vectors
l1_vars <- sort_variables(data, predictors, cluster_variable)$l1_vars
l2_vars <- sort_variables(data, predictors, cluster_variable)$l2_vars
# Step 5) Pull variable names for L1 predictors with random slopes into a variable called random_slope_vars
random_slope_vars <- get_random_slope_vars(model, has_intercept, "lme4")
# Step 6) Determine value of centeredwithincluster
if (is.null(l1_vars)) {
centeredwithincluster <- TRUE
} else {
centeredwithincluster <- get_cwc(l1_vars, cluster_variable, data)
}
# 7a) within_covs (l1 variables)
within <- get_covs(l1_vars, data)
# 7b) pull column numbers for between_covs (l2 variables)
between <- get_covs(l2_vars, data)
# 7c) pull column numbers for random_covs (l1 variables with random slopes)
random <- get_covs(random_slope_vars, data)
# 8a) gamma_w, fixed slopes for L1 variables (from l1_vars list)
gammaw <- c()
i <- 1L
for (variable in l1_vars) {
gammaw[i] <- fixef(model)[variable]
i <- i + 1L
}
# 8b) gamma_b, intercept value if hasintercept = TRUE, and fixed slopes for L2 variables (from between list)
gammab <- c()
if (isTRUE(has_intercept)) {
gammab[1L] <- fixef(model)[1L]
i <- 2L
} else {
i <- 1L
}
for (var in l2_vars) {
gammab[i] <- fixef(model)[var]
i = i + 1L
}
# Step 9) Tau matrix, results from VarCorr(model)
vcov <- VarCorr(model)
tau <- as.matrix(Matrix::bdiag(vcov))
# Step 10) Sigma^2 value, Rij
sigma2 <- getME(model, "sigma")^2L
# Step 11) Input everything into r2mlm_manual
r2mlm_manual(as.data.frame(data), within_covs = within, between_covs = between, random_covs = random,
gamma_w = gammaw, gamma_b = gammab, Tau = tau, sigma2 = sigma2, has_intercept = has_intercept, clustermeancentered = centeredwithincluster)
}
#### r2mlm_nlme() Function ####
r2mlm_nlme <- function(model) {
# Visible global function defnition
is <- getME <- NULL
# Step 1) check if model has_intercept
if (isTRUE(attr(terms(model), which = "intercept") == 1L)) {
has_intercept = TRUE
} else {
has_intercept = FALSE
}
# Step 2) Pull all variable names from the formula
all_variables <- all.vars(formula(model))
cluster_variable <- attr(nlme::getGroups(model), which = "label")
# Add the grouping var to list of all variables, and calculate formula length (for later use, to iterate)
all_variables[length(all_variables) + 1L] <- cluster_variable
formula_length <- length(all_variables) # this returns the number of elements in the all_vars list TODO remove this
# Step 3a) Pull and prepare data
data <- prepare_data(model, "nlme", cluster_variable)
# Step 3b) Determine whether data is appropriate format. Only the cluster variable can be a factor, for now
outcome_and_predictors <- all_variables[1L:length(all_variables) - 1L]
for (variable in outcome_and_predictors) {
if (isTRUE(!is(data[[variable]], "integer") && !is(data[[variable]], "numeric"))) {
stop("Your data must be numeric. Only the cluster variable can be a factor.")
