Code for this demo adapted from here: https://nickmichalak.blogspot.com/2016/07/reproducing-hayess-process-models.html

1 Example dataset

The dataset for this example contains standardized test scores on reading, writing, math, science, and social studies. There are also binary indicators (e.g., hisci) that encode whether someone received a high score on a given test. We also want to compute centered versions of each variable to aid interpretation in moderation-related models.

Following Hayes, I will use the convention \(X\) for independent variable/predictor, \(M\) for mediator, \(W\) for moderator, and \(Y\) for outcome.

df <- read_csv("mediation_data.csv")
## Parsed with column specification:
## cols(
##   id = col_integer(),
##   female = col_integer(),
##   ses = col_integer(),
##   prog = col_integer(),
##   read = col_integer(),
##   write = col_integer(),
##   math = col_integer(),
##   science = col_integer(),
##   socst = col_integer(),
##   honors = col_integer(),
##   awards = col_integer(),
##   cid = col_integer(),
##   hiread = col_integer(),
##   hiwrite = col_integer(),
##   hisci = col_integer(),
##   himath = col_integer()
## )
df <- df %>% mutate_at(vars(read, write, math, science, socst), funs(c=scale))

2 Moderation with a continuous moderator

Hayes Process model 1

Does the relationship between performance on reading (\(Y\)) and social studies (\(X\)) tests depend on math ability (\(W\))?

library(lavaan)

# create interaction term between centered X (socst) and W (math)
df <- df %>% mutate(socst_x_math = socst_c * math_c)

# parameters
moderation_model <- '
  # regressions
  read ~ b1*socst_c
  read ~ b2*math_c
  read ~ b3*socst_x_math
  
  # define mean parameter label for centered math for use in simple slopes
  math_c ~ math.mean*1
  
  # define variance parameter label for centered math for use in simple slopes
  math_c ~~ math.var*math_c
  
  # simple slopes for condition effect
  SD.below := b1 + b3*(math.mean - sqrt(math.var))
  mean := b1 + b3*(math.mean)
  SD.above := b1 + b3*(math.mean + sqrt(math.var))
  '

