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Moderated Regression: Categorical Moderators"
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_cat_2w
print(head(dat), digits = 3)
#> x y c1 c2 gp site
#> 1 19.8 24.5 29.5 10.2 Control Site 1
#> 2 24.0 21.6 15.3 10.3 Control Site 1
#> 3 28.4 24.7 29.4 20.4 Control Site 1
#> 4 39.7 26.9 32.4 16.7 Control Site 1
#> 5 33.4 21.1 36.5 10.2 Control Site 1
#> 6 22.7 23.8 27.9 23.5 Control Site 1
This dataset has 6 variables:
-
one outcome variable (y),
-
one predictor (x),
-
two categorical moderators (gp, site),
-
two control variables (c1 and c2).
The moderator gp has two possible
values: "Control" and "Treatment".
The moderator site has three possible
values: "Site 1", "Site 2", and
"Site 3".
We will first consider two models, each
with only one of the two moderators.
One Categorical Moderator: Two Categories
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_gp <- lm(
y ~ gp*x + c1 + c2,
data = dat
)
These are the estimates of the regression coefficients
of the paths:
summary(lm_y_gp)
#>
#> Call:
#> lm(formula = y ~ gp * x + c1 + c2, data = dat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -10.8400 -2.6594 -0.2263 2.6201 15.5956
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 20.26140 1.49550 13.548 < 2e-16 ***
#> gpTreatment -4.96460 1.09316 -4.542 6.77e-06 ***
#> x -0.02738 0.03674 -0.745 0.456
#> c1 -0.02833 0.03843 -0.737 0.461
#> c2 0.39637 0.04438 8.932 < 2e-16 ***
#> gpTreatment:x 0.34746 0.04953 7.015 6.31e-12 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 4.093 on 594 degrees of freedom
#> Multiple R-squared: 0.2698, Adjusted R-squared: 0.2636
#> F-statistic: 43.89 on 5 and 594 DF, p-value: < 2.2e-16
Conditional Effects
We can now use cond_effects() to
estimate the effects of x
on y for
different categories of the moderator gp.
(Refer to vignette("manymome") and the help page
of cond_effects() on the arguments.)
out_gp <- cond_effects(
wlevels = "gp",
x = "x",
y = "y",
fit = lm_y_gp
)
out_gp
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): gp
#> Moderator(s) represented by: gpTreatment
#> Computation Formula:
#> (b.y~x + (b.gpTreatment:x)*(gpTreatment))
#>
#>
#> [gp] (gpTreatment) ind SE Stat pvalue Sig CI.lo CI.hi
#> 1 Control 0 -0.027 0.037 -0.745 0.456 -0.100 0.045
#> 2 Treatment 1 0.320 0.033 9.655 0.000 *** 0.255 0.385
#>
#> - [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 categories of gp.
In the group "Control", the effect of x is
-0.027,
with a 95% confidence interval
[-0.100, 0.045].
In the group "Treatment", the effect of x is
0.320,
with a 95% confidence interval
[0.255, 0.385].
NOTE: The standard error (SE) and
related results are computed using
the pick-a-point approach by
@rogosa_comparing_1980.
Plotting the Conditional Effects
The output of cond_effects() has a plot
method for plotting the conditional effects:
plot(out_gp)
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(out_gp,
graph_type = "tumble")
In this example, the distributions of x
for the groups
are similar.
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
effects from x to y conditional
on gp
can be estimated by setting
standardized_x and standardized_y to TRUE.
This is the output:
std_gp <- cond_effects(
wlevels = "gp",
x = "x",
y = "y",
fit = lm_y_gp,
boot_ci = TRUE,
R = 5000,
seed = 54532,
standardized_x = TRUE,
standardized_y = TRUE
)
#> 19 processes started to run bootstrapping.
std_gp
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): gp
#> Moderator(s) represented by: gpTreatment
#> Computation Formula:
#> (b.y~x + (b.gpTreatment:x)*(gpTreatment))*sd_x/sd_y
#>
#>
#> [gp] (gpTreatment) std CI.lo CI.hi Sig ind
#> 1 Control 0 -0.039 -0.130 0.048 -0.027
#> 2 Treatment 1 0.457 0.363 0.551 Sig 0.320
#>
#> - [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.
In the group "Control", the standardized effect of x is
-0.039,
with a 95% confidence interval
[-0.130, 0.048].
In the group "Treatment", the standardized effect of x is
0.457,
with a 95% confidence interval
[0.363, 0.551].
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_gp)
plot(std_gp,
graph_type = "tumble")
One Categorical Moderator: Three Categories
The steps demonstrated above can be used
for a categorical moderator with any number
of levels.
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 a multiple regression model:
lm_y_site <- lm(
y ~ site*x + c1 + c2,
data = dat
)
These are the estimates of the regression coefficients
of the paths:
summary(lm_y_site)
#>
#> Call:
#> lm(formula = y ~ site * x + c1 + c2, data = dat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -12.2398 -3.0852 0.2666 2.7867 17.3110
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 19.01557 1.99047 9.553 < 2e-16 ***
#> siteSite 2 -6.89570 2.15974 -3.193 0.001484 **
#> siteSite 3 1.30350 1.76713 0.738 0.461027
#> x 0.10205 0.05877 1.737 0.082989 .