}
}
# Step 4a) Define variables you'll be sorting
if (isTRUE(length(outcome_and_predictors) == 1L)) {
predictors <- get_interaction_vars(model)
} else {
predictors <- append(outcome_and_predictors[2:length(outcome_and_predictors)], get_interaction_vars(model))
}
# Step 4b) Create and fill vectors
l1_vars <- sort_variables(data, predictors, cluster_variable)$l1_vars
l2_vars <- sort_variables(data, predictors, cluster_variable)$l2_vars
# Step 5) pull variable names for L1 predictors with random slopes into a variable called random_slope_vars
random_slope_vars <- get_random_slope_vars(model, has_intercept, "nlme")
# Step 6) determine value of centeredwithincluster
if (isTRUE(is.null(l1_vars))) {
centeredwithincluster <- TRUE
} else {
centeredwithincluster <- get_cwc(l1_vars, cluster_variable, data)
}
# 7a) within_covs (l1 variables)
within <- get_covs(l1_vars, data)
# 7b) Pull column numbers for between_covs (l2 variables)
between <- get_covs(l2_vars, data)
# 7c) Pull column numbers for random_covs (l1 variables with random slopes)
random <- get_covs(random_slope_vars, data)
# 8a) gamma_w, fixed slopes for L1 variables (from l1_vars list)
gammaw <- c()
i = 1L
for (variable in l1_vars) {
gammaw[i] <- nlme::fixef(model)[variable]
i = i + 1L
}
# 8b) gamma_b, intercept value if hasintercept = TRUE, and fixed slopes for L2 variables (from between list)
gammab <- c()
if (isTRUE(has_intercept)) {
gammab[1L] <- nlme::fixef(model)[1L]
i = 2L
} else {
i = 1L
}
for (variable in l2_vars) {
gammab[i] <- nlme::fixef(model)[variable]
i = i + 1L
}
# Step 9) Tau matrix
tau <- nlme::getVarCov(model)
# Step 10) sigma^2 value, Rij
sigma2 <- model$sigma^2
# Step 11) Input everything into r2mlm_manual
r2mlm_manual(as.data.frame(data), within_covs = within, between_covs = between, random_covs = random,
gamma_w = gammaw, gamma_b = gammab, Tau = tau, sigma2 = sigma2, has_intercept = has_intercept, clustermeancentered = centeredwithincluster)
}
#### r2mlm_manual() Function ####
r2mlm_manual <- function(data, within_covs, between_covs, random_covs,
gamma_w, gamma_b, Tau, sigma2, has_intercept = TRUE, clustermeancentered = TRUE) {
if (isTRUE(has_intercept)) {
if (isTRUE(length(gamma_b) > 1L)) gamma <- c(1L, gamma_w, gamma_b[2:length(gamma_b)])
if (isTRUE(length(gamma_b) == 1L)) gamma <- c(1L, gamma_w)
if (isTRUE(is.null(within_covs))) gamma_w <- 0L
}
if (isTRUE(!has_intercept)) {
gamma <- c(gamma_w, gamma_b)
if (isTRUE(is.null(within_covs))) gamma_w <- 0L
if (isTRUE(is.null(between_covs))) gamma_b <- 0L
}
if (isTRUE(is.null(gamma))) gamma <- 0L
# Compute phi
phi <- var(cbind(1L, data[, c(within_covs)], data[, c(between_covs)]), na.rm = TRUE)
if (isTRUE(!has_intercept)) phi <- var(cbind(data[, c(within_covs)], data[, c(between_covs)]), na.rm = TRUE)
if (isTRUE(is.null(within_covs) && is.null(within_covs) && !has_intercept)) phi <- 0L
phi_w <- var(data[, within_covs], na.rm = TRUE)
if (isTRUE(is.null(within_covs))) phi_w <- 0L
phi_b <- var(cbind(1L, data[, between_covs]), na.rm = TRUE)
if (isTRUE(is.null(between_covs))) phi_b <- 0L
# Compute psi and kappa
var_randomcovs <- var(cbind(1, data[, c(random_covs)]), na.rm = TRUE)
if (isTRUE(length(Tau) > 1L)) psi <- matrix(c(diag(Tau)), ncol = 1L)
if (isTRUE(length(Tau) == 1L)) psi <- Tau
if (isTRUE(length(Tau) > 1L)) kappa <- matrix(c(Tau[lower.tri(Tau) == TRUE]), ncol = 1)
if (isTRUE(length(Tau) == 1L)) kappa <- 0L
v <- matrix(c(diag(var_randomcovs)), ncol = 1L)
r <- matrix(c(var_randomcovs[lower.tri(var_randomcovs) == TRUE]), ncol = 1L)
if (isTRUE(is.null(random_covs))) {
v <- 0L
r <- 0L