# fit the model using nonparametric bootstrapping (this takes some time)
sem1 <- sem(model = moderation_model,
            data = df,
            se = "bootstrap",
            bootstrap = 1000)
## Warning in lavaan::lavaan(model = moderation_model, data = df, se =
## "bootstrap", : lavaan WARNING: syntax contains parameters involving
## exogenous covariates; switching to fixed.x = FALSE
# fit measures
summary(sem1, fit.measures = TRUE, standardized = TRUE, rsquare = TRUE)
## lavaan (0.5-23.1097) converged normally after  34 iterations
## 
##   Number of observations                           200
## 
##   Estimator                                         ML
##   Minimum Function Test Statistic               76.103
##   Degrees of freedom                                 2
##   P-value (Chi-square)                           0.000
## 
## Model test baseline model:
## 
##   Minimum Function Test Statistic              234.093
##   Degrees of freedom                                 5
##   P-value                                        0.000
## 
## User model versus baseline model:
## 
##   Comparative Fit Index (CFI)                    0.677
##   Tucker-Lewis Index (TLI)                       0.191
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -1509.766
##   Loglikelihood unrestricted model (H1)      -1471.714
## 
##   Number of free parameters                         12
##   Akaike (AIC)                                3043.531
##   Bayesian (BIC)                              3083.111
##   Sample-size adjusted Bayesian (BIC)         3045.094
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.430
##   90 Percent Confidence Interval          0.351  0.516
##   P-value RMSEA <= 0.05                          0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.180
## 
## Parameter Estimates:
## 
##   Information                                 Observed
##   Standard Errors                            Bootstrap
##   Number of requested bootstrap draws             1000
##   Number of successful bootstrap draws            1000
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   read ~                                                                
##     socst_c   (b1)    4.013    0.571    7.029    0.000    4.013    0.437
##     math_c    (b2)    4.503    0.603    7.466    0.000    4.503    0.490
##     scst_x_mt (b3)    1.135    0.503    2.254    0.024    1.135    0.118
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   socst_c ~~                                                            
##     socst_x_math     -0.082    0.101   -0.807    0.419   -0.082   -0.086
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     math_c  (mth.)   -0.000    0.070   -0.000    1.000   -0.000   -0.000
##    .read             51.615    0.585   88.240    0.000   51.615    5.628
##     socst_c          -0.000    0.072   -0.000    1.000   -0.000   -0.000
##     scst_x_           0.542    0.069    7.846    0.000    0.542    0.569
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     math_c  (mth.)    0.995    0.082   12.090    0.000    0.995    1.000
##    .read             47.473    4.698   10.105    0.000   47.473    0.564
##     socst_c           0.995    0.091   10.909    0.000    0.995    1.000
##     scst_x_           0.908    0.106    8.567    0.000    0.908    1.000
## 
## R-Square:
##                    Estimate
##     read              0.436
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     SD.below          2.882    0.724    3.981    0.000    2.882    0.319
##     mean              4.013    0.584    6.875    0.000    4.013    0.437
##     SD.above          5.145    0.812    6.339    0.000    5.145    0.554
#compute bias-corrected estimates of bootstrapped confidence intervals
parameterEstimates(sem1, boot.ci.type = "bca.simple",
                   level = .95, ci = TRUE, standardized = FALSE)
lhs op rhs label est se z pvalue ci.lower ci.upper
read ~ socst_c b1 4.013 0.571 7.029 0.000 2.890 5.253
read ~ math_c b2 4.503 0.603 7.466 0.000 3.282 5.709
read ~ socst_x_math b3 1.135 0.503 2.254 0.024 0.115 2.143
math_c ~1 math.mean 0.000 0.070 0.000 1.000 -0.131 0.147
math_c ~~ math_c math.var 0.995 0.082 12.090 0.000 0.846 1.168
read ~~ read 47.473 4.698 10.105 0.000 39.123 58.338
socst_c ~~ socst_c 0.995 0.091 10.909 0.000 0.827 1.194
socst_c ~~ socst_x_math -0.082 0.101 -0.807 0.419 -0.296 0.096
socst_x_math ~~ socst_x_math 0.908 0.106 8.567 0.000 0.713 1.130
read ~1 51.615 0.585 88.240 0.000 50.541 52.796
socst_c ~1 0.000 0.072 0.000 1.000 -0.138 0.140
socst_x_math ~1 0.542 0.069 7.846 0.000 0.413 0.682
SD.below := b1+b3*(math.mean-sqrt(math.var)) SD.below 2.882 0.724 3.981 0.000 1.523 4.310
mean := b1+b3*(math.mean) mean 4.013 0.584 6.875 0.000 2.868 5.293
SD.above := b1+b3*(math.mean+sqrt(math.var)) SD.above 5.145 0.812 6.339 0.000 3.530 6.693

3 Simple mediation

Hayes Process model 4

Is the relationship between science and math mediated by read?

# parameters
mediation_model <- '
    # direct effect
      science ~ cp*math_c
      direct := cp
  
    # regressions
      read_c ~ a*math_c
      science ~ b*read_c
  
    # indirect effect (a*b)
      indirect := a*b
  
    # total effect
      total := cp + (a*b)
'

# fit model
sem2 <- sem(model = mediation_model, data = df, se = "bootstrap", bootstrap = 1000)
## Warning in lavaan(slotOptions = lavoptions, slotParTable = lavpartable, :
## lavaan WARNING: model has NOT converged!

