Nothing
Serial Mediation Models Corrected for Measurement Error"
In manymome: Mediation, Moderation and Moderated-Mediation After Model Fitting
Introduction
(This document is for Version 0.3.4.25 or later, currently
on GitHub only. The new version is scheduled to be available
on CRAN in June 2026.)
This article is a brief illustration of how to use
the all-in-one "quick" functions from
manymome
(Cheung & Cheung, 2024)
to fit a structural model and
compute and test indirect effects in
one step.
It is similar to using them on observed
scores, such as scaled scores, as
illustrated in this article.
However, that approach does not take into
account measurement errors, which can
be done if a construct is measured by
several indicators.
This article demonstrates how to fit
a model and test indirect effects when
some of the constructs are latent variables
measured by multiple indicators, with the
measurement error taken into account using
structural equation modeling (SEM).
NOTE: This is part of a series of similar articles
with duplicated sections, each dedicated
to one form of models. See a full list
of articles here.
Data Set
The sample dataset data_indicators
will be used in this illustration.
library(manymome)
dat <- data_indicators
print(round(head(dat), 2))
#> x_1 x_2 x_3 x_4 m1_1 m1_2 m1_3 m1_4 m2_1 m2_2 m2_3 m2_4 m3_1
#> 1 -0.15 2.39 0.70 0.31 0.30 1.06 -0.23 0.58 -0.16 -0.09 0.38 -0.89 0.58
#> 2 -1.25 0.59 -2.42 1.27 -0.21 -1.86 -1.78 -1.75 -2.32 -0.84 -1.19 -0.50 -1.95
#> 3 -0.02 -0.82 -0.49 0.48 0.85 -0.19 2.17 0.49 1.49 2.05 -0.84 -0.29 2.59
#> 4 0.24 -1.13 -1.27 1.81 -0.50 -0.91 -2.01 0.41 0.30 1.22 -1.10 -0.33 1.86
#> 5 0.08 1.36 0.32 0.16 1.19 0.34 1.48 0.64 0.43 1.90 -1.08 1.33 3.43
#> 6 -2.20 -1.77 -1.38 0.54 0.31 -1.88 0.16 0.20 -2.74 -2.74 -1.36 -1.06 -0.61
#> m3_2 m3_3 m3_4 y_1 y_2 y_3 y_4 c1_1 c1_2 c1_3 c1_4 c2_1 c2_2
#> 1 -0.12 2.40 2.05 -2.89 -1.33 -1.83 2.11 0.70 0.09 -2.25 0.27 -1.24 -0.60
#> 2 -2.71 -3.50 -2.26 -1.49 -2.13 -0.24 2.34 0.22 0.70 -0.28 1.43 -0.05 -2.23
#> 3 2.80 -0.16 2.28 1.80 -0.56 1.55 -0.90 1.29 -0.08 1.91 1.26 -0.68 -1.04
#> 4 0.95 1.32 1.09 -0.10 0.28 -2.11 0.56 -1.90 -0.61 -0.75 -2.30 -1.80 0.28
#> 5 2.82 2.91 1.56 2.53 1.92 2.62 -2.18 1.51 2.84 2.88 2.42 -0.85 -0.54
#> 6 -0.90 0.52 1.14 0.58 -1.91 0.71 1.20 1.50 -1.79 1.08 0.17 -1.53 -3.94
#> c2_3 c2_4 x m1 m2 m3 y c1 c2
#> 1 -1.69 -1.52 0.66 0.43 -0.19 1.23 -2.04 -0.30 -1.26
#> 2 -2.93 -2.10 -1.09 -1.40 -1.21 -2.60 -1.55 0.52 -1.83
#> 3 -0.89 -0.09 -0.45 0.83 0.60 1.88 0.92 1.09 -0.68
#> 4 0.07 0.62 -0.99 -0.75 0.02 1.31 -0.62 -1.39 -0.21
#> 5 -2.72 -2.63 0.40 0.91 0.64 2.68 2.31 2.41 -1.69
#> 6 -2.48 -0.73 -1.47 -0.30 -1.98 0.04 -0.45 0.24 -2.17
For illustration, only the following variables
will be used:
-
x_1, x_2, x_3, and x_4:
The indicators
of fx, the predictor.
The indicator x_4 is a reverse items.
-
m1_1, m1_2, m1_3, and m1_4:
The indicators
of fm1, the first mediator.
-
m3_1, m3_2, m3_3, and m3_4:
The indicators
of fm3, the second mediator.
-
y_1, y_2, y_3, and y_4:
The indicators
of fy, the outcome variable.
The indicator y_4 is a reverse items.
-
c1_1, c1_2, c1_3, and c1_4:
The indicators
of fc1, a control variable.
-
c2:
An observed control variable. To
illustrate mixing latent and observed
variables in a model.
Serial Mediation Model
Suppose we would like to fit a simple
mediation model with two mediators,
fm1 and fm3,
with fc1 and c2 as the control
variables.
We would to fit the model above,
and compute and test the indirect
effect along the path fx -> fm1 -> fm3 -> fy,
as well as two other paths,
fx -> fm1 -> y and fx -> fm3 -> y,
using nonparametric bootstrapping.
Analysis
We can do that in one single step using
q_serial_mediation():
out <- q_serial_mediation(
x = "fx",
y = "fy",
m = c("fm1", "fm3"),
cov = c("fc1", "c2"),
indicators = list(
fx = c("x_1", "x_2", "x_3", "-x_4"),
fy = c("y_1", "y_2", "y_3", "-y_4"),
fm1 = c("m1_1", "m1_2", "m1_3", "m1_4"),
fm3 = c("m3_1", "m3_2", "m3_3", "m3_4"),
fc1 = c("c1_1", "c1_2", "c1_3", "c1_4")
),
fit_method = "sem",
indicator_method = "measurement_model",
data = dat,
R = 5000,
seed = 1234
)
These are the arguments:
-
x: The name of the predictor ("x" variable).
