Skip to contents

Moderated Regression: Two Predictors and Two Moderators

Shu Fai Cheung & Sing-Hang Cheung

2026-07-02

Source: vignettes/articles/mo_lm_2x2w.Rmd
mo_lm_2x2w.Rmd

Introduction

This article is part of a series of brief illustrations of how to use cond_effects() from the package manymome (Cheung & Cheung, 2024) to estimate the conditional effects when the model parameters are estimate 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_2x2w
print(head(dat), digits = 3)
#>      y   x1   x2   w1   w2   c1   c2
#> 1 0.46 1.55 0.73 2.06 1.13 2.69 0.43
#> 2 0.49 1.54 0.78 2.29 1.56 3.47 0.15
#> 3 0.69 1.45 0.96 1.98 1.91 2.75 0.43
#> 4 0.83 1.81 1.07 2.52 1.60 3.00 0.51
#> 5 0.53 1.36 0.70 1.83 0.98 3.79 0.62
#> 6 0.51 1.78 0.92 1.94 1.66 1.80 0.56

This dataset has 7 variables:

  • one outcome variable (y),

  • two predictors (x1, x2),

  • two moderators (w1, w2),

  • two control variables (c1 and c2).

Suppose this is the model being fitted, with control variables omitted from the plot for readability:

Model
Model

Fit by Regression

The path parameters can be estimated by multiple regression using lm():

lm_y <- lm(
  y ~ w1*x1 + w2*x2 + c1 + c2,
  data = dat
)

These are the estimates of the regression coefficient of the paths:

summary(lm_y)
#> 
#> Call:
#> lm(formula = y ~ w1 * x1 + w2 * x2 + c1 + c2, data = dat)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -0.210561 -0.046282 -0.000192  0.045746  0.217473 
#> 
#> Coefficients:
#>              Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  2.234274   0.294081   7.597 1.31e-12 ***
#> w1          -0.797684   0.120573  -6.616 3.62e-10 ***
#> x1          -1.167658   0.160032  -7.296 7.70e-12 ***
#> w2          -0.236512   0.128024  -1.847   0.0662 .  
#> x2          -0.364031   0.215891  -1.686   0.0934 .  
#> c1           0.006922   0.010930   0.633   0.5273    
#> c2          -0.058162   0.052763  -1.102   0.2717    
#> w1:x1        0.628532   0.074538   8.432 8.18e-15 ***
#> w2:x2        0.356141   0.158749   2.243   0.0260 *  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.07412 on 191 degrees of freedom
#> Multiple R-squared:  0.6038, Adjusted R-squared:  0.5872 
#> F-statistic: 36.38 on 8 and 191 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 moderators, w1 and w2.

Conditional Effects of x1

Suppose we want to estimate the effect from x1 to y, conditional on w1:

(Refer to vignette("manymome") and the help page of cond_effects() on the arguments.)

out1 <- cond_effects(
  wlevels = "w1",
  x = "x1",
  y = "y",
  fit = lm_y
)
out1
#> 
#> == Conditional effects ==
#> 
#>  Path: x1 -> y
#>  Conditional on moderator(s): w1
#>  Moderator(s) represented by: w1
#> 
#>      [w1]  (w1)    ind    SE   Stat pvalue Sig  CI.lo CI.hi
#> 1 M+1.0SD 2.285  0.269 0.023 11.801  0.000 ***  0.224 0.313
#> 2 Mean    2.032  0.110 0.021  5.222  0.000 ***  0.068 0.151
#> 3 M-1.0SD 1.779 -0.050 0.033 -1.513  0.132     -0.114 0.015
#> 
#>  - [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.
#>

The column ind show the effects of x1 on y for different levels of w1.

When w1 is one standard deviation below mean, the effect of x1 is -0.050, with 95% confidence interval [-0.114, 0.015].

When w1 is one standard deviation above mean, the effect of x1 is 0.269, with 95% confidence interval [0.224, 0.313].

NOTE: The standard error (SE) and related results are computed using the pick-a-point approach by Rogosa (1980).

Conditional Effects of x2

The step to compute the conditional effects for the other predictor, x2, for different levels of w2 is similar:

out2 <- cond_effects(
  wlevels = "w2",
  x = "x2",
  y = "y",
  fit = lm_y
)
out2
#> 
#> == Conditional effects ==
#> 
#>  Path: x2 -> y
#>  Conditional on moderator(s): w2
#>  Moderator(s) represented by: w2
#> 
#>      [w2]  (w2)    ind    SE   Stat pvalue Sig  CI.lo CI.hi
#> 1 M+1.0SD 1.666  0.229 0.071  3.218  0.002  **  0.089 0.370
#> 2 Mean    1.332  0.110 0.047  2.358  0.019  *   0.018 0.203
#> 3 M-1.0SD 0.998 -0.008 0.070 -0.120  0.904     -0.147 0.130
#> 
#>  - [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.
#>

When w2 is one standard deviation below mean, the effect of x2 is -0.008, with 95% confidence interval [-0.147, 0.130].