#> c1 -0.03133 0.04062 -0.771 0.440934
#> c2 0.35831 0.04681 7.655 7.91e-14 ***
#> siteSite 2:x 0.32975 0.08710 3.786 0.000169 ***
#> siteSite 3:x -0.08913 0.08673 -1.028 0.304478
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 4.317 on 592 degrees of freedom
#> Multiple R-squared: 0.1905, Adjusted R-squared: 0.1809
#> F-statistic: 19.9 on 7 and 592 DF, p-value: < 2.2e-16
Conditional Effects
The function cond_effects() will determine
the number of categories automatically:
out_site <- cond_effects(
wlevels = "site",
x = "x",
y = "y",
fit = lm_y_site
)
out_site
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): site
#> Moderator(s) represented by: siteSite 2, siteSite 3
#> Computation Formula:
#> (b.y~x + (b.siteSite 2:x)*(siteSite 2) + (b.siteSite 3:x)*(siteSite 3))
#>
#>
#> [site] (siteSite 2) (siteSite 3) ind SE Stat pvalue Sig CI.lo CI.hi
#> 1 Site 1 0 0 0.102 0.059 1.737 0.083 -0.013 0.217
#> 2 Site 2 1 0 0.432 0.064 6.715 0.000 *** 0.306 0.558
#> 3 Site 3 0 1 0.013 0.064 0.203 0.840 -0.112 0.138
#>
#> - [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.
In the site "Site 1", the effect of x is
0.102,
with a 95% confidence interval
[-0.013, 0.217].
In the site "Site 2", the effect of x is
0.432,
with a 95% confidence interval
[0.306, 0.558].
In the site "Site 3", the effect of x is
0.013,
with a 95% confidence interval
[-0.112, 0.138].
Plotting the Conditional Effects
These are the plots of the conditional effects:
plot(out_site)
plot(out_site,
graph_type = "tumble")
This example demonstrates the advantage of
the tumble graph. The distribution of
x in "Site 3" has a mean lower than
those in the other two sites. The
conventional plot using the same range
of x in all three groups will give
the incorrect impression of a cross-over
of the lines
in the samples (though it is possible that
such a cross-over of the lines happens in the
populations).
Standardized Conditional Effects
This is the output of the standardized
conditional effects, with bootstrap
confidence intervals:
std_site <- cond_effects(
wlevels = "site",
x = "x",
y = "y",
fit = lm_y_site,
boot_ci = TRUE,
R = 5000,
seed = 54532,
standardized_x = TRUE,
standardized_y = TRUE
)
#> 19 processes started to run bootstrapping.
std_site
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): site
#> Moderator(s) represented by: siteSite 2, siteSite 3
#> Computation Formula:
#> (b.y~x + (b.siteSite 2:x)*(siteSite 2) + (b.siteSite 3:x)*(siteSite 3))*sd_x/sd_y
#>
#>
#> [site] (siteSite 2) (siteSite 3) std CI.lo CI.hi Sig ind
#> 1 Site 1 0 0 0.146 -0.023 0.317 0.102
#> 2 Site 2 1 0 0.616 0.414 0.818 Sig 0.432
#> 3 Site 3 0 1 0.018 -0.157 0.201 0.013
#>
#> - [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.
Plot Standardized Conditional Effects
These are the plots of the standardized
conditional effects:
plot(std_site)
plot(std_site,
graph_type = "tumble")
Two Categorical Moderators
The steps demonstrated above can be used
in a regression model with any number of
moderators.
Suppose this is the model being fitted,
with control variables omitted from the
plot for readability, both gp and site
included, but no interaction between them:
Fit by Regression
We first fit the regression model as usual:
lm_y_gp_site <- lm(
y ~ gp*x + site*x + c1 + c2,
data = dat
)
These are the estimates of the regression coefficients
of the paths:
summary(lm_y_gp_site)
#>
#> Call:
#> lm(formula = y ~ gp * x + site * x + c1 + c2, data = dat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -11.4604 -2.5856 -0.0874 2.5927 14.1720
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 21.09442 1.89659 11.122 < 2e-16 ***
#> gpTreatment -5.24348 1.07451 -4.880 1.37e-06 ***
#> x -0.08388 0.05955 -1.409 0.159472
#> siteSite 2 -6.51680 2.00106 -3.257 0.001192 **
#> siteSite 3 1.39984 1.64547 0.851 0.395268
#> c1 -0.01832 0.03766 -0.486 0.626860
#> c2 0.38081 0.04345 8.764 < 2e-16 ***
#> gpTreatment:x 0.35510 0.04857 7.312 8.64e-13 ***
#> x:siteSite 2 0.31277 0.08070 3.876 0.000118 ***
#> x:siteSite 3 -0.08921 0.08091 -1.103 0.270626
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 3.999 on 590 degrees of freedom
#> Multiple R-squared: 0.3077, Adjusted R-squared: 0.2971
#> F-statistic: 29.13 on 9 and 590 DF, p-value: < 2.2e-16
Conditional Effects
The function cond_effects() can be used
for any number of moderators, as long as
they are listed in wlevels:
out_gp_site <- cond_effects(
wlevels = c("gp", "site"),
x = "x",
y = "y",
fit = lm_y_gp_site
)
out_gp_site
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): gp, site
#> Moderator(s) represented by: gpTreatment, siteSite 2, siteSite 3
#> Computation Formula:
#> (b.y~x + (b.gpTreatment:x)*(gpTreatment) + (b.x:siteSite 2)*(siteSite 2) + (b.x:siteSite 3)*(siteSite
#> 3))
#>
#>
#> [gp] [site] (gpTreatment) (siteSite 2) (siteSite 3) ind SE Stat pvalue Sig CI.lo CI.hi
#> 1 Control Site 1 0 0 0 -0.084 0.060 -1.409 0.159 -0.201 0.033
#> 2 Control Site 2 0 1 0 0.229 0.065 3.527 0.000 *** 0.101 0.356
#> 3 Control Site 3 0 0 1 -0.173 0.067 -2.592 0.010 ** -0.304 -0.042
#> 4 Treatment Site 1 1 0 0 0.271 0.060 4.543 0.000 *** 0.154 0.388
#> 5 Treatment Site 2 1 1 0 0.584 0.064 9.146 0.000 *** 0.459 0.709
#> 6 Treatment Site 3 1 0 1 0.182 0.062 2.936 0.003 ** 0.060 0.304
#>
#> - [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.