## Warning in lavaan(slotOptions = lavoptions, slotParTable = lavpartable, :
## lavaan WARNING: model has NOT converged!
## Warning in bootstrap.internal(object = NULL, lavmodel. = lavmodel,
## lavsamplestats. = lavsamplestats, : lavaan WARNING: only 997 bootstrap
## draws were successful
# fit measures
summary(sem2, fit.measures = TRUE, standardize = TRUE, rsquare = TRUE)
## lavaan (0.5-23.1097) converged normally after  27 iterations
## 
##   Number of observations                           200
## 
##   Estimator                                         ML
##   Minimum Function Test Statistic                0.000
##   Degrees of freedom                                 0
## 
## Model test baseline model:
## 
##   Minimum Function Test Statistic              245.569
##   Degrees of freedom                                 3
##   P-value                                        0.000
## 
## User model versus baseline model:
## 
##   Comparative Fit Index (CFI)                    1.000
##   Tucker-Lewis Index (TLI)                       1.000
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -1185.600
##   Loglikelihood unrestricted model (H1)      -1185.600
## 
##   Number of free parameters                          5
##   Akaike (AIC)                                2381.200
##   Bayesian (BIC)                              2397.691
##   Sample-size adjusted Bayesian (BIC)         2381.851
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.000
##   90 Percent Confidence Interval          0.000  0.000
##   P-value RMSEA <= 0.05                             NA
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.000
## 
## Parameter Estimates:
## 
##   Information                                 Observed
##   Standard Errors                            Bootstrap
##   Number of requested bootstrap draws             1000
##   Number of successful bootstrap draws             997
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   science ~                                                             
##     math_c    (cp)    3.763    0.770    4.887    0.000    3.763    0.380
##   read_c ~                                                              
##     math_c     (a)    0.662    0.050   13.163    0.000    0.662    0.662
##   science ~                                                             
##     read_c     (b)    3.747    0.767    4.885    0.000    3.747    0.378
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .science          50.894    4.935   10.314    0.000   50.894    0.522
##    .read_c            0.559    0.054   10.389    0.000    0.559    0.561
## 
## R-Square:
##                    Estimate
##     science           0.478
##     read_c            0.439
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     direct            3.763    0.770    4.885    0.000    3.763    0.380
##     indirect          2.481    0.501    4.950    0.000    2.481    0.251
##     total             6.245    0.518   12.052    0.000    6.245    0.631

4 First-stage moderated mediation

Does the indirect effect of math (\(X\)) on science (\(Y\)) via read (\(M\)) depend on write (\(W\))? More specifically, does writing ability moderate the relationship betwen math and reading? For example, perhaps only people with high writing and math ability tend to score higher on a reading test. If this is true, then the indirect effect of math on science via reading depends on writing as well.

#compute math x writing interaction term
df <- df %>% mutate(math_x_write = math_c*write_c)

moderated_mediation_1 <- '
  # regressions
    read_c ~ a1*math_c
    science ~ b*read_c
    read_c ~ a2*write_c
    read_c ~ a3*math_x_write
    science ~ cdash*math_c

  # mean of centered write (for use in simple slopes)
    write_c ~ write.mean*1

  # variance of centered write (for use in simple slopes)
    write_c ~~ write.var*write_c

  # index of moderated mediation
    imm := a3*b    

  # indirect effects conditional on moderator (a1 + a3*a2.value)*b
    indirect.SDbelow := a1*b + a3*-sqrt(write.var)*b
    indirect.mean := a1*b + a3*write.mean*b
    indirect.SDabove := a1*b + a3*sqrt(write.var)*b

'

