Specifying user-defined mediation models • wsMed

Use wsMed(form = "UD", paths = ...) when the mediator connections do not match a predefined model. Supply paired raw measurements, then list the arrows among mediators and from mediators to the outcome. This tutorial uses the package’s simulated data and the default inference and reporting settings.

data(example_data)

Mediator numbering

Mediator labels are assigned according to the order of the variables supplied to M_C1 and M_C2. For example:

Construct	Condition 1	Condition 2	Model label
A	`A1`	`A2`	`M1`
B	`B1`	`B2`	`M2`
C	`C1`	`C2`	`M3`
D	`D1`	`D2`	`Y`

Therefore, "M1 -> M3" represents a path from construct A to construct C. Mediator numbering identifies variables; it does not determine path direction.

Specifying a custom model

Write one directed connection per string in paths. Use the model labels M1, M2, and so on, plus Y, rather than raw column names such as A1. For example, to connect A to C, C to the outcome, and B to the outcome:

custom_paths <- c(
  "M1 -> M3",  # A to C
  "M3 -> Y",   # C to the outcome
  "M2 -> Y"    # B to the outcome
)

Entry in `paths`	Connection in this example
`"M1 -> M3"`	A predicts C
`"M3 -> Y"`	C predicts the outcome D
`"M2 -> Y"`	B predicts the outcome D

Each string represents one arrow, not a complete mediation route. For example, specify "M1 -> M3" and "M3 -> Y" separately instead of writing "M1 -> M3 -> Y". Spaces around -> are optional.

Only the listed mediator connections are included. In this example there is no direct path from M1 to Y, and no directed path between M1 and M2. Paths from the condition to every mediator, and the direct path from the condition to the outcome, are added automatically. Do not add X to the path vector.

Fitting the complete model with wsMed

Pass the path vector to wsMed() along with the paired measurements:

result_custom <- wsMed(
  data = example_data,
  M_C1 = c("A1", "B1", "C1"),
  M_C2 = c("A2", "B2", "C2"),
  Y_C1 = "D1",
  Y_C2 = "D2",
  form = "UD",
  paths = custom_paths
)
print(result_custom)
#> wsMed analysis
#> Model: UD | DE | Observations: 100 
#> Effects:
#> wsMed all | raw scale | mc 
#> 95% percentile intervals; 20000 joint draws; SE from effect draws
#> Condition contrast: C2 - C1 
#>                                        Estimate Std. Error  CI lower  CI upper
#> cp: Direct effect                     -3.00e-02   1.68e-02 -6.31e-02  2.90e-03
#> indirect_1_3: M1 -> M3                -2.25e-05   3.74e-04 -8.64e-04  7.36e-04
#> indirect_2: M2                        -1.40e-03   2.75e-03 -8.43e-03  2.98e-03
#> indirect_3: M3                        -1.85e-04   2.17e-03 -5.11e-03  4.38e-03
#> total_indirect: Total indirect effect -1.60e-03   3.54e-03 -9.79e-03  4.91e-03
#> total: Total effect                   -3.16e-02   1.69e-02 -6.48e-02  1.61e-03
#> Inference draws: 20000 retained of 20000 ; 0 excluded before effect extraction.
#> Use as.data.frame() for the full-precision table and metadata.
#> Post-fit admissibility: 1 of 1 ; excluded rows: 0