IMPORTANT: Even though this model does not
have a three-way interaction, the conditional
effects still need to consider both
moderators. This is because the effect of
x depends on all moderators, whether
there is 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 = "gp",
x = "x",
y = "y",
fit = lm_y_gp_site
)
#> Warning in (function (xi, yi, yiname, digits = 3, y, wvalues = NULL, warn = TRUE, : siteSite 2, siteSite 3 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, : siteSite 2, siteSite 3 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): gp
#> Moderator(s) represented by: gpTreatment
#> Computation Formula:
#> (b.y~x + (b.gpTreatment:x)*(gpTreatment) + (b.x:siteSite 2)*(siteSite 2) + (b.x:siteSite 3)*(siteSite
#> 3))
#>
#>
#> [gp] (gpTreatment) ind SE Stat pvalue Sig CI.lo CI.hi
#> 1 Control 0 -0.084 0.060 -1.409 0.159 -0.201 0.033
#> 2 Treatment 1 0.271 0.060 4.543 0.000 *** 0.154 0.388
#>
#> - [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
These are the plots of the conditional effects:
plot(out_gp_site)
For two or more moderators, it is not
easy to visualize the conditional effects
if all lines are plotted on the same graph.
The argument facet_grid_cols can be used
to plot the effect of one moderator for
each category of the other moderator.
This is the plot of effects for "Control"
and "Treatment":
plot(out_gp_site,
facet_grid_cols = "gp")
This is the plot of effects for each site:
plot(out_gp_site,
facet_grid_cols = "site")
Note that, without three-way interaction,
the moderating effect of gp is the same
in all three sites, and the moderating effect
of site is the same in all two
groups. The six lines are different simply
because the effect of x depends on both
gp and site. They do not denote
three-way interaction (because it is not in the
regression model).
Tumble Plots
We already know the distributions of x
are not the same in all three sites.
Therefore, the tumble graph is a better
way to visualize the effects:
plot(out_gp_site,
facet_grid_cols = "gp",
graph_type = "tumble")
plot(out_gp_site,
facet_grid_cols = "site",
graph_type = "tumble")
Standardized Conditional Effects
This is the output of the standardized
conditional effects, with bootstrap
confidence intervals:
std_gp_site <- cond_effects(
wlevels = c("gp", "site"),
x = "x",
y = "y",
fit = lm_y_gp_site,
boot_ci = TRUE,
R = 5000,
seed = 54532,
standardized_x = TRUE,
standardized_y = TRUE
)
#> 19 processes started to run bootstrapping.
std_gp_site
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): gp, site
#> Moderator(s) represented by: gpTreatment, siteSite 2, siteSite 3
#> Computation Formula:
#> (b.y~x + (b.gpTreatment:x)*(gpTreatment) + (b.x:siteSite 2)*(siteSite 2) + (b.x:siteSite 3)*(siteSite
#> 3))*sd_x/sd_y
#>
#>
#> [gp] [site] (gpTreatment) (siteSite 2) (siteSite 3) std CI.lo CI.hi Sig ind
#> 1 Control Site 1 0 0 0 -0.120 -0.275 0.035 -0.084
#> 2 Control Site 2 0 1 0 0.327 0.153 0.495 Sig 0.229
#> 3 Control Site 3 0 0 1 -0.247 -0.460 -0.027 Sig -0.173
#> 4 Treatment Site 1 1 0 0 0.387 0.220 0.549 Sig 0.271
#> 5 Treatment Site 2 1 1 0 0.833 0.650 1.009 Sig 0.584
#> 6 Treatment Site 3 1 0 1 0.260 0.068 0.466 Sig 0.182
#>
#> - [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.
Tumble Plots of Standardized Conditional Effects
These are the plots of the standardized
conditional effects, with facet_grid_cols
set:
plot(std_gp_site,
facet_grid_cols = "gp",
graph_type = "tumble")
plot(std_gp_site,
facet_grid_cols = "site",
graph_type = "tumble")
Two Categorical Moderators, with Three-Way Interaction
Suppose that we suspect that the two
categorical moderators interact with each
other. That is, the group difference in
the effect of x may not be the same in
all three sites, or the site difference
in the effect of x may not be the same
in the two groups.
The steps demonstrated above can also be used
in this regression model:
lm_y_gp_x_site <- lm(
y ~ x*site*gp + c1 + c2,
data = dat
)
The test of the difference between this
model and the previous model with no three-way
interaction supports that this model fits
better:
anova(
lm_y_gp_site,
lm_y_gp_x_site
)
#> Analysis of Variance Table
#>
#> Model 1: y ~ gp * x + site * x + c1 + c2
#> Model 2: y ~ x * site * gp + c1 + c2
#> Res.Df RSS Df Sum of Sq F Pr(>F)
#> 1 590 9436.6
#> 2 586 9108.5 4 328.07 5.2766 0.0003516 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
These are the estimates of the regression coefficients
of this model:
summary(lm_y_gp_x_site)
#>
#> Call:
#> lm(formula = y ~ x * site * gp + c1 + c2, data = dat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -11.0954 -2.5367 -0.0151 2.4875 14.3118
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 22.077128 2.213474 9.974 < 2e-16 ***
#> x -0.128074 0.074369 -1.722 0.08557 .
