Nothing
Moderating Effect by Multiple Regression"
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_2w
print(head(dat), digits = 3)
#> y x w1 w2 c1 c2
#> 1 3.85 5.00 6.11 4.22 3.82 7.80
#> 2 6.58 6.75 6.20 2.98 7.09 5.47
#> 3 4.89 5.29 5.82 5.20 4.98 6.70
#> 4 8.06 6.79 6.95 4.21 5.97 5.31
#> 5 7.50 6.88 7.01 5.70 6.20 5.67
#> 6 5.92 5.63 9.45 4.75 4.78 6.39
This dataset has 6 variables:
-
one outcome variable (y),
-
one predictor (x),
-
two moderators (w1, w2),
-
two control variables (c1 and c2).
We will start with a model with only one
moderator.
One Moderator
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_w1 <- lm(
y ~ w1*x + c1 + c2,
data = dat
)
These are the estimates of the regression coefficients
of the paths:
summary(lm_y_w1)
#>
#> Call:
#> lm(formula = y ~ w1 * x + c1 + c2, data = dat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2.29074 -0.62350 -0.05257 0.64493 2.60438
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 6.14284 3.32593 1.847 0.066277 .
#> w1 -0.52086 0.46251 -1.126 0.261486
#> x -0.95008 0.60613 -1.567 0.118636
#> c1 0.08857 0.06496 1.363 0.174331
#> c2 0.18836 0.05498 3.426 0.000747 ***
#> w1:x 0.18154 0.08380 2.166 0.031508 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.9566 on 194 degrees of freedom
#> Multiple R-squared: 0.3322, Adjusted R-squared: 0.315
#> F-statistic: 19.3 on 5 and 194 DF, p-value: 1.41e-15
Conditional Effects
We can now use cond_effects() to
estimate the effect of x on y for
different levels of the moderator (w1).
Suppose we want to estimate the
effect from x to y,
conditional on w1:
(Refer to vignette("manymome") and the help page
of cond_effects() on the arguments.)
out_xy_on_w1 <- cond_effects(
wlevels = "w1",
x = "x",
y = "y",
fit = lm_y_w1
)
out_xy_on_w1
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): w1
#> Moderator(s) represented by: w1
#> Computation Formula:
#> (b.y~x + (b.w1:x)*(w1))
#>
#>
#> [w1] (w1) ind SE Stat pvalue Sig CI.lo CI.hi
#> 1 M+1.0SD 7.942 0.492 0.101 4.853 0.000 *** 0.292 0.692
#> 2 Mean 6.878 0.299 0.082 3.643 0.000 *** 0.137 0.460
#> 3 M-1.0SD 5.813 0.105 0.138 0.762 0.447 -0.167 0.378
#>
#> - [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
x on y for different levels of w1.
When w1 is one standard deviation
below the mean, the effect of x1 is
0.105,
with a 95% confidence interval
[-0.167, 0.378].
When w1 is one standard deviation
above the mean, the effect of x1 is
0.492,
with a 95% confidence interval
[0.292, 0.692].
NOTE: The standard error (SE) and
related results are computed using
the pick-a-point approach by
@rogosa_comparing_1980.
Plotting the Conditional Effects
Conventional Plot
The output of cond_effects() has a plot
method for plotting the conditional effects
(also called simple effects):
plot(out_xy_on_w1)
By default, the lines
span the range of one standard deviation below
and above the mean of the x variable.
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(out_xy_on_w1,
graph_type = "tumble")
In this example, the distribution of the predictor
varies slightly as the level of the moderator changes:
-
lower with a smaller variation when w1 is
one standard deviation below its mean, and
-
higher with a larger variation when
w1 is one standard deviation above its
mean.
Therefore, the tumble graph is a better
way to visualize the moderating effect
of w1.
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 x to y conditional
on w1
can be estimated by setting
standardized_x and standardized_y to TRUE.
std_xy_on_w1 <- cond_effects(
wlevels = "w1",
x = "x",
y = "y",
fit = lm_y_w1,
boot_ci = TRUE,
R = 5000,
seed = 54532,
standardized_x = TRUE,
standardized_y = TRUE
)
#> 19 processes started to run bootstrapping.
std_xy_on_w1
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): w1
#> Moderator(s) represented by: w1
#> Computation Formula:
#> (b.y~x + (b.w1:x)*(w1))*sd_x/sd_y
#>
#>
#> [w1] (w1) std CI.lo CI.hi Sig ind
#> 1 M+1.0SD 7.942 0.374 0.204 0.518 Sig 0.492
#> 2 Mean 6.878 0.227 0.094 0.352 Sig 0.299
#> 3 M-1.0SD 5.813 0.080 -0.136 0.280 0.105
#>
#> - [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 w1 is one standard deviation below
its mean, the standardized effect is
0.080,
with a 95% confidence interval
[-0.136, 0.280].