-
y: The name of the outcome ("y" variable).
-
m: A vector of the names of the mediators
("m" variables). The order of the names represents
the order of the mediators, from the
first name to the last name. Therefore,
if the path is x -> fm1 -> fm3 -> y,
set this to c("fm1", "fm3").
-
cov: The name(s) of the control variable(s)
(covariate(s)). By default, they predict both
the mediator and the outcome.
-
indicators: The indicators for latent
variables. It should be a named list of
variable names. The name of each element
is the name of the latent variable. If
an indicator is a reverse item, add
a minus sign, -, before its name. It
will be reverse-coded (multiplied by -1)
before fitting the model. If a variable
in x, y, m, or cov is not one
of the latent factor, it is an observed
variable.
-
fit_method and indicator_method:
To fit a model with the measurement part,
using the information from indicators,
set fit_method to "sem" and
indicator_method to "measurement_model"
-
data: The data frame for the model.
-
R: The number of bootstrap samples.
Should be at least 5000 but can be
larger for stable results.
-
seed: The seed for the random number
generator. Should always be set to an
integer to make the results reproducible.
The computation may take some time to run,
especially if missing data is presence,
due to the processing time combining
bootstrapping and full-information maximum
likelihood (FIML), the default way to
handle missing data.
Results
We can then just print the output:
out
The printout is long. They will be
discussed section-by-section below.
These are the main sections of the
default results:
Basic Information
This is the section Basic Information:
#> ===================================================
#> | Basic Information |
#> ===================================================
#>
#> Predictor(x): fx
#> Outcome(y): fy
#> Mediator(s)(m): fm1, fm3
#> Model: Serial Mediation Model
#> Indicators Method: Fit a measurement model
#>
#> The path model fitted:
#>
#> fm1 ~ fx + fc1 + c2
#> fm3 ~ fx + fm1 + fc1 + c2
#> fy ~ fm1 + fm3 + fx + fc1 + c2
#> fx =~ x_1 + x_2 + x_3 + x_4
#> fy =~ y_1 + y_2 + y_3 + y_4
#> fm1 =~ m1_1 + m1_2 + m1_3 + m1_4
#> fm3 =~ m3_1 + m3_2 + m3_3 + m3_4
#> fc1 =~ c1_1 + c1_2 + c1_3 + c1_4
#>
#> The original number of cases: 600
#> The number of cases in the analysis: 600
#> Missing data handling: FIML (full information maximum likelihood)
It shows the variables, the models, and the
number of cases. It also shows the
measurement model in the model, in
lavaan syntax:
-
The name on the left-hand side of =~
is the latent variable.
-
The name on the right-hand side of =~,
joined by +, are the indicators.
When the model is fitted by SEM, the default
method to handle missing data is
full-information maximum likelihood (FIML).
Indicator Information
#> ===================================================
#> | Indicator Information |
#> ===================================================
#>
#> The indicators for the following variable(s):
#>
#> fx: x_1, x_2, x_3, -x_4
#> fy: y_1, y_2, y_3, -y_4
#> fm1: m1_1, m1_2, m1_3, m1_4
#> fm3: m3_1, m3_2, m3_3, m3_4
#> fc1: c1_1, c1_2, c1_3, c1_4
#>
#> Note:
#> - '-' denotes revserse-coded items.
#>
#> The standardized factor loadings:
#>
#> fx:
#> Reliability: 0.7222
#> Loading
#> x_1 0.5686
#> x_2 0.5981
#> x_3 0.6588
#> x_4 0.6791
#>
#> fy:
#> Reliability: 0.8813
#> Loading
#> y_1 0.8295
#> y_2 0.7811
#> y_3 0.7991
#> y_4 0.8134
#>
#> fm1:
#> Reliability: 0.7746
#> Loading
#> m1_1 0.6862
#> m1_2 0.6212
#> m1_3 0.7230
#> m1_4 0.6820
#>
#> fm3:
#> Reliability: 0.8186
#> Loading
#> m3_1 0.7658
#> m3_2 0.7265
#> m3_3 0.7217
#> m3_4 0.6964
#>
#> fc1:
#> Reliability: 0.7150
#> Loading
#> c1_1 0.6025
#> c1_2 0.6565
#> c1_3 0.5576
#> c1_4 0.6624
#>
#> Note:
#> - Revserse-coded items have been reverse-coded when estimating the
#> loadings.
#> - If the loading of an item is negative, its coding (revserse or
#> non-reverse) may be incorrect.
#> - Reliability estimated by omage coefficients.
The section Indicator Information shows
results related to each latent factor:
-
The indicators for each latent variable.
-
The standardized factor loadings from
the SEM results.
- Note that, after recoding, all
loadings should be positive. If
any loading is negative, it may be
a reverse indicator not specified
in
indicators.
-
The reliability coefficient for each
latent factor, based on the factor loadings.
Omega coefficient is used, computed by
semTools::compRelSEM().
Model Fit
The section Structural Equation Modeling Results
print the results from lavaan for the model.
#> Model Test User Model:
#>
#> Test statistic 186.674
#> Degrees of freedom 177
#> P-value (Chi-square) 0.294
#>
#> Model Test Baseline Model:
#>
#> Test statistic 4333.127
#> Degrees of freedom 210
#> P-value 0.000
#>
#> User Model versus Baseline Model:
#>
#> Comparative Fit Index (CFI) 0.998
#> Tucker-Lewis Index (TLI) 0.997
#>
#> Root Mean Square Error of Approximation:
#>
#> RMSEA 0.010
#> Confidence interval - lower 0.000
#> Confidence interval - upper 0.021
#> P-value H_0: RMSEA <= 0.050 1.000
#> P-value H_0: RMSEA >= 0.080 0.000
#>
#> Standardized Root Mean Square Residual:
#>
#> SRMR 0.029
First, results on goodness-of-fit are printed:
-
Model Test User Model reports the
model $\chi^2$ and its df and p-value.