This call uses the default Monte Carlo CIs and unstandardized output. Use the usual summary method for effect and path-coefficient tables:

summary(result_custom)
#> wsMed statistical summary
#> Model: UD | Missing: DE | Observations: 100 
#> Effects:
#> wsMed all | raw scale | mc 
#> 95% percentile intervals; 20000 joint draws; SE from effect draws
#> Condition contrast: C2 - C1 
#>                                        Estimate Std. Error  CI lower  CI upper
#> cp: Direct effect                     -3.00e-02   1.68e-02 -6.31e-02  2.90e-03
#> indirect_1_3: M1 -> M3                -2.25e-05   3.74e-04 -8.64e-04  7.36e-04
#> indirect_2: M2                        -1.40e-03   2.75e-03 -8.43e-03  2.98e-03
#> indirect_3: M3                        -1.85e-04   2.17e-03 -5.11e-03  4.38e-03
#> total_indirect: Total indirect effect -1.60e-03   3.54e-03 -9.79e-03  4.91e-03
#> total: Total effect                   -3.16e-02   1.69e-02 -6.48e-02  1.61e-03
#> Inference draws: 20000 retained of 20000 ; 0 excluded before effect extraction.
#> Use as.data.frame() for the full-precision table and metadata.
#> 
#> Path coefficients:
#>       Estimate Std. Error CI lower CI upper
#> cp    -0.03000    0.01684 -0.06307  0.00290
#> b2    -0.09632    0.09303 -0.27662  0.08915
#> d2    -0.01377    0.08708 -0.18534  0.15458
#> b3    -0.01350    0.10245 -0.21287  0.18842
#> d3    -0.16325    0.10038 -0.35988  0.03574
#> a1    -0.02712    0.01757 -0.06167  0.00727
#> a2     0.01448    0.01798 -0.02044  0.04994
#> a3     0.01368    0.01641 -0.01819  0.04573
#> b_1_3 -0.06142    0.09198 -0.23967  0.12025
#> d_1_3 -0.07843    0.08705 -0.24852  0.09208
#> Post-fit admissibility: 1 of 1 ; excluded rows: 0 
#> Tables: $effects, $coefficients; analysis settings: $info.

To include a direct M1-to-outcome connection, add its arrow to the vector:

expanded_paths <- c(
  custom_paths,
  "M1 -> Y"
)

Refit the same call with paths = expanded_paths to include that connection. This changes the fitted model and adds the indirect route through M1 alone.

Input requirements

Reference only mediator labels defined by the supplied column pairs.
Use one arrow per string, starting at a mediator and ending at another mediator or Y. At least one arrow must end at Y.
Do not repeat an arrow or use a self-loop such as "M1 -> M1".
Do not create a cycle, such as both "M1 -> M2" and "M2 -> M1".

Every mediator intended to contribute an indirect effect needs a route to Y, either directly or through other mediators. The package warns if a supplied mediator has no such route.

Indirect effects

wsMed() identifies all indirect routes implied by the specified arrows. In the following notation, X denotes the within-subject condition; it is not an entry to add to paths. The model defined by custom_paths contains three indirect effects:

Route	Parameter label	Product
`X -> M1 -> M3 -> Y`	`indirect_1_3`	`a1 * b_1_3 * b3`
`X -> M2 -> Y`	`indirect_2`	`a2 * b2`
`X -> M3 -> Y`	`indirect_3`	`a3 * b3`

The last effect exists because the path from the condition to every mediator, including $M_3$ , is added automatically. All identified indirect effects are summed to form total_indirect, and the total effect is defined as cp + total_indirect.

Parameter labels

The generated model uses the following labels:

Path	Difference component	Level component
$X \rightarrow M_i$	`ai`	–
$M_i \rightarrow Y$	`bi`	`di`
$M_j \rightarrow M_i$	`b_j_i`	`d_j_i`
$X \rightarrow Y$	`cp`	–

For example, the path $M_1 \rightarrow M_3$ uses b_1_3 and d_1_3, whereas $M_3 \rightarrow Y$ uses b3 and d3.

Add a moderator

Keep form = "UD" and the same custom_paths. Add W to identify the between-subject moderator column and MP to select the path coefficients to moderate. You do not need to construct product terms yourself.

