Skip to contents

To perform mediation, moderation, and conditional process (moderated mediation) analyses, people may use software like Mplus, SPSS "PROCESS" macro, and SPSS "MLmed" macro. Some R packages can also perform such analyses separately and in a complex way, including R package "mediation", R package "interactions", and R package "lavaan". Some other R packages or scripts/modules have been further developed to improve the convenience, including jamovi module "jAMM" (by Marcello Gallucci, based on the lavaan package), R package "processR" (by Keon-Woong Moon, not official, also based on the lavaan package), and R script file "process.R" (the official PROCESS R code by Andrew F. Hayes, but it is not yet an R package and has some bugs and limitations).

Here, the bruceR::PROCESS() function provides an alternative to performing mediation/moderation analyses in R. This function supports a total of 24 kinds of SPSS PROCESS models (Hayes, 2018) and also supports multilevel mediation/moderation analyses. Overall, it supports the most frequently used types of mediation, moderation, moderated moderation (3-way interaction), and moderated mediation (conditional indirect effect) analyses for (generalized) linear or linear mixed models.

Specifically, the bruceR::PROCESS() function fits regression models based on the data, variable names, and a few other arguments that users input (with no need to specify the PROCESS model number and no need to manually mean-center the variables). The function can automatically judge the model number/type and also conduct grand-mean centering before model building (using the bruceR::grand_mean_center() function).

This automatic grand-mean centering can be turned off by setting center=FALSE.

Note that this automatic grand-mean centering (1) makes the results of main effects accurate for interpretation; (2) does not change any results of model fit (it only affects the interpretation of main effects); (3) is only conducted in "PART 1" (for an accurate estimate of main effects) but not in "PART 2" because it is more intuitive and interpretable to use the raw values of variables for the simple-slope tests in "PART 2"; (4) is not optional to users because mean-centering should always be done when there is an interaction; (5) is not conflicted with group-mean centering because after group-mean centering the grand mean of a variable will also be 0, such that the automatic grand-mean centering (with mean = 0) will not change any values of the variable.

If you need to do group-mean centering, please do this before using PROCESS. bruceR::group_mean_center() is a useful function of group-mean centering. Remember that the automatic grand-mean centering in PROCESS never affects the values of a group-mean centered variable, which already has a grand mean of 0.

The bruceR::PROCESS() function uses:

  1. the interactions::sim_slopes() function to estimate simple slopes (and conditional direct effects) in moderation, moderated moderation, and moderated mediation models (PROCESS Models 1, 2, 3, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 58, 59, 72, 73, 75, 76).

  2. the mediation::mediate() function to estimate (conditional) indirect effects in (moderated) mediation models (PROCESS Models 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 58, 59, 72, 73, 75, 76).

  3. the lavaan::sem() function to perform serial multiple mediation analysis (PROCESS Model 6).

If you use this function in your research and report its results in your paper, please cite not only bruceR but also the other R packages it uses internally (mediation, interactions, and/or lavaan).

Two parts of results are printed:

PART 1. Regression model summary (using bruceR::model_summary() to summarize the models)

PART 2. Mediation/moderation effect estimates (using one or a combination of the above packages and functions to estimate the effects)

To organize the PART 2 output, the results of Simple Slopes are titled in green, whereas the results of Indirect Path are titled in blue.

Disclaimer: Although this function is named after PROCESS, Andrew F. Hayes has no role in its design, and its development is independent from the official SPSS PROCESS macro and "process.R" script. Any error or limitation should be attributed to the three R packages/functions that bruceR::PROCESS() uses internally. Moreover, as mediation analyses include random processes (i.e., bootstrap resampling or Monte Carlo simulation), the results of mediation analyses are unlikely to be exactly the same across different software (even if you set the same random seed in different software).

Usage

PROCESS(
  data,
  y = "",
  x = "",
  meds = c(),
  mods = c(),
  covs = c(),
  clusters = c(),
  hlm.re.m = "",
  hlm.re.y = "",
  hlm.type = c("1-1-1", "2-1-1", "2-2-1"),
  med.type = c("parallel", "serial"),
  mod.type = c("2-way", "3-way"),
  mod.path = c("x-y", "x-m", "m-y", "all"),
  cov.path = c("y", "m", "both"),
  mod1.val = NULL,
  mod2.val = NULL,
  ci = c("boot", "bc.boot", "bca.boot", "mcmc"),
  nsim = 100,
  seed = NULL,
  center = TRUE,
  std = FALSE,
  digits = 3,
  file = NULL
)

Arguments

data

Data frame.

y, x

Variable name of outcome (Y) and predictor (X).

It supports both continuous (numeric) and dichotomous (factor) variables.

meds

Variable name(s) of mediator(s) (M). Use c() to combine multiple mediators.

It supports both continuous (numeric) and dichotomous (factor) variables.

It allows an infinite number of mediators in parallel or 2~4 mediators in serial.

* Order matters when med.type="serial" (PROCESS Model 6: serial mediation).

mods

Variable name(s) of 0~2 moderator(s) (W). Use c() to combine multiple moderators.

It supports all types of variables: continuous (numeric), dichotomous (factor), and multicategorical (factor).

* Order matters when mod.type="3-way" (PROCESS Models 3, 5.3, 11, 12, 18, 19, 72, and 73).

** Do not set this argument when med.type="serial" (PROCESS Model 6).

covs

Variable name(s) of covariate(s) (i.e., control variables). Use c() to combine multiple covariates. It supports all types of (and an infinite number of) variables.

clusters

HLM (multilevel) cluster(s): e.g., "School", c("Prov", "City"), c("Sub", "Item").

hlm.re.m, hlm.re.y

HLM (multilevel) random effect term of M model and Y model. By default, it converts clusters to lme4 syntax of random intercepts: e.g., "(1 | School)" or "(1 | Sub) + (1 | Item)".

You may specify these arguments to include more complex terms: e.g., random slopes "(X | School)", or 3-level random effects "(1 | Prov/City)".

hlm.type

HLM (multilevel) mediation type (levels of "X-M-Y"): "1-1-1" (default), "2-1-1" (indeed the same as "1-1-1" in a mixed model), or "2-2-1" (currently not fully supported, as limited by the mediation package). In most cases, no need to set this argument.

med.type

Type of mediator: "parallel" (default) or "serial" (only relevant to PROCESS Model 6). Partial matches of "p" or "s" also work. In most cases, no need to set this argument.

mod.type

Type of moderator: "2-way" (default) or "3-way" (relevant to PROCESS Models 3, 5.3, 11, 12, 18, 19, 72, and 73). Partial matches of "2" or "3" also work.

mod.path

Which path(s) do the moderator(s) influence? "x-y", "x-m", "m-y", or any combination of them (use c() to combine), or "all" (i.e., all of them). No default value.

cov.path

Which path(s) do the control variable(s) influence? "y", "m", or "both" (default).

mod1.val, mod2.val

By default (NULL), it uses Mean +/- SD of a continuous moderator (numeric) or all levels of a dichotomous/multicategorical moderator (factor) to perform simple slope analyses and/or conditional mediation analyses. You may manually specify a vector of certain values: e.g., mod1.val=c(1, 3, 5) or mod1.val=c("A", "B", "C").

ci

Method for estimating the standard error (SE) and 95% confidence interval (CI) of indirect effect(s). Defaults to "boot" for (generalized) linear models or "mcmc" for (generalized) linear mixed models (i.e., multilevel models).

"boot"

Percentile Bootstrap

"bc.boot"

Bias-Corrected Percentile Bootstrap

"bca.boot"

Bias-Corrected and Accelerated (BCa) Percentile Bootstrap

"mcmc"

Markov Chain Monte Carlo (Quasi-Bayesian)

* Note that these methods never apply to the estimates of simple slopes. You should not report the 95% CIs of simple slopes as Bootstrap or Monte Carlo CIs, because they are just standard CIs without any resampling method.

nsim

Number of simulation samples (bootstrap resampling or Monte Carlo simulation) for estimating SE and 95% CI. Defaults to 100 for running examples faster. In formal analyses, however, nsim=1000 (or larger) is strongly suggested!

seed

Random seed for obtaining reproducible results. Defaults to NULL. You may set to any number you prefer (e.g., seed=1234, just an uncountable number).