-
Basic fit measures (CFI, TLI, RMSEA,
and SRMR) are also printed.
Regression Results
The next few sections print the
regression coefficients when predicting
the mediator (fm in this model) and
the outcome variable (fy in this model).
These are the results in predicting fm1
and fm3:
#> ----------------
#> Predicting fm1 :
#> ----------------
#>
#> Model:
#> fm1 ~ fx + fc1 + c2
#> Estimate CI.lo CI.hi betaS Std. Error z value Pr(>|z|)
#> (Intercept) 0.0000 0.0000 0.0000 ---- 0.0000 ---- ----
#> fx 0.6809 0.4794 0.8825 0.4904 0.1028 6.622 <2e-16 ***
#> fc1 0.0759 -0.0811 0.2329 0.0577 0.0801 0.948 0.343
#> c2 0.0446 -0.0464 0.1356 0.0415 0.0464 0.961 0.336
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> R-square: 0.2706
#>
#> Chi-Squared Difference Test for the R-square
#>
#> Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
#> fit1 177 38319 38640 186.67
#> fit0 180 38419 38727 292.55 105.87 3 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> - BetaS are standardized coefficients with (a) only numeric variables
#> standardized and (b) product terms formed after standardization.
#> Variable(s) standardized is/are: fm1, fx, fc1, c2
#> - CI.lo and CI.hi are the 95.0% confidence levels of 'Estimate'
#> computed from the z values and standard errors.
#>
#> ----------------
#> Predicting fm3 :
#> ----------------
#>
#> Model:
#> fm3 ~ fx + fm1 + fc1 + c2
#> Estimate CI.lo CI.hi betaS Std. Error z value Pr(>|z|)
#> (Intercept) 0.0000 0.0000 0.0000 ---- 0.0000 ---- ----
#> fx -0.1030 -0.3051 0.0991 -0.0677 0.1031 -0.999 0.318
#> fm1 0.7181 0.5681 0.8681 0.6556 0.0765 9.383 <2e-16 ***
#> fc1 0.0218 -0.1364 0.1800 0.0151 0.0807 0.270 0.787
#> c2 0.0369 -0.0563 0.1302 0.0314 0.0476 0.776 0.438
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> R-square: 0.3960
#>
#> Chi-Squared Difference Test for the R-square
#>
#> Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
#> fit1 177 38319 38640 186.67
#> fit0 181 38486 38790 361.96 175.28 4 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> - BetaS are standardized coefficients with (a) only numeric variables
#> standardized and (b) product terms formed after standardization.
#> Variable(s) standardized is/are: fm3, fx, fm1, fc1, c2
#> - CI.lo and CI.hi are the 95.0% confidence levels of 'Estimate'
#> computed from the z values and standard errors.
These are the results in predicting y:
#> ---------------
#> Predicting fy :
#> ---------------
#>
#> Model:
#> fy ~ fm1 + fm3 + fx + fc1 + c2
#> Estimate CI.lo CI.hi betaS Std. Error z value Pr(>|z|)
#> (Intercept) 0.0000 0.0000 0.0000 ---- 0.0000 ---- ----
#> fm1 0.0202 -0.1662 0.2066 0.0143 0.0951 0.212 0.8318
#> fm3 0.6423 0.4904 0.7943 0.4976 0.0775 8.284 <2e-16 ***
#> fx 0.5571 0.3214 0.7928 0.2838 0.1203 4.632 <2e-16 ***
#> fc1 0.3280 0.1491 0.5070 0.1763 0.0913 3.593 0.0003 ***
#> c2 0.1384 0.0360 0.2408 0.0911 0.0523 2.649 0.0081 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> R-square: 0.5392
#>
#> Chi-Squared Difference Test for the R-square
#>
#> Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
#> fit1 177 38319 38640 186.67
#> fit0 182 38632 38931 509.31 322.64 5 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> - BetaS are standardized coefficients with (a) only numeric variables
#> standardized and (b) product terms formed after standardization.
#> Variable(s) standardized is/are: fy, fm1, fm3, fx, fc1, c2
#> - CI.lo and CI.hi are the 95.0% confidence levels of 'Estimate'
#> computed from the z values and standard errors.
The results from lavaan::sem() are
reformatted to make them similar to those
by regression analysis.
If desired, the output of lavaan::sem()
can be extracted from the output by
get_fit().
Indirect Effect Results
By default, this section prints the estimated
indirect effect, confidence interval,
and asymmetric bootstrap p-value.
Four sections will be printed:
- The original indirect effect.
#> == Indirect Effect(s) ==
#>
#> ind CI.lo CI.hi Sig pvalue SE
#> fx -> fm1 -> fy 0.0138 -0.1409 0.1421 0.8632 0.0710
#> fx -> fm1 -> fm3 -> fy 0.3141 0.1998 0.4747 Sig 0.0000 0.0698
#> fx -> fm3 -> fy -0.0661 -0.2091 0.0554 0.2936 0.0666
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 5000 samples.
#> - [pvalue] are asymmetric bootstrap p-values.
#> - [SE] are standard errors.
#> - The 'ind' column shows the indirect effect(s).
- The indirect effect with the predictor ("x") standardized.
#> == Indirect Effect(s) (x-variable(s) Standardized) ==
#>
#> std CI.lo CI.hi Sig pvalue SE
#> fx -> fm1 -> fy 0.0099 -0.0989 0.1002 0.8632 0.0504
#> fx -> fm1 -> fm3 -> fy 0.2253 0.1476 0.3289 Sig 0.0000 0.0465
#> fx -> fm3 -> fy -0.0474 -0.1447 0.0400 0.2936 0.0466
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 5000 samples.