Argument	What to supply	Example
`paths`	The mediator connections in the model	`custom_paths`
`W`	The moderator’s column name in the data	`"D3"`
`MP`	Coefficient labels for the selected paths	`c("b_1_3", "b3")`

Choose the moderated paths

paths uses arrows such as "M1 -> M3", whereas MP uses coefficient labels such as "b_1_3". For the current custom model:

Path to moderate	Entry in `MP`	Meaning
M1 to M3	`"b_1_3"`	The M1-difference slope in the M3-difference equation varies with W.
M3 to the outcome	`"b3"`	The M3-difference slope in the outcome equation varies with W.
M2 to the outcome	`"b2"`	The M2-difference slope in the outcome equation varies with W.

Thus, MP = "b3" selects only the M3-to-outcome slope, while MP = c("b_1_3", "b3") selects both slopes along the M1-to-M3-to-outcome route. To add moderation of the M2-to-outcome slope, use MP = c("b_1_3", "b3", "b2"). Each label must exist in the model: "b1" is unavailable here because "M1 -> Y" is absent from custom_paths. Do not put an indirect-effect label such as "indirect_1_3" in MP; select the coefficients that form that effect.

The corresponding average-component slopes can be selected with "d_1_3", "d3", and "d2". These are distinct coefficients: selecting "b3" does not also select "d3".

The condition-to-mediator labels "a1", "a2", "a3", and the direct-effect label "cp" are also accepted. With W present, wsMed includes its main effect in every mediator-difference and outcome-difference equation. Consequently, these condition effects can vary with W even when their labels are omitted from MP. For the b and d slopes, MP controls which interaction terms are added.

Fit and read the results

Here, D3 is a continuous moderator. This call selects the M1-to-M3 and M3-to-outcome slopes and keeps the default Monte Carlo CIs and unstandardized output:

custom_moderated <- wsMed(
  data = example_data,
  M_C1 = c("A1", "B1", "C1"),
  M_C2 = c("A2", "B2", "C2"),
  Y_C1 = "D1",
  Y_C2 = "D2",
  form = "UD",
  paths = custom_paths,
  W = "D3",
  MP = c("b_1_3", "b3")
)
print(custom_moderated)
#> wsMed analysis
#> Model: UD | DE | Observations: 100 
#> Effects:
#> wsMed all | raw scale | mc 
#> 95% percentile intervals; 20000 joint draws; SE from effect draws
#> Condition contrast: C2 - C1 
#> Moderator: D3 in raw units; centering reference = 0.452271 
#>                                                       Estimate Std. Error
#> cp@1: Direct effect [D3 = 0.452]                     -0.029850   0.016991
#> indirect_1_3@1: M1 -> M3 [D3 = 0.452]                -0.000022   0.000367
#> indirect_2@1: M2 [D3 = 0.452]                        -0.001400   0.002744
#> indirect_3@1: M3 [D3 = 0.452]                        -0.000183   0.002168
#> total_indirect@1: Total indirect effect [D3 = 0.452] -0.001605   0.003581
#> total@1: Total effect [D3 = 0.452]                   -0.031455   0.017152
#>                                                       CI lower  CI upper
#> cp@1: Direct effect [D3 = 0.452]                     -0.063464  0.003219
#> indirect_1_3@1: M1 -> M3 [D3 = 0.452]                -0.000838  0.000719
#> indirect_2@1: M2 [D3 = 0.452]                        -0.008152  0.003015
#> indirect_3@1: M3 [D3 = 0.452]                        -0.005150  0.004257
#> total_indirect@1: Total indirect effect [D3 = 0.452] -0.009945  0.004960
#> total@1: Total effect [D3 = 0.452]                   -0.065383  0.002083
#> Inference draws: 20000 retained of 20000 ; 0 excluded before effect extraction.
#> Use as.data.frame() for the full-precision table and metadata.
#> Post-fit admissibility: 1 of 1 ; excluded rows: 0

print() reports the conditional effects. summary() also shows the conditional path coefficients:

