Indirect and Total Effects

‘lavEffects’ computes various ‘effects’ that are functions of the estimated model parameters of a fitted lavaan object. In this version, the focus is on the classic (LISREL-style) total, indirect and direct effects among the variables that appear in the structural part of the model (that is, the variables that are involved in a regression). Both observed and latent variables are supported.

This avoids the need to pre-specify all indirect (and total) effects manually in the model syntax (using the := operator), which can be tedious when there are many of them.

Usage

lavEffects(object, effects = c("total", "indirect"),
    se_def = NULL, level = 0.95, monte_carlo = NULL,
    boot_ci_type = "perc", zstat = TRUE, pvalue = TRUE, ci = TRUE,
    standardized = FALSE, cov_std = TRUE,
    add_class = TRUE, output = "data.frame", ...)

Arguments

object

An object of class lavaan.

effects

Character vector. One or more of "total", "indirect" and "direct". The default is c("total", "indirect"). The order of the elements does not matter; effects are always reported in the order total, indirect, direct.

se_def

Character (or NULL). The method used to compute standard errors (and confidence intervals) for the effects. If "monte_carlo", the Monte Carlo method is used: parameters are drawn from a multivariate normal distribution with mean the parameter estimates and covariance matrix the estimated covariance matrix of the parameters, and the effects are recomputed for each draw (percentile confidence intervals). If "delta", the delta method is used (based on a numerical Jacobian of the effects with respect to the free parameters; symmetric normal-theory confidence intervals). If "bootstrap", the bootstrap draws that are stored in the fitted object (when the model was fitted with se = "bootstrap") are used to compute bootstrap standard errors and bootstrap percentile confidence intervals. If "none", no standard errors are computed.

If se_def = NULL (the default), the choice is inherited from the fitted object: if the model was fitted with se = "bootstrap" (and bootstrap draws are available), the bootstrap method is used; else if the model was fitted with se_def = "monte.carlo", the Monte Carlo method is used; otherwise the Monte Carlo method is used as the default. An explicit value always overrides this inheritance.

level

Numeric. The confidence level for the confidence intervals.

monte_carlo

List (or NULL). Settings for the Monte Carlo method (only used when se_def = "monte.carlo"), with elements R (the number of Monte Carlo draws) and seed (the random seed). If NULL (the default), the monte_carlo settings that were used when the model was fitted are inherited (which themselves default to list(R = 20000L, seed = NULL)). The state of the global random number generator is saved and restored, so the seed used here does not affect the user's random number stream.

boot_ci_type

Character. Only used when se_def = "bootstrap". The type of bootstrap confidence interval, as in parameterEstimates: "norm" (normal-theory using the bootstrap bias and standard error), "basic" (basic bootstrap), "perc" (the default; percentile method), "bca.simple" (bias-corrected percentile method, without the acceleration term) or "bca" (the adjusted bootstrap percentile method, with both bias and acceleration correction).

zstat

Logical. If TRUE, an extra column is added with the \(z\)-statistic for each effect (the ratio of the estimate to its standard error).

pvalue

Logical. If TRUE, an extra column is added with the (two-sided) p-value, computed using the standard normal distribution.

ci

Logical. If TRUE (the default), two extra columns (ci.lower and ci.upper) are added with the lower and upper bounds of the confidence interval. If FALSE, the confidence interval bounds are not shown.

standardized

Logical, character vector, or vector of (observed) variable names, behaving as in parameterEstimates. If TRUE, standardized effects are added as extra columns (point estimates only): std.lv (only the latent variables are standardized), std.all (both observed and latent variables are standardized) and, if there are exogenous observed variables and fixed.x = TRUE, std.nox (like std.all, but the exogenous observed variables are not standardized). A character vector selects specific types (a subset of "std.lv", "std.all", "std.nox"); a vector of observed variable names standardizes only those observed variables (plus the latent variables), and the result is reported in a std.user column. The standardized effect of \(j\) on \(i\) equals the unstandardized effect times the ratio of the model-implied standard deviations \(sd_j / sd_i\).

cov_std

Logical. See parameterEstimates. (Has no influence on the standardized regression-type effects reported here.)

add_class

Logical. If TRUE, the result (when output = "data.frame") is given the class c("lavaan.effects", "lavaan.data.frame", "data.frame"), so that a dedicated print method is used.

output

Character. If "data.frame" (the default), the result is a (classed) data.frame with one row per effect. If "list", the result is a list of matrices (one per effect type, and one per block in the multilevel/multigroup case), where element \([i,j]\) contains the effect of variable \(j\) on variable \(i\).

...

Only old names of arguments - with dots - are allowed here.

Details

All effects are derived from the reduced-form matrix \((I - B)^{-1}\), where \(B\) is the matrix of (direct) regression coefficients among the structural variables (the beta matrix in the LISREL representation that lavaan uses internally). Writing \(B[i,j]\) for the direct effect of variable \(j\) on variable \(i\), we have:

the direct effect of \(j\) on \(i\) equals \(B[i,j]\);
the total effect of \(j\) on \(i\) equals \(\left((I - B)^{-1} - I\right)[i,j]\);
the indirect effect of \(j\) on \(i\) equals the total effect minus the direct effect.