#> - [pvalue] are asymmetric bootstrap p-values.
#> - [SE] are standard errors.
#> - std: The partially standardized indirect effect(s).
#> - x-variable(s) standardized.
- The indirect effect with the outcome ("y") standardized.
#> == Indirect Effect(s) (y-variable(s) Standardized) ==
#>
#> std CI.lo CI.hi Sig pvalue SE
#> fx -> fm1 -> fy 0.0098 -0.0995 0.1006 0.8632 0.0506
#> fx -> fm1 -> fm3 -> fy 0.2231 0.1446 0.3355 Sig 0.0000 0.0489
#> fx -> fm3 -> fy -0.0470 -0.1494 0.0396 0.2936 0.0474
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 5000 samples.
#> - [pvalue] are asymmetric bootstrap p-values.
#> - [SE] are standard errors.
#> - std: The partially standardized indirect effect(s).
#> - y-variable(s) standardized.
- The indirect effect with both
x and y standardized.
#> == Indirect Effect(s) (Both x-variable(s) and y-variable(s) Standardized) ==
#>
#> std CI.lo CI.hi Sig pvalue SE
#> fx -> fm1 -> fy 0.0070 -0.0689 0.0723 0.8632 0.0359
#> fx -> fm1 -> fm3 -> fy 0.1600 0.1060 0.2316 Sig 0.0000 0.0322
#> fx -> fm3 -> fy -0.0337 -0.1034 0.0285 0.2936 0.0332
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 5000 samples.
#> - [pvalue] are asymmetric bootstrap p-values.
#> - [SE] are standard errors.
#> - std: The standardized indirect effect(s).
The indirect effect with both the predictor
("x") and the outcome ("y") standardized,
also called the completely
standardized indirect effect or
simply the standardized indirect
effect.
These four versions of the results are
printed by default so that users can
select and interpret sections as they
see fit.
Direct Effect Results
The section Direct Effect Results prints the estimated
direct effect (from the predictor "x" to
the outcome "y", not mediated), confidence interval,
and asymmetric bootstrap p-value.
#> ===================================================
#> | Direct Effect Results |
#> ===================================================
#>
#> ----------------------------------------------------------------
#>
#> == Effect(s) ==
#>
#> ind CI.lo CI.hi Sig pvalue SE
#> fx -> fy 0.5571 0.3234 0.8378 Sig 0.0000 0.1302
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 5000 samples.
#> - [pvalue] are asymmetric bootstrap p-values.
#> - [SE] are standard errors.
#> - The 'ind' column shows the effect(s).
#>
#> ----------------------------------------------------------------
#>
#> == Effect(s) (x-variable(s) Standardized) ==
#>
#> std CI.lo CI.hi Sig pvalue SE
#> fx -> fy 0.3995 0.2342 0.5737 Sig 0.0000 0.0869
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 5000 samples.
#> - [pvalue] are asymmetric bootstrap p-values.
#> - [SE] are standard errors.
#> - std: The partially standardized effect(s).
#> - x-variable(s) standardized.
#>
#> ----------------------------------------------------------------
#>
#> == Effect(s) (y-variable(s) Standardized) ==
#>
#> std CI.lo CI.hi Sig pvalue SE
#> fx -> fy 0.3957 0.2324 0.5950 Sig 0.0000 0.0910
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 5000 samples.
#> - [pvalue] are asymmetric bootstrap p-values.
#> - [SE] are standard errors.
#> - std: The partially standardized effect(s).
#> - y-variable(s) standardized.
#>
#> ----------------------------------------------------------------
#>
#> == Effect(s) (Both x-variable(s) and y-variable(s) Standardized) ==
#>
#> std CI.lo CI.hi Sig pvalue SE
#> fx -> fy 0.2838 0.1677 0.4048 Sig 0.0000 0.0603
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 5000 samples.
#> - [pvalue] are asymmetric bootstrap p-values.
#> - [SE] are standard errors.
The z-test of the direct effect is
already available in section
Structural Equation Modeling Results.
The bootstrap results are printed in
case users prefer using the same
confidence interval method for all
the effects.
Additional Issues
Customize Control Variables
If the control variables for the
regression models are different, we can
set cov to a named list. The names are
the variables with control variables,
and the element under each name is
a character vector of the control variables.
For example,
- If we set
cov to list(m1 = "c1", m2 = "c2", y = c("c1", "c2")),
then only c1 is included in predicting m1,
only c2 is included in predicting m2,
while both c1 and c2 are included
in predicting y.
A variable that does not appear in the
list does not have control variables.
Customize The Printout
The print method of the output of
the quick mediation functions have
arguments for customizing the output.
These are arguments that likely may be
used:
-
digits: The number of digits after
the decimal place for most results.
Default is 4.
-
pvalue_digits: The number of digits
after the decimal place for p-values.
Default is 4.
See the help page of print.q_mediation()
for other arguments.
Speed and Parallel Processing
By default, parallel processing is used.
If this failed for some reasons,
add parallel to FALSE. It will take
longer, sometimes much long, to run
if parallel is set to FALSE. Therefore,
use parallel processing whenever possible.
Progress Bar
By default, a progress bar will be displayed
when doing bootstrapping. This can be
disabled by adding progress = FALSE.
Workflow
The quick functions are simply functions
to do the following tasks internally:
-
Fit all the models by SEM
using lavaan::sem().
-
Call all_indirect_paths() to
identify all indirect paths.
-
Call many_indirect_effects() to
compute all indirect effects and
form their confidence intervals.
-
Call total_indirect_effect() to
compute the total indirect effect.
Therefore, all the tasks they do can be
done manually by the functions above.
These all-in-one functions are developed just
as convenient functions to do all these
tasks in one call.
