MMM WT 2023/24: Exercise 8

Author

Affiliation

Susanne Adler

Institute for Marketing, Ludwig-Maximilians-University Munich

Set-up

Load relevant packages.

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library(seminr)
library(dplyr)
library(ggplot2)

data <- openxlsx::read.xlsx("MMM_influencer_data.xlsx")

(If necessary) do all relevant steps from the last exercise.

• transform WTP to a numeric variable
• set up the model (mm and sm)
• estimate the model (model)
• summarize the estimated model (model_sum)
• bootstrap and summarize the model (model_boot and model_boot_sum)

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data$WP01_01_num <- gsub(pattern = ",", replacement = ".", data$WP01_01) %>% 
  as.numeric()

# Model set up

## Measurement model

mm <- constructs(
  composite("SIC", multi_items("SC02_0", 1:7)),
  composite("PL", multi_items("PL01_0", c(1, 4, 6, 7))),
  composite("PQ", multi_items("PQ01_0", 1:4)),
  composite("PI", multi_items("PI01_0", c(1, 2, 4, 5, 6))),
  composite("WTP", single_item("WP01_01_num"))
  )

## Structural model

sm <- relationships(
  paths(from = "SIC", to = c("PL", "PQ", "PI")),
  paths(from = "PL", to = "PI"),
  paths(from = "PQ", to = "PI"),
  paths(from = "PI", to = "WTP"))

# Estimate the model

model <- estimate_pls(data = data,
                      measurement_model = mm,
                      structural_model  = sm,
                      inner_weights = path_weighting)

Generating the seminr model

All 223 observations are valid.

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# Summarize the model
model_sum <- summary(model)

# Bootstrap

model_boot <- bootstrap_model(
  seminr_model = model,
  nboot = 1000, # number of bootstrap iterations
  cores = parallel::detectCores(), # use all cores
  seed = 1001)

Bootstrapping model using seminr...

SEMinR Model successfully bootstrapped

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model_boot_sum <- summary(model_boot)

Mediation

reminder: indirect paths

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model_sum$total_indirect_effects

      SIC    PL    PQ    PI   WTP
SIC 0.000 0.000 0.000 0.355 0.176
PL  0.000 0.000 0.000 0.000 0.288
PQ  0.000 0.000 0.000 0.000 0.064
PI  0.000 0.000 0.000 0.000 0.000
WTP 0.000 0.000 0.000 0.000 0.000

test specific indirect paths (SIC -> PL -> PI and SIC -> PQ -> PI)

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specific_effect_significance(model_boot, 
                             from = "SIC", 
                             through = "PL", 
                             to = "PI", 
                             alpha = 0.05)

 Original Est. Bootstrap Mean   Bootstrap SD        T Stat.        2.5% CI 
    0.29694860     0.29837003     0.03927164     7.56140087     0.22694566 
      97.5% CI 
    0.37820012 

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specific_effect_significance(model_boot, 
                             from = "SIC", 
                             through = "PQ", 
                             to = "PI", 
                             alpha = 0.05)

 Original Est. Bootstrap Mean   Bootstrap SD        T Stat.        2.5% CI 
    0.05778152     0.05795351     0.02399739     2.40782492     0.01172313 
      97.5% CI 
    0.10580581 

Show an interpretation

# SIC -> PL -> PI: indirect effect positive and significant (indirect effect = 0.297, t = 7.56, CI= [0.227;0.378])

# SIC -> PQ -> PI: indirect effect positive and significant (indirect effect = 0.058, t = 2.41, CI= [0.011;0.106])

# indirect effect for SIC -> PL -> PI larger than for SIC -> PQ -> PI.

Recap: bootstrapped direct path significance

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model_boot_sum$bootstrapped_paths

            Original Est. Bootstrap Mean Bootstrap SD T Stat. 2.5% CI 97.5% CI
SIC  ->  PL         0.454          0.457        0.050   9.146   0.356    0.548
SIC  ->  PQ         0.398          0.401        0.053   7.468   0.292    0.501
SIC  ->  PI         0.044          0.043        0.058   0.761  -0.069    0.155
PL  ->  PI          0.654          0.654        0.057  11.568   0.540    0.771
PQ  ->  PI          0.145          0.144        0.056   2.587   0.030    0.244
PI  ->  WTP         0.441          0.445        0.051   8.715   0.344    0.540

Mediation type

Show an interpretation

# p1*p2 are significant (see above)
# p3 is n.s. with SIC -> PI (path coefficient = 0.044, t = 0.761, CI95% = [-0.069; 0.155])

## --> indirect only mediation

If p.3. would have been significant

Evaluate sign p1 * p2 * p3

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model_sum$paths["SIC", "PL"] *
  model_sum$paths["PL","PI"] * 
  model_sum$paths["SIC","PI"]

[1] 0.0130681

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model_sum$paths["SIC", "PQ"] *
  model_sum$paths["PQ","PI"] * 
  model_sum$paths["SIC","PI"]

[1] 0.002542846

Show an interpretation

# p3 is positive
# thus both indirect and direct effect point to the same direction
# --> complementary, partial mediation (applies to both mediations)

:::

Moderation

Model set up and estimation

Fist of all:

Influencer group is a character / string variable and cannot be used as a numeric variable in the seminr model.

Therefore, we have to specify influencer group as a numeric (0/1) variable to use it in the model.

To specify:

• convert the InfluencerGroup (Emma or Franklin) variable to a numeric variable (1st to a factor then to a numeric variable)

• use -1 to get 0 and 1 as numeric values for the InfluencerGroup_num variable.