When w is one standard deviation above
its mean, the standardized effect is
0.374,
with a 95% confidence interval
[0.204, 0.518].
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(std_xy_on_w1)
This is the tumble graph:
plot(std_xy_on_w1,
graph_type = "tumble")
Two Moderators
Suppose we would like to examine the
moderating effects of two moderators,
w1 and w2, on the effect of x on
y. This is the model being fitted,
with control variables omitted from the
plot for readability:
The path parameters
can be estimated by the following multiple
regression model:
lm_y_w1w2 <- lm(
y ~ x*w1 + x*w2 + c1 + c2,
data = dat
)
These are the estimates of the regression coefficients
of the paths:
summary(lm_y_w1w2)
#>
#> Call:
#> lm(formula = y ~ x * w1 + x * w2 + c1 + c2, data = dat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2.29403 -0.65294 -0.01277 0.64934 2.58194
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 6.88344 3.64126 1.890 0.060210 .
#> x -1.17277 0.66638 -1.760 0.080015 .
#> w1 -0.57214 0.46589 -1.228 0.220932
#> w2 -0.08857 0.49859 -0.178 0.859190
#> c1 0.08432 0.06481 1.301 0.194794
#> c2 0.19407 0.05509 3.523 0.000534 ***
#> x:w1 0.19117 0.08432 2.267 0.024482 *
#> x:w2 0.03787 0.08633 0.439 0.661378
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.9529 on 192 degrees of freedom
#> Multiple R-squared: 0.3442, Adjusted R-squared: 0.3203
#> F-statistic: 14.4 on 7 and 192 DF, p-value: 5.341e-15
Although w2 does not significantly moderate
the effect of x, we still proceed to examine
its moderating effect, just for illustration.
Conditional Effects
The function cond_effects() can be
used again, even with two moderators:
out_xy_on_w1w2 <- cond_effects(
wlevels = c("w1", "w2"),
x = "x",
y = "y",
fit = lm_y_w1w2
)
out_xy_on_w1w2
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): w1, w2
#> Moderator(s) represented by: w1, w2
#> Computation Formula:
#> (b.y~x + (b.x:w1)*(w1) + (b.x:w2)*(w2))
#>
#>
#> [w1] [w2] (w1) (w2) ind SE Stat pvalue Sig CI.lo CI.hi
#> 1 M+1.0SD M+1.0SD 7.942 4.829 0.528 0.122 4.343 0.000 *** 0.288 0.768
#> 2 M+1.0SD M-1.0SD 7.942 2.897 0.455 0.141 3.220 0.002 ** 0.176 0.734
#> 3 M-1.0SD M+1.0SD 5.813 4.829 0.121 0.163 0.744 0.458 -0.201 0.444
#> 4 M-1.0SD M-1.0SD 5.813 2.897 0.048 0.159 0.303 0.762 -0.266 0.363
#>
#> - [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
x on y for different levels of w1
and w2.
IMPORTANT: Even though this model does not
have three-way interaction, the conditional
effects still need to consider both
moderators. It is because the effect of
x depends on all moderators, whether
there is a higher-order interaction or not.
If one or more moderators are omitted,
a warning message will be issued. This
is an example:
cond_effects(
wlevels = "w1",
x = "x",
y = "y",
fit = lm_y_w1w2
)
#> Warning in (function (xi, yi, yiname, digits = 3, y, wvalues = NULL, warn =
#> TRUE, : w2 modelled as moderator(s) for the path from y~x to y but not included
#> in 'wvalues'. They will be set to zero in computing the conditional effect,
#> which may not be meaningful. Please check.
#> Warning in (function (xi, yi, yiname, digits = 3, y, wvalues = NULL, warn =
#> TRUE, : w2 modelled as moderator(s) for the path from y~x to y but not included
#> in 'wvalues'. They will be set to zero in computing the conditional effect,
#> which may not be meaningful. Please check.
