Moderated Regression: Numerical and Categorical Moderators

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_cat_num_2w
print(head(dat), digits = 3)
#>      x    w    y   c1   c2   city
#> 1 17.4 19.2 18.4 22.5 27.6 City A
#> 2 18.0 18.1 30.8 24.1 17.5 City A
#> 3 19.1 22.7 17.7 29.0 12.0 City A
#> 4 19.5 13.7 24.2 24.6 18.4 City A
#> 5 16.6 24.3 22.0 22.2 20.8 City A
#> 6 17.9 15.6 21.3 24.1 19.8 City A

This dataset has 6 variables:

one outcome variable (y),
one predictor (x),
one numerical moderator (w).
one categorical moderator (city),
two control variables (c1 and c2).

The moderator city has two possible values: "City A" and "City B".

Models with only numerical moderators or only categorical moderators have been covered in other articles of this series. Therefore, only two models will be considered: a model with no three-way interaction and a model with three-way interaction.

One Numerical Moderator and One Categorical Moderator

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

Model

Fit by Regression

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

lm_y <- lm(
  y ~ w*x + city*x + c1 + c2,
  data = dat
)

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

summary(lm_y)
#> 
#> Call:
#> lm(formula = y ~ w * x + city * x + c1 + c2, data = dat)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -8.9852 -2.6831 -0.2673  2.8285 12.6873 
#> 
#> Coefficients:
#>              Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  40.71044    8.09286   5.030 1.12e-06 ***
#> w            -1.32342    0.36592  -3.617 0.000381 ***
#> x            -1.42616    0.42465  -3.358 0.000945 ***
#> cityCity B   -8.43872    4.81058  -1.754 0.080992 .  
#> c1           -0.04418    0.06852  -0.645 0.519863    
#> c2            0.22908    0.06722   3.408 0.000797 ***
#> w:x           0.08828    0.02020   4.370 2.03e-05 ***
#> x:cityCity B  0.57437    0.26404   2.175 0.030832 *  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 4.169 on 192 degrees of freedom
#> Multiple R-squared:  0.4438, Adjusted R-squared:  0.4235 
#> F-statistic: 21.89 on 7 and 192 DF,  p-value: < 2.2e-16

Conditional Effects

We can now use cond_effects() to estimate the effects of x on y for different levels of w and different cities.

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

out <- cond_effects(
  wlevels = c("city", "w"),
  x = "x",
  y = "y",
  fit = lm_y
)
out
#> 
#> == Conditional effects ==
#> 
#>  Path: x -> y
#>  Conditional on moderator(s): city, w
#>  Moderator(s) represented by: cityCity B, w
#> 
#>   [city]     [w] (cityCity B)    (w)    ind    SE   Stat pvalue Sig  CI.lo CI.hi
#> 1 City A M+1.0SD            0 24.200  0.710 0.267  2.660  0.008 **   0.184 1.237
#> 2 City A M-1.0SD            0 13.353 -0.247 0.247 -1.001  0.318     -0.735 0.240
#> 3 City B M+1.0SD            1 24.200  1.285 0.151  8.491  0.000 ***  0.986 1.583
#> 4 City B M-1.0SD            1 13.353  0.327 0.170  1.927  0.055     -0.008 0.662
#> 
#>  - [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 x on y for combinations of the levels of the moderators.

IMPORTANT: Even though this model does not have a 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 = "w",
  x = "x",
  y = "y",
  fit = lm_y
)
#> Warning in (function (xi, yi, yiname, digits = 3, y, wvalues = NULL, warn = TRUE, : cityCity B 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, : cityCity B 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, : cityCity B 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): w
#>  Moderator(s) represented by: w
#> 
#>       [w]    (w)    ind    SE   Stat pvalue Sig  CI.lo CI.hi
#> 1 M+1.0SD 24.200  0.710 0.267  2.660  0.008  **  0.184 1.237
#> 2 Mean    18.777  0.231 0.233  0.994  0.321     -0.228 0.691
#> 3 M-1.0SD 13.353 -0.247 0.247 -1.001  0.318     -0.735 0.240
#> 
#>  - [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.
#>

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

Plotting the Conditional Effects

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

plot(out)

Conventional Plot of Conditional Effects

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.

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 level of the other moderator.

In this case, it is natural to plot the moderating effect of w in each city:

plot(out,
     facet_grid_cols = "city")

Conventional Plot of Conditional Effects (by city)

Note that, without three-way interaction, the moderating effect of w is the same in all cities. The lines are different simply because the effect of x depends on both w and city. They do not denote a three-way interaction (because it is not in the regression model).

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(out,
     facet_grid_cols = "city",
     graph_type = "tumble")

Tumble Plot of Conditional Effects (by city)

In this example, the distributions of x for the two cities are different: The standard deviations of x are larger in City B. Therefore, the tumble graph is more appropriate than the conventional graph.

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 x to y conditional on w and city can be estimated by setting standardized_x and standardized_y to TRUE.

