Nothing
Moderated Regression: Two Predictors and One Moderator"
In manymome: Mediation, Moderation and Moderated-Mediation After Model Fitting
Introduction
This article is part of a series of
brief illustrations of how
to use
cond_effects()
from the package
manymome
[@cheung_manymome_2024]
to estimate the conditional
effects
when the model parameters are estimated by
ordinary least squares (OLS) multiple regression
using lm(). For moderated mediation
tested by OLS regression, please refer
to this article.
(Articles in this series had duplicated
sections to make each of them self-contained.)
Data Set and Model
This is the sample data set used for
illustration:
library(manymome)
dat <- data_mod_2x1w
print(head(dat), digits = 3)
#> y x1 x2 w c1 c2
#> 1 2.05 1.80 2.75 2.09 1.83 1.60
#> 2 1.41 1.87 2.78 1.94 1.31 1.22
#> 3 1.94 1.74 3.06 2.39 1.50 1.29
#> 4 0.84 1.73 2.56 1.13 1.41 1.18
#> 5 1.33 0.95 2.86 1.32 1.45 1.11
#> 6 1.31 2.12 2.82 1.67 1.66 1.36
This dataset has 6 variables:
-
one outcome variable (y),
-
two predictors (x1, x2),
-
one moderator (w),
-
two control variables (c1 and c2).
Suppose this is the model being fitted,
with control variables omitted from the
plot for readability:
Fit by Regression
The path parameters
can be estimated by multiple regression
using lm():
lm_y <- lm(
y ~ w*x1 + w*x2 + c1 + c2,
data = dat
)
These are the estimates of the regression coefficients
of the paths:
summary(lm_y)
#>
#> Call:
#> lm(formula = y ~ w * x1 + w * x2 + c1 + c2, data = dat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.46049 -0.12989 -0.00299 0.11899 0.54333
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 3.91686 0.46699 8.387 1.06e-14 ***
#> w -1.94839 0.25990 -7.497 2.35e-12 ***
#> x1 -0.63573 0.18248 -3.484 0.000612 ***
#> x2 -0.89621 0.13362 -6.707 2.16e-10 ***
#> c1 0.17521 0.05892 2.974 0.003318 **
#> c2 0.12061 0.05786 2.085 0.038421 *
#> w:x1 0.45617 0.10450 4.365 2.07e-05 ***
#> w:x2 0.56611 0.07950 7.121 2.09e-11 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.2023 on 192 degrees of freedom
#> Multiple R-squared: 0.4886, Adjusted R-squared: 0.47
#> F-statistic: 26.21 on 7 and 192 DF, p-value: < 2.2e-16
Conditional Effects
We can now use cond_effects() to
estimate the effects of x1 and x2
on y for
different levels of the moderator (w).
Conditional Effects of x1
Suppose we want to estimate the
effect from x1 to y,
conditional on w:
(Refer to vignette("manymome") and the help page
of cond_effects() on the arguments.)
out1 <- cond_effects(
wlevels = "w",
x = "x1",
y = "y",
fit = lm_y
)
out1
#>
#> == Conditional effects ==
#>
#> Path: x1 -> y
#> Conditional on moderator(s): w
#> Moderator(s) represented by: w
#> Computation Formula:
#> (b.y~x1 + (b.w:x1)*(w))
#>
#>
#> [w] (w) ind SE Stat pvalue Sig CI.lo CI.hi
#> 1 M+1.0SD 2.176 0.357 0.069 5.168 0.000 *** 0.221 0.493
#> 2 Mean 1.717 0.148 0.047 3.160 0.002 ** 0.055 0.240
#> 3 M-1.0SD 1.258 -0.062 0.065 -0.953 0.342 -0.189 0.066
#>
#> - [SE] are regression standard errors.
#> - [Stat] are the t statistics used to test the effects.
#> - [pvalue] are p-values computed from 'Stat'.
#> - [Sig]: 0 '***' 0.001 '**' 0.01 '*' 0.05 ' ' 1.
#> - [CI.lo to CI.hi] are 95.0% confidence interval computed from
#> regression standard errors.
#> - The 'ind' column shows the conditional effects.
#> - Call 'print_all_cond_indirect_effects()' to print the detailed
#> outputs of all levels.
The column ind shows the effects of
x1 on y for different levels of w.
When w is one standard deviation
below the mean, the effect of x1 is
-0.062,
with a 95% confidence interval
[-0.189, 0.066].
When w is one standard deviation
above the mean, the effect of x1 is
0.357,
with a 95% confidence interval
[0.221, 0.493].