#> siteSite 2 -5.276352 2.748471 -1.920 0.05538 .
#> siteSite 3 -3.063916 2.296016 -1.334 0.18258
#> gpTreatment -8.501074 2.708983 -3.138 0.00179 **
#> c1 -0.002987 0.037316 -0.080 0.93622
#> c2 0.385877 0.042952 8.984 < 2e-16 ***
#> x:siteSite 2 0.222148 0.112091 1.982 0.04796 *
#> x:siteSite 3 0.154193 0.113159 1.363 0.17352
#> x:gpTreatment 0.453702 0.107574 4.218 2.86e-05 ***
#> siteSite 2:gpTreatment -1.970235 3.954140 -0.498 0.61848
#> siteSite 3:gpTreatment 8.743721 3.254470 2.687 0.00742 **
#> x:siteSite 2:gpTreatment 0.157510 0.159449 0.988 0.32364
#> x:siteSite 3:gpTreatment -0.485461 0.160220 -3.030 0.00255 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 3.943 on 586 degrees of freedom
#> Multiple R-squared: 0.3317, Adjusted R-squared: 0.3169
#> F-statistic: 22.38 on 13 and 586 DF, p-value: < 2.2e-16
Conditional Effects
The function cond_effects() can be
used in exactly the same way, whether
the moderators interact with each other
or not:
out_gp_x_site <- cond_effects(
wlevels = c("site", "gp"),
x = "x",
y = "y",
fit = lm_y_gp_x_site
)
out_gp_x_site
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): site, gp
#> Moderator(s) represented by: siteSite 2, siteSite 3, gpTreatment
#> Computation Formula:
#> (b.y~x + (b.x:siteSite 2)*(siteSite 2) + (b.x:siteSite 3)*(siteSite 3) + (b.x:gpTreatment)*(gpTreatment)
#> + (b.x:siteSite 2:gpTreatment)*(siteSite 2*gpTreatment) + (b.x:siteSite 3:gpTreatment)*(siteSite
#> 3*gpTreatment))
#>
#>
#> [site] [gp] (siteSite 2) (siteSite 3) (gpTreatment) ind SE Stat pvalue Sig CI.lo CI.hi
#> 1 Site 1 Control 0 0 0 -0.128 0.074 -1.722 0.086 -0.274 0.018
#> 2 Site 1 Treatment 0 0 1 0.326 0.078 4.190 0.000 *** 0.173 0.478
#> 3 Site 2 Control 1 0 0 0.094 0.084 1.122 0.262 -0.071 0.259
#> 4 Site 2 Treatment 1 0 1 0.705 0.083 8.543 0.000 *** 0.543 0.867
#> 5 Site 3 Control 0 1 0 0.026 0.085 0.307 0.759 -0.141 0.193
#> 6 Site 3 Treatment 0 1 1 -0.006 0.082 -0.069 0.945 -0.167 0.156
#>
#> - [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
These are the tumble plots of the conditional effects,
with facet_grid_cols set:
plot(out_gp_x_site,
facet_grid_cols = "gp",
graph_type = "tumble")
plot(out_gp_x_site,
facet_grid_cols = "site",
graph_type = "tumble")
Standardized Conditional Effects
This is the output of the standardized
conditional effects, with bootstrap
confidence intervals:
std_gp_x_site <- cond_effects(
wlevels = c("gp", "site"),
x = "x",
y = "y",
fit = lm_y_gp_x_site,
boot_ci = TRUE,
R = 5000,
seed = 54532,
standardized_x = TRUE,
standardized_y = TRUE
)
#> 19 processes started to run bootstrapping.
std_gp_x_site
#>
#> == Conditional effects ==
#>
#> Path: x -> y
#> Conditional on moderator(s): gp, site
#> Moderator(s) represented by: gpTreatment, siteSite 2, siteSite 3
#> Computation Formula:
#> (b.y~x + (b.x:siteSite 2)*(siteSite 2) + (b.x:siteSite 3)*(siteSite 3) + (b.x:gpTreatment)*(gpTreatment)
#> + (b.x:siteSite 2:gpTreatment)*(siteSite 2*gpTreatment) + (b.x:siteSite 3:gpTreatment)*(siteSite
#> 3*gpTreatment))*sd_x/sd_y
#>
#>
#> [gp] [site] (gpTreatment) (siteSite 2) (siteSite 3) std CI.lo CI.hi Sig ind
#> 1 Control Site 1 0 0 0 -0.183 -0.374 0.003 -0.128
#> 2 Control Site 2 0 1 0 0.134 -0.045 0.298 0.094
#> 3 Control Site 3 0 0 1 0.037 -0.252 0.316 0.026
#> 4 Treatment Site 1 1 0 0 0.465 0.226 0.692 Sig 0.326
#> 5 Treatment Site 2 1 1 0 1.006 0.740 1.269 Sig 0.705
#> 6 Treatment Site 3 1 0 1 -0.008 -0.229 0.231 -0.006
#>
#> - [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.