* Note that all mediation models include random processes (i.e., bootstrap resampling or Monte Carlo simulation). To get exactly the same results between runs, you need to set a random seed. However, even if you set the same seed number, it is unlikely to get exactly the same results across different R packages (e.g., lavaan vs. mediation) and software (e.g., SPSS, Mplus, R, jamovi).

center

Centering numeric (continuous) predictors? Defaults to TRUE (suggested).

std

Standardizing variables to get standardized coefficients? Defaults to FALSE. If TRUE, it will standardize all numeric (continuous) variables before building regression models. However, it is not suggested to set std=TRUE for generalized linear (mixed) models.

digits

Number of decimal places of output. Defaults to 3.

file

File name of MS Word (.doc). Currently, only regression model summary can be saved.

Value

Invisibly return a list of results:

process.id

PROCESS model number.

process.type

PROCESS model type.

model.m

"Mediator" (M) models (a list of multiple models).

model.y

"Outcome" (Y) model.

results

Effect estimates and other results (unnamed list object).

Details

For more details and illustrations, see PROCESS-bruceR-SPSS (PDF and Markdown files).

References

Hayes, A. F. (2018). Introduction to mediation, moderation, and conditional process analysis (second edition): A regression-based approach. Guilford Press.

Yzerbyt, V., Muller, D., Batailler, C., & Judd, C. M. (2018). New recommendations for testing indirect effects in mediational models: The need to report and test component paths. Journal of Personality and Social Psychology, 115(6), 929--943.

Examples

#### NOTE ####
## In the following examples, I set nsim=100 to save time.
## In formal analyses, nsim=1000 (or larger) is suggested!

#### Demo Data ####
# ?mediation::student
data = mediation::student %>%
  dplyr::select(SCH_ID, free, smorale, pared, income,
                gender, work, attachment, fight, late, score)
names(data)[2:3] = c("SCH_free", "SCH_morale")
names(data)[4:7] = c("parent_edu", "family_inc", "gender", "partjob")
data$gender01 = 1 - data$gender  # 0 = female, 1 = male
# dichotomous X: as.factor()
data$gender = factor(data$gender01, levels=0:1, labels=c("Female", "Male"))
# dichotomous Y: as.factor()
data$pass = as.factor(ifelse(data$score>=50, 1, 0))

#### Descriptive Statistics and Correlation Analyses ####
Freq(data$gender)
#> Frequency Statistics:
#> ─────────────────
#>            N    %
#> ─────────────────
#> Female  5044 52.1
#> Male    4635 47.9
#> ─────────────────
#> Total N = 9,679
Freq(data$pass)
#> Frequency Statistics:
#> ────────────
#>       N    %
#> ────────────
#> 0  3856 39.8
#> 1  5823 60.2
#> ────────────
#> Total N = 9,679
Describe(data)     # file="xxx.doc"
#> Descriptive Statistics:
#> ──────────────────────────────────────────────────────────────────────
#>                N   Mean     SD | Median   Min    Max Skewness Kurtosis
#> ──────────────────────────────────────────────────────────────────────
#> SCH_ID      9679 285.50 164.45 | 285.00  1.00 568.00    -0.00    -1.21
#> SCH_free    9679   2.99   1.86 |   3.00  1.00   7.00     0.47    -0.97
#> SCH_morale  9679   4.02   0.75 |   4.00  2.00   5.00    -0.45    -0.08
#> parent_edu  9679   0.44   0.50 |   0.00  0.00   1.00     0.26    -1.93
#> family_inc  9679   9.26   2.34 |  10.00  1.00  16.00    -0.79     0.72
#> gender*     9679   1.48   0.50 |   1.00  1.00   2.00     0.08    -1.99
#> partjob     9679   0.39   0.49 |   0.00  0.00   1.00     0.47    -1.78
#> attachment  9679   0.89   0.32 |   1.00  0.00   1.00    -2.42     3.87
#> fight       9679   0.13   0.33 |   0.00  0.00   1.00     2.26     3.10
#> late        9679   2.24   1.13 |   2.00  1.00   5.00     0.90     0.29
#> score       9679  51.91   9.69 |  52.00 19.00  87.00    -0.11    -0.12
#> gender01    9679   0.48   0.50 |   0.00  0.00   1.00     0.08    -1.99
#> pass*       9679   1.60   0.49 |   2.00  1.00   2.00    -0.42    -1.83
#> ──────────────────────────────────────────────────────────────────────
#> 
#> NOTE: `gender`, `pass` transformed to numeric.
Corr(data[,4:11])  # file="xxx.doc"
#> NOTE: `gender` transformed to numeric.
#> 
#> Pearson's r and 95% confidence intervals:
#> ──────────────────────────────────────────────────────────
#>                            r       [95% CI]     p        N
#> ──────────────────────────────────────────────────────────
#> parent_edu-family_inc   0.38 [ 0.36,  0.40] <.001 *** 9679
#> parent_edu-gender       0.01 [-0.01,  0.03]  .320     9679
#> parent_edu-partjob     -0.02 [-0.04,  0.00]  .071 .   9679
#> parent_edu-attachment   0.03 [ 0.01,  0.05]  .012 *   9679
#> parent_edu-fight       -0.08 [-0.10, -0.06] <.001 *** 9679
#> parent_edu-late        -0.04 [-0.06, -0.02] <.001 *** 9679
#> parent_edu-score        0.28 [ 0.27,  0.30] <.001 *** 9679
#> family_inc-gender       0.03 [ 0.01,  0.05]  .008 **  9679
#> family_inc-partjob      0.02 [ 0.00,  0.04]  .044 *   9679
#> family_inc-attachment   0.00 [-0.02,  0.02]  .763     9679
#> family_inc-fight       -0.10 [-0.12, -0.08] <.001 *** 9679
#> family_inc-late        -0.07 [-0.09, -0.05] <.001 *** 9679
#> family_inc-score        0.33 [ 0.32,  0.35] <.001 *** 9679
#> gender-partjob          0.00 [-0.02,  0.02]  .842     9679
#> gender-attachment      -0.11 [-0.13, -0.09] <.001 *** 9679
#> gender-fight            0.18 [ 0.16,  0.20] <.001 *** 9679
#> gender-late             0.01 [-0.01,  0.03]  .582     9679
#> gender-score            0.08 [ 0.06,  0.10] <.001 *** 9679
#> partjob-attachment     -0.03 [-0.05, -0.01]  .001 **  9679
#> partjob-fight           0.06 [ 0.04,  0.08] <.001 *** 9679
#> partjob-late            0.04 [ 0.02,  0.06] <.001 *** 9679
#> partjob-score          -0.02 [-0.04, -0.00]  .047 *   9679
#> attachment-fight       -0.16 [-0.18, -0.14] <.001 *** 9679
#> attachment-late        -0.16 [-0.18, -0.14] <.001 *** 9679
#> attachment-score        0.06 [ 0.04,  0.08] <.001 *** 9679
#> fight-late              0.17 [ 0.15,  0.18] <.001 *** 9679
#> fight-score            -0.16 [-0.18, -0.14] <.001 *** 9679
#> late-score             -0.14 [-0.16, -0.12] <.001 *** 9679
#> ──────────────────────────────────────────────────────────
#> 

