std.coef: Standardized Coefficients for Linear, Multilevel and Mixed-Effects Models

Description

This function computes standardized coefficients for linear models estimated by using the lm() function and for multilevel and linear mixed-effects models estimated by using the lmer() or lme() function from the lme4 or nlme package.

Usage

std.coef(model, print = c("all", "stdx", "stdy", "stdyx"), digits = 3, p.digits = 3,
         write = NULL, append = TRUE, check = TRUE, output = TRUE)

Value

Returns an object of class misty.object, which is a list with following entries:

call: function call
type: type of analysis
data: data frame with variables used in the analysis
model: model specified in model
args: specification of function arguments
result: list with result tables, i.e., coef for the regression table including standardized coefficients and sd for the standard deviation of the outcome and predictor(s)

Arguments

model: a fitted model of class "lm", "lmerMod", "lmerModLmerTest" or "lme".
print: a character vector indicating which results to show, i.e. "all", for all results, "stdx" for standardizing only the predictor, "stdy" for for standardizing only the criterion, and "stdyx" for for standardizing both the predictor and the criterion. Note that the default setting is depending on the level of measurement of the predictors, i.e., if all predictors are continuous, the default setting is print = "stdyx"; if all predictors are binary, the default setting is print = "stdy", and if predictors are continuous and binary, the default setting is print = c("stdy", "stdyx").
digits: an integer value indicating the number of decimal places to be used for displaying results.
p.digits: an integer value indicating the number of decimal places to be used for displaying the p-value.
write: a character string naming a file for writing the output into either a text file with file extension ".txt" (e.g., "Output.txt") or Excel file with file extension ".xlsx" (e.g., "Output.xlsx"). If the file name does not contain any file extension, an Excel file will be written.
append: logical: if TRUE (default), output will be appended to an existing text file with extension .txt specified in write, if FALSE existing text file will be overwritten.
check: logical: if TRUE (default), argument specification is checked.
output: logical: if TRUE (default), output is shown on the console.

Author

Takuya Yanagida takuya.yanagida@univie.ac.at

Details

Linear Regression Model

The linear regression model is expressed as follows:

$$y_i = \beta_0 + \beta_1x_i + \epsilon_i$$

where $y_i$ is the outcome variable for individual $i$, $\beta_0$ is the intercept, $\beta_1$ is the slope (aka regression coefficient), $x_i$ is the predictor for individual $i$, and $\epsilon_i$ is the residual for individual $i$.

The slope $\beta_1$ estimated by using the lm() function can be standardized with respect to only $x$, only $y$, or both $y$ and $x$:

StdX Standardization: $StdX(\beta_1)$ standardizes with respect to $x$ only and is interpreted as expected difference in $y$ between individuals that differ one standard deviation referred to as $SD(x)$:

$$StdX(\beta_1) = \beta_1 SD(x)$$
StdY Standardization: $StdY(\beta_1)$ standardizes with respect to $y$ only and is interpreted as expected difference in $y$ standard deviation units, referred to as $SD(y)$, between individuals that differ one unit in $x$:

$$StdY(\beta_1) = \frac{\beta_1}{SD(x)}$$
StdYX Standardization: $StdYX(\beta_1)$ standardizes with respect to both $y$ and $x$ and is interpreted as expected difference in $y$ standard deviation units between individuals that differ one standard deviation in $x$:

$$StdYX(\beta_1) = \beta_1 \frac{SD(x)}{SD(y)}$$

Note that the $StdYX(\beta_1)$ and the $StdY(\beta_1)$ standardizations are not suitable for the slope of a binary predictor because a one standard deviation change in a binary variable is generally not of interest (Muthen et al, 2016). Accordingly, the function does not provide the $StdYX(\beta_1)$ and the $StdY(\beta_1)$ standardizations whenever a binary vector, factor, or character vector is specified for the predictor variable.

Moderated Regression Model

The moderated regression model is expressed as follows:

$$y_i = \beta_0 + \beta_1x_{1i} + \beta_2x_{2i} + \beta_3x_{1i}x_{2i} + \epsilon_i$$

where $\beta_3$ is the slope for the interaction variable $x_1x_2$.

The slope $\beta_3$ is standardized by using the product of standard deviations $SD(x_1)SD(x_2)$ rather than the standard deviation of the product $SD(x_1 x_2)$ for the interaction variable $x_1x_2$ as discussed in Wen et al. (2010).

Note that the function does not use binary variables in the interaction term in standardizing the interaction variable. For example, when standardizing the interaction term $x1:x2:x3$ with $x2$ being binary, the product $SD(x_1)SD(x_3)$ while excluding binary predictor x2 is used to standardize the interaction term.

Polynomial Regression Model

The polynomial regression model is expressed as follows:

$$y_i = \beta_0 + \beta_1x_{i} + \beta_2x^2_{i} + \epsilon_i$$

where $\beta_2$ is the slope for the quadratic term $x^2$.

The slope $\beta_3$ is standardized by using the product of standard deviations $SD(x)SD(x)$ rather than the standard deviation of the product $SD(x x)$ for the quadratic term $x^2$.