These are the plots of the standardized
conditional effects, with facet_grid_cols
set:
plot(std_gp_x_site,
facet_grid_cols = "gp",
graph_type = "tumble")
plot(std_gp_x_site,
facet_grid_cols = "site",
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_cat_2w print(head(dat), digits = 3) #> x y c1 c2 gp site #> 1 19.8 24.5 29.5 10.2 Control Site 1 #> 2 24.0 21.6 15.3 10.3 Control Site 1 #> 3 28.4 24.7 29.4 20.4 Control Site 1 #> 4 39.7 26.9 32.4 16.7 Control Site 1 #> 5 33.4 21.1 36.5 10.2 Control Site 1 #> 6 22.7 23.8 27.9 23.5 Control Site 1
This dataset has 6 variables:
-
one outcome variable (
y), -
one predictor (
x), -
two categorical moderators (
gp,site), -
two control variables (
c1andc2).
The moderator gp has two possible
values: "Control" and "Treatment".
The moderator site has three possible
values: "Site 1", "Site 2", and
"Site 3".
We will first consider two models, each with only one of the two moderators.
One Categorical Moderator: Two Categories
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_gp <- lm( y ~ gp*x + c1 + c2, data = dat )
These are the estimates of the regression coefficients of the paths:
summary(lm_y_gp) #> #> Call: #> lm(formula = y ~ gp * x + c1 + c2, data = dat) #> #> Residuals: #> Min 1Q Median 3Q Max #> -10.8400 -2.6594 -0.2263 2.6201 15.5956 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 20.26140 1.49550 13.548 < 2e-16 *** #> gpTreatment -4.96460 1.09316 -4.542 6.77e-06 *** #> x -0.02738 0.03674 -0.745 0.456 #> c1 -0.02833 0.03843 -0.737 0.461 #> c2 0.39637 0.04438 8.932 < 2e-16 *** #> gpTreatment:x 0.34746 0.04953 7.015 6.31e-12 *** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 4.093 on 594 degrees of freedom #> Multiple R-squared: 0.2698, Adjusted R-squared: 0.2636 #> F-statistic: 43.89 on 5 and 594 DF, p-value: < 2.2e-16
Conditional Effects
We can now use cond_effects() to
estimate the effects of x
on y for
different categories of the moderator gp.
(Refer to vignette("manymome") and the help page
of cond_effects() on the arguments.)
out_gp <- cond_effects( wlevels = "gp", x = "x", y = "y", fit = lm_y_gp ) out_gp #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): gp #> Moderator(s) represented by: gpTreatment #> Computation Formula: #> (b.y~x + (b.gpTreatment:x)*(gpTreatment)) #> #> #> [gp] (gpTreatment) ind SE Stat pvalue Sig CI.lo CI.hi #> 1 Control 0 -0.027 0.037 -0.745 0.456 -0.100 0.045 #> 2 Treatment 1 0.320 0.033 9.655 0.000 *** 0.255 0.385 #> #> - [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 categories of gp.
In the group "Control", the effect of x is
-0.027,
with a 95% confidence interval
[-0.100, 0.045].
In the group "Treatment", the effect of x is
0.320,
with a 95% confidence interval
[0.255, 0.385].
NOTE: The standard error (SE) and
related results are computed using
the pick-a-point approach by
@rogosa_comparing_1980.
Plotting the Conditional Effects
The output of cond_effects() has a plot
method for plotting the conditional effects:
plot(out_gp)
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(out_gp, graph_type = "tumble")
In this example, the distributions of x
for the groups
are similar.
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
effects from x to y conditional
on gp
can be estimated by setting
standardized_x and standardized_y to TRUE.
This is the output:
std_gp <- cond_effects( wlevels = "gp", x = "x", y = "y", fit = lm_y_gp, boot_ci = TRUE, R = 5000, seed = 54532, standardized_x = TRUE, standardized_y = TRUE ) #> 19 processes started to run bootstrapping. std_gp #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): gp #> Moderator(s) represented by: gpTreatment #> Computation Formula: #> (b.y~x + (b.gpTreatment:x)*(gpTreatment))*sd_x/sd_y #> #> #> [gp] (gpTreatment) std CI.lo CI.hi Sig ind #> 1 Control 0 -0.039 -0.130 0.048 -0.027 #> 2 Treatment 1 0.457 0.363 0.551 Sig 0.320 #> #> - [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.
In the group "Control", the standardized effect of x is
-0.039,
with a 95% confidence interval
[-0.130, 0.048].
In the group "Treatment", the standardized effect of x is
0.457,
with a 95% confidence interval
[0.363, 0.551].
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_gp)
plot(std_gp, graph_type = "tumble")
One Categorical Moderator: Three Categories
The steps demonstrated above can be used for a categorical moderator with any number of levels.