#> Correlation matrix is displayed in the RStudio `Plots` Pane.
#> 

#### PROCESS Analyses ####

## Model 1 ##
PROCESS(data, y="score", x="late", mods="gender")  # continuous Y
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 1 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Simple Moderation
#> -    Outcome (Y) : score
#> -  Predictor (X) : late
#> -  Mediators (M) : -
#> - Moderators (W) : gender
#> - Covariates (C) : -
#> -   HLM Clusters : -
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Outcome:
#> -    score ~ late*gender
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ───────────────────────────────────────────
#>                  (1) score     (2) score   
#> ───────────────────────────────────────────
#> (Intercept)        51.912 ***    51.174 ***
#>                    (0.098)       (0.135)   
#> late               -1.174 ***    -0.947 ***
#>                    (0.087)       (0.122)   
#> genderMale                        1.545 ***
#>                                  (0.195)   
#> late:genderMale                  -0.462 ** 
#>                                  (0.173)   
#> ───────────────────────────────────────────
#> R^2                 0.019         0.026    
#> Adj. R^2            0.018         0.025    
#> Num. obs.        9679          9679        
#> ───────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘interactions’ (v1.1.5)
#> Effect Type : Simple Moderation (Model 1)
#> Sample Size : 9679
#> Random Seed : -
#> Simulations : -
#> 
#> Interaction Effect on "score" (Y)
#> ──────────────────────────────────────
#>                   F df1  df2     p    
#> ──────────────────────────────────────
#> late * gender  7.14   1 9675  .008 ** 
#> ──────────────────────────────────────
#> 
#> Simple Slopes: "late" (X) ==> "score" (Y)
#> ───────────────────────────────────────────────────────────
#>  "gender" Effect    S.E.       t     p             [95% CI]
#> ───────────────────────────────────────────────────────────
#>  Female   -0.947 (0.122)  -7.772 <.001 *** [-1.186, -0.708]
#>  Male     -1.409 (0.122) -11.513 <.001 *** [-1.649, -1.169]
#> ───────────────────────────────────────────────────────────
#> 
PROCESS(data, y="pass", x="late", mods="gender")   # dichotomous Y
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 1 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Simple Moderation
#> -    Outcome (Y) : pass
#> -  Predictor (X) : late
#> -  Mediators (M) : -
#> - Moderators (W) : gender
#> - Covariates (C) : -
#> -   HLM Clusters : -
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Outcome:
#> -    pass ~ late*gender
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ──────────────────────────────────────────────
#>                   (1) pass       (2) pass     
#> ──────────────────────────────────────────────
#> (Intercept)           0.418 ***      0.306 ***
#>                      (0.021)        (0.029)   
#> late                 -0.232 ***     -0.201 ***
#>                      (0.019)        (0.026)   
#> genderMale                           0.238 ***
#>                                     (0.042)   
#> late:genderMale                     -0.066    
#>                                     (0.037)   
#> ──────────────────────────────────────────────
#> McFadden's R^2        0.012          0.015    
#> Nagelkerke's R^2      0.022          0.027    
#> AIC               12859.980      12829.489    
#> BIC               12874.335      12858.200    
#> Num. obs.          9679           9679        
#> ──────────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘interactions’ (v1.1.5)
#> Effect Type : Simple Moderation (Model 1)
#> Sample Size : 9679
#> Random Seed : -
#> Simulations : -
#> 
#> Interaction Effect on "pass" (Y)
#> ─────────────────────────────────
#>                Chisq df     p    
#> ─────────────────────────────────
#> late * gender   3.18  1  .075 .  
#> ─────────────────────────────────
#> 
#> Simple Slopes: "late" (X) ==> "pass" (Y)
#> ───────────────────────────────────────────────────────────
#>  "gender" Effect    S.E.       z     p             [95% CI]
#> ───────────────────────────────────────────────────────────
#>  Female   -0.201 (0.026)  -7.769 <.001 *** [-0.252, -0.151]
#>  Male     -0.268 (0.027) -10.082 <.001 *** [-0.320, -0.216]
#> ───────────────────────────────────────────────────────────
#> 

# (multilevel moderation)
PROCESS(data, y="score", x="late", mods="gender",  # continuous Y (LMM)
        clusters="SCH_ID")
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 1 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Simple Moderation
#> -    Outcome (Y) : score
#> -  Predictor (X) : late
#> -  Mediators (M) : -
#> - Moderators (W) : gender
#> - Covariates (C) : -
#> -   HLM Clusters : SCH_ID
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Outcome:
#> -    score ~ late*gender + (1 | SCH_ID)
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ─────────────────────────────────────────────────────
#>                          (1) score      (2) score    
#> ─────────────────────────────────────────────────────
#> (Intercept)                 51.705 ***     50.986 ***
#>                             (0.204)        (0.221)   
#> late                        -0.932 ***     -0.763 ***
#>                             (0.082)        (0.114)   
#> genderMale                                  1.509 ***
#>                                            (0.182)   
#> late:genderMale                            -0.346 *  
#>                                            (0.160)   
#> ─────────────────────────────────────────────────────
#> Marginal R^2                 0.012          0.018    
#> Conditional R^2              0.209          0.214    
#> AIC                      70092.182      70027.142    
#> BIC                      70120.893      70070.208    
#> Num. obs.                 9679           9679        
#> Num. groups: SCH_ID        568            568        
#> Var: SCH_ID (Intercept)     18.549         18.392    
#> Var: Residual               74.298         73.764    
#> ─────────────────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘interactions’ (v1.1.5)
#> Effect Type : Simple Moderation (Model 1)
#> Sample Size : 9679
#> Random Seed : -
#> Simulations : -
#> 
#> Interaction Effect on "score" (Y)
#> ──────────────────────────────────────
#>                   F df1  df2     p    
#> ──────────────────────────────────────
#> late * gender  4.70   1 9356  .030 *  
#> ──────────────────────────────────────
#> 
#> Simple Slopes: "late" (X) ==> "score" (Y)
#> ──────────────────────────────────────────────────────────
#>  "gender" Effect    S.E.      t     p             [95% CI]
#> ──────────────────────────────────────────────────────────
#>  Female   -0.763 (0.114) -6.676 <.001 *** [-0.987, -0.539]
#>  Male     -1.109 (0.114) -9.688 <.001 *** [-1.333, -0.885]
#> ──────────────────────────────────────────────────────────
#> 
PROCESS(data, y="pass", x="late", mods="gender",   # dichotomous Y (GLMM)
        clusters="SCH_ID")
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 1 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Simple Moderation
#> -    Outcome (Y) : pass
#> -  Predictor (X) : late
#> -  Mediators (M) : -
#> - Moderators (W) : gender
#> - Covariates (C) : -
#> -   HLM Clusters : SCH_ID
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Outcome:
#> -    pass ~ late*gender + (1 | SCH_ID)
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ─────────────────────────────────────────────────────
#>                          (1) pass       (2) pass     
#> ─────────────────────────────────────────────────────
#> (Intercept)                  0.441 ***      0.314 ***
#>                             (0.044)        (0.049)   
#> late                        -0.224 ***     -0.202 ***
#>                             (0.021)        (0.029)   
#> genderMale                                  0.271 ***
#>                                            (0.047)   
#> late:genderMale                            -0.049    
#>                                            (0.041)   
#> ─────────────────────────────────────────────────────
#> AIC                      12227.697      12197.299    
#> BIC                      12249.231      12233.188    
#> Num. obs.                 9679           9679        
#> Num. groups: SCH_ID        568            568        
#> Var: SCH_ID (Intercept)      0.754          0.756    
#> ─────────────────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘interactions’ (v1.1.5)
#> Effect Type : Simple Moderation (Model 1)
#> Sample Size : 9679
#> Random Seed : -
#> Simulations : -
#> 
#> Interaction Effect on "pass" (Y)
#> ─────────────────────────────────
#>                Chisq df     p    
#> ─────────────────────────────────
#> late * gender   1.43  1  .232    
#> ─────────────────────────────────
#> 
#> Simple Slopes: "late" (X) ==> "pass" (Y)
#> ──────────────────────────────────────────────────────────
#>  "gender" Effect    S.E.      z     p             [95% CI]
#> ──────────────────────────────────────────────────────────
#>  Female   -0.202 (0.029) -6.997 <.001 *** [-0.259, -0.145]
#>  Male     -0.251 (0.029) -8.541 <.001 *** [-0.308, -0.193]
#> ──────────────────────────────────────────────────────────
#> 

