std.coef: Standardized Coefficients for Linear, Multilevel and...

In misty: Miscellaneous Functions 'T. Yanagida'

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)

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.

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.

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)

Author(s)

Takuya Yanagida [e-mail]

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

#----------------------------------------------------------------------------
# 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)

## Not run: 
#----------------------------------------------------------------------------
# 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)
## End(Not run)

misty documentation built on June 8, 2025, 1:35 p.m.