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 a multiple regression model:
lm_y_site <- lm( y ~ site*x + c1 + c2, data = dat )
These are the estimates of the regression coefficients of the paths:
summary(lm_y_site) #> #> Call: #> lm(formula = y ~ site * x + c1 + c2, data = dat) #> #> Residuals: #> Min 1Q Median 3Q Max #> -12.2398 -3.0852 0.2666 2.7867 17.3110 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 19.01557 1.99047 9.553 < 2e-16 *** #> siteSite 2 -6.89570 2.15974 -3.193 0.001484 ** #> siteSite 3 1.30350 1.76713 0.738 0.461027 #> x 0.10205 0.05877 1.737 0.082989 . #> c1 -0.03133 0.04062 -0.771 0.440934 #> c2 0.35831 0.04681 7.655 7.91e-14 *** #> siteSite 2:x 0.32975 0.08710 3.786 0.000169 *** #> siteSite 3:x -0.08913 0.08673 -1.028 0.304478 #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 4.317 on 592 degrees of freedom #> Multiple R-squared: 0.1905, Adjusted R-squared: 0.1809 #> F-statistic: 19.9 on 7 and 592 DF, p-value: < 2.2e-16
Conditional Effects
The function cond_effects() will determine
the number of categories automatically:
out_site <- cond_effects( wlevels = "site", x = "x", y = "y", fit = lm_y_site ) out_site #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): site #> Moderator(s) represented by: siteSite 2, siteSite 3 #> Computation Formula: #> (b.y~x + (b.siteSite 2:x)*(siteSite 2) + (b.siteSite 3:x)*(siteSite 3)) #> #> #> [site] (siteSite 2) (siteSite 3) ind SE Stat pvalue Sig CI.lo CI.hi #> 1 Site 1 0 0 0.102 0.059 1.737 0.083 -0.013 0.217 #> 2 Site 2 1 0 0.432 0.064 6.715 0.000 *** 0.306 0.558 #> 3 Site 3 0 1 0.013 0.064 0.203 0.840 -0.112 0.138 #> #> - [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.
In the site "Site 1", the effect of x is
0.102,
with a 95% confidence interval
[-0.013, 0.217].
In the site "Site 2", the effect of x is
0.432,
with a 95% confidence interval
[0.306, 0.558].
In the site "Site 3", the effect of x is
0.013,
with a 95% confidence interval
[-0.112, 0.138].
Plotting the Conditional Effects
These are the plots of the conditional effects:
plot(out_site)
plot(out_site, graph_type = "tumble")
This example demonstrates the advantage of
the tumble graph. The distribution of
x in "Site 3" has a mean lower than
those in the other two sites. The
conventional plot using the same range
of x in all three groups will give
the incorrect impression of a cross-over
of the lines
in the samples (though it is possible that
such a cross-over of the lines happens in the
populations).
Standardized Conditional Effects
This is the output of the standardized conditional effects, with bootstrap confidence intervals:
std_site <- cond_effects( wlevels = "site", x = "x", y = "y", fit = lm_y_site, boot_ci = TRUE, R = 5000, seed = 54532, standardized_x = TRUE, standardized_y = TRUE ) #> 19 processes started to run bootstrapping. std_site #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): site #> Moderator(s) represented by: siteSite 2, siteSite 3 #> Computation Formula: #> (b.y~x + (b.siteSite 2:x)*(siteSite 2) + (b.siteSite 3:x)*(siteSite 3))*sd_x/sd_y #> #> #> [site] (siteSite 2) (siteSite 3) std CI.lo CI.hi Sig ind #> 1 Site 1 0 0 0.146 -0.023 0.317 0.102 #> 2 Site 2 1 0 0.616 0.414 0.818 Sig 0.432 #> 3 Site 3 0 1 0.018 -0.157 0.201 0.013 #> #> - [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.
Plot Standardized Conditional Effects
These are the plots of the standardized conditional effects:
plot(std_site)
plot(std_site, graph_type = "tumble")
Two Categorical Moderators
The steps demonstrated above can be used in a regression model with any number of moderators.
Suppose this is the model being fitted,
with control variables omitted from the
plot for readability, both gp and site
included, but no interaction between them:
Fit by Regression
We first fit the regression model as usual:
lm_y_gp_site <- lm( y ~ gp*x + site*x + c1 + c2, data = dat )
These are the estimates of the regression coefficients of the paths:
summary(lm_y_gp_site) #> #> Call: #> lm(formula = y ~ gp * x + site * x + c1 + c2, data = dat) #> #> Residuals: #> Min 1Q Median 3Q Max #> -11.4604 -2.5856 -0.0874 2.5927 14.1720 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 21.09442 1.89659 11.122 < 2e-16 *** #> gpTreatment -5.24348 1.07451 -4.880 1.37e-06 *** #> x -0.08388 0.05955 -1.409 0.159472 #> siteSite 2 -6.51680 2.00106 -3.257 0.001192 ** #> siteSite 3 1.39984 1.64547 0.851 0.395268 #> c1 -0.01832 0.03766 -0.486 0.626860 #> c2 0.38081 0.04345 8.764 < 2e-16 *** #> gpTreatment:x 0.35510 0.04857 7.312 8.64e-13 *** #> x:siteSite 2 0.31277 0.08070 3.876 0.000118 *** #> x:siteSite 3 -0.08921 0.08091 -1.103 0.270626 #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 3.999 on 590 degrees of freedom #> Multiple R-squared: 0.3077, Adjusted R-squared: 0.2971 #> F-statistic: 29.13 on 9 and 590 DF, p-value: < 2.2e-16
Conditional Effects
The function cond_effects() can be used
for any number of moderators, as long as
they are listed in wlevels:
out_gp_site <- cond_effects( wlevels = c("gp", "site"), x = "x", y = "y", fit = lm_y_gp_site ) out_gp_site #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): gp, site #> Moderator(s) represented by: gpTreatment, siteSite 2, siteSite 3 #> Computation Formula: #> (b.y~x + (b.gpTreatment:x)*(gpTreatment) + (b.x:siteSite 2)*(siteSite 2) + (b.x:siteSite 3)*(siteSite #> 3)) #> #> #> [gp] [site] (gpTreatment) (siteSite 2) (siteSite 3) ind SE Stat pvalue Sig CI.lo CI.hi #> 1 Control Site 1 0 0 0 -0.084 0.060 -1.409 0.159 -0.201 0.033 #> 2 Control Site 2 0 1 0 0.229 0.065 3.527 0.000 *** 0.101 0.356 #> 3 Control Site 3 0 0 1 -0.173 0.067 -2.592 0.010 ** -0.304 -0.042 #> 4 Treatment Site 1 1 0 0 0.271 0.060 4.543 0.000 *** 0.154 0.388 #> 5 Treatment Site 2 1 1 0 0.584 0.064 9.146 0.000 *** 0.459 0.709 #> 6 Treatment Site 3 1 0 1 0.182 0.062 2.936 0.003 ** 0.060 0.304 #> #> - [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.