# fit model
sem3 <- sem(model = moderated_mediation_1, data = df, se = "bootstrap", bootstrap = 1000)
## Warning in lavaan::lavaan(model = moderated_mediation_1, data = df, se
## = "bootstrap", : lavaan WARNING: syntax contains parameters involving
## exogenous covariates; switching to fixed.x = FALSE
# fit measures
summary(sem3, fit.measures = TRUE, standardize = TRUE, rsquare = TRUE)
## lavaan (0.5-23.1097) converged normally after  32 iterations
## 
##   Number of observations                           200
## 
##   Estimator                                         ML
##   Minimum Function Test Statistic              119.179
##   Degrees of freedom                                 4
##   P-value (Chi-square)                           0.000
## 
## Model test baseline model:
## 
##   Minimum Function Test Statistic              386.622
##   Degrees of freedom                                 9
##   P-value                                        0.000
## 
## User model versus baseline model:
## 
##   Comparative Fit Index (CFI)                    0.695
##   Tucker-Lewis Index (TLI)                       0.314
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -1718.688
##   Loglikelihood unrestricted model (H1)      -1659.098
## 
##   Number of free parameters                         16
##   Akaike (AIC)                                3469.376
##   Bayesian (BIC)                              3522.149
##   Sample-size adjusted Bayesian (BIC)         3471.459
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.379
##   90 Percent Confidence Interval          0.323  0.440
##   P-value RMSEA <= 0.05                          0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.199
## 
## Parameter Estimates:
## 
##   Information                                 Observed
##   Standard Errors                            Bootstrap
##   Number of requested bootstrap draws             1000
##   Number of successful bootstrap draws            1000
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   read_c ~                                                              
##     math_c    (a1)    0.464    0.068    6.794    0.000    0.464    0.511
##   science ~                                                             
##     read_c     (b)    3.747    0.782    4.792    0.000    3.747    0.358
##   read_c ~                                                              
##     write_c   (a2)    0.315    0.065    4.841    0.000    0.315    0.347
##     mth_x_w   (a3)    0.039    0.055    0.713    0.476    0.039    0.039
##   science ~                                                             
##     math_c  (cdsh)    3.763    0.786    4.786    0.000    3.763    0.397
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   math_c ~~                                                             
##     math_x_write      0.110    0.086    1.283    0.200    0.110    0.123
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     write_c (wrt.)    0.000    0.072    0.000    1.000    0.000    0.000
##    .read_c           -0.024    0.066   -0.363    0.717   -0.024   -0.026
##    .science          51.850    0.513  101.080    0.000   51.850    5.477
##     math_c           -0.000    0.071   -0.000    1.000   -0.000   -0.000
##     mth_x_w           0.614    0.063    9.788    0.000    0.614    0.684
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     write_c (wrt.)    0.995    0.080   12.390    0.000    0.995    1.000
##    .read_c            0.501    0.048   10.421    0.000    0.501    0.612
##    .science          50.894    5.095    9.988    0.000   50.894    0.568
##     math_c            0.995    0.080   12.482    0.000    0.995    1.000
##     mth_x_w           0.806    0.084    9.574    0.000    0.806    1.000
## 
## R-Square:
##                    Estimate
##     read_c            0.388
##     science           0.432
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     imm               0.146    0.211    0.690    0.490    0.146    0.014
##     indirect.SDblw    1.592    0.482    3.300    0.001    1.592    0.169
##     indirect.mean     1.737    0.393    4.425    0.000    1.737    0.183
##     indirect.SDabv    1.883    0.406    4.631    0.000    1.883    0.197
#compute bias-corrected confidence intervals
parameterEstimates(sem3, boot.ci.type = "bca.simple", level = .95, ci = TRUE, standardized = FALSE)
lhs op rhs label est se z pvalue ci.lower ci.upper
read_c ~ math_c a1 0.464 0.068 6.794 0.000 0.327 0.587
science ~ read_c b 3.747 0.782 4.792 0.000 2.073 5.198
read_c ~ write_c a2 0.315 0.065 4.841 0.000 0.194 0.448
read_c ~ math_x_write a3 0.039 0.055 0.713 0.476 -0.071 0.143
science ~ math_c cdash 3.763 0.786 4.786 0.000 2.308 5.410
write_c ~1 write.mean 0.000 0.072 0.000 1.000 -0.137 0.143
write_c ~~ write_c write.var 0.995 0.080 12.390 0.000 0.838 1.160
read_c ~~ read_c 0.501 0.048 10.421 0.000 0.418 0.603
science ~~ science 50.894 5.095 9.988 0.000 42.058 62.922
math_c ~~ math_c 0.995 0.080 12.482 0.000 0.850 1.167
math_c ~~ math_x_write 0.110 0.086 1.283 0.200 -0.056 0.274
math_x_write ~~ math_x_write 0.806 0.084 9.574 0.000 0.648 0.976
read_c ~1 -0.024 0.066 -0.363 0.717 -0.150 0.097
science ~1 51.850 0.513 101.080 0.000 50.838 52.796
math_c ~1 0.000 0.071 0.000 1.000 -0.143 0.142
math_x_write ~1 0.614 0.063 9.788 0.000 0.500 0.743
imm := a3*b imm 0.146 0.211 0.690 0.490 -0.259 0.600
indirect.SDbelow := a1b+a3-sqrt(write.var)*b indirect.SDbelow 1.592 0.482 3.300 0.001 0.858 2.728
indirect.mean := a1b+a3write.mean*b indirect.mean 1.737 0.393 4.425 0.000 1.026 2.575
indirect.SDabove := a1b+a3sqrt(write.var)*b indirect.SDabove 1.883 0.406 4.631 0.000 1.086 2.729