#> Warning in (function (xi, yi, yiname, digits = 3, y, wvalues = NULL, warn =
#> TRUE, : w2 modelled as moderator(s) for the path from y~x to y but not included
#> in 'wvalues'. They will be set to zero in computing the conditional effect,
#> which may not be meaningful. Please check.
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): w1
#> Moderator(s) represented by: w1
#> Computation Formula:
#> (b.y~x + (b.x:w1)*(w1) + (b.x:w2)*(w2))
#>
#>
#> [w1] (w1) ind SE Stat pvalue Sig CI.lo CI.hi
#> 1 M+1.0SD 7.942 0.346 0.363 0.951 0.343 -0.371 1.062
#> 2 Mean 6.878 0.142 0.349 0.407 0.684 -0.546 0.831
#> 3 M-1.0SD 5.813 -0.061 0.357 -0.172 0.864 -0.766 0.643
#>
#> - [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.
Plotting the Conditional Effects
Conventional Plots
The plot() method can be used directly
even with two moderators:
plot(out_xy_on_w1w2)
This plot is not easy to read when the
model has two or more moderators.
The argument
facet_grid_cols can be used to generate
one plot for each level of one of the moderators,
presented in one row, side-by-side.
For example, suppose we would like to
generate one graph for each level of w2,
we add facet_grid_cols = "w2":
plot(out_xy_on_w1w2,
facet_grid_cols = "w2")
We can do the same for w1:
plot(out_xy_on_w1w2,
facet_grid_cols = "w1")
Tumble Plots
We can also use graph_type = "tumble"
to generate tumble graphs:
plot(out_xy_on_w1w2,
facet_grid_cols = "w2",
graph_type = "tumble")
We can do the same for w1:
plot(out_xy_on_w1w2,
facet_grid_cols = "w1",
graph_type = "tumble")
Standardized Conditional Effects
The standardized conditional
effect from x to y conditional
on w1 and w2
can be estimated by setting
standardized_x and standardized_y to TRUE,
with bootstrap confidence intervals:
std_xy_on_w1w2 <- cond_effects(
wlevels = c("w1", "w2"),
x = "x",
y = "y",
fit = lm_y_w1w2,
boot_ci = TRUE,
R = 5000,
seed = 54532,
standardized_x = TRUE,
standardized_y = TRUE
)
#> 19 processes started to run bootstrapping.
std_xy_on_w1w2
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): w1, w2
#> Moderator(s) represented by: w1, w2
#> Computation Formula:
#> (b.y~x + (b.x:w1)*(w1) + (b.x:w2)*(w2))*sd_x/sd_y
#>
#>
#> [w1] [w2] (w1) (w2) std CI.lo CI.hi Sig ind
#> 1 M+1.0SD M+1.0SD 7.942 4.829 0.402 0.203 0.560 Sig 0.528
#> 2 M+1.0SD M-1.0SD 7.942 2.897 0.346 0.143 0.586 Sig 0.455
#> 3 M-1.0SD M+1.0SD 5.813 4.829 0.092 -0.169 0.323 0.121
#> 4 M-1.0SD M-1.0SD 5.813 2.897 0.037 -0.210 0.264 0.048
#>
#> - [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.
The plot() method can also be used
on the standardized conditional effects:
plot(std_xy_on_w1w2,
facet_grid_cols = "w2")
plot(std_xy_on_w1w2,
facet_grid_cols = "w1")
These are the tumble graphs:
plot(std_xy_on_w1w2,
facet_grid_cols = "w2",
graph_type = "tumble")
plot(std_xy_on_w1w2,
facet_grid_cols = "w1",
graph_type = "tumble")
Two Moderators, with Three-Way Interaction
Suppose that we suspect that the two
moderators interact with each
other. That is, the moderating effect of w1 on
the effect of x may not be the same for
different levels of w2, or the moderating
effect of w2 on
the effect of x may depend on the level
of w1.
The steps demonstrated above can also be used
in this regression model:
lm_y_w1_x_w2 <- lm(
y ~ w1*w2*x + c1 + c2,
data = dat
)
These are the estimates of the regression coefficients
of this model:
summary(lm_y_w1_x_w2)
#>
#> Call:
#> lm(formula = y ~ w1 * w2 * x + c1 + c2, data = dat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2.27658 -0.63668 -0.03106 0.61117 2.55406
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) -23.46400 14.16783 -1.656 0.09934 .