Only effects that are structurally non-zero (that is, for which a directed path from \(j\) to \(i\) exists) are reported. For indirect effects, this means that at least one path of length two or more must exist.

The delta and Monte Carlo methods only require the parameter estimates and the estimated covariance matrix of the parameters. The delta method produces (symmetric) normal-theory confidence intervals; the Monte Carlo method produces percentile-based confidence intervals, which need not be symmetric around the point estimate. The Monte Carlo method typically behaves better than the delta method for effects that are products of parameters (such as indirect effects), in particular in smaller samples. When the model was fitted with se = "bootstrap", the bootstrap draws are reused to obtain bootstrap standard errors and bootstrap percentile confidence intervals (this requires no additional model fitting).

Models fitted with conditional.x = TRUE are also supported: in that case the structural model is \(eta = B eta + Gamma x + zeta\), where the exogenous covariates \(x\) are stored in the gamma matrix. The effects of these covariates on the endogenous variables are then computed as \((I - B)^{-1} Gamma\) (total), \(Gamma\) (direct) and \(((I - B)^{-1} - I) Gamma\) (indirect), and reported alongside the effects among the beta variables.

Value

If output = "data.frame" (the default), a data.frame (of class lavaan.effects) with the following columns: effect (the type of effect: total, indirect or direct), lhs (the outcome variable), op (always "~"), rhs (the predictor variable), and, depending on the arguments, group/level/block, est, se, z, pvalue, ci.lower, ci.upper and (if standardized is requested) one or more of std.lv, std.all, std.nox and std.user. The effect in a given row is the effect of rhs on lhs.

If output = "list", a (possibly nested) list of matrices, where element \([i,j]\) contains the effect of variable \(j\) on variable \(i\).

Examples

# a simple mediation model
set.seed(1234)
X <- rnorm(300)
M <- 0.5 * X + rnorm(300)
Y <- 0.4 * M + 0.3 * X + rnorm(300)
Data <- data.frame(X = X, Y = Y, M = M)
model <- ' # direct effect
             Y ~ c*X
           # mediator
             M ~ a*X
             Y ~ b*M
         '
fit <- sem(model, data = Data)

# total and indirect effects, with Monte Carlo standard errors
lavEffects(fit)
#> 
#> Effects:
#> 
#>   Standard errors                      Monte Carlo
#>   Monte Carlo draws                          20000
#>   Confidence level                            0.95
#> 
#> Total Effects:
#>           Estimate  Std.Err  z-value  P(>|z|) ci_lower ci_upper
#>     Y ~ M    0.408    0.053    7.762    0.000    0.305    0.511
#>     Y ~ X    0.533    0.059    9.036    0.000    0.417    0.649
#>     M ~ X    0.521    0.060    8.749    0.000    0.405    0.638
#> 
#> Indirect Effects:
#>           Estimate  Std.Err  z-value  P(>|z|) ci_lower ci_upper
#>     Y ~ X    0.213    0.037    5.795    0.000    0.146    0.289
#> 

# all effects, using the delta method
lavEffects(fit, effects = c("total", "indirect", "direct"), se_def = "delta")
#> 
#> Effects:
#> 
#>   Standard errors                            Delta
#>   Confidence level                            0.95
#> 
#> Total Effects:
#>           Estimate  Std.Err  z-value  P(>|z|) ci_lower ci_upper
#>     Y ~ M    0.408    0.053    7.764    0.000    0.305    0.511
#>     Y ~ X    0.533    0.059    9.001    0.000    0.417    0.649
#>     M ~ X    0.521    0.059    8.772    0.000    0.404    0.637
#> 
#> Indirect Effects:
#>           Estimate  Std.Err  z-value  P(>|z|) ci_lower ci_upper
#>     Y ~ X    0.213    0.037    5.814    0.000    0.141    0.284
#> 
#> Direct Effects:
#>           Estimate  Std.Err  z-value  P(>|z|) ci_lower ci_upper
#>     Y ~ M    0.408    0.053    7.764    0.000    0.305    0.511
#>     Y ~ X    0.321    0.061    5.291    0.000    0.202    0.440
#>     M ~ X    0.521    0.059    8.772    0.000    0.404    0.637
#> 

# add standardized (total and indirect) effects
lavEffects(fit, se_def = "delta", standardized = TRUE)
#> 
#> Effects:
#> 
#>   Standard errors                            Delta
#>   Confidence level                            0.95
#> 
#> Total Effects:
#>           Estimate  Std.Err  z-value  P(>|z|) ci_lower ci_upper   std.lv
#>     Y ~ M    0.408    0.053    7.764    0.000    0.305    0.511    0.408
#>     Y ~ X    0.533    0.059    9.001    0.000    0.417    0.649    0.533
#>     M ~ X    0.521    0.059    8.772    0.000    0.404    0.637    0.521
#>            std.all  std.nox
#>     Y ~ M    0.407    0.407
#>     Y ~ X    0.461    0.460
#>     M ~ X    0.452    0.451
#> 
#> Indirect Effects:
#>           Estimate  Std.Err  z-value  P(>|z|) ci_lower ci_upper   std.lv
#>     Y ~ X    0.213    0.037    5.814    0.000    0.141    0.284    0.213
#>            std.all  std.nox
#>     Y ~ X    0.184    0.183
#>

Usage

Arguments

Details

Value

See also

Examples