In doomlab/learnSTATS: Introduction to Statistics (ANLY 500)

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
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What is Mediation?

• Refers to a situation when the relationship between a predictor variable and outcome variable can be explained by their relationship to a third variable (the mediator).

knitr::include_graphics("pictures/medmod/1.png")

Baron & Kenny (1986)

• Mediation is tested through three regression models:

  • Predicting the outcome from the predictor variable.
  • Predicting the mediator from the predictor variable.
  • Predicting the outcome from both the predictor variable and the mediator.

Baron & Kenny (1986)

• Four conditions of mediation:

  • The predictor must significantly predict the outcome variable.*
  • The predictor must significantly predict the mediator.
  • The mediator must significantly predict the outcome variable.
  • The predictor variable must predict the outcome variable less strongly in model 3 than in model 1.

Baron & Kenny (1986): Limitations

• How much of a reduction in the relationship between the predictor and outcome is necessary to infer mediation?
• People tend to look for a change in significance, which can lead to the 'all or nothing' thinking that p-values encourage.

• Two solutions:

  • The Sobel test
  • Bootstrapping the indirect effect and confidence intervals

Sobel Test

• An alternative is to estimate the indirect effect and its significance using the Sobel test (Sobel, 1982).
• If the Sobel test is significant, there is significant mediation
• We are going to use the Aroian Test version of Sobel which corrects for the standard error of the a and b paths.

Bootstrapping

• What is bootstrapping?

  • A procedure where you run an analysis many times to create an average effect and confidence interval of the effect.
  • Bootstrapped datasets are created by randomly picking a sample from your dataset with replacement
  • Think about this like the lottery, if they put the balls back.
  • If the confidence interval of the indirect (mediation) effect does not cross zero, then that implies that mediation has occurred.

Mediation: Data Screening

• Mediation models usually occur in stages - meaning separate models are run with different IV / DV combinations.
• Sometimes people call these "steps", but do not confuse this with hierarchical regression.

• So, what does that do for data screening?

  • Screen both the IV and the Mediator at the same time predicting the DV.
  • This way you are screening the final anticipated stage of the mediation model and are screening all the variables.
  • You can also use the fake regression procedure to do all three variables at once.

Mediation: Example

• Does Facebook use mediate the relationship between previous knowledge and exam scores?
• Implying that the overuse of Facebook prevents people from studying, so they do differently on their exam?
• Important not to have missing data, so you are comparing models that all use the same data points.

library(rio)
master <- import("data/mediation.sav")
str(master)

summary(master)
master <- na.omit(master)

Mediation: c Path

• The c path: X --> Y: $$b = 0.32, t(237) = 3.63, p < .001$$

model1 <- lm(exam ~ previous, data = master)
summary(model1)

Mediation: a Path

• The a path: X --> M: $$b = -0.09, t(237) = -2.17, p = .031$$

model2 <- lm(facebook ~ previous, data = master)
summary(model2)

Mediation: b, c' Path

• Add in the b and c' paths: X + M --> Y
• b Path: $$b = -0.61, t(237) = -4.77, p < .001$$
• c' Path: $$b = 0.26, t(237) = 3.09, p = .002$$

model3 <- lm(exam ~ previous + facebook, data = master)
summary(model3)

Mediation: Interpretation

• Previous experience positively impacts exam scores (c path = 0.32).
• Previous experience negatively impacts Facebook time (a path = -0.09).
• Controlling for previous experience, Facebook time negatively impacts exam scores (b path = -0.61).
• Controlling for Facebook scores, previous experience positively impacts exam scores (c' path = 0.26).
• So, did mediation happen? Is a change from 0.31 to 0.26 important?

Mediation: Sobel Test

• The Aroian Test: $Z = \frac{a \times b}{\sqrt{b^2 \times SD_a^2 + a^2 \times SD_b^2 + SD_a^2 \times SD_b^2}}$

• If the indirect effect is larger than the error, we would conclude that the addition of the M variable changed the c path.

  • Fill in the correct coefficient numbers
  • Note if you use this code provided and fill in X and M in the correct order, you will not need to change the Sobel code.

#aroian sobel
a <- coef(model2)[2]
b <- coef(model3)[3]
SEa <- summary(model2)$coefficients[2,2]
SEb <- summary(model3)$coefficients[3,2]
zscore <- (a*b)/(sqrt((b^2*SEa^2)+(a^2*SEb^2)+(SEa^2*SEb^2)))
zscore
#two tailed test 
pnorm(abs(zscore), lower.tail = F)*2

Mediation: Sobel Test

• Z = 1.93, p = .053
• We would conclude that no mediation had occurred.

Mediation: All Together + Bootstrapping

#devtools::install_github("doomlab/MeMoBootR")
library(MeMoBootR)

#no missing data allowed
med_results <- mediation1(y = "exam",
                          x = "previous", 
                          m = "facebook", 
                          df = master)

Mediation: MeMoBootR

• The MeMoBootR package gives you data screening, each step of the mediation, and the bootstrapping results!
• The data screening does not include accuracy or missing data, so that should be completed first.

head(med_results$datascreening$fulldata)
med_results$datascreening$correl
med_results$datascreening$linearity
med_results$datascreening$normality
med_results$datascreening$homogen

Mediation: MeMoBootR

• For each of our stages of mediation, you can print out the models:

summary(med_results$model1)
summary(med_results$model2)
summary(med_results$model3)

Mediation: MeMoBootR

• Next, you can get the Sobel test results:

med_results$indirect.effect
med_results$z.score
med_results$p.value

Mediation: MeMoBootR

• Last, let's get the bootstrapped results:

• The indirect effect would be reported as: $0.06, 95\% CI[-0.01, 0.12]$

med_results$boot.results
med_results$boot.ci
med_results$diagram

Moderation

• The combined effect of two variables on another is known conceptually as moderation, and in statistical terms as an interaction effect.