NOTE: The standard error (SE) and
related results are computed using
the pick-a-point approach by
@rogosa_comparing_1980.
Conditional Effects of x2
The step to compute the conditional
effects for the other predictor, x2,
is similar:
out2 <- cond_effects(
wlevels = "w",
x = "x2",
y = "y",
fit = lm_y
)
out2
#>
#> == Conditional effects ==
#>
#> Path: x2 -> y
#> Conditional on moderator(s): w
#> Moderator(s) represented by: w
#> Computation Formula:
#> (b.y~x2 + (b.w:x2)*(w))
#>
#>
#> [w] (w) ind SE Stat pvalue Sig CI.lo CI.hi
#> 1 M+1.0SD 2.176 0.336 0.069 4.862 0.000 *** 0.200 0.472
#> 2 Mean 1.717 0.076 0.050 1.505 0.134 -0.024 0.175
#> 3 M-1.0SD 1.258 -0.184 0.055 -3.363 0.001 *** -0.292 -0.076
#>
#> - [SE] are regression standard errors.
#> - [Stat] are the t statistics used to test the effects.
#> - [pvalue] are p-values computed from 'Stat'.
#> - [Sig]: 0 '***' 0.001 '**' 0.01 '*' 0.05 ' ' 1.
#> - [CI.lo to CI.hi] are 95.0% confidence interval computed from
#> regression standard errors.
#> - The 'ind' column shows the conditional effects.
#> - Call 'print_all_cond_indirect_effects()' to print the detailed
#> outputs of all levels.
When w is one standard deviation
below the mean, the effect of x2 is
-0.184,
with a 95% confidence interval
[-0.292, -0.076].
When w is one standard deviation
above the mean, the effect of x2 is
0.336,
with a 95% confidence interval
[0.200, 0.472].
Plotting the Conditional Effects
Conventional Plot
The output of cond_effects() has a plot
method for plotting the conditional effects:
plot(out1)
plot(out2)
By default, the lines
span the range of one standard deviation below
and above the mean of the predictor.
The plot can be customized in a lot of ways.
Please refer to the help page of
plot.cond_indirect_effects() for available
options.
Tumble Plot
If the distribution
of the x variable may vary for different
levels of the moderators, a version of
tumble graph proposed by @bodner_tumble_2016
can be plotted by adding graph_type = "tumble":
plot(out1,
graph_type = "tumble")
In this example, the distributions of x1
for the three levels of moderator w
are similar.
plot(out2,
graph_type = "tumble")
The distributions of x2
vary as the level of the moderator w changes:
-
the mean of x2 is lower when w is
one standard deviation below its mean,
and,
-
the mean of x2 is higher when
w is one standard deviation above its
mean.
Therefore, the tumble graph is a better
way to visualize the moderating effect
of w on the effect of x2. For x1,
the conventional graph is sufficient.
Standardized Conditional Effects
Although OLS can be used to estimate and
test the
unstandardized effects, it is inappropriate
for forming the confidence intervals for the
standardized effects. See
@yuan_biases_2011 on the issue of standardized
regression coefficients.
To form nonparametric bootstrap confidence intervals for
effects to be computed, add boot_ci = TRUE,
R to the number of bootstrap samples
(should be 5000 or even 10000, for
multiple regression), and seed (set
it to an integer to ensure the results are
reproducible).
The standardized conditional
effect from x1 and x2 to y conditional
on w
can be estimated by setting
standardized_x and standardized_y to TRUE.
This is the output for x1:
std1 <- cond_effects(
wlevels = "w",
x = "x1",
y = "y",
fit = lm_y,
boot_ci = TRUE,
R = 5000,
seed = 54532,
standardized_x = TRUE,
standardized_y = TRUE
)
#> 19 processes started to run bootstrapping.
std1
#>
#> == Conditional effects ==
#>
#> Path: x1 -> y
#> Conditional on moderator(s): w
#> Moderator(s) represented by: w
#> Computation Formula:
#> (b.y~x1 + (b.w:x1)*(w))*sd_x1/sd_y
#>
#>
#> [w] (w) std CI.lo CI.hi Sig ind
#> 1 M+1.0SD 2.176 0.401 0.267 0.533 Sig 0.357
#> 2 Mean 1.717 0.166 0.077 0.254 Sig 0.148
#> 3 M-1.0SD 1.258 -0.069 -0.191 0.060 -0.062
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 5000 samples.
#> - std: The standardized conditional effects.
#> - ind: The unstandardized conditional effects.
#> - Call 'print_all_cond_indirect_effects()' to print the detailed
#> outputs of all levels.