# (Johnson-Neyman (J-N) interval and plot)
PROCESS(data, y="score", x="gender", mods="late") -> P
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 1 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Simple Moderation
#> -    Outcome (Y) : score
#> -  Predictor (X) : gender (recoded: Female=0, Male=1)
#> -  Mediators (M) : -
#> - Moderators (W) : late
#> - Covariates (C) : -
#> -   HLM Clusters : -
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Outcome:
#> -    score ~ gender*late
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ───────────────────────────────────────
#>              (1) score     (2) score   
#> ───────────────────────────────────────
#> (Intercept)    51.912 ***    51.913 ***
#>                (0.098)       (0.097)   
#> gender          1.530 ***     1.545 ***
#>                (0.196)       (0.195)   
#> late                         -1.169 ***
#>                              (0.086)   
#> gender:late                  -0.462 ** 
#>                              (0.173)   
#> ───────────────────────────────────────
#> R^2             0.006         0.026    
#> Adj. R^2        0.006         0.025    
#> Num. obs.    9679          9679        
#> ───────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘interactions’ (v1.1.5)
#> Effect Type : Simple Moderation (Model 1)
#> Sample Size : 9679
#> Random Seed : -
#> Simulations : -
#> 
#> Interaction Effect on "score" (Y)
#> ──────────────────────────────────────
#>                   F df1  df2     p    
#> ──────────────────────────────────────
#> gender * late  7.14   1 9675  .008 ** 
#> ──────────────────────────────────────
#> 
#> Simple Slopes: "gender" (X) ==> "score" (Y)
#> ───────────────────────────────────────────────────────────
#>  "late"       Effect    S.E.     t     p           [95% CI]
#> ───────────────────────────────────────────────────────────
#>  1.116 (- SD)  2.064 (0.275) 7.504 <.001 *** [1.525, 2.604]
#>  2.242 (Mean)  1.545 (0.195) 7.938 <.001 *** [1.163, 1.926]
#>  3.367 (+ SD)  1.025 (0.275) 3.727 <.001 *** [0.486, 1.564]
#> ───────────────────────────────────────────────────────────
#> 
P$results[[1]]$jn[[1]]       # Johnson-Neyman interval
#> JOHNSON-NEYMAN INTERVAL 
#> 
#> When late is OUTSIDE the interval [4.03, 14.93], the slope of gender
#> is p < .05.
#> 
#> Note: The range of observed values of late is [1.00, 5.00]
#> 
P$results[[1]]$jn[[1]]$plot  # Johnson-Neyman plot (ggplot object)

GLM_summary(P$model.y)       # detailed results of regression
#> 
#> General Linear Model (OLS Regression)
#> 
#> Model Fit:
#> F(3, 9675) = 84.92, p = 3e-54 ***
#> R² = 0.02566 (Adjusted R² = 0.02535)
#> 
#> Unstandardized Coefficients:
#> Outcome Variable: score
#> N = 9679
#> ────────────────────────────────────────────────────────────────────
#>                   b    S.E.       t     p        [95% CI of b]   VIF
#> ────────────────────────────────────────────────────────────────────
#> (Intercept)  51.913 (0.097) 534.035 <.001 *** [51.723, 52.104]      
#> gender        1.545 (0.195)   7.938 <.001 *** [ 1.163,  1.926] 1.000
#> late         -1.169 (0.086) -13.519 <.001 *** [-1.338, -0.999] 1.001
#> gender:late  -0.462 (0.173)  -2.673  .008 **  [-0.800, -0.123] 1.001
#> ────────────────────────────────────────────────────────────────────
#> 
#> Standardized Coefficients (β):
#> Outcome Variable: score
#> N = 9679
#> ─────────────────────────────────────────────────────────────────────────────────
#>                   β    S.E.       t     p        [95% CI of β] r(partial) r(part)
#> ─────────────────────────────────────────────────────────────────────────────────
#> gender        0.080 (0.010)   7.938 <.001 *** [ 0.060,  0.099]      0.080   0.080
#> late         -0.136 (0.010) -13.519 <.001 *** [-0.155, -0.116]     -0.136  -0.136
#> gender:late  -0.027 (0.010)  -2.673  .008 **  [-0.047, -0.007]     -0.027  -0.027
#> ─────────────────────────────────────────────────────────────────────────────────
#> 

# (allows multicategorical moderator)
d = airquality
d$Month = as.factor(d$Month)  # moderator: factor with levels "5"~"9"
PROCESS(d, y="Temp", x="Solar.R", mods="Month")
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 1 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Simple Moderation
#> -    Outcome (Y) : Temp
#> -  Predictor (X) : Solar.R
#> -  Mediators (M) : -
#> - Moderators (W) : Month
#> - Covariates (C) : -
#> -   HLM Clusters : -
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Outcome:
#> -    Temp ~ Solar.R*Month
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ────────────────────────────────────────
#>                 (1) Temp     (2) Temp   
#> ────────────────────────────────────────
#> (Intercept)      78.116 ***   66.012 ***
#>                  (0.736)      (1.232)   
#> Solar.R           0.028 ***    0.027 *  
#>                  (0.008)      (0.011)   
#> Month6                        12.967 ***
#>                               (1.699)   
#> Month7                        17.366 ***
#>                               (1.742)   
#> Month8                        18.235 ***
#>                               (1.741)   
#> Month9                        11.129 ***
#>                               (1.721)   
#> Solar.R:Month6                 0.002    
#>                               (0.017)   
#> Solar.R:Month7                -0.009    
#>                               (0.018)   
#> Solar.R:Month8                 0.009    
#>                               (0.019)   
#> Solar.R:Month9                -0.014    
#>                               (0.019)   
#> ────────────────────────────────────────
#> R^2               0.076        0.549    
#> Adj. R^2          0.070        0.519    
#> Num. obs.       146          146        
#> ────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘interactions’ (v1.1.5)
#> Effect Type : Simple Moderation (Model 1)
#> Sample Size : 146 (7 missing observations deleted)
#> Random Seed : -
#> Simulations : -
#> 
#> Interaction Effect on "Temp" (Y)
#> ───────────────────────────────────────
#>                     F df1 df2     p    
#> ───────────────────────────────────────
#> Solar.R * Month  0.36   4 136  .838    
#> ───────────────────────────────────────
#> 
#> Simple Slopes: "Solar.R" (X) ==> "Temp" (Y)
#> ───────────────────────────────────────────────────────
#>  "Month" Effect    S.E.     t     p            [95% CI]
#> ───────────────────────────────────────────────────────
#>  5        0.027 (0.011) 2.432  .016 *   [ 0.005, 0.048]
#>  6        0.029 (0.013) 2.242  .027 *   [ 0.003, 0.054]
#>  7        0.017 (0.014) 1.186  .238     [-0.011, 0.046]
#>  8        0.035 (0.016) 2.202  .029 *   [ 0.004, 0.067]
#>  9        0.013 (0.015) 0.865  .389     [-0.017, 0.043]
#> ───────────────────────────────────────────────────────
#> 