See this
article
for computing and testing indirect effects
for more complicated models.
Final Remarks
For details on the all-in-one functions,
please refer to the help page of
q_mediation().
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Introduction
(This document is for Version 0.3.4.25 or later, currently on GitHub only. The new version is scheduled to be available on CRAN in June 2026.)
This article is a brief illustration of how to use the all-in-one "quick" functions from manymome (Cheung & Cheung, 2024) to fit a structural model and compute and test indirect effects in one step.
It is similar to using them on observed scores, such as scaled scores, as illustrated in this article. However, that approach does not take into account measurement errors, which can be done if a construct is measured by several indicators.
This article demonstrates how to fit a model and test indirect effects when some of the constructs are latent variables measured by multiple indicators, with the measurement error taken into account using structural equation modeling (SEM).
NOTE: This is part of a series of similar articles with duplicated sections, each dedicated to one form of models. See a full list of articles here.
Data Set
The sample dataset data_indicators
will be used in this illustration.
library(manymome) dat <- data_indicators print(round(head(dat), 2)) #> x_1 x_2 x_3 x_4 m1_1 m1_2 m1_3 m1_4 m2_1 m2_2 m2_3 m2_4 m3_1 #> 1 -0.15 2.39 0.70 0.31 0.30 1.06 -0.23 0.58 -0.16 -0.09 0.38 -0.89 0.58 #> 2 -1.25 0.59 -2.42 1.27 -0.21 -1.86 -1.78 -1.75 -2.32 -0.84 -1.19 -0.50 -1.95 #> 3 -0.02 -0.82 -0.49 0.48 0.85 -0.19 2.17 0.49 1.49 2.05 -0.84 -0.29 2.59 #> 4 0.24 -1.13 -1.27 1.81 -0.50 -0.91 -2.01 0.41 0.30 1.22 -1.10 -0.33 1.86 #> 5 0.08 1.36 0.32 0.16 1.19 0.34 1.48 0.64 0.43 1.90 -1.08 1.33 3.43 #> 6 -2.20 -1.77 -1.38 0.54 0.31 -1.88 0.16 0.20 -2.74 -2.74 -1.36 -1.06 -0.61 #> m3_2 m3_3 m3_4 y_1 y_2 y_3 y_4 c1_1 c1_2 c1_3 c1_4 c2_1 c2_2 #> 1 -0.12 2.40 2.05 -2.89 -1.33 -1.83 2.11 0.70 0.09 -2.25 0.27 -1.24 -0.60 #> 2 -2.71 -3.50 -2.26 -1.49 -2.13 -0.24 2.34 0.22 0.70 -0.28 1.43 -0.05 -2.23 #> 3 2.80 -0.16 2.28 1.80 -0.56 1.55 -0.90 1.29 -0.08 1.91 1.26 -0.68 -1.04 #> 4 0.95 1.32 1.09 -0.10 0.28 -2.11 0.56 -1.90 -0.61 -0.75 -2.30 -1.80 0.28 #> 5 2.82 2.91 1.56 2.53 1.92 2.62 -2.18 1.51 2.84 2.88 2.42 -0.85 -0.54 #> 6 -0.90 0.52 1.14 0.58 -1.91 0.71 1.20 1.50 -1.79 1.08 0.17 -1.53 -3.94 #> c2_3 c2_4 x m1 m2 m3 y c1 c2 #> 1 -1.69 -1.52 0.66 0.43 -0.19 1.23 -2.04 -0.30 -1.26 #> 2 -2.93 -2.10 -1.09 -1.40 -1.21 -2.60 -1.55 0.52 -1.83 #> 3 -0.89 -0.09 -0.45 0.83 0.60 1.88 0.92 1.09 -0.68 #> 4 0.07 0.62 -0.99 -0.75 0.02 1.31 -0.62 -1.39 -0.21 #> 5 -2.72 -2.63 0.40 0.91 0.64 2.68 2.31 2.41 -1.69 #> 6 -2.48 -0.73 -1.47 -0.30 -1.98 0.04 -0.45 0.24 -2.17
For illustration, only the following variables will be used:
-
x_1,x_2,x_3, andx_4: The indicators offx, the predictor. The indicatorx_4is a reverse items. -
m1_1,m1_2,m1_3, andm1_4: The indicators offm1, the first mediator. -
m3_1,m3_2,m3_3, andm3_4: The indicators offm3, the second mediator. -
y_1,y_2,y_3, andy_4: The indicators offy, the outcome variable. The indicatory_4is a reverse items. -
c1_1,c1_2,c1_3, andc1_4: The indicators offc1, a control variable. -
c2: An observed control variable. To illustrate mixing latent and observed variables in a model.
Serial Mediation Model
Suppose we would like to fit a simple
mediation model with two mediators,
fm1 and fm3,
with fc1 and c2 as the control
variables.
We would to fit the model above,
and compute and test the indirect
effect along the path fx -> fm1 -> fm3 -> fy,
as well as two other paths,
fx -> fm1 -> y and fx -> fm3 -> y,
using nonparametric bootstrapping.