5 Second-stage moderated mediation

Mediated effects could also be moderated at what Hayes calls the ‘second stage’ (i.e., the relationship between \(M\) and \(Y\)). In the Hayes Process model world, this is also called Model 14. For example, does the indirect effect of math (\(X\)) on science (\(Y\)) via read (\(M\)) depend on write (\(W\)) because it moderates the relationship betwen math and reading? Perhaps only people with high writing and reading ability tend to do well in science. If this is true, then the indirect effect of math on science via reading depends on writing as well.

#compute math x writing interaction term
df <- df %>% mutate(read_x_write = read_c*write_c)

moderated_mediation_stage2 <- '
  # regressions
    read_c ~ a*math_c
    science ~ b1*read_c
    science ~ b2*write_c
    science ~ b3*read_x_write
    science ~ cdash*math_c

  # mean of centered write (moderator; for use in simple slopes)
    write_c ~ write.mean*1

  # variance of centered write (moderator; for use in simple slopes)
    write_c ~~ write.var*write_c

  #index of moderated mediation
    imm := a*b3

  # indirect effects conditional on moderator (a1 + a3*b2.value)*a
    indirect.SDbelow := a*b1 + a*-sqrt(write.var)*b3
    indirect.mean := a*b1 + a*write.mean*b3
    indirect.SDabove := a*b1 + a*sqrt(write.var)*b3
'