## Model 2 ##
PROCESS(data, y="score", x="late",
        mods=c("gender", "family_inc"),
        mod.type="2-way")  # or omit "mod.type", default is "2-way"
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 2 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Parallel Moderation (2 mods; 2-way)
#> -    Outcome (Y) : score
#> -  Predictor (X) : late
#> -  Mediators (M) : -
#> - Moderators (W) : gender, family_inc
#> - Covariates (C) : -
#> -   HLM Clusters : -
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Outcome:
#> -    score ~ late*gender + late*family_inc
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ───────────────────────────────────────────
#>                  (1) score     (2) score   
#> ───────────────────────────────────────────
#> (Intercept)        51.912 ***    51.255 ***
#>                    (0.098)       (0.127)   
#> late               -1.174 ***    -0.836 ***
#>                    (0.087)       (0.115)   
#> genderMale                        1.375 ***
#>                                  (0.184)   
#> family_inc                        1.339 ***
#>                                  (0.040)   
#> late:genderMale                  -0.301    
#>                                  (0.163)   
#> late:family_inc                   0.007    
#>                                  (0.034)   
#> ───────────────────────────────────────────
#> R^2                 0.019         0.129    
#> Adj. R^2            0.018         0.129    
#> Num. obs.        9679          9679        
#> ───────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘interactions’ (v1.1.5)
#> Effect Type : Parallel Moderation (2 mods; 2-way) (Model 2)
#> Sample Size : 9679
#> Random Seed : -
#> Simulations : -
#> 
#> Interaction Effects on "score" (Y)
#> ───────────────────────────────────────────
#>                        F df1  df2     p    
#> ───────────────────────────────────────────
#> late * gender       3.40   1 9673  .065 .  
#> late * family_inc   0.04   1 9673  .849    
#> (All Interactions)  1.72   2 9673  .179    
#> ───────────────────────────────────────────
#> 
#> Simple Slopes: "late" (X) ==> "score" (Y)
#> ────────────────────────────────────────────────────────────────────────
#>  "family_inc"  "gender" Effect    S.E.      t     p             [95% CI]
#> ────────────────────────────────────────────────────────────────────────
#>  6.923 (- SD)  Female   -0.851 (0.137) -6.219 <.001 *** [-1.119, -0.583]
#>  6.923 (- SD)  Male     -1.152 (0.138) -8.369 <.001 *** [-1.422, -0.882]
#>  9.258 (Mean)  Female   -0.836 (0.115) -7.238 <.001 *** [-1.062, -0.609]
#>  9.258 (Mean)  Male     -1.137 (0.116) -9.792 <.001 *** [-1.365, -0.909]
#>  11.594 (+ SD) Female   -0.820 (0.144) -5.699 <.001 *** [-1.103, -0.538]
#>  11.594 (+ SD) Male     -1.122 (0.144) -7.778 <.001 *** [-1.405, -0.839]
#> ────────────────────────────────────────────────────────────────────────
#> 

## Model 3 ##
PROCESS(data, y="score", x="late",
        mods=c("gender", "family_inc"),
        mod.type="3-way")
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 3 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Moderated Moderation (2 mods; 3-way)
#> -    Outcome (Y) : score
#> -  Predictor (X) : late
#> -  Mediators (M) : -
#> - Moderators (W) : gender, family_inc
#> - Covariates (C) : -
#> -   HLM Clusters : -
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Outcome:
#> -    score ~ late*gender*family_inc
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ──────────────────────────────────────────────────────
#>                             (1) score     (2) score   
#> ──────────────────────────────────────────────────────
#> (Intercept)                   51.912 ***    51.266 ***
#>                               (0.098)       (0.127)   
#> late                          -1.174 ***    -0.818 ***
#>                               (0.087)       (0.116)   
#> genderMale                                   1.348 ***
#>                                             (0.185)   
#> family_inc                                   1.375 ***
#>                                             (0.055)   
#> late:genderMale                             -0.339 *  
#>                                             (0.164)   
#> late:family_inc                              0.088    
#>                                             (0.050)   
#> genderMale:family_inc                       -0.069    
#>                                             (0.079)   
#> late:genderMale:family_inc                  -0.153 *  
#>                                             (0.069)   
#> ──────────────────────────────────────────────────────
#> R^2                            0.019         0.130    
#> Adj. R^2                       0.018         0.129    
#> Num. obs.                   9679          9679        
#> ──────────────────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘interactions’ (v1.1.5)
#> Effect Type : Moderated Moderation (2 mods; 3-way) (Model 3)
#> Sample Size : 9679
#> Random Seed : -
#> Simulations : -
#> 
#> Interaction Effects on "score" (Y)
#> ───────────────────────────────────────────────────
#>                                F df1  df2     p    
#> ───────────────────────────────────────────────────
#> late * gender               4.26   1 9671  .039 *  
#> late * family_inc           3.11   1 9671  .078 .  
#> gender * family_inc         0.75   1 9671  .386    
#> late * gender * family_inc  4.95   1 9671  .026 *  
#> (All Interactions)          2.33   4 9671  .054 .  
#> ───────────────────────────────────────────────────
#> 
#> Conditional Interaction Effects on "score" (Y)
#> ────────────────────────────────────────────────────
#>  "family_inc"    Interaction    F df1  df2     p    
#> ────────────────────────────────────────────────────
#>  6.923 (- SD)  late * gender 0.01   1 9671  .938    
#>  9.258 (Mean)  late * gender 4.26   1 9671  .039 *  
#>  11.594 (+ SD) late * gender 8.56   1 9671  .003 ** 
#> ────────────────────────────────────────────────────
#> 
#> Simple Slopes: "late" (X) ==> "score" (Y)
#> ────────────────────────────────────────────────────────────────────────
#>  "family_inc"  "gender" Effect    S.E.      t     p             [95% CI]
#> ────────────────────────────────────────────────────────────────────────
#>  6.923 (- SD)  Female   -1.022 (0.157) -6.506 <.001 *** [-1.330, -0.714]
#>  6.923 (- SD)  Male     -1.005 (0.155) -6.470 <.001 *** [-1.309, -0.700]
#>  9.258 (Mean)  Female   -0.818 (0.116) -7.068 <.001 *** [-1.044, -0.591]
#>  9.258 (Mean)  Male     -1.157 (0.117) -9.927 <.001 *** [-1.385, -0.928]
#>  11.594 (+ SD) Female   -0.613 (0.170) -3.604 <.001 *** [-0.947, -0.280]
#>  11.594 (+ SD) Male     -1.308 (0.166) -7.894 <.001 *** [-1.633, -0.983]
#> ────────────────────────────────────────────────────────────────────────
#> 
PROCESS(data, y="pass", x="gender",
        mods=c("late", "family_inc"),
        mod1.val=c(1, 3, 5),     # moderator 1: late
        mod2.val=seq(1, 15, 2),  # moderator 2: family_inc
        mod.type="3-way")
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 3 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Moderated Moderation (2 mods; 3-way)
#> -    Outcome (Y) : pass
#> -  Predictor (X) : gender (recoded: Female=0, Male=1)
#> -  Mediators (M) : -
#> - Moderators (W) : late, family_inc
#> - Covariates (C) : -
#> -   HLM Clusters : -
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Outcome:
#> -    pass ~ gender*late*family_inc
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ────────────────────────────────────────────────────
#>                         (1) pass       (2) pass     
#> ────────────────────────────────────────────────────
#> (Intercept)                 0.414 ***      0.442 ***
#>                            (0.021)        (0.022)   
#> gender                      0.228 ***      0.219 ***
#>                            (0.042)        (0.044)   
#> late                                      -0.216 ***
#>                                           (0.019)   
#> family_inc                                 0.259 ***
#>                                           (0.010)   
#> gender:late                               -0.046    
#>                                           (0.039)   
#> gender:family_inc                         -0.022    
#>                                           (0.020)   
#> late:family_inc                            0.005    
#>                                           (0.009)   
#> gender:late:family_inc                    -0.032    
#>                                           (0.018)   
#> ────────────────────────────────────────────────────
#> McFadden's R^2              0.002          0.073    
#> Nagelkerke's R^2            0.004          0.127    
#> AIC                     12989.483      12080.387    
#> BIC                     13003.839      12137.809    
#> Num. obs.                9679           9679        
#> ────────────────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘interactions’ (v1.1.5)
#> Effect Type : Moderated Moderation (2 mods; 3-way) (Model 3)
#> Sample Size : 9679
#> Random Seed : -
#> Simulations : -
#> 
#> Interaction Effects on "pass" (Y)
#> ──────────────────────────────────────────────
#>                             Chisq df     p    
#> ──────────────────────────────────────────────
#> gender * late                1.43  1  .231    
#> gender * family_inc          1.17  1  .279    
#> late * family_inc            0.32  1  .573    
#> gender * late * family_inc   3.18  1  .075 .  
#> (All Interactions)           5.72  4  .221    
#> ──────────────────────────────────────────────
#> 
#> Conditional Interaction Effects on "pass" (Y)
#> ──────────────────────────────────────────────
#>  "family_inc"   Interaction Chisq df     p    
#> ──────────────────────────────────────────────
#>  1.000        gender * late  2.08  1  .149    
#>  3.000        gender * late  1.73  1  .188    
#>  5.000        gender * late  1.15  1  .285    
#>  7.000        gender * late  0.22  1  .642    
#>  9.000        gender * late  0.97  1  .323    
#>  11.000       gender * late  3.93  1  .047 *  
#>  13.000       gender * late  4.34  1  .037 *  
#>  15.000       gender * late  4.19  1  .041 *  
#> ──────────────────────────────────────────────
#> 
#> Simple Slopes: "gender" (X) ==> "pass" (Y)
#> ────────────────────────────────────────────────────────────────────
#>  "family_inc" "late" Effect    S.E.      z     p            [95% CI]
#> ────────────────────────────────────────────────────────────────────
#>  1.000        1.000   0.131 (0.253)  0.518  .605     [-0.365, 0.627]
#>  1.000        3.000   0.561 (0.201)  2.795  .005 **  [ 0.168, 0.954]
#>  1.000        5.000   0.991 (0.441)  2.247  .025 *   [ 0.127, 1.855]
#>  3.000        1.000   0.166 (0.196)  0.850  .395     [-0.217, 0.550]
#>  3.000        3.000   0.470 (0.155)  3.029  .002 **  [ 0.166, 0.774]
#>  3.000        5.000   0.773 (0.341)  2.270  .023 *   [ 0.106, 1.440]
#>  5.000        1.000   0.202 (0.141)  1.435  .151     [-0.074, 0.477]
#>  5.000        3.000   0.378 (0.111)  3.402 <.001 *** [ 0.160, 0.596]
#>  5.000        5.000   0.555 (0.244)  2.275  .023 *   [ 0.077, 1.033]
#>  7.000        1.000   0.237 (0.092)  2.576  .010 **  [ 0.057, 0.418]
#>  7.000        3.000   0.287 (0.072)  3.962 <.001 *** [ 0.145, 0.429]
#>  7.000        5.000   0.337 (0.159)  2.124  .034 *   [ 0.026, 0.649]
#>  9.000        1.000   0.272 (0.066)  4.109 <.001 *** [ 0.143, 0.402]
#>  9.000        3.000   0.196 (0.052)  3.775 <.001 *** [ 0.094, 0.298]
#>  9.000        5.000   0.120 (0.114)  1.049  .294     [-0.104, 0.343]
#>  11.000       1.000   0.308 (0.087)  3.544 <.001 *** [ 0.138, 0.478]
#>  11.000       3.000   0.105 (0.069)  1.529  .126     [-0.030, 0.239]
#>  11.000       5.000  -0.098 (0.151) -0.650  .515     [-0.395, 0.198]
#>  13.000       1.000   0.343 (0.134)  2.564  .010 *   [ 0.081, 0.606]
#>  13.000       3.000   0.014 (0.106)  0.128  .898     [-0.194, 0.221]
#>  13.000       5.000  -0.316 (0.234) -1.351  .177     [-0.775, 0.143]
#>  15.000       1.000   0.379 (0.188)  2.009  .045 *   [ 0.009, 0.748]
#>  15.000       3.000  -0.078 (0.150) -0.520  .603     [-0.371, 0.215]
#>  15.000       5.000  -0.534 (0.330) -1.619  .105     [-1.181, 0.113]
#> ────────────────────────────────────────────────────────────────────
#> 