summary(custom_moderated)
#> wsMed statistical summary
#> Model: UD | Missing: DE | Observations: 100 
#> Effects:
#> wsMed all | raw scale | mc 
#> 95% percentile intervals; 20000 joint draws; SE from effect draws
#> Condition contrast: C2 - C1 
#> Moderator: D3 in raw units; centering reference = 0.452271 
#>                                                       Estimate Std. Error
#> cp@1: Direct effect [D3 = 0.452]                     -0.029850   0.016991
#> indirect_1_3@1: M1 -> M3 [D3 = 0.452]                -0.000022   0.000367
#> indirect_2@1: M2 [D3 = 0.452]                        -0.001400   0.002744
#> indirect_3@1: M3 [D3 = 0.452]                        -0.000183   0.002168
#> total_indirect@1: Total indirect effect [D3 = 0.452] -0.001605   0.003581
#> total@1: Total effect [D3 = 0.452]                   -0.031455   0.017152
#>                                                       CI lower  CI upper
#> cp@1: Direct effect [D3 = 0.452]                     -0.063464  0.003219
#> indirect_1_3@1: M1 -> M3 [D3 = 0.452]                -0.000838  0.000719
#> indirect_2@1: M2 [D3 = 0.452]                        -0.008152  0.003015
#> indirect_3@1: M3 [D3 = 0.452]                        -0.005150  0.004257
#> total_indirect@1: Total indirect effect [D3 = 0.452] -0.009945  0.004960
#> total@1: Total effect [D3 = 0.452]                   -0.065383  0.002083
#> Inference draws: 20000 retained of 20000 ; 0 excluded before effect extraction.
#> Use as.data.frame() for the full-precision table and metadata.
#> 
#> Path coefficients:
#>                             Estimate Std. Error CI lower CI upper
#> cp@1: cp [D3 = 0.452]       -0.02985    0.01699 -0.06346  0.00322
#> b2@1: b2 [D3 = 0.452]       -0.09668    0.09286 -0.27706  0.08467
#> d2@1: d2 [D3 = 0.452]       -0.00641    0.08965 -0.18213  0.16843
#> b3@1: b3 [D3 = 0.452]       -0.01370    0.10379 -0.21420  0.19224
#> d3@1: d3 [D3 = 0.452]       -0.15680    0.10278 -0.35544  0.04381
#> a1@1: a1 [D3 = 0.452]       -0.02712    0.01728 -0.06113  0.00670
#> a2@1: a2 [D3 = 0.452]        0.01448    0.01791 -0.02029  0.04996
#> a3@1: a3 [D3 = 0.452]        0.01336    0.01623 -0.01834  0.04508
#> b_1_3@1: b_1_3 [D3 = 0.452] -0.05915    0.09216 -0.24131  0.12014
#> d_1_3@1: d_1_3 [D3 = 0.452] -0.05761    0.09046 -0.23376  0.11900
#> Post-fit admissibility: 1 of 1 ; excluded rows: 0 
#> Tables: $effects, $coefficients; analysis settings: $info.

For this continuous moderator, the default tables evaluate effects at the centering reference shown in the output. The default plot instead shows the indirect effects across the observed moderator range:

plot(custom_moderated)

plot of chunk custom-moderation-plot

Each indirect effect uses all coefficients along its route, evaluated at the same moderator value. For example, indirect_1_3 combines the condition-to-M1, M1-to-M3, and M3-to-outcome coefficients. The total indirect effect includes all routes in custom_paths, including those without a selected b-path interaction. A model that includes moderation need not show evidence that an effect varies; interpret the estimates and their uncertainty.