# fit model
sem4 <- sem(model = moderated_mediation_stage2, data = df, se = "bootstrap", bootstrap = 1000)
## Warning in lavaan::lavaan(model = moderated_mediation_stage2, data = df, :
## lavaan WARNING: syntax contains parameters involving exogenous covariates;
## switching to fixed.x = FALSE
# fit measures
summary(sem4, fit.measures = TRUE, standardize = TRUE, rsquare = TRUE)
## lavaan (0.5-23.1097) converged normally after  43 iterations
## 
##   Number of observations                           200
## 
##   Estimator                                         ML
##   Minimum Function Test Statistic              133.763
##   Degrees of freedom                                 4
##   P-value (Chi-square)                           0.000
## 
## Model test baseline model:
## 
##   Minimum Function Test Statistic              390.617
##   Degrees of freedom                                 9
##   P-value                                        0.000
## 
## User model versus baseline model:
## 
##   Comparative Fit Index (CFI)                    0.660
##   Tucker-Lewis Index (TLI)                       0.235
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -1729.435
##   Loglikelihood unrestricted model (H1)      -1662.553
## 
##   Number of free parameters                         16
##   Akaike (AIC)                                3490.870
##   Bayesian (BIC)                              3543.643
##   Sample-size adjusted Bayesian (BIC)         3492.953
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.403
##   90 Percent Confidence Interval          0.346  0.463
##   P-value RMSEA <= 0.05                          0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.220
## 
## Parameter Estimates:
## 
##   Information                                 Observed
##   Standard Errors                            Bootstrap
##   Number of requested bootstrap draws             1000
##   Number of successful bootstrap draws            1000
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   read_c ~                                                              
##     math_c     (a)    0.662    0.051   13.097    0.000    0.662    0.662
##   science ~                                                             
##     read_c    (b1)    3.207    0.755    4.250    0.000    3.207    0.348
##     write_c   (b2)    1.636    0.700    2.337    0.019    1.636    0.178
##     rd_x_wr   (b3)   -0.933    0.526   -1.772    0.076   -0.933   -0.093
##     math_c  (cdsh)    3.157    0.806    3.916    0.000    3.157    0.343
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   math_c ~~                                                             
##     read_x_write      0.049    0.072    0.675    0.500    0.049    0.053
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     write_c (wrt.)    0.000    0.070    0.000    1.000    0.000    0.000
##    .read_c           -0.000    0.053   -0.000    1.000   -0.000   -0.000
##    .science          52.404    0.589   88.954    0.000   52.404    5.701
##     math_c           -0.000    0.070   -0.000    1.000   -0.000   -0.000
##     rd_x_wr           0.594    0.064    9.296    0.000    0.594    0.647
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     write_c (wrt.)    0.995    0.078   12.725    0.000    0.995    1.000
##    .read_c            0.559    0.055   10.249    0.000    0.559    0.561
##    .science          48.102    5.201    9.249    0.000   48.102    0.569
##     math_c            0.995    0.077   12.992    0.000    0.995    1.000
##     rd_x_wr           0.841    0.107    7.853    0.000    0.841    1.000
## 
## R-Square:
##                    Estimate
##     read_c            0.439
##     science           0.431
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     imm              -0.618    0.353   -1.751    0.080   -0.618   -0.062
##     indirect.SDblw    2.740    0.623    4.395    0.000    2.740    0.292
##     indirect.mean     2.124    0.485    4.375    0.000    2.124    0.230
##     indirect.SDabv    1.507    0.569    2.647    0.008    1.507    0.169
#compute bias-corrected confidence intervals
parameterEstimates(sem3, boot.ci.type = "bca.simple", level = .95, ci = TRUE, standardized = FALSE)
lhs op rhs label est se z pvalue ci.lower ci.upper
read_c ~ math_c a1 0.464 0.068 6.794 0.000 0.327 0.587
science ~ read_c b 3.747 0.782 4.792 0.000 2.073 5.198
read_c ~ write_c a2 0.315 0.065 4.841 0.000 0.194 0.448
read_c ~ math_x_write a3 0.039 0.055 0.713 0.476 -0.071 0.143
science ~ math_c cdash 3.763 0.786 4.786 0.000 2.308 5.410
write_c ~1 write.mean 0.000 0.072 0.000 1.000 -0.137 0.143
write_c ~~ write_c write.var 0.995 0.080 12.390 0.000 0.838 1.160
read_c ~~ read_c 0.501 0.048 10.421 0.000 0.418 0.603
science ~~ science 50.894 5.095 9.988 0.000 42.058 62.922
math_c ~~ math_c 0.995 0.080 12.482 0.000 0.850 1.167
math_c ~~ math_x_write 0.110 0.086 1.283 0.200 -0.056 0.274
math_x_write ~~ math_x_write 0.806 0.084 9.574 0.000 0.648 0.976
read_c ~1 -0.024 0.066 -0.363 0.717 -0.150 0.097
science ~1 51.850 0.513 101.080 0.000 50.838 52.796
math_c ~1 0.000 0.071 0.000 1.000 -0.143 0.142
math_x_write ~1 0.614 0.063 9.788 0.000 0.500 0.743
imm := a3*b imm 0.146 0.211 0.690 0.490 -0.259 0.600
indirect.SDbelow := a1b+a3-sqrt(write.var)*b indirect.SDbelow 1.592 0.482 3.300 0.001 0.858 2.728
indirect.mean := a1b+a3write.mean*b indirect.mean 1.737 0.393 4.425 0.000 1.026 2.575
indirect.SDabove := a1b+a3sqrt(write.var)*b indirect.SDabove 1.883 0.406 4.631 0.000 1.086 2.729

6 Multiple-groups moderation

In a two-group case, treating moderator as a 0/1 variable is an easy way to handle mediation in SEM. When a moderating variable, \(W\), is categorical, but has many levels (e.g., 5 groups), setting up dummy codes and computing all of the relevant interactions is very painful and laborious.

The better route is to treat the moderator \(W\) as a grouping variable such that any path in the model could be moderated by group. This allows one to test categorical moderation of parameters not often considered in regression, such as latent variable variance estimates across groups.