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data$InfluencerGroup_num <- as.numeric(as.factor(data$InfluencerGroup))-1

table(data$InfluencerGroup_num, data$InfluencerGroup)

   
    Emma Franklin
  0  107        0
  1    0      116

Model set up

To do a moderation analysis, we have to set up the model again specifying the moderator

• as an interaction term in the measurement model and

• as a part of the structural model

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# Measurement model

mm_moderation <- constructs(
  composite("SIC", multi_items("SC02_0", 1:7)),
  composite("PL", multi_items("PL01_0", c(1, 4, 6, 7))),
  composite("PQ", multi_items("PQ01_0", 1:4)),
  composite("PI", multi_items("PI01_0", c(1, 2, 4, 5, 6))),
  composite("WTP", single_item("WP01_01_num")),
  composite("Influencer", single_item("InfluencerGroup_num")),
  interaction_term(iv = "SIC", moderator = "Influencer", method = two_stage)
)

# Structural model

sm_moderation <- relationships(
  paths(from = c("SIC", "Influencer", "SIC*Influencer"), to = c("PL", "PQ", "PI")),
  paths(from = "PL", to = "PI"),
  paths(from = "PQ", to = "PI"),
  paths(from = "PI", to = "WTP"))

Estimate and summarize the model

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model_moderation <- estimate_pls(data = data,
                      measurement_model = mm_moderation,
                      structural_model  = sm_moderation,
                      inner_weights = path_weighting)

Generating the seminr model

All 223 observations are valid.

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model_moderation_sum <- summary(model_moderation)

model_moderation_boot <- bootstrap_model(
  seminr_model = model_moderation,
  nboot = 1000, # number of bootstrap iterations
  cores = parallel::detectCores(), # use all cores
  seed = 1001)

Bootstrapping model using seminr...

SEMinR Model successfully bootstrapped

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model_moderation_boot_sum <- summary(model_moderation_boot)

Assess moderating paths

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model_moderation_boot_sum$bootstrapped_paths

                       Original Est. Bootstrap Mean Bootstrap SD T Stat.
SIC  ->  PL                    0.434          0.436        0.054   8.060
SIC  ->  PQ                    0.375          0.378        0.056   6.633
SIC  ->  PI                    0.043          0.042        0.057   0.743
Influencer  ->  PL            -0.239         -0.239        0.060  -3.996
Influencer  ->  PQ            -0.335         -0.335        0.054  -6.228
Influencer  ->  PI            -0.021         -0.027        0.047  -0.457
SIC*Influencer  ->  PL         0.068          0.065        0.055   1.253
SIC*Influencer  ->  PQ         0.146          0.144        0.060   2.438
SIC*Influencer  ->  PI        -0.031         -0.029        0.049  -0.620
PL  ->  PI                     0.653          0.654        0.058  11.241
PQ  ->  PI                     0.140          0.136        0.060   2.333
PI  ->  WTP                    0.441          0.445        0.051   8.707
                       2.5% CI 97.5% CI
SIC  ->  PL              0.321    0.536
SIC  ->  PQ              0.264    0.481
SIC  ->  PI             -0.071    0.150
Influencer  ->  PL      -0.353   -0.127
Influencer  ->  PQ      -0.437   -0.231
Influencer  ->  PI      -0.119    0.066
SIC*Influencer  ->  PL  -0.039    0.170
SIC*Influencer  ->  PQ   0.012    0.256
SIC*Influencer  ->  PI  -0.125    0.066
PL  ->  PI               0.534    0.771
PQ  ->  PI               0.015    0.247
PI  ->  WTP              0.344    0.540

Show an interpretation

# Focusing on the moderating effects (i.e., SIC*Influencer)
## There is no significant effect of SIC*Influencer on PL or PI. Specifically,
### SIC*Influencer -> PL (path coeffiecient = 0.068, t =  1.253, CI95% = [-0.039, 0.170]
### SIC*Influencer -> PI (path coeffiecient = -0.031, t = -0.620, CI95% = [-0.125, 0.066]

## However, we find a significant interaction effect for SIC*Influencer on PQ
### path coefficient = 0.146, t = 2.438, CI 95% = [0.012, 0.256]
## the path coefficient is positive which means that the relationship between PQ and SIC is higher for higher values of the moderator --> relationship is higher for Franklin (InfluencerGroup_num = 1) than for Emma (InfluencerGroup_num = 0).

Plot interaction effects

To plot the interaction effect, we extract the latent variable scores and plot the relationship for SIC to each variable (PQ, PI, PL) for both influencer groups.

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data$PQ_LV <- model_moderation$construct_scores[, "PQ"]
data$PL_LV <- model_moderation$construct_scores[, "PL"]
data$PI_LV <- model_moderation$construct_scores[, "PI"]
data$SIC_LV <- model_moderation$construct_scores[, "SIC"]

PQ

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ggplot(data,
       aes(SIC_LV, PQ_LV, color = InfluencerGroup)) +
  geom_count() + # use to get colored points per group
  geom_smooth(method = "lm") + # use to get the regression line
  theme_minimal()

`geom_smooth()` using formula = 'y ~ x'

PL

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ggplot(data,
       aes(SIC_LV, PL_LV, color = InfluencerGroup)) +
  geom_count() +
  geom_smooth(method = "lm") +
  theme_minimal()

`geom_smooth()` using formula = 'y ~ x'

PI

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ggplot(data,
       aes(SIC_LV, PI_LV, color = InfluencerGroup)) +
  geom_count() +
  geom_smooth(method = "lm") +
  theme_minimal()

`geom_smooth()` using formula = 'y ~ x'