#> w1 3.68387 1.98858 1.853 0.06550 .
#> w2 7.85339 3.63294 2.162 0.03189 *
#> x 4.46108 2.56813 1.737 0.08399 .
#> c1 0.08359 0.06427 1.301 0.19500
#> c2 0.19658 0.05465 3.597 0.00041 ***
#> w1:w2 -1.11035 0.50635 -2.193 0.02953 *
#> w1:x -0.59907 0.35889 -1.669 0.09672 .
#> w2:x -1.43247 0.65486 -2.187 0.02993 *
#> w1:w2:x 0.20562 0.09102 2.259 0.02501 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.9449 on 190 degrees of freedom
#> Multiple R-squared: 0.3618, Adjusted R-squared: 0.3316
#> F-statistic: 11.97 on 9 and 190 DF, p-value: 7.009e-15
The significant three-way interaction
term, w1:w2:x, suggests that, although
w2 does not moderate the effect of x
on y, it does moderate the moderating
effect of w1.
Conditional Effects
The function cond_effects() can be
used in exactly the same way, whether
the moderators interact with each other
or not:
out_xy_on_w1_x_w2 <- cond_effects(
wlevels = c("w1", "w2"),
x = "x",
y = "y",
fit = lm_y_w1_x_w2
)
out_xy_on_w1_x_w2
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): w1, w2
#> Moderator(s) represented by: w1, w2
#> Computation Formula:
#> (b.y~x + (b.w1:x)*(w1) + (b.w2:x)*(w2) + (b.w1:w2:x)*(w1*w2))
#>
#>
#> [w1] [w2] (w1) (w2) ind SE Stat pvalue Sig CI.lo CI.hi
#> 1 M+1.0SD M+1.0SD 7.942 4.829 0.672 0.137 4.903 0.000 *** 0.402 0.942
#> 2 M+1.0SD M-1.0SD 7.942 2.897 0.284 0.161 1.768 0.079 -0.033 0.602
#> 3 M-1.0SD M+1.0SD 5.813 4.829 -0.167 0.207 -0.806 0.421 -0.574 0.241
#> 4 M-1.0SD M-1.0SD 5.813 2.897 0.292 0.191 1.530 0.128 -0.084 0.667
#>
#> - [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.
As shown above, among the levels examined,
x has a significant effect on y
only when both w1 and w2 are one
standard deviation above their means.
Plotting the Conditional Effects
These are the tumble plots of the conditional effects,
with facet_grid_cols set:
plot(out_xy_on_w1_x_w2,
facet_grid_cols = "w2",
graph_type = "tumble")
plot(out_xy_on_w1_x_w2,
facet_grid_cols = "w1",
graph_type = "tumble")
Standardized Conditional Effects
This is the output of the standardized
conditional effects, with bootstrap
confidence intervals:
std_xy_on_w1_x_w2 <- cond_effects(
wlevels = c("w1", "w2"),
x = "x",
y = "y",
fit = lm_y_w1_x_w2,
boot_ci = TRUE,
R = 5000,
seed = 54532,
standardized_x = TRUE,
standardized_y = TRUE
)
#> 19 processes started to run bootstrapping.
std_xy_on_w1_x_w2
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): w1, w2
#> Moderator(s) represented by: w1, w2
#> Computation Formula:
#> (b.y~x + (b.w1:x)*(w1) + (b.w2:x)*(w2) +
#> (b.w1:w2:x)*(w1*w2))*sd_x/sd_y
#>
#>
#> [w1] [w2] (w1) (w2) std CI.lo CI.hi Sig ind
#> 1 M+1.0SD M+1.0SD 7.942 4.829 0.511 0.360 0.760 Sig 0.672
#> 2 M+1.0SD M-1.0SD 7.942 2.897 0.216 -0.015 0.456 0.284
#> 3 M-1.0SD M+1.0SD 5.813 4.829 -0.127 -0.462 0.177 -0.167
#> 4 M-1.0SD M-1.0SD 5.813 2.897 0.222 -0.062 0.501 0.292
#>
#> - [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.