When w is one standard deviation below
its mean, the standardized effect of x1 is
-0.069,
with a 95% confidence interval
[-0.191, 0.060].
When w is one standard deviation above
its mean, the standardized effect of x1 is
0.401,
with a 95% confidence interval
[0.267, 0.533].
This is the output for x2:
std2 <- cond_effects(
wlevels = "w",
x = "x2",
y = "y",
fit = lm_y,
boot_ci = TRUE,
R = 5000,
seed = 54532,
standardized_x = TRUE,
standardized_y = TRUE
)
#> 19 processes started to run bootstrapping.
std2
#>
#> == Conditional effects ==
#>
#> Path: x2 -> y
#> Conditional on moderator(s): w
#> Moderator(s) represented by: w
#> Computation Formula:
#> (b.y~x2 + (b.w:x2)*(w))*sd_x2/sd_y
#>
#>
#> [w] (w) std CI.lo CI.hi Sig ind
#> 1 M+1.0SD 2.176 0.420 0.265 0.586 Sig 0.336
#> 2 Mean 1.717 0.095 -0.026 0.211 0.076
#> 3 M-1.0SD 1.258 -0.230 -0.362 -0.104 Sig -0.184
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 5000 samples.
#> - std: The standardized conditional effects.
#> - ind: The unstandardized conditional effects.
#> - Call 'print_all_cond_indirect_effects()' to print the detailed
#> outputs of all levels.
When w is one standard deviation below
its mean, the standardized effect of x2 is
-0.230,
with a 95% confidence interval
[-0.362, -0.104].
When w is one standard deviation above
its mean, the standardized effect of x2 is
0.420,
with a 95% confidence interval
[0.265, 0.586].
Plot Standardized Conditional Effects
The plot() method can also be used
on the standardized conditional effects,
although the only differences are the
values displayed on the axes:
plot(std1)
plot(std2)
These are the tumble graphs:
plot(std1,
graph_type = "tumble")
plot(std2,
graph_type = "tumble")
Other Moderated Regression Models
The function cond_effects() has no limit
on the number of moderators and the number
of predictors with their effects moderated.
The demonstrations of other moderated
regression models can be found in
the list of articles.
The levels
for the moderators are controlled by mod_levels()
and related functions in the same way whether a
model is fitted by lavaan::sem() or lm().
Please refer to other articles (e.g.,
vignette("manymome") and vignette("mod_levels"))
on how to estimate effects in other models analyzed by
multiple regression.
References
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Introduction
This article is part of a series of
brief illustrations of how
to use
cond_effects()
from the package
manymome
[@cheung_manymome_2024]
to estimate the conditional
effects
when the model parameters are estimated by
ordinary least squares (OLS) multiple regression
using lm(). For moderated mediation
tested by OLS regression, please refer
to this article.
(Articles in this series had duplicated sections to make each of them self-contained.)
Data Set and Model
This is the sample data set used for illustration:
library(manymome) dat <- data_mod_2x1w print(head(dat), digits = 3) #> y x1 x2 w c1 c2 #> 1 2.05 1.80 2.75 2.09 1.83 1.60 #> 2 1.41 1.87 2.78 1.94 1.31 1.22 #> 3 1.94 1.74 3.06 2.39 1.50 1.29 #> 4 0.84 1.73 2.56 1.13 1.41 1.18 #> 5 1.33 0.95 2.86 1.32 1.45 1.11 #> 6 1.31 2.12 2.82 1.67 1.66 1.36
This dataset has 6 variables:
-
one outcome variable (
y), -
two predictors (
x1,x2), -
one moderator (
w), -
two control variables (
c1andc2).
Suppose this is the model being fitted, with control variables omitted from the plot for readability:
Fit by Regression
The path parameters
can be estimated by multiple regression
using lm():
lm_y <- lm( y ~ w*x1 + w*x2 + c1 + c2, data = dat )
These are the estimates of the regression coefficients of the paths:
summary(lm_y) #> #> Call: #> lm(formula = y ~ w * x1 + w * x2 + c1 + c2, data = dat) #> #> Residuals: #> Min 1Q Median 3Q Max #> -0.46049 -0.12989 -0.00299 0.11899 0.54333 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 3.91686 0.46699 8.387 1.06e-14 *** #> w -1.94839 0.25990 -7.497 2.35e-12 *** #> x1 -0.63573 0.18248 -3.484 0.000612 *** #> x2 -0.89621 0.13362 -6.707 2.16e-10 *** #> c1 0.17521 0.05892 2.974 0.003318 ** #> c2 0.12061 0.05786 2.085 0.038421 * #> w:x1 0.45617 0.10450 4.365 2.07e-05 *** #> w:x2 0.56611 0.07950 7.121 2.09e-11 *** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 0.2023 on 192 degrees of freedom #> Multiple R-squared: 0.4886, Adjusted R-squared: 0.47 #> F-statistic: 26.21 on 7 and 192 DF, p-value: < 2.2e-16
Conditional Effects
We can now use cond_effects() to
estimate the effects of x1 and x2
on y for
different levels of the moderator (w).