Multilevel and Mixed-Effects Model

The random intercept and slope model in the multiple-equation notation is expressed as follows:

Level 1: $$y_{ij} = \beta_{0j} + \beta_{1j}x_{ij} + r_{ij}$$
Level 2: $$\beta_{0j} = \gamma_{00} + \gamma_{01}z_{j} + u_{0j}$$ $$\beta_{1j} = \gamma_{10} + u_{1j}$$

The model expressed in the single-equation notation is as follows:

$$y_{ij} = \gamma_{00} + \gamma_{10}x_{ij} + \gamma_{01}z_{j} + u_{0j} + u_{1j}x_{ij} + r_{ij}$$

where $y_{ij}$ is the outcome variable for individual $i$ in group $j$, $\gamma_{00}$ is the fixed-effect average intercept, $\gamma_{10}$ is the fixed-effect average slope for the Level-1 predictor $x$, and $\gamma_{01}$ is the fixed-effect slope for the Level-2 predictor $z$.

The slopes $\gamma_{10}$ and $\gamma_{01}$ are standardized according to the within- and between-group or within-and between-person standard deviations, i.e., slopes are standardizes with respect to the $x$ and $y$ standard deviation relevant for the level of the fixed effect of interest. The resulting standardized slopes are called pseudo-standardized coefficients (Hoffman 2015, p. 342). The StdYX Standardization for $\gamma_{10}$ and $\gamma_{10}$ is expressed as follows:

Level-1 Predictor: $$StdYX(\gamma_{10}) = \gamma_{10} \frac{SD(x_{ij})}{SD(y_{ij})}$$
Level-2 Predictor: $$StdYX(\gamma_{01}) = \gamma_{01} \frac{SD(x_{j})}{SD(y_{j})}$$

where $SD(x_{ij})$ and $SD(x_{j})$ are the standard deviations of the predictors at each analytic level, $SD(y_{ij})$ is the square root of the Level-1 residual variance $\sigma^2_{r}$ and $SD(y_{j})$ is square root of the Level-2 intercept variance $\sigma^2_{u_0}$ which are estimated in a null model using the lmer function in the lme4 package using the restricted maximum likelihood estimation method.

The function uses the square root of the Level-1 residual variance $\sigma^2_{r}$ to standardize the slope of the cross-level interaction though it should be noted that it is unclear whether this is the correct approach to standardize the slope of the cross-level interaction.

References

Hoffman, L. (2015). Longitudinal Analysis: Modeling Within-Person Fluctuation and Change. Routledge.

Muthen, B. O., Muthen, L. K., & Asparouhov, T. (2016). Regression and mediation analysis using Mplus. Muthen & Muthen.

Wen, Z., Marsh, H. W., & Hau, K.-T. (2010). Structural equation models of latent interactions: An appropriate standardized solution and its scale-free properties. Structural Equation Modeling: A Multidisciplinary Journal, 17, 1-22. https://doi.org/10.1080/10705510903438872

Examples

Run this code

#----------------------------------------------------------------------------
# Linear Model

# Example 1a: Continuous predictors
mod.lm1 <- lm(mpg ~ cyl + disp, data = mtcars)
std.coef(mod.lm1)

# Example 1b: Print all standardized coefficients
std.coef(mod.lm1, print = "all")

# Example 1c: Binary predictor
mod.lm2 <- lm(mpg ~ vs, data = mtcars)
std.coef(mod.lm2)

# Example 1d: Continuous and binary predictors
mod.lm3 <- lm(mpg ~ disp + vs, data = mtcars)
std.coef(mod.lm3)

# Example 1e: Continuous predictors with interaction term
mod.lm4 <- lm(mpg ~ cyl*disp, data = mtcars)
std.coef(mod.lm4)

# Example 1f: Continuous and binary predictor with interaction term
mod.lm5 <- lm(mpg ~ cyl*vs, data = mtcars)
std.coef(mod.lm5)

# Example 1g: Continuous predictor with a quadratic term
mod.lm6 <- lm(mpg ~ cyl + I(cyl^2), data = mtcars)
std.coef(mod.lm6)

if (FALSE) {
#----------------------------------------------------------------------------
# Multilevel and Linear Mixed-Effects Model

# Load lme4, nlme, and ggplot2 package
misty::libraries(lme4, nlme)

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

# Cluster mean centering, center() from the misty package
Demo.twolevel <- center(Demo.twolevel, x2, type = "CWC", cluster = "cluster")

# Grand mean centering, center() from the misty package
Demo.twolevel <- center(Demo.twolevel, w1, type = "CGM", cluster = "cluster")

# Estimate models using the lme4 package
mod1a <- lmer(y1 ~ x2.c + w1.c + (1 + x2.c | cluster), data = Demo.twolevel,
              REML = FALSE)
mod2a <- lmer(y1 ~ x2.c + w1.c + x2.c:w1.c + (1 + x2.c | cluster),
              data = Demo.twolevel, REML = FALSE)

# Estimate models using the nlme package
mod1b <- lme(y1 ~ x2.c + w1.c, random = ~ 1 + x2.c | cluster, data = Demo.twolevel,
             method = "ML")
mod2b <- lme(y1 ~ x2.c + w1.c + x2.c:w1.c, random = ~ 1 + x2.c | cluster,
             data = Demo.twolevel, method = "ML")

# Example 2: Continuous predictors
std.coef(mod1a)
std.coef(mod1b)

# Example 2: Continuous predictors with cross-level interaction
std.coef(mod2a)
std.coef(mod2b)

#----------------------------------------------------------------------------
# Example 3: Write Results into a text or Excel file

# Example 3a: Text file
std.coef(mod.lm1, write = "Std_Coef.txt", output = FALSE, check = FALSE)

# Example 3b: Excel file
std.coef(mod.lm1, write = "Std_Coef.xlsx", output = FALSE, check = FALSE)}

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