m <- matrix(1L, ncol = 1L)
}
if (isTRUE(length(random_covs) > 0L)) m <- matrix(c(colMeans(cbind(1, data[, c(random_covs)]), na.rm = TRUE)), ncol = 1L)
# Total variance
totalvar_notdecomp <- t(v)%*%psi + 2L*(t(r)%*%kappa) + t(gamma)%*%phi%*%gamma + t(m)%*%Tau%*%m + sigma2
totalwithinvar <- (t(gamma_w)%*%phi_w%*%gamma_w) + (t(v)%*%psi + 2L*(t(r)%*%kappa)) + sigma2
totalbetweenvar <- (t(gamma_b)%*%phi_b%*%gamma_b) + Tau[1L]
totalvar <- totalwithinvar + totalbetweenvar
# Total decomp
decomp_fixed_notdecomp <- (t(gamma)%*%phi%*%gamma) / totalvar_notdecomp
decomp_varslopes_notdecomp <- (t(v)%*%psi + 2L*(t(r)%*%kappa)) / totalvar_notdecomp
decomp_varmeans_notdecomp <- (t(m)%*%Tau%*%m) / totalvar_notdecomp
decomp_sigma_notdecomp <- sigma2/totalvar_notdecomp
decomp_fixed_within <- (t(gamma_w)%*%phi_w%*%gamma_w) / totalvar
decomp_fixed_between <- (t(gamma_b)%*%phi_b%*%gamma_b) / totalvar
decomp_fixed <- decomp_fixed_within + decomp_fixed_between
decomp_varslopes <- (t(v)%*%psi + 2L*(t(r)%*%kappa)) / totalvar
decomp_varmeans <- (t(m)%*%Tau%*%m) / totalvar
decomp_sigma <- sigma2 / totalvar
# Within decomp
decomp_fixed_within_w <- (t(gamma_w)%*%phi_w%*%gamma_w) / totalwithinvar
decomp_varslopes_w <- (t(v)%*%psi + 2L*(t(r)%*%kappa)) / totalwithinvar
decomp_sigma_w <- sigma2 / totalwithinvar
# Between decomp
decomp_fixed_between_b <- (t(gamma_b)%*%phi_b%*%gamma_b) / totalbetweenvar
decomp_varmeans_b <- Tau[1L] / totalbetweenvar
# New measures
if (isTRUE(clustermeancentered)) {
R2_f <- decomp_fixed
R2_f1 <- decomp_fixed_within
R2_f2 <- decomp_fixed_between
R2_fv <- decomp_fixed + decomp_varslopes
R2_fvm <- decomp_fixed + decomp_varslopes + decomp_varmeans
R2_v <- decomp_varslopes
R2_m <- decomp_varmeans
R2_f_w <- decomp_fixed_within_w
R2_f_b <- decomp_fixed_between_b
R2_fv_w <- decomp_fixed_within_w + decomp_varslopes_w
R2_v_w <- decomp_varslopes_w
R2_m_b <- decomp_varmeans_b
}
if (isTRUE(!clustermeancentered)) {
R2_f <- decomp_fixed_notdecomp
R2_fv <- decomp_fixed_notdecomp + decomp_varslopes_notdecomp
R2_fvm <- decomp_fixed_notdecomp + decomp_varslopes_notdecomp + decomp_varmeans_notdecomp
R2_v <- decomp_varslopes_notdecomp
R2_m <- decomp_varmeans_notdecomp
}
if (isTRUE(clustermeancentered)) {
decomp_table <- matrix(c(decomp_fixed_within, decomp_fixed_between, decomp_varslopes, decomp_varmeans, decomp_sigma,
decomp_fixed_within_w, "NA", decomp_varslopes_w, "NA", decomp_sigma_w,
"NA", decomp_fixed_between_b, "NA", decomp_varmeans_b, "NA"), ncol = 3L)
decomp_table <- suppressWarnings(apply(decomp_table, 2, as.numeric))
rownames(decomp_table) <- c("fixed, within", "fixed, between", "slope variation", "mean variation", "sigma2")
colnames(decomp_table) <- c("total", "within", "between")
R2_table <- matrix(c(R2_f1, R2_f2, R2_v, R2_m, R2_f, R2_fv, R2_fvm,
R2_f_w, "NA", R2_v_w, "NA", "NA", R2_fv_w, "NA",
"NA", R2_f_b, "NA", R2_m_b, "NA", "NA", "NA"), ncol = 3L)
R2_table <- suppressWarnings(apply(R2_table, 2, as.numeric)) # make values numeric, not character
rownames(R2_table) <- c("f1", "f2", "v", "m", "f", "fv", "fvm")
colnames(R2_table) <- c("total", "within", "between")
}
if (isTRUE(!clustermeancentered)) {
decomp_table <- matrix(c(decomp_fixed_notdecomp, decomp_varslopes_notdecomp, decomp_varmeans_notdecomp, decomp_sigma_notdecomp), ncol = 1L)
decomp_table <- suppressWarnings(apply(decomp_table, 2L, as.numeric))
rownames(decomp_table) <- c("fixed", "slope variation", "mean variation", "sigma2")
colnames(decomp_table) <- c("total")
R2_table <- matrix(c(R2_f, R2_v, R2_m, R2_fv, R2_fvm), ncol = 1L)