IMPORTANT: Even though this model does not
have a three-way interaction, the conditional
effects still need to consider both
moderators. This is because the effect of
x depends on all moderators, whether
there is 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 = "gp", x = "x", y = "y", fit = lm_y_gp_site ) #> Warning in (function (xi, yi, yiname, digits = 3, y, wvalues = NULL, warn = TRUE, : siteSite 2, siteSite 3 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, : siteSite 2, siteSite 3 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): gp #> Moderator(s) represented by: gpTreatment #> Computation Formula: #> (b.y~x + (b.gpTreatment:x)*(gpTreatment) + (b.x:siteSite 2)*(siteSite 2) + (b.x:siteSite 3)*(siteSite #> 3)) #> #> #> [gp] (gpTreatment) ind SE Stat pvalue Sig CI.lo CI.hi #> 1 Control 0 -0.084 0.060 -1.409 0.159 -0.201 0.033 #> 2 Treatment 1 0.271 0.060 4.543 0.000 *** 0.154 0.388 #> #> - [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
These are the plots of the conditional effects:
plot(out_gp_site)
For two or more moderators, it is not easy to visualize the conditional effects if all lines are plotted on the same graph.
The argument facet_grid_cols can be used
to plot the effect of one moderator for
each category of the other moderator.
This is the plot of effects for "Control"
and "Treatment":
plot(out_gp_site, facet_grid_cols = "gp")
This is the plot of effects for each site:
plot(out_gp_site, facet_grid_cols = "site")
Note that, without three-way interaction,
the moderating effect of gp is the same
in all three sites, and the moderating effect
of site is the same in all two
groups. The six lines are different simply
because the effect of x depends on both
gp and site. They do not denote
three-way interaction (because it is not in the
regression model).
Tumble Plots
We already know the distributions of x
are not the same in all three sites.
Therefore, the tumble graph is a better
way to visualize the effects:
plot(out_gp_site, facet_grid_cols = "gp", graph_type = "tumble")
plot(out_gp_site, facet_grid_cols = "site", graph_type = "tumble")
Standardized Conditional Effects
This is the output of the standardized conditional effects, with bootstrap confidence intervals:
std_gp_site <- cond_effects( wlevels = c("gp", "site"), x = "x", y = "y", fit = lm_y_gp_site, boot_ci = TRUE, R = 5000, seed = 54532, standardized_x = TRUE, standardized_y = TRUE ) #> 19 processes started to run bootstrapping. std_gp_site #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): gp, site #> Moderator(s) represented by: gpTreatment, siteSite 2, siteSite 3 #> Computation Formula: #> (b.y~x + (b.gpTreatment:x)*(gpTreatment) + (b.x:siteSite 2)*(siteSite 2) + (b.x:siteSite 3)*(siteSite #> 3))*sd_x/sd_y #> #> #> [gp] [site] (gpTreatment) (siteSite 2) (siteSite 3) std CI.lo CI.hi Sig ind #> 1 Control Site 1 0 0 0 -0.120 -0.275 0.035 -0.084 #> 2 Control Site 2 0 1 0 0.327 0.153 0.495 Sig 0.229 #> 3 Control Site 3 0 0 1 -0.247 -0.460 -0.027 Sig -0.173 #> 4 Treatment Site 1 1 0 0 0.387 0.220 0.549 Sig 0.271 #> 5 Treatment Site 2 1 1 0 0.833 0.650 1.009 Sig 0.584 #> 6 Treatment Site 3 1 0 1 0.260 0.068 0.466 Sig 0.182 #> #> - [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.
Tumble Plots of Standardized Conditional Effects
These are the plots of the standardized
conditional effects, with facet_grid_cols
set:
plot(std_gp_site, facet_grid_cols = "gp", graph_type = "tumble")
plot(std_gp_site, facet_grid_cols = "site", graph_type = "tumble")
Two Categorical Moderators, with Three-Way Interaction
Suppose that we suspect that the two
categorical moderators interact with each
other. That is, the group difference in
the effect of x may not be the same in
all three sites, or the site difference
in the effect of x may not be the same
in the two groups.