Analysis
We can do that in one single step using
q_serial_mediation():
out <- q_serial_mediation( x = "fx", y = "fy", m = c("fm1", "fm3"), cov = c("fc1", "c2"), indicators = list( fx = c("x_1", "x_2", "x_3", "-x_4"), fy = c("y_1", "y_2", "y_3", "-y_4"), fm1 = c("m1_1", "m1_2", "m1_3", "m1_4"), fm3 = c("m3_1", "m3_2", "m3_3", "m3_4"), fc1 = c("c1_1", "c1_2", "c1_3", "c1_4") ), fit_method = "sem", indicator_method = "measurement_model", data = dat, R = 5000, seed = 1234 )
These are the arguments:
-
x: The name of the predictor ("x" variable). -
y: The name of the outcome ("y" variable). -
m: A vector of the names of the mediators ("m" variables). The order of the names represents the order of the mediators, from the first name to the last name. Therefore, if the path isx -> fm1 -> fm3 -> y, set this toc("fm1", "fm3"). -
cov: The name(s) of the control variable(s) (covariate(s)). By default, they predict both the mediator and the outcome. -
indicators: The indicators for latent variables. It should be a named list of variable names. The name of each element is the name of the latent variable. If an indicator is a reverse item, add a minus sign,-, before its name. It will be reverse-coded (multiplied by-1) before fitting the model. If a variable inx,y,m, orcovis not one of the latent factor, it is an observed variable. -
fit_methodandindicator_method: To fit a model with the measurement part, using the information fromindicators, setfit_methodto"sem"andindicator_methodto"measurement_model" -
data: The data frame for the model. -
R: The number of bootstrap samples. Should be at least 5000 but can be larger for stable results. -
seed: The seed for the random number generator. Should always be set to an integer to make the results reproducible.
The computation may take some time to run, especially if missing data is presence, due to the processing time combining bootstrapping and full-information maximum likelihood (FIML), the default way to handle missing data.
Results
We can then just print the output:
out
The printout is long. They will be discussed section-by-section below.
These are the main sections of the default results:
Basic Information
This is the section Basic Information:
#> =================================================== #> | Basic Information | #> =================================================== #> #> Predictor(x): fx #> Outcome(y): fy #> Mediator(s)(m): fm1, fm3 #> Model: Serial Mediation Model #> Indicators Method: Fit a measurement model #> #> The path model fitted: #> #> fm1 ~ fx + fc1 + c2 #> fm3 ~ fx + fm1 + fc1 + c2 #> fy ~ fm1 + fm3 + fx + fc1 + c2 #> fx =~ x_1 + x_2 + x_3 + x_4 #> fy =~ y_1 + y_2 + y_3 + y_4 #> fm1 =~ m1_1 + m1_2 + m1_3 + m1_4 #> fm3 =~ m3_1 + m3_2 + m3_3 + m3_4 #> fc1 =~ c1_1 + c1_2 + c1_3 + c1_4 #> #> The original number of cases: 600 #> The number of cases in the analysis: 600 #> Missing data handling: FIML (full information maximum likelihood)
It shows the variables, the models, and the
number of cases. It also shows the
measurement model in the model, in
lavaan syntax:
-
The name on the left-hand side of
=~is the latent variable. -
The name on the right-hand side of
=~, joined by+, are the indicators.
When the model is fitted by SEM, the default method to handle missing data is full-information maximum likelihood (FIML).
Indicator Information
#> =================================================== #> | Indicator Information | #> =================================================== #> #> The indicators for the following variable(s): #> #> fx: x_1, x_2, x_3, -x_4 #> fy: y_1, y_2, y_3, -y_4 #> fm1: m1_1, m1_2, m1_3, m1_4 #> fm3: m3_1, m3_2, m3_3, m3_4 #> fc1: c1_1, c1_2, c1_3, c1_4 #> #> Note: #> - '-' denotes revserse-coded items. #> #> The standardized factor loadings: #> #> fx: #> Reliability: 0.7222 #> Loading #> x_1 0.5686 #> x_2 0.5981 #> x_3 0.6588 #> x_4 0.6791 #> #> fy: #> Reliability: 0.8813 #> Loading #> y_1 0.8295 #> y_2 0.7811 #> y_3 0.7991 #> y_4 0.8134 #> #> fm1: #> Reliability: 0.7746 #> Loading #> m1_1 0.6862 #> m1_2 0.6212 #> m1_3 0.7230 #> m1_4 0.6820 #> #> fm3: #> Reliability: 0.8186 #> Loading #> m3_1 0.7658 #> m3_2 0.7265 #> m3_3 0.7217 #> m3_4 0.6964 #> #> fc1: #> Reliability: 0.7150 #> Loading #> c1_1 0.6025 #> c1_2 0.6565 #> c1_3 0.5576 #> c1_4 0.6624 #> #> Note: #> - Revserse-coded items have been reverse-coded when estimating the #> loadings. #> - If the loading of an item is negative, its coding (revserse or #> non-reverse) may be incorrect. #> - Reliability estimated by omage coefficients.
The section Indicator Information shows
results related to each latent factor:
-
The indicators for each latent variable.
-
The standardized factor loadings from the SEM results.
- Note that, after recoding, all
loadings should be positive. If
any loading is negative, it may be
a reverse indicator not specified
in
indicators.
- Note that, after recoding, all
loadings should be positive. If
any loading is negative, it may be
a reverse indicator not specified
in
-
The reliability coefficient for each latent factor, based on the factor loadings. Omega coefficient is used, computed by
semTools::compRelSEM().
Model Fit
The section Structural Equation Modeling Results
print the results from lavaan for the model.
#> Model Test User Model: #> #> Test statistic 186.674 #> Degrees of freedom 177 #> P-value (Chi-square) 0.294 #> #> Model Test Baseline Model: #> #> Test statistic 4333.127 #> Degrees of freedom 210 #> P-value 0.000 #> #> User Model versus Baseline Model: #> #> Comparative Fit Index (CFI) 0.998 #> Tucker-Lewis Index (TLI) 0.997 #> #> Root Mean Square Error of Approximation: #> #> RMSEA 0.010 #> Confidence interval - lower 0.000 #> Confidence interval - upper 0.021 #> P-value H_0: RMSEA <= 0.050 1.000 #> P-value H_0: RMSEA >= 0.080 0.000 #> #> Standardized Root Mean Square Residual: #> #> SRMR 0.029
First, results on goodness-of-fit are printed:
-
Model Test User Modelreports the model $\chi^2$ and its df and p-value. -
Basic fit measures (CFI, TLI, RMSEA, and SRMR) are also printed.