## Model 4 ##
PROCESS(data, y="score", x="parent_edu",
        meds="family_inc", covs="gender",
        ci="boot", nsim=100, seed=1)
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 4 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Simple Mediation
#> -    Outcome (Y) : score
#> -  Predictor (X) : parent_edu
#> -  Mediators (M) : family_inc
#> - Moderators (W) : -
#> - Covariates (C) : gender
#> -   HLM Clusters : -
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Mediator:
#> -    family_inc ~ gender + parent_edu
#> Formula of Outcome:
#> -    score ~ gender + parent_edu + family_inc
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ───────────────────────────────────────────────────────
#>              (1) score     (2) family_inc  (3) score   
#> ───────────────────────────────────────────────────────
#> (Intercept)    51.206 ***     9.207 ***      51.262 ***
#>                (0.130)       (0.030)         (0.126)   
#> genderMale      1.474 ***     0.108 *         1.358 ***
#>                (0.188)       (0.044)         (0.182)   
#> parent_edu      5.537 ***     1.793 ***       3.594 ***
#>                (0.190)       (0.044)         (0.199)   
#> family_inc                                    1.083 ***
#>                                              (0.042)   
#> ───────────────────────────────────────────────────────
#> R^2             0.087         0.146           0.145    
#> Adj. R^2        0.086         0.146           0.145    
#> Num. obs.    9679          9679            9679        
#> ───────────────────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘mediation’ (v4.5.0)
#> Effect Type : Simple Mediation (Model 4)
#> Sample Size : 9679
#> Random Seed : set.seed(1)
#> Simulations : 100 (Bootstrap)
#> 
#> Warning: nsim=1000 (or larger) is suggested!
#> 
#> Running 100 simulations...
#> Indirect Path: "parent_edu" (X) ==> "family_inc" (M) ==> "score" (Y)
#> ─────────────────────────────────────────────────────────────
#>                Effect    S.E.      z     p      [Boot 95% CI]
#> ─────────────────────────────────────────────────────────────
#> Indirect (ab)   1.943 (0.089) 21.759 <.001 *** [1.752, 2.095]
#> Direct (c')     3.594 (0.220) 16.319 <.001 *** [3.155, 3.981]
#> Total (c)       5.537 (0.203) 27.249 <.001 *** [5.084, 5.898]
#> ─────────────────────────────────────────────────────────────
#> Percentile Bootstrap Confidence Interval
#> (SE and CI are estimated based on 100 Bootstrap samples.)
#> 
#> Note. The results based on bootstrapping or other random processes
#> are unlikely identical to other statistical software (e.g., SPSS).
#> To make results reproducible, you need to set a seed (any number).
#> Please see the help page for details: help(PROCESS)
#> Ignore this note if you have already set a seed. :)
#> 