R2_table <- suppressWarnings(apply(R2_table, 2L, as.numeric))
rownames(R2_table) <- c("f", "v", "m", "fv", "fvm")
colnames(R2_table) <- c("total")
}
Output <- list(noquote(decomp_table), noquote(R2_table))
names(Output) <- c("Decompositions", "R2s")
return(Output)
}
#### prepare_data() Function ####
prepare_data <- function(model, calling_function, cluster_variable, second_model = NULL) {
# Step 1a: Pull dataframe associated with model
if (isTRUE(calling_function == "lme4")) {
data <- model@frame
} else {
data <- model[["data"]]
}
# Step 2a) pull interaction terms into list
interaction_vars <- get_interaction_vars(model)
if (isTRUE(!is.null(second_model))) {
interaction_vars_2 <- get_interaction_vars(second_model)
interaction_vars <- unique(append(interaction_vars, interaction_vars_2))
}
# Step 2b) split interaction terms into halves, multiply halves to create new columns in dataframe
data <- add_interaction_vars_to_data(data, interaction_vars)
return(data)
}
#### add_interaction_vars_to_data() Function ####
add_interaction_vars_to_data <- function(data, interaction_vars) {
for (whole in interaction_vars) {
half1 <- unlist(strsplit(whole, ":"))[1L]
half2 <- unlist(strsplit(whole, ":"))[2L]
newcol <- data[[half1]] * data[[half2]]
data <- within(data, assign(whole, newcol))
}
return(data)
}
#### get_covs() Function ####
get_covs <- function(variable_list, data) {
cov_list <- c()
i <- 1L
for (variable in variable_list) {
tmp <- match(variable, names(data))
cov_list[i] <- tmp
i <- i + 1L
}
return(cov_list)
}
#### get_random_slope_vars() Function ####
get_random_slope_vars <- function(model, has_intercept, calling_function) {
# Visible global function defnition
ranef <- NULL
if (isTRUE(calling_function == "lme4")) {
temp_cov_list <- ranef(model)[[1L]]
} else if (calling_function == "nlme") {
temp_cov_list <- nlme::ranef(model)
}
if (isTRUE(has_intercept == 1L)) {
running_count <- 2L
} else {
running_count <- 1L
}
random_slope_vars <- c()
x <- 1L
while (running_count <= length(temp_cov_list)) {
random_slope_vars[x] <- names(temp_cov_list[running_count])
x <- x + 1L
running_count <- running_count + 1L
}
return(random_slope_vars)
}
#### get_cwc() Function ####
get_cwc <- function(l1_vars, cluster_variable, data) {
for (variable in l1_vars) {
t <- tapply(data[, variable], data[, cluster_variable], sum, na.rm = TRUE)
temp_tracker <- 0L
# sum all of the sums
for (i in t) { temp_tracker <- temp_tracker + i }
if (abs(temp_tracker) < 0.0000001) {
centeredwithincluster <- TRUE
} else {
centeredwithincluster <- FALSE
break
}
}
return(centeredwithincluster)
}
#### get_interaction_vars() Function ####
get_interaction_vars <- function(model) {
interaction_vars <- c()
x <- 1L
for (term in attr(terms(model), "term.labels")) {
if (isTRUE(grepl(":", term))) {
interaction_vars[x] <- term
x <- x + 1L
}
}
return(interaction_vars)
}
#### sort_variables() Function ####
sort_variables <- function(data, predictors, cluster_variable) {
l1_vars <- c()
l2_vars <- c()
l1_counter <- 1L
l2_counter <- 1L
for (variable in predictors) {
t <- tapply(data[, variable], data[, cluster_variable], var, na.rm = TRUE)
counter <- 1L
while (counter <= length(t)) {
if (is.na(t[[counter]])) { t[[counter]] <- 0L }
counter <- counter + 1L
}
variance_tracker <- 0L
for (i in t) { variance_tracker <- variance_tracker + i }
if (isTRUE(variance_tracker == 0L)) {
l2_vars[l2_counter] <- variable
l2_counter <- l2_counter + 1L
} else {
l1_vars[l1_counter] <- variable
l1_counter <- l1_counter + 1L
}
}
return(list("l1_vars" = l1_vars, "l2_vars" = l2_vars))
}
#...................