Regression Results
The next few sections print the
regression coefficients when predicting
the mediator (fm in this model) and
the outcome variable (fy in this model).
These are the results in predicting fm1
and fm3:
#> ---------------- #> Predicting fm1 : #> ---------------- #> #> Model: #> fm1 ~ fx + fc1 + c2 #> Estimate CI.lo CI.hi betaS Std. Error z value Pr(>|z|) #> (Intercept) 0.0000 0.0000 0.0000 ---- 0.0000 ---- ---- #> fx 0.6809 0.4794 0.8825 0.4904 0.1028 6.622 <2e-16 *** #> fc1 0.0759 -0.0811 0.2329 0.0577 0.0801 0.948 0.343 #> c2 0.0446 -0.0464 0.1356 0.0415 0.0464 0.961 0.336 #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> R-square: 0.2706 #> #> Chi-Squared Difference Test for the R-square #> #> Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq) #> fit1 177 38319 38640 186.67 #> fit0 180 38419 38727 292.55 105.87 3 < 2.2e-16 *** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> - BetaS are standardized coefficients with (a) only numeric variables #> standardized and (b) product terms formed after standardization. #> Variable(s) standardized is/are: fm1, fx, fc1, c2 #> - CI.lo and CI.hi are the 95.0% confidence levels of 'Estimate' #> computed from the z values and standard errors. #> #> ---------------- #> Predicting fm3 : #> ---------------- #> #> Model: #> fm3 ~ fx + fm1 + fc1 + c2 #> Estimate CI.lo CI.hi betaS Std. Error z value Pr(>|z|) #> (Intercept) 0.0000 0.0000 0.0000 ---- 0.0000 ---- ---- #> fx -0.1030 -0.3051 0.0991 -0.0677 0.1031 -0.999 0.318 #> fm1 0.7181 0.5681 0.8681 0.6556 0.0765 9.383 <2e-16 *** #> fc1 0.0218 -0.1364 0.1800 0.0151 0.0807 0.270 0.787 #> c2 0.0369 -0.0563 0.1302 0.0314 0.0476 0.776 0.438 #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> R-square: 0.3960 #> #> Chi-Squared Difference Test for the R-square #> #> Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq) #> fit1 177 38319 38640 186.67 #> fit0 181 38486 38790 361.96 175.28 4 < 2.2e-16 *** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> - BetaS are standardized coefficients with (a) only numeric variables #> standardized and (b) product terms formed after standardization. #> Variable(s) standardized is/are: fm3, fx, fm1, fc1, c2 #> - CI.lo and CI.hi are the 95.0% confidence levels of 'Estimate' #> computed from the z values and standard errors.
These are the results in predicting y:
#> --------------- #> Predicting fy : #> --------------- #> #> Model: #> fy ~ fm1 + fm3 + fx + fc1 + c2 #> Estimate CI.lo CI.hi betaS Std. Error z value Pr(>|z|) #> (Intercept) 0.0000 0.0000 0.0000 ---- 0.0000 ---- ---- #> fm1 0.0202 -0.1662 0.2066 0.0143 0.0951 0.212 0.8318 #> fm3 0.6423 0.4904 0.7943 0.4976 0.0775 8.284 <2e-16 *** #> fx 0.5571 0.3214 0.7928 0.2838 0.1203 4.632 <2e-16 *** #> fc1 0.3280 0.1491 0.5070 0.1763 0.0913 3.593 0.0003 *** #> c2 0.1384 0.0360 0.2408 0.0911 0.0523 2.649 0.0081 ** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> R-square: 0.5392 #> #> Chi-Squared Difference Test for the R-square #> #> Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq) #> fit1 177 38319 38640 186.67 #> fit0 182 38632 38931 509.31 322.64 5 < 2.2e-16 *** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> - BetaS are standardized coefficients with (a) only numeric variables #> standardized and (b) product terms formed after standardization. #> Variable(s) standardized is/are: fy, fm1, fm3, fx, fc1, c2 #> - CI.lo and CI.hi are the 95.0% confidence levels of 'Estimate' #> computed from the z values and standard errors.
The results from lavaan::sem() are
reformatted to make them similar to those
by regression analysis.
If desired, the output of lavaan::sem()
can be extracted from the output by
get_fit().
Indirect Effect Results
By default, this section prints the estimated indirect effect, confidence interval, and asymmetric bootstrap p-value.
Four sections will be printed:
- The original indirect effect.
#> == Indirect Effect(s) == #> #> ind CI.lo CI.hi Sig pvalue SE #> fx -> fm1 -> fy 0.0138 -0.1409 0.1421 0.8632 0.0710 #> fx -> fm1 -> fm3 -> fy 0.3141 0.1998 0.4747 Sig 0.0000 0.0698 #> fx -> fm3 -> fy -0.0661 -0.2091 0.0554 0.2936 0.0666 #> #> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by #> nonparametric bootstrapping with 5000 samples. #> - [pvalue] are asymmetric bootstrap p-values. #> - [SE] are standard errors. #> - The 'ind' column shows the indirect effect(s).
- The indirect effect with the predictor ("x") standardized.
#> == Indirect Effect(s) (x-variable(s) Standardized) == #> #> std CI.lo CI.hi Sig pvalue SE #> fx -> fm1 -> fy 0.0099 -0.0989 0.1002 0.8632 0.0504 #> fx -> fm1 -> fm3 -> fy 0.2253 0.1476 0.3289 Sig 0.0000 0.0465 #> fx -> fm3 -> fy -0.0474 -0.1447 0.0400 0.2936 0.0466 #> #> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by #> nonparametric bootstrapping with 5000 samples. #> - [pvalue] are asymmetric bootstrap p-values. #> - [SE] are standard errors. #> - std: The partially standardized indirect effect(s). #> - x-variable(s) standardized.