The plot() method can also be used
on the standardized conditional effects,
in the presence of three-way interaction:
plot(std_xy_on_w1_x_w2,
facet_grid_cols = "w2",
graph_type = "tumble")
plot(std_xy_on_w1_x_w2,
facet_grid_cols = "w1",
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_2w print(head(dat), digits = 3) #> y x w1 w2 c1 c2 #> 1 3.85 5.00 6.11 4.22 3.82 7.80 #> 2 6.58 6.75 6.20 2.98 7.09 5.47 #> 3 4.89 5.29 5.82 5.20 4.98 6.70 #> 4 8.06 6.79 6.95 4.21 5.97 5.31 #> 5 7.50 6.88 7.01 5.70 6.20 5.67 #> 6 5.92 5.63 9.45 4.75 4.78 6.39
This dataset has 6 variables:
-
one outcome variable (
y), -
one predictor (
x), -
two moderators (
w1,w2), -
two control variables (
c1andc2).
We will start with a model with only one moderator.
One Moderator
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_w1 <- lm( y ~ w1*x + c1 + c2, data = dat )
These are the estimates of the regression coefficients of the paths:
summary(lm_y_w1) #> #> Call: #> lm(formula = y ~ w1 * x + c1 + c2, data = dat) #> #> Residuals: #> Min 1Q Median 3Q Max #> -2.29074 -0.62350 -0.05257 0.64493 2.60438 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 6.14284 3.32593 1.847 0.066277 . #> w1 -0.52086 0.46251 -1.126 0.261486 #> x -0.95008 0.60613 -1.567 0.118636 #> c1 0.08857 0.06496 1.363 0.174331 #> c2 0.18836 0.05498 3.426 0.000747 *** #> w1:x 0.18154 0.08380 2.166 0.031508 * #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 0.9566 on 194 degrees of freedom #> Multiple R-squared: 0.3322, Adjusted R-squared: 0.315 #> F-statistic: 19.3 on 5 and 194 DF, p-value: 1.41e-15
Conditional Effects
We can now use cond_effects() to
estimate the effect of x on y for
different levels of the moderator (w1).
Suppose we want to estimate the
effect from x to y,
conditional on w1:
(Refer to vignette("manymome") and the help page
of cond_effects() on the arguments.)
out_xy_on_w1 <- cond_effects( wlevels = "w1", x = "x", y = "y", fit = lm_y_w1 ) out_xy_on_w1 #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): w1 #> Moderator(s) represented by: w1 #> Computation Formula: #> (b.y~x + (b.w1:x)*(w1)) #> #> #> [w1] (w1) ind SE Stat pvalue Sig CI.lo CI.hi #> 1 M+1.0SD 7.942 0.492 0.101 4.853 0.000 *** 0.292 0.692 #> 2 Mean 6.878 0.299 0.082 3.643 0.000 *** 0.137 0.460 #> 3 M-1.0SD 5.813 0.105 0.138 0.762 0.447 -0.167 0.378 #> #> - [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
x on y for different levels of w1.
When w1 is one standard deviation
below the mean, the effect of x1 is
0.105,
with a 95% confidence interval
[-0.167, 0.378].
When w1 is one standard deviation
above the mean, the effect of x1 is
0.492,
with a 95% confidence interval
[0.292, 0.692].
NOTE: The standard error (SE) and
related results are computed using
the pick-a-point approach by
@rogosa_comparing_1980.
Plotting the Conditional Effects
Conventional Plot
The output of cond_effects() has a plot
method for plotting the conditional effects
(also called simple effects):
plot(out_xy_on_w1)
By default, the lines
span the range of one standard deviation below
and above the mean of the x variable.
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(out_xy_on_w1, graph_type = "tumble")
In this example, the distribution of the predictor varies slightly as the level of the moderator changes:
-
lower with a smaller variation when
w1is one standard deviation below its mean, and -
higher with a larger variation when
w1is one standard deviation above its mean.
Therefore, the tumble graph is a better
way to visualize the moderating effect
of w1.