Conditional Effects of x1
Suppose we want to estimate the
effect from x1 to y,
conditional on w:
(Refer to vignette("manymome") and the help page
of cond_effects() on the arguments.)
out1 <- cond_effects( wlevels = "w", x = "x1", y = "y", fit = lm_y ) out1 #> #> == Conditional effects == #> #> Path: x1 -> y #> Conditional on moderator(s): w #> Moderator(s) represented by: w #> Computation Formula: #> (b.y~x1 + (b.w:x1)*(w)) #> #> #> [w] (w) ind SE Stat pvalue Sig CI.lo CI.hi #> 1 M+1.0SD 2.176 0.357 0.069 5.168 0.000 *** 0.221 0.493 #> 2 Mean 1.717 0.148 0.047 3.160 0.002 ** 0.055 0.240 #> 3 M-1.0SD 1.258 -0.062 0.065 -0.953 0.342 -0.189 0.066 #> #> - [SE] are regression standard errors. #> - [Stat] are the t statistics used to test the effects. #> - [pvalue] are p-values computed from 'Stat'. #> - [Sig]: 0 '***' 0.001 '**' 0.01 '*' 0.05 ' ' 1. #> - [CI.lo to CI.hi] are 95.0% confidence interval computed from #> regression standard errors. #> - The 'ind' column shows the conditional effects. #> - Call 'print_all_cond_indirect_effects()' to print the detailed #> outputs of all levels.
The column ind shows the effects of
x1 on y for different levels of w.
When w is one standard deviation
below the mean, the effect of x1 is
-0.062,
with a 95% confidence interval
[-0.189, 0.066].
When w is one standard deviation
above the mean, the effect of x1 is
0.357,
with a 95% confidence interval
[0.221, 0.493].
NOTE: The standard error (SE) and
related results are computed using
the pick-a-point approach by
@rogosa_comparing_1980.
Conditional Effects of x2
The step to compute the conditional
effects for the other predictor, x2,
is similar:
out2 <- cond_effects( wlevels = "w", x = "x2", y = "y", fit = lm_y ) out2 #> #> == Conditional effects == #> #> Path: x2 -> y #> Conditional on moderator(s): w #> Moderator(s) represented by: w #> Computation Formula: #> (b.y~x2 + (b.w:x2)*(w)) #> #> #> [w] (w) ind SE Stat pvalue Sig CI.lo CI.hi #> 1 M+1.0SD 2.176 0.336 0.069 4.862 0.000 *** 0.200 0.472 #> 2 Mean 1.717 0.076 0.050 1.505 0.134 -0.024 0.175 #> 3 M-1.0SD 1.258 -0.184 0.055 -3.363 0.001 *** -0.292 -0.076 #> #> - [SE] are regression standard errors. #> - [Stat] are the t statistics used to test the effects. #> - [pvalue] are p-values computed from 'Stat'. #> - [Sig]: 0 '***' 0.001 '**' 0.01 '*' 0.05 ' ' 1. #> - [CI.lo to CI.hi] are 95.0% confidence interval computed from #> regression standard errors. #> - The 'ind' column shows the conditional effects. #> - Call 'print_all_cond_indirect_effects()' to print the detailed #> outputs of all levels.
When w is one standard deviation
below the mean, the effect of x2 is
-0.184,
with a 95% confidence interval
[-0.292, -0.076].
When w is one standard deviation
above the mean, the effect of x2 is
0.336,
with a 95% confidence interval
[0.200, 0.472].
Plotting the Conditional Effects
Conventional Plot
The output of cond_effects() has a plot
method for plotting the conditional effects:
plot(out1)
plot(out2)
By default, the lines span the range of one standard deviation below and above the mean of the predictor.
The plot can be customized in a lot of ways.
Please refer to the help page of
plot.cond_indirect_effects() for available
options.
Tumble Plot
If the distribution
of the x variable may vary for different
levels of the moderators, a version of
tumble graph proposed by @bodner_tumble_2016
can be plotted by adding graph_type = "tumble":
plot(out1, graph_type = "tumble")
In this example, the distributions of x1
for the three levels of moderator w
are similar.
plot(out2, graph_type = "tumble")
The distributions of x2
vary as the level of the moderator w changes:
-
the mean of
x2is lower whenwis one standard deviation below its mean, and, -
the mean of
x2is higher whenwis one standard deviation above its mean.