- The indirect effect with the outcome ("y") standardized.
#> == Indirect Effect(s) (y-variable(s) Standardized) == #> #> std CI.lo CI.hi Sig pvalue SE #> fx -> fm1 -> fy 0.0098 -0.0995 0.1006 0.8632 0.0506 #> fx -> fm1 -> fm3 -> fy 0.2231 0.1446 0.3355 Sig 0.0000 0.0489 #> fx -> fm3 -> fy -0.0470 -0.1494 0.0396 0.2936 0.0474 #> #> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by #> nonparametric bootstrapping with 5000 samples. #> - [pvalue] are asymmetric bootstrap p-values. #> - [SE] are standard errors. #> - std: The partially standardized indirect effect(s). #> - y-variable(s) standardized.
- The indirect effect with both
xandystandardized.
#> == Indirect Effect(s) (Both x-variable(s) and y-variable(s) Standardized) == #> #> std CI.lo CI.hi Sig pvalue SE #> fx -> fm1 -> fy 0.0070 -0.0689 0.0723 0.8632 0.0359 #> fx -> fm1 -> fm3 -> fy 0.1600 0.1060 0.2316 Sig 0.0000 0.0322 #> fx -> fm3 -> fy -0.0337 -0.1034 0.0285 0.2936 0.0332 #> #> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by #> nonparametric bootstrapping with 5000 samples. #> - [pvalue] are asymmetric bootstrap p-values. #> - [SE] are standard errors. #> - std: The standardized indirect effect(s).
The indirect effect with both the predictor ("x") and the outcome ("y") standardized, also called the completely standardized indirect effect or simply the standardized indirect effect.
These four versions of the results are printed by default so that users can select and interpret sections as they see fit.
Direct Effect Results
The section Direct Effect Results prints the estimated
direct effect (from the predictor "x" to
the outcome "y", not mediated), confidence interval,
and asymmetric bootstrap p-value.
#> =================================================== #> | Direct Effect Results | #> =================================================== #> #> ---------------------------------------------------------------- #> #> == Effect(s) == #> #> ind CI.lo CI.hi Sig pvalue SE #> fx -> fy 0.5571 0.3234 0.8378 Sig 0.0000 0.1302 #> #> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by #> nonparametric bootstrapping with 5000 samples. #> - [pvalue] are asymmetric bootstrap p-values. #> - [SE] are standard errors. #> - The 'ind' column shows the effect(s). #> #> ---------------------------------------------------------------- #> #> == Effect(s) (x-variable(s) Standardized) == #> #> std CI.lo CI.hi Sig pvalue SE #> fx -> fy 0.3995 0.2342 0.5737 Sig 0.0000 0.0869 #> #> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by #> nonparametric bootstrapping with 5000 samples. #> - [pvalue] are asymmetric bootstrap p-values. #> - [SE] are standard errors. #> - std: The partially standardized effect(s). #> - x-variable(s) standardized. #> #> ---------------------------------------------------------------- #> #> == Effect(s) (y-variable(s) Standardized) == #> #> std CI.lo CI.hi Sig pvalue SE #> fx -> fy 0.3957 0.2324 0.5950 Sig 0.0000 0.0910 #> #> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by #> nonparametric bootstrapping with 5000 samples. #> - [pvalue] are asymmetric bootstrap p-values. #> - [SE] are standard errors. #> - std: The partially standardized effect(s). #> - y-variable(s) standardized. #> #> ---------------------------------------------------------------- #> #> == Effect(s) (Both x-variable(s) and y-variable(s) Standardized) == #> #> std CI.lo CI.hi Sig pvalue SE #> fx -> fy 0.2838 0.1677 0.4048 Sig 0.0000 0.0603 #> #> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by #> nonparametric bootstrapping with 5000 samples. #> - [pvalue] are asymmetric bootstrap p-values. #> - [SE] are standard errors.
The z-test of the direct effect is already available in section Structural Equation Modeling Results. The bootstrap results are printed in case users prefer using the same confidence interval method for all the effects.
Additional Issues
Customize Control Variables
If the control variables for the
regression models are different, we can
set cov to a named list. The names are
the variables with control variables,
and the element under each name is
a character vector of the control variables.
For example,
- If we set
covtolist(m1 = "c1", m2 = "c2", y = c("c1", "c2")), then onlyc1is included in predictingm1, onlyc2is included in predictingm2, while bothc1andc2are included in predictingy.
A variable that does not appear in the list does not have control variables.
Customize The Printout
The print method of the output of
the quick mediation functions have
arguments for customizing the output.
These are arguments that likely may be
used:
-
digits: The number of digits after the decimal place for most results. Default is 4. -
pvalue_digits: The number of digits after the decimal place for p-values. Default is 4.
See the help page of print.q_mediation()
for other arguments.
Speed and Parallel Processing
By default, parallel processing is used.
If this failed for some reasons,
add parallel to FALSE. It will take
longer, sometimes much long, to run
if parallel is set to FALSE. Therefore,
use parallel processing whenever possible.
Progress Bar
By default, a progress bar will be displayed
when doing bootstrapping. This can be
disabled by adding progress = FALSE.
Workflow
The quick functions are simply functions to do the following tasks internally:
-
Fit all the models by SEM using
lavaan::sem(). -
Call
all_indirect_paths()to identify all indirect paths. -
Call
many_indirect_effects()to compute all indirect effects and form their confidence intervals. -
Call
total_indirect_effect()to compute the total indirect effect.
Therefore, all the tasks they do can be done manually by the functions above. These all-in-one functions are developed just as convenient functions to do all these tasks in one call.
See this article for computing and testing indirect effects for more complicated models.
Final Remarks
For details on the all-in-one functions,
please refer to the help page of
q_mediation().
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