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 x to y conditional
on w1
can be estimated by setting
standardized_x and standardized_y to TRUE.
std_xy_on_w1 <- cond_effects( wlevels = "w1", x = "x", y = "y", fit = lm_y_w1, boot_ci = TRUE, R = 5000, seed = 54532, standardized_x = TRUE, standardized_y = TRUE ) #> 19 processes started to run bootstrapping. std_xy_on_w1 #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): w1 #> Moderator(s) represented by: w1 #> Computation Formula: #> (b.y~x + (b.w1:x)*(w1))*sd_x/sd_y #> #> #> [w1] (w1) std CI.lo CI.hi Sig ind #> 1 M+1.0SD 7.942 0.374 0.204 0.518 Sig 0.492 #> 2 Mean 6.878 0.227 0.094 0.352 Sig 0.299 #> 3 M-1.0SD 5.813 0.080 -0.136 0.280 0.105 #> #> - [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 w1 is one standard deviation below
its mean, the standardized effect is
0.080,
with a 95% confidence interval
[-0.136, 0.280].
When w is one standard deviation above
its mean, the standardized effect is
0.374,
with a 95% confidence interval
[0.204, 0.518].
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(std_xy_on_w1)
This is the tumble graph:
plot(std_xy_on_w1, graph_type = "tumble")
Two Moderators
Suppose we would like to examine the
moderating effects of two moderators,
w1 and w2, on the effect of x on
y. This is the model being fitted,
with control variables omitted from the
plot for readability:
The path parameters can be estimated by the following multiple regression model:
lm_y_w1w2 <- lm( y ~ x*w1 + x*w2 + c1 + c2, data = dat )
These are the estimates of the regression coefficients of the paths:
summary(lm_y_w1w2) #> #> Call: #> lm(formula = y ~ x * w1 + x * w2 + c1 + c2, data = dat) #> #> Residuals: #> Min 1Q Median 3Q Max #> -2.29403 -0.65294 -0.01277 0.64934 2.58194 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 6.88344 3.64126 1.890 0.060210 . #> x -1.17277 0.66638 -1.760 0.080015 . #> w1 -0.57214 0.46589 -1.228 0.220932 #> w2 -0.08857 0.49859 -0.178 0.859190 #> c1 0.08432 0.06481 1.301 0.194794 #> c2 0.19407 0.05509 3.523 0.000534 *** #> x:w1 0.19117 0.08432 2.267 0.024482 * #> x:w2 0.03787 0.08633 0.439 0.661378 #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 0.9529 on 192 degrees of freedom #> Multiple R-squared: 0.3442, Adjusted R-squared: 0.3203 #> F-statistic: 14.4 on 7 and 192 DF, p-value: 5.341e-15
Although w2 does not significantly moderate
the effect of x, we still proceed to examine
its moderating effect, just for illustration.
Conditional Effects
The function cond_effects() can be
used again, even with two moderators:
out_xy_on_w1w2 <- cond_effects( wlevels = c("w1", "w2"), x = "x", y = "y", fit = lm_y_w1w2 ) out_xy_on_w1w2 #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): w1, w2 #> Moderator(s) represented by: w1, w2 #> Computation Formula: #> (b.y~x + (b.x:w1)*(w1) + (b.x:w2)*(w2)) #> #> #> [w1] [w2] (w1) (w2) ind SE Stat pvalue Sig CI.lo CI.hi #> 1 M+1.0SD M+1.0SD 7.942 4.829 0.528 0.122 4.343 0.000 *** 0.288 0.768 #> 2 M+1.0SD M-1.0SD 7.942 2.897 0.455 0.141 3.220 0.002 ** 0.176 0.734 #> 3 M-1.0SD M+1.0SD 5.813 4.829 0.121 0.163 0.744 0.458 -0.201 0.444 #> 4 M-1.0SD M-1.0SD 5.813 2.897 0.048 0.159 0.303 0.762 -0.266 0.363 #> #> - [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
x on y for different levels of w1
and w2.
IMPORTANT: Even though this model does not
have three-way interaction, the conditional
effects still need to consider both
moderators. It is because the effect of
x depends on all moderators, whether
there is a higher-order interaction or not.