Therefore, the tumble graph is a better
way to visualize the moderating effect
of w on the effect of x2. For x1,
the conventional graph is sufficient.
Standardized Conditional Effects
Although OLS can be used to estimate and test the unstandardized effects, it is inappropriate for forming the confidence intervals for the standardized effects. See @yuan_biases_2011 on the issue of standardized regression coefficients.
To form nonparametric bootstrap confidence intervals for
effects to be computed, add boot_ci = TRUE,
R to the number of bootstrap samples
(should be 5000 or even 10000, for
multiple regression), and seed (set
it to an integer to ensure the results are
reproducible).
The standardized conditional
effect from x1 and x2 to y conditional
on w
can be estimated by setting
standardized_x and standardized_y to TRUE.
This is the output for x1:
std1 <- cond_effects( wlevels = "w", x = "x1", y = "y", fit = lm_y, boot_ci = TRUE, R = 5000, seed = 54532, standardized_x = TRUE, standardized_y = TRUE ) #> 19 processes started to run bootstrapping. std1 #> #> == Conditional effects == #> #> Path: x1 -> y #> Conditional on moderator(s): w #> Moderator(s) represented by: w #> Computation Formula: #> (b.y~x1 + (b.w:x1)*(w))*sd_x1/sd_y #> #> #> [w] (w) std CI.lo CI.hi Sig ind #> 1 M+1.0SD 2.176 0.401 0.267 0.533 Sig 0.357 #> 2 Mean 1.717 0.166 0.077 0.254 Sig 0.148 #> 3 M-1.0SD 1.258 -0.069 -0.191 0.060 -0.062 #> #> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by #> nonparametric bootstrapping with 5000 samples. #> - std: The standardized conditional effects. #> - ind: The unstandardized conditional effects. #> - Call 'print_all_cond_indirect_effects()' to print the detailed #> outputs of all levels.
When w is one standard deviation below
its mean, the standardized effect of x1 is
-0.069,
with a 95% confidence interval
[-0.191, 0.060].
When w is one standard deviation above
its mean, the standardized effect of x1 is
0.401,
with a 95% confidence interval
[0.267, 0.533].
This is the output for x2:
std2 <- cond_effects( wlevels = "w", x = "x2", y = "y", fit = lm_y, boot_ci = TRUE, R = 5000, seed = 54532, standardized_x = TRUE, standardized_y = TRUE ) #> 19 processes started to run bootstrapping. std2 #> #> == Conditional effects == #> #> Path: x2 -> y #> Conditional on moderator(s): w #> Moderator(s) represented by: w #> Computation Formula: #> (b.y~x2 + (b.w:x2)*(w))*sd_x2/sd_y #> #> #> [w] (w) std CI.lo CI.hi Sig ind #> 1 M+1.0SD 2.176 0.420 0.265 0.586 Sig 0.336 #> 2 Mean 1.717 0.095 -0.026 0.211 0.076 #> 3 M-1.0SD 1.258 -0.230 -0.362 -0.104 Sig -0.184 #> #> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by #> nonparametric bootstrapping with 5000 samples. #> - std: The standardized conditional effects. #> - ind: The unstandardized conditional effects. #> - Call 'print_all_cond_indirect_effects()' to print the detailed #> outputs of all levels.
When w is one standard deviation below
its mean, the standardized effect of x2 is
-0.230,
with a 95% confidence interval
[-0.362, -0.104].
When w is one standard deviation above
its mean, the standardized effect of x2 is
0.420,
with a 95% confidence interval
[0.265, 0.586].
Plot Standardized Conditional Effects
The plot() method can also be used
on the standardized conditional effects,
although the only differences are the
values displayed on the axes:
plot(std1)
plot(std2)
These are the tumble graphs:
plot(std1, graph_type = "tumble")
plot(std2, graph_type = "tumble")
Other Moderated Regression Models
The function cond_effects() has no limit
on the number of moderators and the number
of predictors with their effects moderated.
The demonstrations of other moderated regression models can be found in the list of articles.
The levels
for the moderators are controlled by mod_levels()
and related functions in the same way whether a
model is fitted by lavaan::sem() or lm().
Please refer to other articles (e.g.,
vignette("manymome") and vignette("mod_levels"))
on how to estimate effects in other models analyzed by
multiple regression.
References
Try the manymome package in your browser
Any scripts or data that you put into this service are public.
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We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
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