### R-squared measures ####
r2mlm.out <- r2mlm(model)
rs <- suppressWarnings(list(decomp = matrix(apply(r2mlm.out$Decomposition, 2L, as.numeric),
ncol = ncol(r2mlm.out$Decomposition),
dimnames = list(rownames(r2mlm.out$Decompositions),
colnames(r2mlm.out$Decompositions))),
r2 = matrix(apply(r2mlm.out$R2s, 2L, as.numeric), ncol = ncol(r2mlm.out$R2s),
dimnames = list(rownames(r2mlm.out$R2s), colnames(r2mlm.out$R2s)))))
#...................
### Plot ####
part <- NULL
# Predictors are not cluster-mean-centered
if (isTRUE(ncol(rs$decomp) == 1L)) {
df <- data.frame(var = factor(rep("Total", times = 4L)),
part = factor(c("Fixed Slopes", "Slope Variation", "Intercept Variation", "Residual"),
levels = c("Residual", "Intercept Variation", "Slope Variation", "Fixed Slopes")),
y = as.vector(rs$decomp))
# Predictors are cluster-mean-centered
} else {
df <- data.frame(var = factor(rep(c("Total", "Within", "Between"), each = 5L),
levels = c("Total", "Within", "Between")),
part = factor(c("Fixed Slopes (Within)", "Fixed Slopes (Between)","Slope Variation (Within)", "Intercept Variation (Between)", "Residual (Within)"),
levels = c("Residual (Within)", "Intercept Variation (Between)", "Slope Variation (Within)", "Fixed Slopes (Between)", "Fixed Slopes (Within)")),
y = as.vector(rs$decomp))
}
p <- ggplot2::ggplot(df, ggplot2::aes(x = var, y = y, fill = part)) +
ggplot2::geom_bar(stat = "identity") +
ggplot2::scale_y_continuous(name = "Proportion of Variance",
breaks = seq(0L, 1L, by = 0.1)) +
ggplot2::theme_bw() +
ggplot2::theme(axis.title.x = ggplot2::element_blank(),
axis.ticks.x = ggplot2::element_blank(),
legend.title = ggplot2::element_blank(),
legend.position = "bottom",
legend.box.margin = ggplot2::margin(-10L, 6L, 6L, 6L)) +
ggplot2::guides(fill = ggplot2::guide_legend(nrow = 2L, reverse = TRUE))
# Gray color scales
if (isTRUE(gray)) {
p <- p + ggplot2::scale_fill_grey(start = end, end = start)
} else {
p <- p + ggplot2::scale_fill_manual(values = rev(color))
}
# Print plot
if (isTRUE(plot)) { suppressWarnings(print(p)) }
} else {
p <- NULL
rs <- list(decomp = matrix(NA, ncol = 3L, nrow = 5L,
dimnames = list(c("fixed, within", "fixed, between", "slope variation", "mean variation", "sigma2"),
c("total", "within", "between"))),
r2 = matrix(NA, ncol = 3L, nrow = 7L,
dimnames = list(c("f1", "f2", "v", "m", "f", "fv", "fvm"),
c("total", "within", "between"))))
}
#_____________________________________________________________________________
#
# Return object --------------------------------------------------------------
object <- list(call = match.call(),
type = "multilevel.r2",
model = model,
plot = p,
args = list(print = print, digits = digits, plot = plot, gray = gray,
start = start, end = end, color = color, check = check,
output = output),
result = list(rb = data.frame(rb1 = rb1, rb2 = rb2),
sb = data.frame(sb1 = sb1, sb2 = sb2),
ns = data.frame(marg = marg, cond = cond),
rs = list(decomp = rs$decomp,
total = data.frame(f1 = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "f1", "total"], NA),
f2 = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "f2", "total"], NA),
f = rs$r2[row.names(rs$r2) == "f", "total"],
v = rs$r2[row.names(rs$r2) == "v", "total"],
m = rs$r2[row.names(rs$r2) == "m", "total"],
fv = rs$r2[row.names(rs$r2) == "fv", "total"],
fvm = rs$r2[row.names(rs$r2) == "fvm", "total"]),
within = data.frame(f1 = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "f1", "within"], NA),
v = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "v", "within"], NA),
f1v = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "fv", "within"], NA)),
between = data.frame(f2 = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "f2", "between"], NA),
m = ifelse(ncol(rs$r2) > 1L, rs$r2[row.names(rs$r2) == "m", "between"], NA)))))
class(object) <- "misty.object"
#_____________________________________________________________________________
#
# Output ---------------------------------------------------------------------
if (isTRUE(output)) { print(object, check = FALSE) }
return(invisible(object))
}
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