Choose moderator values

To request particular values, add at when reporting results. The following uses the 25th, 50th, and 75th percentiles of D3 as illustrative probes:

probe_values <- quantile(
  example_data$D3,
  probs = c(0.25, 0.50, 0.75)
)
summary(
  custom_moderated,
  type = "indirect",
  at = list(D3 = probe_values)
)
#> wsMed statistical summary
#> Model: UD | Missing: DE | Observations: 100 
#> Effects:
#> wsMed indirect | raw scale | mc 
#> 95% percentile intervals; 20000 joint draws; SE from effect draws
#> Condition contrast: C2 - C1 
#> Moderator: D3 in raw units; centering reference = 0.452271 
#>                                        Estimate Std. Error  CI lower  CI upper
#> indirect_1_3@1: M1 -> M3 [D3 = 0.288] -7.61e-05   7.42e-04 -1.79e-03  1.41e-03
#> indirect_2@1: M2 [D3 = 0.288]         -2.31e-03   3.68e-03 -1.14e-02  3.47e-03
#> indirect_3@1: M3 [D3 = 0.288]         -4.06e-04   3.35e-03 -7.90e-03  6.61e-03
#> indirect_1_3@2: M1 -> M3 [D3 = 0.462] -1.99e-05   3.62e-04 -8.21e-04  7.13e-04
#> indirect_2@2: M2 [D3 = 0.462]         -1.35e-03   2.72e-03 -8.05e-03  3.09e-03
#> indirect_3@2: M3 [D3 = 0.462]         -1.73e-04   2.16e-03 -5.11e-03  4.25e-03
#> indirect_1_3@3: M1 -> M3 [D3 = 0.605] -4.42e-08   4.89e-04 -1.01e-03  1.05e-03
#> indirect_2@3: M2 [D3 = 0.605]         -5.60e-04   2.92e-03 -7.43e-03  5.28e-03
#> indirect_3@3: M3 [D3 = 0.605]         -5.49e-05   2.79e-03 -6.26e-03  5.93e-03
#> Inference draws: 20000 retained of 20000 ; 0 excluded before effect extraction.
#> Use as.data.frame() for the full-precision table and metadata.
#> 
#> Path coefficients:
#>                              Estimate Std. Error  CI lower  CI upper
#> cp@1: cp [D3 = 0.288]       -0.025502   0.021699 -0.068032  0.017184
#> b2@1: b2 [D3 = 0.288]       -0.096683   0.092856 -0.277064  0.084672
#> d2@1: d2 [D3 = 0.288]       -0.006408   0.089646 -0.182134  0.168425
#> b3@1: b3 [D3 = 0.288]       -0.020509   0.118303 -0.248059  0.213903
#> d3@1: d3 [D3 = 0.288]       -0.156803   0.102775 -0.355442  0.043808
#> a1@1: a1 [D3 = 0.288]       -0.030228   0.021483 -0.072126  0.012219
#> a2@1: a2 [D3 = 0.288]        0.023870   0.022105 -0.019499  0.067463
#> a3@1: a3 [D3 = 0.288]        0.019788   0.020417 -0.019758  0.059997
#> b_1_3@1: b_1_3 [D3 = 0.288] -0.122735   0.115591 -0.349926  0.102452
#> d_1_3@1: d_1_3 [D3 = 0.288] -0.057606   0.090456 -0.233756  0.118996
#> cp@2: cp [D3 = 0.462]       -0.030101   0.016985 -0.063845  0.002871
#> b2@2: b2 [D3 = 0.462]       -0.096683   0.092856 -0.277064  0.084672
#> d2@2: d2 [D3 = 0.462]       -0.006408   0.089646 -0.182134  0.168425
#> b3@2: b3 [D3 = 0.462]       -0.013311   0.104543 -0.215033  0.194371
#> d3@2: d3 [D3 = 0.462]       -0.156803   0.102775 -0.355442  0.043808
#> a1@2: a1 [D3 = 0.462]       -0.026945   0.017289 -0.060974  0.006947
#> a2@2: a2 [D3 = 0.462]        0.013941   0.017923 -0.020877  0.049342
#> a3@2: a3 [D3 = 0.462]        0.012991   0.016235 -0.018635  0.044618
#> b_1_3@2: b_1_3 [D3 = 0.462] -0.055471   0.092270 -0.237402  0.124112
#> d_1_3@2: d_1_3 [D3 = 0.462] -0.057606   0.090456 -0.233756  0.118996
#> cp@3: cp [D3 = 0.605]       -0.033877   0.020525 -0.074733  0.005715
#> b2@3: b2 [D3 = 0.605]       -0.096683   0.092856 -0.277064  0.084672
#> d2@3: d2 [D3 = 0.605]       -0.006408   0.089646 -0.182134  0.168425
#> b3@3: b3 [D3 = 0.605]       -0.007402   0.133446 -0.271143  0.254797
#> d3@3: d3 [D3 = 0.605]       -0.156803   0.102775 -0.355442  0.043808
#> a1@3: a1 [D3 = 0.605]       -0.024249   0.020896 -0.065307  0.016484
#> a2@3: a2 [D3 = 0.605]        0.005790   0.021547 -0.035794  0.048020
#> a3@3: a3 [D3 = 0.605]        0.007411   0.019648 -0.031067  0.045667
#> b_1_3@3: b_1_3 [D3 = 0.605] -0.000246   0.113022 -0.218961  0.222366
#> d_1_3@3: d_1_3 [D3 = 0.605] -0.057606   0.090456 -0.233756  0.118996
#> Post-fit admissibility: 1 of 1 ; excluded rows: 0 
#> Tables: $effects, $coefficients; analysis settings: $info.