# (allows an infinite number of multiple mediators in parallel)
PROCESS(data, y="score", x="parent_edu",
        meds=c("family_inc", "late"),
        covs=c("gender", "partjob"),
        ci="boot", nsim=100, seed=1)
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 4 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Parallel Multiple Mediation (2 meds)
#> -    Outcome (Y) : score
#> -  Predictor (X) : parent_edu
#> -  Mediators (M) : family_inc, late
#> - Moderators (W) : -
#> - Covariates (C) : gender, partjob
#> -   HLM Clusters : -
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Mediator:
#> -    family_inc ~ gender + partjob + parent_edu
#> -    late ~ gender + partjob + parent_edu
#> Formula of Outcome:
#> -    score ~ gender + partjob + parent_edu + family_inc + late
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ─────────────────────────────────────────────────────────────────────
#>              (1) score     (2) family_inc  (3) late      (4) score   
#> ─────────────────────────────────────────────────────────────────────
#> (Intercept)    51.206 ***     9.207 ***       2.235 ***    51.254 ***
#>                (0.130)       (0.030)         (0.016)       (0.125)   
#> genderMale      1.475 ***     0.107 *         0.013         1.374 ***
#>                (0.188)       (0.044)         (0.023)       (0.181)   
#> partjob        -0.301         0.132 **        0.091 ***    -0.353    
#>                (0.193)       (0.045)         (0.023)       (0.186)   
#> parent_edu      5.531 ***     1.796 ***      -0.092 ***     3.545 ***
#>                (0.190)       (0.044)         (0.023)       (0.197)   
#> family_inc                                                  1.057 ***
#>                                                            (0.042)   
#> late                                                       -0.957 ***
#>                                                            (0.081)   
#> ─────────────────────────────────────────────────────────────────────
#> R^2             0.087         0.146           0.003         0.158    
#> Adj. R^2        0.087         0.146           0.003         0.157    
#> Num. obs.    9679          9679            9679          9679        
#> ─────────────────────────────────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘mediation’ (v4.5.0)
#> Effect Type : Parallel Multiple Mediation (2 meds) (Model 4)
#> Sample Size : 9679
#> Random Seed : set.seed(1)
#> Simulations : 100 (Bootstrap)
#> 
#> Warning: nsim=1000 (or larger) is suggested!
#> 
#> Running 100 simulations...
#> Indirect Path: "parent_edu" (X) ==> "family_inc" (M) ==> "score" (Y)
#> ─────────────────────────────────────────────────────────────
#>                Effect    S.E.      z     p      [Boot 95% CI]
#> ─────────────────────────────────────────────────────────────
#> Indirect (ab)   1.898 (0.089) 21.262 <.001 *** [1.711, 2.055]
#> Direct (c')     3.545 (0.218) 16.280 <.001 *** [3.089, 3.937]
#> ─────────────────────────────────────────────────────────────
#> Percentile Bootstrap Confidence Interval
#> (SE and CI are estimated based on 100 Bootstrap samples.)
#> 
#> Running 100 simulations...
#> Indirect Path: "parent_edu" (X) ==> "late" (M) ==> "score" (Y)
#> ─────────────────────────────────────────────────────────────
#>                Effect    S.E.      z     p      [Boot 95% CI]
#> ─────────────────────────────────────────────────────────────
#> Indirect (ab)   0.088 (0.022)  3.946 <.001 *** [0.054, 0.136]
#> Direct (c')     3.545 (0.218) 16.280 <.001 *** [3.089, 3.937]
#> ─────────────────────────────────────────────────────────────
#> Percentile Bootstrap Confidence Interval
#> (SE and CI are estimated based on 100 Bootstrap samples.)
#> 
#> Note. The results based on bootstrapping or other random processes
#> are unlikely identical to other statistical software (e.g., SPSS).
#> To make results reproducible, you need to set a seed (any number).
#> Please see the help page for details: help(PROCESS)
#> Ignore this note if you have already set a seed. :)
#> 

# (multilevel mediation)
PROCESS(data, y="score", x="SCH_free",
        meds="late", clusters="SCH_ID",
        ci="mcmc", nsim=100, seed=1)
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 4 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Simple Mediation
#> -    Outcome (Y) : score
#> -  Predictor (X) : SCH_free
#> -  Mediators (M) : late
#> - Moderators (W) : -
#> - Covariates (C) : -
#> -   HLM Clusters : SCH_ID
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Mediator:
#> -    late ~ SCH_free + (1 | SCH_ID)
#> Formula of Outcome:
#> -    score ~ SCH_free + late + (1 | SCH_ID)
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ────────────────────────────────────────────────────────────────────
#>                          (1) score      (2) late       (3) score    
#> ────────────────────────────────────────────────────────────────────
#> (Intercept)                 51.853 ***      2.245 ***     51.858 ***
#>                             (0.162)        (0.017)        (0.159)   
#> SCH_free                    -1.611 ***      0.049 ***     -1.566 ***
#>                             (0.085)        (0.009)        (0.084)   
#> late                                                      -0.902 ***
#>                                                           (0.082)   
#> ────────────────────────────────────────────────────────────────────
#> Marginal R^2                 0.095          0.007          0.106    
#> Conditional R^2              0.201          0.073          0.208    
#> AIC                      69944.810      29498.368      69828.530    
#> BIC                      69973.521      29527.079      69864.419    
#> Num. obs.                 9679           9679           9679        
#> Num. groups: SCH_ID        568            568            568        
#> Var: SCH_ID (Intercept)      9.935          0.084          9.592    
#> Var: Residual               75.201          1.176         74.339    
#> ────────────────────────────────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘mediation’ (v4.5.0)
#> Effect Type : Simple Mediation (Model 4)
#> Sample Size : 9679
#> Random Seed : set.seed(1)
#> Simulations : 100 (Monte Carlo)
#> 
#> Warning: nsim=1000 (or larger) is suggested!
#> 
#> Running 100 simulations...
#> Indirect Path: "SCH_free" (X) ==> "late" (M) ==> "score" (Y)
#> ────────────────────────────────────────────────────────────────
#>                Effect    S.E.       z     p        [MCMC 95% CI]
#> ────────────────────────────────────────────────────────────────
#> Indirect (ab)  -0.044 (0.009)  -5.155 <.001 *** [-0.061, -0.030]
#> Direct (c')    -1.573 (0.078) -20.230 <.001 *** [-1.720, -1.440]
#> Total (c)      -1.618 (0.078) -20.624 <.001 *** [-1.769, -1.480]
#> ────────────────────────────────────────────────────────────────
#> Monte Carlo (Quasi-Bayesian) Confidence Interval
#> (Effect, SE, and CI are estimated based on 100 Monte Carlo samples.)
#> 
#> Note. The results based on bootstrapping or other random processes
#> are unlikely identical to other statistical software (e.g., SPSS).
#> To make results reproducible, you need to set a seed (any number).
#> Please see the help page for details: help(PROCESS)
#> Ignore this note if you have already set a seed. :)
#> 