If one or more moderators are omitted, a warning message will be issued. This is an example:
cond_effects( wlevels = "w1", x = "x", y = "y", fit = lm_y_w1w2 ) #> Warning in (function (xi, yi, yiname, digits = 3, y, wvalues = NULL, warn = #> TRUE, : w2 modelled as moderator(s) for the path from y~x to y but not included #> in 'wvalues'. They will be set to zero in computing the conditional effect, #> which may not be meaningful. Please check. #> Warning in (function (xi, yi, yiname, digits = 3, y, wvalues = NULL, warn = #> TRUE, : w2 modelled as moderator(s) for the path from y~x to y but not included #> in 'wvalues'. They will be set to zero in computing the conditional effect, #> which may not be meaningful. Please check. #> Warning in (function (xi, yi, yiname, digits = 3, y, wvalues = NULL, warn = #> TRUE, : w2 modelled as moderator(s) for the path from y~x to y but not included #> in 'wvalues'. They will be set to zero in computing the conditional effect, #> which may not be meaningful. Please check. #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): w1 #> Moderator(s) represented by: w1 #> Computation Formula: #> (b.y~x + (b.x:w1)*(w1) + (b.x:w2)*(w2)) #> #> #> [w1] (w1) ind SE Stat pvalue Sig CI.lo CI.hi #> 1 M+1.0SD 7.942 0.346 0.363 0.951 0.343 -0.371 1.062 #> 2 Mean 6.878 0.142 0.349 0.407 0.684 -0.546 0.831 #> 3 M-1.0SD 5.813 -0.061 0.357 -0.172 0.864 -0.766 0.643 #> #> - [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.
Plotting the Conditional Effects
Conventional Plots
The plot() method can be used directly
even with two moderators:
plot(out_xy_on_w1w2)
This plot is not easy to read when the
model has two or more moderators.
The argument
facet_grid_cols can be used to generate
one plot for each level of one of the moderators,
presented in one row, side-by-side.
For example, suppose we would like to
generate one graph for each level of w2,
we add facet_grid_cols = "w2":
plot(out_xy_on_w1w2, facet_grid_cols = "w2")
We can do the same for w1:
plot(out_xy_on_w1w2, facet_grid_cols = "w1")
Tumble Plots
We can also use graph_type = "tumble"
to generate tumble graphs:
plot(out_xy_on_w1w2, facet_grid_cols = "w2", graph_type = "tumble")
We can do the same for w1:
plot(out_xy_on_w1w2, facet_grid_cols = "w1", graph_type = "tumble")
Standardized Conditional Effects
The standardized conditional
effect from x to y conditional
on w1 and w2
can be estimated by setting
standardized_x and standardized_y to TRUE,
with bootstrap confidence intervals:
std_xy_on_w1w2 <- cond_effects( wlevels = c("w1", "w2"), x = "x", y = "y", fit = lm_y_w1w2, boot_ci = TRUE, R = 5000, seed = 54532, standardized_x = TRUE, standardized_y = TRUE ) #> 19 processes started to run bootstrapping. std_xy_on_w1w2 #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): w1, w2 #> Moderator(s) represented by: w1, w2 #> Computation Formula: #> (b.y~x + (b.x:w1)*(w1) + (b.x:w2)*(w2))*sd_x/sd_y #> #> #> [w1] [w2] (w1) (w2) std CI.lo CI.hi Sig ind #> 1 M+1.0SD M+1.0SD 7.942 4.829 0.402 0.203 0.560 Sig 0.528 #> 2 M+1.0SD M-1.0SD 7.942 2.897 0.346 0.143 0.586 Sig 0.455 #> 3 M-1.0SD M+1.0SD 5.813 4.829 0.092 -0.169 0.323 0.121 #> 4 M-1.0SD M-1.0SD 5.813 2.897 0.037 -0.210 0.264 0.048 #> #> - [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.
The plot() method can also be used
on the standardized conditional effects:
plot(std_xy_on_w1w2, facet_grid_cols = "w2")
plot(std_xy_on_w1w2, facet_grid_cols = "w1")
These are the tumble graphs:
plot(std_xy_on_w1w2, facet_grid_cols = "w2", graph_type = "tumble")
plot(std_xy_on_w1w2, facet_grid_cols = "w1", graph_type = "tumble")
Two Moderators, with Three-Way Interaction
Suppose that we suspect that the two
moderators interact with each
other. That is, the moderating effect of w1 on
the effect of x may not be the same for
different levels of w2, or the moderating
effect of w2 on
the effect of x may depend on the level
of w1.