When w2 is one standard deviation above mean, the effect of x2 is 0.229, with 95% confidence interval [0.089, 0.370].

Plotting the Conditional Effects

Conventional Plot

The output of cond_effects() has a plot method for plotting the conditional effects:

plot(out1)
Conventional Plot of Conditional Effects (w1)
Conventional Plot of Conditional Effects (w1)
plot(out2)
Conventional Plot of Conditional Effects (w2)
Conventional Plot of Conditional Effects (w2)

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 way. 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 (2016) can be plotted by adding graph_type = "tumble":

plot(out1,
     graph_type = "tumble")
Tumble Plot of Conditional Effects (w1)
Tumble Plot of Conditional Effects (w1)

The distributions of x1 vary as the level of the moderator w1 changes: the mean of x1 is lower and the standard deviation smaller when w1 is one standard deviation below its mean, and has higher mean and larger standard deviation when w1 is one standard deviation above its mean.

Therefore, the tumble graph is a better way to visualize the moderating effect of w1 on the effect of x1.

plot(out2,
     graph_type = "tumble")
Tumble Plot of Conditional Effects (w2)
Tumble Plot of Conditional Effects (w2)

In this example, the distributions of x2 for the three levels of moderator w2 are similar. For x2, 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 & Chan (2011) on the issue on standardized regression coefficients.

To form nonparametric bootstrap confidence interval 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 x1 and x2 to y conditional on w1 and w2 can be estimated by setting standardized_x and standardized_y to TRUE.

This is the output for x1:

std1 <- cond_effects(
  wlevels = "w1",
  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): w1
#>  Moderator(s) represented by: w1
#> 
#>      [w1]  (w1)    std  CI.lo CI.hi Sig    ind
#> 1 M+1.0SD 2.285  0.635  0.529 0.741 Sig  0.269
#> 2 Mean    2.032  0.259  0.158 0.350 Sig  0.110
#> 3 M-1.0SD 1.779 -0.117 -0.272 0.029     -0.050
#> 
#>  - [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.
#>

When w1 is one standard deviation below its mean, the standardized effect of x1 is -0.117, with 95% confidence interval [-0.272, 0.029].

When w1 is one standard deviation above its mean, the standardized effect of x1 is 0.635, with 95% confidence interval [0.529, 0.741].

This is the output for x2:

std2 <- cond_effects(
  wlevels = "w2",
  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): w2
#>  Moderator(s) represented by: w2
#> 
#>      [w2]  (w2)    std  CI.lo CI.hi Sig    ind
#> 1 M+1.0SD 1.666  0.227  0.111 0.344 Sig  0.229
#> 2 Mean    1.332  0.109  0.030 0.196 Sig  0.110
#> 3 M-1.0SD 0.998 -0.008 -0.132 0.136     -0.008
#> 
#>  - [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.
#>

When w2 is one standard deviation below its mean, the standardized effect of x2 is -0.008, with 95% confidence interval [-0.132, 0.136].

When w2 is one standard deviation above its mean, the standardized effect of x2 is 0.227, with 95% confidence interval [0.111, 0.344].

Conventional 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)
Conventional Plot of Standardized Conditional Effects (w1)
Conventional Plot of Standardized Conditional Effects (w1)
plot(std2)
Conventional Plot of Standardized Conditional Effects (w2)
Conventional Plot of Standardized Conditional Effects (w2)

Tumble Plot Standardized Conditional Effects

These are the tumble graphs:

plot(std1,
     graph_type = "tumble")
Tumble Plot of Standardized Conditional Effects (w1)
Tumble Plot of Standardized Conditional Effects (w1)
plot(std2,
     graph_type = "tumble")
Tumble Plot of Standardized Conditional Effects (w2)
Tumble Plot of Standardized Conditional Effects (w2)

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 from 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 model analyzed by multiple regression.

References

Bodner, T. E. (2016). Tumble graphs: Avoiding misleading end point extrapolation when graphing interactions from a moderated multiple regression analysis. Journal of Educational and Behavioral Statistics, 41(6), 593–604. https://doi.org/10.3102/1076998616657080
Cheung, S. F., & Cheung, S.-H. (2024). Manymome: An R package for computing the indirect effects, conditional effects, and conditional indirect effects, standardized or unstandardized, and their bootstrap confidence intervals, in many (though not all) models. Behavior Research Methods, 56(5), 4862–4882. https://doi.org/10.3758/s13428-023-02224-z
Rogosa, D. (1980). Comparing nonparallel regression lines. Psychological Bulletin, 88(2), 307–321. https://doi.org/10.1037/0033-2909.88.2.307
Yuan, K.-H., & Chan, W. (2011). Biases and standard errors of standardized regression coefficients. Psychometrika, 76(4), 670–690. https://doi.org/10.1007/s11336-011-9224-6