The name D3 in at must match the column supplied to W. Supply values on the original scale, even though wsMed centers continuous moderators internally. Choose values that are meaningful for your study and supported by the data. Changing at reuses the fitted model and inference draws. See Read and export results for selecting and comparing effects.

Use a categorical moderator

The same paths and MP specification works with a binary or multicategory moderator. Store it as an unordered factor with the desired reference group as its first level, then supply its column name through W; for example, use W = "Group" if that factor is named Group. The default tables report effects for all observed groups, and plot() displays group-specific effects with CIs. The categorical predictor tutorial shows how to declare the factor and compare groups.

Standardized moderated custom models

After the default analysis, request standardized results with:

summary(
  custom_moderated,
  standardized = TRUE
)
#> wsMed statistical summary
#> Model: UD | Missing: DE | Observations: 100 
#> Effects:
#> wsMed all | marginal scale | mc 
#> 95% percentile intervals; 20000 joint draws; SE from effect draws
#> Condition contrast: C2 - C1 
#> Moderator: D3 in raw units; centering reference = 0.452271 
#> Standardization: effect / marginal model-implied SD(D2 - D1); condition contrast is unscaled.
#> Plug-in marginal outcome-difference SD: 0.169551 
#> The denominator is not a moderator-specific conditional SD.
#>                                                      Estimate Std. Error
#> cp@1: Direct effect [D3 = 0.452]                     -0.17605    0.09921
#> indirect_1_3@1: M1 -> M3 [D3 = 0.452]                -0.00013    0.00210
#> indirect_2@1: M2 [D3 = 0.452]                        -0.00826    0.01566
#> indirect_3@1: M3 [D3 = 0.452]                        -0.00108    0.01242
#> total_indirect@1: Total indirect effect [D3 = 0.452] -0.00947    0.02047
#> total@1: Total effect [D3 = 0.452]                   -0.18552    0.10010
#>                                                      CI lower CI upper
#> cp@1: Direct effect [D3 = 0.452]                     -0.37298  0.01861
#> indirect_1_3@1: M1 -> M3 [D3 = 0.452]                -0.00481  0.00410
#> indirect_2@1: M2 [D3 = 0.452]                        -0.04678  0.01749
#> indirect_3@1: M3 [D3 = 0.452]                        -0.02938  0.02437
#> total_indirect@1: Total indirect effect [D3 = 0.452] -0.05665  0.02834
#> total@1: Total effect [D3 = 0.452]                   -0.38315  0.01184
#> Inference draws: 20000 retained of 20000 ; 0 excluded before effect extraction.
#> Use as.data.frame() for the full-precision table and metadata.
#> 
#> Path coefficients:
#>                             Estimate Std. Error CI lower CI upper
#> cp@1: cp [D3 = 0.452]       -0.17605    0.09921 -0.37298  0.01861
#> b2@1: b2 [D3 = 0.452]       -0.10242    0.09570 -0.28644  0.08721
#> d2@1: d2 [D3 = 0.452]       -0.00762    0.10379 -0.21175  0.19580
#> b3@1: b3 [D3 = 0.452]       -0.01312    0.09834 -0.20399  0.18269
#> d3@1: d3 [D3 = 0.452]       -0.16174    0.10200 -0.35126  0.04594
#> a1@1: a1 [D3 = 0.452]       -0.15463    0.09977 -0.35401  0.03847
#> a2@1: a2 [D3 = 0.452]        0.08064    0.10039 -0.11299  0.28036
#> a3@1: a3 [D3 = 0.452]        0.08230    0.09945 -0.11224  0.27691
#> b_1_3@1: b_1_3 [D3 = 0.452] -0.06390    0.09788 -0.25431  0.12881
#> d_1_3@1: d_1_3 [D3 = 0.452] -0.06531    0.10064 -0.25939  0.13360
#> Post-fit admissibility: 1 of 1 ; excluded rows: 0 
#> Tables: $effects, $coefficients; analysis settings: $info.