The steps demonstrated above can also be used in this regression model:
lm_y_gp_x_site <- lm( y ~ x*site*gp + c1 + c2, data = dat )
The test of the difference between this model and the previous model with no three-way interaction supports that this model fits better:
anova( lm_y_gp_site, lm_y_gp_x_site ) #> Analysis of Variance Table #> #> Model 1: y ~ gp * x + site * x + c1 + c2 #> Model 2: y ~ x * site * gp + c1 + c2 #> Res.Df RSS Df Sum of Sq F Pr(>F) #> 1 590 9436.6 #> 2 586 9108.5 4 328.07 5.2766 0.0003516 *** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
These are the estimates of the regression coefficients of this model:
summary(lm_y_gp_x_site) #> #> Call: #> lm(formula = y ~ x * site * gp + c1 + c2, data = dat) #> #> Residuals: #> Min 1Q Median 3Q Max #> -11.0954 -2.5367 -0.0151 2.4875 14.3118 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 22.077128 2.213474 9.974 < 2e-16 *** #> x -0.128074 0.074369 -1.722 0.08557 . #> siteSite 2 -5.276352 2.748471 -1.920 0.05538 . #> siteSite 3 -3.063916 2.296016 -1.334 0.18258 #> gpTreatment -8.501074 2.708983 -3.138 0.00179 ** #> c1 -0.002987 0.037316 -0.080 0.93622 #> c2 0.385877 0.042952 8.984 < 2e-16 *** #> x:siteSite 2 0.222148 0.112091 1.982 0.04796 * #> x:siteSite 3 0.154193 0.113159 1.363 0.17352 #> x:gpTreatment 0.453702 0.107574 4.218 2.86e-05 *** #> siteSite 2:gpTreatment -1.970235 3.954140 -0.498 0.61848 #> siteSite 3:gpTreatment 8.743721 3.254470 2.687 0.00742 ** #> x:siteSite 2:gpTreatment 0.157510 0.159449 0.988 0.32364 #> x:siteSite 3:gpTreatment -0.485461 0.160220 -3.030 0.00255 ** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 3.943 on 586 degrees of freedom #> Multiple R-squared: 0.3317, Adjusted R-squared: 0.3169 #> F-statistic: 22.38 on 13 and 586 DF, p-value: < 2.2e-16
Conditional Effects
The function cond_effects() can be
used in exactly the same way, whether
the moderators interact with each other
or not:
out_gp_x_site <- cond_effects( wlevels = c("site", "gp"), x = "x", y = "y", fit = lm_y_gp_x_site ) out_gp_x_site #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): site, gp #> Moderator(s) represented by: siteSite 2, siteSite 3, gpTreatment #> Computation Formula: #> (b.y~x + (b.x:siteSite 2)*(siteSite 2) + (b.x:siteSite 3)*(siteSite 3) + (b.x:gpTreatment)*(gpTreatment) #> + (b.x:siteSite 2:gpTreatment)*(siteSite 2*gpTreatment) + (b.x:siteSite 3:gpTreatment)*(siteSite #> 3*gpTreatment)) #> #> #> [site] [gp] (siteSite 2) (siteSite 3) (gpTreatment) ind SE Stat pvalue Sig CI.lo CI.hi #> 1 Site 1 Control 0 0 0 -0.128 0.074 -1.722 0.086 -0.274 0.018 #> 2 Site 1 Treatment 0 0 1 0.326 0.078 4.190 0.000 *** 0.173 0.478 #> 3 Site 2 Control 1 0 0 0.094 0.084 1.122 0.262 -0.071 0.259 #> 4 Site 2 Treatment 1 0 1 0.705 0.083 8.543 0.000 *** 0.543 0.867 #> 5 Site 3 Control 0 1 0 0.026 0.085 0.307 0.759 -0.141 0.193 #> 6 Site 3 Treatment 0 1 1 -0.006 0.082 -0.069 0.945 -0.167 0.156 #> #> - [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
These are the tumble plots of the conditional effects,
with facet_grid_cols set:
plot(out_gp_x_site, facet_grid_cols = "gp", graph_type = "tumble")
plot(out_gp_x_site, facet_grid_cols = "site", graph_type = "tumble")
Standardized Conditional Effects
This is the output of the standardized conditional effects, with bootstrap confidence intervals:
std_gp_x_site <- cond_effects( wlevels = c("gp", "site"), x = "x", y = "y", fit = lm_y_gp_x_site, boot_ci = TRUE, R = 5000, seed = 54532, standardized_x = TRUE, standardized_y = TRUE ) #> 19 processes started to run bootstrapping. std_gp_x_site #> #> == Conditional effects == #> #> Path: x -> y #> Conditional on moderator(s): gp, site #> Moderator(s) represented by: gpTreatment, siteSite 2, siteSite 3 #> Computation Formula: #> (b.y~x + (b.x:siteSite 2)*(siteSite 2) + (b.x:siteSite 3)*(siteSite 3) + (b.x:gpTreatment)*(gpTreatment) #> + (b.x:siteSite 2:gpTreatment)*(siteSite 2*gpTreatment) + (b.x:siteSite 3:gpTreatment)*(siteSite #> 3*gpTreatment))*sd_x/sd_y #> #> #> [gp] [site] (gpTreatment) (siteSite 2) (siteSite 3) std CI.lo CI.hi Sig ind #> 1 Control Site 1 0 0 0 -0.183 -0.374 0.003 -0.128 #> 2 Control Site 2 0 1 0 0.134 -0.045 0.298 0.094 #> 3 Control Site 3 0 0 1 0.037 -0.252 0.316 0.026 #> 4 Treatment Site 1 1 0 0 0.465 0.226 0.692 Sig 0.326 #> 5 Treatment Site 2 1 1 0 1.006 0.740 1.269 Sig 0.705 #> 6 Treatment Site 3 1 0 1 -0.008 -0.229 0.231 -0.006 #> #> - [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.
These are the plots of the standardized
conditional effects, with facet_grid_cols
set:
plot(std_gp_x_site, facet_grid_cols = "gp", graph_type = "tumble")
plot(std_gp_x_site, facet_grid_cols = "site", 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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