This is the output:

std <- cond_effects(
  wlevels = c("city", "w"),
  x = "x",
  y = "y",
  fit = lm_y,
  boot_ci = TRUE,
  R = 5000,
  seed = 54532,
  standardized_x = TRUE,
  standardized_y = TRUE
)
#> 19 processes started to run bootstrapping.
std
#> 
#> == Conditional effects ==
#> 
#>  Path: x -> y
#>  Conditional on moderator(s): city, w
#>  Moderator(s) represented by: cityCity B, w
#> 
#>   [city]     [w] (cityCity B)    (w)    std  CI.lo CI.hi Sig    ind
#> 1 City A M+1.0SD            0 24.200  0.383  0.090 0.679 Sig  0.710
#> 2 City A M-1.0SD            0 13.353 -0.133 -0.492 0.194     -0.247
#> 3 City B M+1.0SD            1 24.200  0.692  0.538 0.851 Sig  1.285
#> 4 City B M-1.0SD            1 13.353  0.176 -0.016 0.350      0.327
#> 
#>  - [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.
#>

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,
     facet_grid_cols = "city",
     graph_type = "tumble")

Tumble Plot of Standardized Conditional Effects (by city)

One Numerical Moderator and One Categorical Moderator, with Three-Way Interaction

Suppose that we suspect that the two moderators interact with each other. That is, the moderating effect of w on the effect of x may not be the same in the two cities.

The steps demonstrated above can also be used in this regression model:

lm_y_city_x_w <- lm(
  y ~ x*city*w + c1 + c2,
  data = dat
)

These are the estimates of the regression coefficient of this model:

summary(lm_y_city_x_w)
#> 
#> Call:
#> lm(formula = y ~ x * city * w + c1 + c2, data = dat)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -8.5552 -2.6543 -0.2097  2.5379 13.0804 
#> 
#> Coefficients:
#>                Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)     8.00474   13.73622   0.583 0.560754    
#> x               0.56284    0.74693   0.754 0.452058    
#> cityCity B     33.04780   15.79870   2.092 0.037785 *  
#> w               0.55856    0.71716   0.779 0.437037    
#> c1             -0.04241    0.06614  -0.641 0.522120    
#> c2              0.23012    0.06622   3.475 0.000632 ***
#> x:cityCity B   -2.02718    0.87260  -2.323 0.021229 *  
#> x:w            -0.02640    0.04021  -0.657 0.512205    
#> cityCity B:w   -2.33308    0.82853  -2.816 0.005377 ** 
#> x:cityCity B:w  0.14589    0.04601   3.171 0.001773 ** 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 4.022 on 190 degrees of freedom
#> Multiple R-squared:  0.4876, Adjusted R-squared:  0.4634 
#> F-statistic: 20.09 on 9 and 190 DF,  p-value: < 2.2e-16

The three-way interaction term, x:city City B:w, is significant, suggesting a three-way interaction.

Conditional Effects

The function cond_effects() can be used in exactly the same way, whether the moderators interact with each other or not:

out_city_x_w <- cond_effects(
  wlevels = c("city", "w"),
  x = "x",
  y = "y",
  fit = lm_y_city_x_w
)
out_city_x_w
#> 
#> == Conditional effects ==
#> 
#>  Path: x -> y
#>  Conditional on moderator(s): city, w
#>  Moderator(s) represented by: cityCity B, w
#> 
#>   [city]     [w] (cityCity B)    (w)    ind    SE   Stat pvalue Sig  CI.lo CI.hi
#> 1 City A M+1.0SD            0 24.200 -0.076 0.344 -0.221  0.825     -0.754 0.602
#> 2 City A M-1.0SD            0 13.353  0.210 0.285  0.738  0.461     -0.352 0.772
#> 3 City B M+1.0SD            1 24.200  1.427 0.156  9.170  0.000 ***  1.120 1.734
#> 4 City B M-1.0SD            1 13.353  0.131 0.176  0.745  0.457     -0.216 0.479
#> 
#>  - [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 results show that, within one standard deviation of the mean of w, x has significant effects only in City B.

Plotting the Conditional Effects

These are the tumble plots of the conditional effects, with facet_grid_cols set:

plot(out_city_x_w,
     facet_grid_cols = "city",
     graph_type = "tumble")

Tumble Plot of Conditional Effects (By City)

Standardized Conditional Effects

This is the output of the standardized conditional effects, with bootstrap confidence intervals:

std_city_x_w <- cond_effects(
  wlevels = c("city", "w"),
  x = "x",
  y = "y",
  fit = lm_y_city_x_w,
  boot_ci = TRUE,
  R = 5000,
  seed = 54532,
  standardized_x = TRUE,
  standardized_y = TRUE
)
#> 19 processes started to run bootstrapping.
std_city_x_w
#> 
#> == Conditional effects ==
#> 
#>  Path: x -> y
#>  Conditional on moderator(s): city, w
#>  Moderator(s) represented by: cityCity B, w
#> 
#>   [city]     [w] (cityCity B)    (w)    std  CI.lo CI.hi Sig    ind
#> 1 City A M+1.0SD            0 24.200 -0.041 -0.458 0.343     -0.076
#> 2 City A M-1.0SD            0 13.353  0.113 -0.372 0.433      0.210
#> 3 City B M+1.0SD            1 24.200  0.769  0.626 0.925 Sig  1.427
#> 4 City B M-1.0SD            1 13.353  0.071 -0.127 0.216      0.131
#> 
#>  - [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.
#>

These are the plots of the standardized conditional effects, with facet_grid_cols set:

plot(std_city_x_w,
     facet_grid_cols = "city",
     graph_type = "tumble")

Tumble Plot of Standardized Conditional Effects (By City)

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

Shu Fai Cheung & Sing-Hang Cheung

2026-07-03

Introduction

Data Set and Model

One Numerical Moderator and One Categorical Moderator

Fit by Regression

Conditional Effects

Plotting the Conditional Effects

Tumble Plot

Standardized Conditional Effects

Plot Standardized Conditional Effects

One Numerical Moderator and One Categorical Moderator, with Three-Way Interaction

Conditional Effects

Plotting the Conditional Effects

Standardized Conditional Effects

Other Moderated Regression Models

References