## Model 6 ##
PROCESS(data, y="score", x="parent_edu",
        meds=c("family_inc", "late"),
        covs=c("gender", "partjob"),
        med.type="serial",
        ci="boot", nsim=100, seed=1)
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 6 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Serial Multiple Mediation (2 meds)
#> -    Outcome (Y) : score
#> -  Predictor (X) : parent_edu
#> -  Mediators (M) : family_inc, late
#> - Moderators (W) : -
#> - Covariates (C) : gender, partjob
#> -   HLM Clusters : -
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Mediator:
#> -    family_inc ~ gender + partjob + parent_edu
#> -    late ~ gender + partjob + parent_edu + family_inc
#> Formula of Outcome:
#> -    score ~ gender + partjob + parent_edu + family_inc + late
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ─────────────────────────────────────────────────────────────────────
#>              (1) score     (2) family_inc  (3) late      (4) score   
#> ─────────────────────────────────────────────────────────────────────
#> (Intercept)    51.206 ***     9.207 ***       2.516 ***    51.254 ***
#>                (0.130)       (0.030)         (0.051)       (0.125)   
#> genderMale      1.475 ***     0.107 *         0.017         1.374 ***
#>                (0.188)       (0.044)         (0.023)       (0.181)   
#> partjob        -0.301         0.132 **        0.096 ***    -0.353    
#>                (0.193)       (0.045)         (0.023)       (0.186)   
#> parent_edu      5.531 ***     1.796 ***      -0.037         3.545 ***
#>                (0.190)       (0.044)         (0.025)       (0.197)   
#> family_inc                                   -0.030 ***     1.057 ***
#>                                              (0.005)       (0.042)   
#> late                                                       -0.957 ***
#>                                                            (0.081)   
#> ─────────────────────────────────────────────────────────────────────
#> R^2             0.087         0.146           0.007         0.158    
#> Adj. R^2        0.087         0.146           0.006         0.157    
#> Num. obs.    9679          9679            9679          9679        
#> ─────────────────────────────────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘lavaan’ (v0.6.18)
#> Effect Type : Serial Multiple Mediation (2 meds) (Model 6)
#> Sample Size : 9679
#> Random Seed : set.seed(1)
#> Simulations : 100 (Bootstrap)
#> 
#> Warning: nsim=1000 (or larger) is suggested!
#> 
#> Running 100 simulations (lavaan model)...
#> LAVAAN Syntax:
#> family_inc ~ gender + partjob + a1*parent_edu
#> late ~ gender + partjob + a2*parent_edu + d12*family_inc
#> score ~ gender + partjob + c.*parent_edu + b1*family_inc + b2*late
#> Indirect_All := a1*b1 + a2*b2 + a1*d12*b2
#> Ind_X_M1_Y := a1*b1
#> Ind_X_M2_Y := a2*b2
#> Ind_X_M1_M2_Y := a1*d12*b2
#> Direct := c.
#> Total := c. + a1*b1 + a2*b2 + a1*d12*b2
#> ────────────────────────────────────────────────────────────────────────
#>                  Estimate    S.E.      z     p       [Boot 95% CI]  Beta
#> ────────────────────────────────────────────────────────────────────────
#>   Indirect_All      1.986 (0.088) 22.644 <.001 *** [ 1.798, 2.135] 0.102
#>   Ind_X_M1_Y        1.898 (0.083) 22.816 <.001 *** [ 1.724, 2.058] 0.097
#>   Ind_X_M2_Y        0.036 (0.027)  1.327  .185     [-0.011, 0.086] 0.002
#>   Ind_X_M1_M2_Y     0.052 (0.011)  4.774 <.001 *** [ 0.034, 0.078] 0.003
#>   Direct            3.545 (0.178) 19.879 <.001 *** [ 3.179, 3.825] 0.182
#>   Total             5.531 (0.171) 32.290 <.001 *** [ 5.130, 5.857] 0.283
#> ────────────────────────────────────────────────────────────────────────
#> Percentile Bootstrap Confidence Interval
#> (SE and CI are estimated based on 100 Bootstrap samples.)
#> 
#> Note. The results based on bootstrapping or other random processes
#> are unlikely identical to other statistical software (e.g., SPSS).
#> To make results reproducible, you need to set a seed (any number).
#> Please see the help page for details: help(PROCESS)
#> Ignore this note if you have already set a seed. :)
#> 

## Model 8 ##
PROCESS(data, y="score", x="fight",
        meds="late",
        mods="gender",
        mod.path=c("x-m", "x-y"),
        ci="boot", nsim=100, seed=1)
#> 
#> ****************** PART 1. Regression Model Summary ******************
#> 
#> PROCESS Model Code : 8 (Hayes, 2018; www.guilford.com/p/hayes3)
#> PROCESS Model Type : Moderated Mediation
#> -    Outcome (Y) : score
#> -  Predictor (X) : fight
#> -  Mediators (M) : late
#> - Moderators (W) : gender
#> - Covariates (C) : -
#> -   HLM Clusters : -
#> 
#> All numeric predictors have been grand-mean centered.
#> (For details, please see the help page of PROCESS.)
#> 
#> Formula of Mediator:
#> -    late ~ fight*gender
#> Formula of Outcome:
#> -    score ~ fight*gender + late
#> 
#> CAUTION:
#>   Fixed effect (coef.) of a predictor involved in an interaction
#>   denotes its "simple effect/slope" at the other predictor = 0.
#>   Only when all predictors in an interaction are mean-centered
#>   can the fixed effect denote the "main effect"!
#>   
#> Model Summary
#> 
#> ──────────────────────────────────────────────────────────
#>                   (1) score     (2) late      (3) score   
#> ──────────────────────────────────────────────────────────
#> (Intercept)         51.912 ***     2.272 ***    50.824 ***
#>                     (0.097)       (0.016)       (0.136)   
#> fight               -4.585 ***     0.646 ***    -6.188 ***
#>                     (0.293)       (0.062)       (0.527)   
#> genderMale                        -0.057 *       2.129 ***
#>                                   (0.023)       (0.196)   
#> fight:genderMale                  -0.103         2.305 ***
#>                                   (0.074)       (0.633)   
#> late                                            -0.950 ***
#>                                                 (0.087)   
#> ──────────────────────────────────────────────────────────
#> R^2                  0.025         0.028         0.050    
#> Adj. R^2             0.025         0.028         0.049    
#> Num. obs.         9679          9679          9679        
#> ──────────────────────────────────────────────────────────
#> Note. * p < .05, ** p < .01, *** p < .001.
#> 
#> ************ PART 2. Mediation/Moderation Effect Estimate ************
#> 
#> Package Use : ‘mediation’ (v4.5.0), ‘interactions’ (v1.1.5)
#> Effect Type : Moderated Mediation (Model 8)
#> Sample Size : 9679
#> Random Seed : set.seed(1)
#> Simulations : 100 (Bootstrap)
#> 
#> Warning: nsim=1000 (or larger) is suggested!
#> 
#> Interaction Effect on "score" (Y)
#> ────────────────────────────────────────
#>                     F df1  df2     p    
#> ────────────────────────────────────────
#> fight * gender  13.24   1 9674 <.001 ***
#> ────────────────────────────────────────
#> 
#> Simple Slopes: "fight" (X) ==> "score" (Y)
#> (Conditional Direct Effects [c'] of X on Y)
#> ───────────────────────────────────────────────────────────
#>  "gender" Effect    S.E.       t     p             [95% CI]
#> ───────────────────────────────────────────────────────────
#>  Female   -6.188 (0.527) -11.742 <.001 *** [-7.221, -5.155]
#>  Male     -3.883 (0.359) -10.821 <.001 *** [-4.587, -3.180]
#> ───────────────────────────────────────────────────────────
#> 
#> Interaction Effect on "late" (M)
#> ───────────────────────────────────────
#>                    F df1  df2     p    
#> ───────────────────────────────────────
#> fight * gender  1.90   1 9675  .168    
#> ───────────────────────────────────────
#> 
#> Simple Slopes: "fight" (X) ==> "late" (M)
#> (Conditional Effects [a] of X on M)
#> ────────────────────────────────────────────────────────
#>  "gender" Effect    S.E.      t     p           [95% CI]
#> ────────────────────────────────────────────────────────
#>  Female    0.646 (0.062) 10.485 <.001 *** [0.525, 0.766]
#>  Male      0.543 (0.042) 12.989 <.001 *** [0.461, 0.625]
#> ────────────────────────────────────────────────────────
#> 
#> Running 100 * 2 simulations...
#> Indirect Path: "fight" (X) ==> "late" (M) ==> "score" (Y)
#> (Conditional Indirect Effects [ab] of X through M on Y)
#> ──────────────────────────────────────────────────────────
#>  "gender" Effect    S.E.      z     p        [Boot 95% CI]
#> ──────────────────────────────────────────────────────────
#>  Female   -0.614 (0.086) -7.117 <.001 *** [-0.807, -0.477]
#>  Male     -0.516 (0.060) -8.538 <.001 *** [-0.645, -0.411]
#> ──────────────────────────────────────────────────────────
#> Percentile Bootstrap Confidence Interval
#> (SE and CI are estimated based on 100 Bootstrap samples.)
#> 
#> Note. The results based on bootstrapping or other random processes
#> are unlikely identical to other statistical software (e.g., SPSS).
#> To make results reproducible, you need to set a seed (any number).
#> Please see the help page for details: help(PROCESS)
#> Ignore this note if you have already set a seed. :)
#> 

## For more examples and details, see the "note" subfolder at:
## https://github.com/psychbruce/bruceR/tree/main/note