Alternatively, include standardized = TRUE in the original wsMed() call to make standardized reporting the default. at still uses raw moderator values. See standardized effects for the scaling rules and CIs.

Optional: inspect the generated equations

The call to wsMed() performs data preparation and model generation for you. Use printGM(result_custom) to inspect the fitted model specification. If you want to generate the equations without fitting, the lower-level functions below provide that option.

PrepareData() converts the raw condition-specific variables into mediator difference and centered level components and an outcome difference score.

prepared_data <- PrepareData(
  data = example_data,
  M_C1 = c("A1", "B1", "C1"),
  M_C2 = c("A2", "B2", "C2"),
  Y_C1 = "D1",
  Y_C2 = "D2"
)

names(prepared_data)
#> [1] "Ydiff"  "M1diff" "M2diff" "M3diff" "M1avg"  "M2avg"  "M3avg"

For mediator $M_i$ , the two components are

$M_{i,\mathrm{diff}} = M_{i2} - M_{i1}$

and

$M_{i,\mathrm{avg}} = \widetilde M_{i,\mathrm{avg}}-\overline{\widetilde M}_{i,\mathrm{avg}}, \qquad \widetilde M_{i,\mathrm{avg}}=\frac{M_{i1} + M_{i2}}{2},$

where the bar is the sample mean used by PrepareData() to center the average component. The difference component is not centered.

The corresponding model syntax is generated as follows:

custom_model <- GenerateModelCustom(
  prepared_data = prepared_data,
  paths = custom_paths
)

cat(custom_model)
#> Ydiff ~ cp*1 + b2*M2diff + d2*M2avg + b3*M3diff + d3*M3avg
#> M1diff ~ a1*1
#> M2diff ~ a2*1
#> M3diff ~ a3*1 + b_1_3*M1diff + d_1_3*M1avg
#> indirect_1_3 := a1 * b_1_3 * b3
#> indirect_2 := a2 * b2
#> indirect_3 := a3 * b3
#> total_indirect := indirect_1_3 + indirect_2 + indirect_3
#> total_effect := cp + total_indirect

For every specified structural edge, the generated regression includes both the difference and level components of the predictor mediator. Thus, "M1 -> M3" contributes

b_1_3*M1diff + d_1_3*M1avg

to the regression equation for M3diff.

Only specified structural paths are added. Because "M1 -> Y" is absent from custom_paths, M1diff and M1avg do not appear directly in the regression equation for Ydiff.