The steps demonstrated above can also be used in this regression model:
lm_y_w1_x_w2 <- lm( y ~ w1*w2*x + c1 + c2, data = dat )
These are the estimates of the regression coefficients of this model:
summary(lm_y_w1_x_w2) #> #> Call: #> lm(formula = y ~ w1 * w2 * x + c1 + c2, data = dat) #> #> Residuals: #> Min 1Q Median 3Q Max #> -2.27658 -0.63668 -0.03106 0.61117 2.55406 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) -23.46400 14.16783 -1.656 0.09934 . #> w1 3.68387 1.98858 1.853 0.06550 . #> w2 7.85339 3.63294 2.162 0.03189 * #> x 4.46108 2.56813 1.737 0.08399 . #> c1 0.08359 0.06427 1.301 0.19500 #> c2 0.19658 0.05465 3.597 0.00041 *** #> w1:w2 -1.11035 0.50635 -2.193 0.02953 * #> w1:x -0.59907 0.35889 -1.669 0.09672 . #> w2:x -1.43247 0.65486 -2.187 0.02993 * #> w1:w2:x 0.20562 0.09102 2.259 0.02501 * #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 0.9449 on 190 degrees of freedom #> Multiple R-squared: 0.3618, Adjusted R-squared: 0.3316 #> F-statistic: 11.97 on 9 and 190 DF, p-value: 7.009e-15
The significant three-way interaction
term, w1:w2:x, suggests that, although
w2 does not moderate the effect of x
on y, it does moderate the moderating
effect of w1.
Conditional Effects
The function cond_effects() can be
used in exactly the same way, whether
the moderators interact with each other
or not:
out_xy_on_w1_x_w2 <- cond_effects( wlevels = c("w1", "w2"), x = "x", y = "y", fit = lm_y_w1_x_w2 ) out_xy_on_w1_x_w2 #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): w1, w2 #> Moderator(s) represented by: w1, w2 #> Computation Formula: #> (b.y~x + (b.w1:x)*(w1) + (b.w2:x)*(w2) + (b.w1:w2:x)*(w1*w2)) #> #> #> [w1] [w2] (w1) (w2) ind SE Stat pvalue Sig CI.lo CI.hi #> 1 M+1.0SD M+1.0SD 7.942 4.829 0.672 0.137 4.903 0.000 *** 0.402 0.942 #> 2 M+1.0SD M-1.0SD 7.942 2.897 0.284 0.161 1.768 0.079 -0.033 0.602 #> 3 M-1.0SD M+1.0SD 5.813 4.829 -0.167 0.207 -0.806 0.421 -0.574 0.241 #> 4 M-1.0SD M-1.0SD 5.813 2.897 0.292 0.191 1.530 0.128 -0.084 0.667 #> #> - [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.
As shown above, among the levels examined,
x has a significant effect on y
only when both w1 and w2 are one
standard deviation above their means.
Plotting the Conditional Effects
These are the tumble plots of the conditional effects,
with facet_grid_cols set:
plot(out_xy_on_w1_x_w2, facet_grid_cols = "w2", graph_type = "tumble")
plot(out_xy_on_w1_x_w2, facet_grid_cols = "w1", graph_type = "tumble")
Standardized Conditional Effects
This is the output of the standardized conditional effects, with bootstrap confidence intervals:
std_xy_on_w1_x_w2 <- cond_effects( wlevels = c("w1", "w2"), x = "x", y = "y", fit = lm_y_w1_x_w2, boot_ci = TRUE, R = 5000, seed = 54532, standardized_x = TRUE, standardized_y = TRUE ) #> 19 processes started to run bootstrapping. std_xy_on_w1_x_w2 #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): w1, w2 #> Moderator(s) represented by: w1, w2 #> Computation Formula: #> (b.y~x + (b.w1:x)*(w1) + (b.w2:x)*(w2) + #> (b.w1:w2:x)*(w1*w2))*sd_x/sd_y #> #> #> [w1] [w2] (w1) (w2) std CI.lo CI.hi Sig ind #> 1 M+1.0SD M+1.0SD 7.942 4.829 0.511 0.360 0.760 Sig 0.672 #> 2 M+1.0SD M-1.0SD 7.942 2.897 0.216 -0.015 0.456 0.284 #> 3 M-1.0SD M+1.0SD 5.813 4.829 -0.127 -0.462 0.177 -0.167 #> 4 M-1.0SD M-1.0SD 5.813 2.897 0.222 -0.062 0.501 0.292 #> #> - [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.
The plot() method can also be used
on the standardized conditional effects,
in the presence of three-way interaction:
plot(std_xy_on_w1_x_w2, facet_grid_cols = "w2", graph_type = "tumble")
plot(std_xy_on_w1_x_w2, facet_grid_cols = "w1", 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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