• Do violent video games make people aggressive?

  • DV: Aggression

  • IV:

    • Callous unemotional traits
    • Number of hours spent playing video games per week
    • Interaction

Conceptual Moderation Model

• If callous-unemotional traits were a moderator then we're saying that the strength or direction of the relationship between game playing and aggression depends on the strength of callous-unemotional traits.

knitr::include_graphics("pictures/medmod/4.png")

Conceptual Moderation Model

• If the X variable was categorical, you are basically checking if the separate groups (levels) have different slopes for the non-categorical variable.

knitr::include_graphics("pictures/medmod/5.png")

Conceptual Moderation Model

• However, if X is continuous, we have to find a new way to figure out the shape of the interaction.

knitr::include_graphics("pictures/medmod/6.png")

Statistical Moderation Model

knitr::include_graphics("pictures/medmod/7.png")

Centering variables

• The interaction term makes the bs for the main predictors uninterpretable in many situations.
• Centering refers to the process of transforming a variable into deviations around a fixed point.

• Centering solves two problems:

  • Interpretation
  • Multicollinearity

Centering variables

• Easiest way to center to make them useful:

  • Score - Mean for that variable.
  • What does that do?

• Centered variables for main effects have two interpretations

  • Effect of that predictor at the mean value for the sample
  • Average effect of the predictor across a range of scores for the other predictor

Moderation: Simple Slopes

• If the interaction is significant, then you usually ignore the main effects (each variable by itself).
• So what do I do if my interaction is significant? A simple slope analysis
• If variables are categorical, then you are simply looking at the differences across one variable or another

Moderation: Simple Slopes

• For continuous variables, you "create" low, average, and high groups.

  • Low groups are people who are one SD below the mean
  • Average groups are people are at the mean
  • High groups are people who are one SD above the mean
  • However, creating groups is backwards from what you think.

Moderation: Simple Slopes

• We are examining the interaction between hours of video games and unemotional traits to predict aggression
• Think about which variable you want to know the differences in (i.e., low, average, high).
• So at different levels of callousness, we want to examine the relationship between hours of video games and aggression.

master <- import("data/moderation.sav")
str(master)

Moderation: Centering

• Center your variables using scale()

  • Using scale = F turns off dividing by SD, which would create z-scores.
  • You can use z-score centering if you wanted the coefficients to be beta. It is likely easier to interpret the b values though.

#center the X and M variable (NOT THE DV)
#when you create these use the final dataset 
master$zvid = scale(master$Vid_Games, scale = F) #mean center, not z score
master$zcal = scale(master$CaUnTs, scale = F)

Moderation: Running

• For data screening, we would screen the model with the interaction, and all other considerations are the same as regression.

• Run the model:

  • You do not have to create an interaction column, to multiply does that automatically for you.
  • Also includes the main effects of each variable by itself.

#run the model to see if the moderation is significant
#use the z score variables!
#be sure to do X*M so the graph is right 
modmodel <- lm(Aggression ~ zvid*zcal, data = master)

Moderation: Interpretation

• Overall model is significant, $F(3, 438) = 88.46, p < .001, R^2 = .38$
• Interaction effect is significant, $b = 0.03, t(438) = 3.88, p < .001$

summary(modmodel)

Moderation: Simple Slopes

• So, how do we create groups when we have a continuous variable?

  • Most people look at "low" as one SD below the mean and "high" as one SD above the mean.
  • Lots of combinations are possible though.

Moderation: Simple Slopes

• Create the low and high versions of the M variable (callousness in this example)

  • Low created by ADDING a SD
  • High created by SUBTRACTING a SD
  • The rule is that we have to bring them to the middle because we centered so that zero is the middle

#create the low and high z score variables 
master$zcallow <- master$zcal + sd(master$zcal) #bring them up
master$zcalhigh <- master$zcal - sd(master$zcal) #bring them down
summary(master)

Moderation: Simple Slopes

• Then run the model with the new low and high versions of the variable.
• What do I do with that?
• You look at the other variable because the coefficients for M and X*M will not change.

#run the models
#be sure to X*M
modmodellow <- lm(Aggression ~ zvid*zcallow, data = master)
modmodelhigh <- lm(Aggression ~ zvid*zcalhigh, data = master)

Moderation: Simple Slopes

• At low levels of callousness, video game hours is unrelated to aggression, b = -0.09 ...

summary(modmodellow) #low slope

Moderation: Simple Slopes

• At average levels of callousness, hours of video games predicts increasing aggression, b = 0.17 ...

summary(modmodel) #average slope

Moderation: Simple Slopes

• At high levels of callousness, the strength of hours of video games predicting aggression is the strongest, b = 0.43 ...

summary(modmodelhigh) #high slope

Moderation: Simple Slopes

• Significant interactions will create fan effects of slopes that either get stronger/weaker in strength or switch directions.
• Here, we find that increasing levels of callousness leads to increasing effects of video game time on aggression.

Moderation: Graphing

• We can use the ggplot to create a graph of the moderation effect.
• But we are going to manually draw the lines on a scatterplot graph.

library(ggplot2)
cleanup <- theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(), 
                panel.background = element_blank(), 
                axis.line.x = element_line(color = "black"),
                axis.line.y = element_line(color = "black"),
                legend.key = element_rect(fill = "white"),
                text = element_text(size = 15))

modgraph <- ggplot(master, aes(zvid, Aggression))

##change Cal to the new moderator label
##change xlab for the new X label
modgraph + 
  xlab("Centered Video Games") + 
  geom_point(color = "gray") +
  geom_abline(aes(intercept = modmodellow$coefficients[1],
                  slope = modmodellow$coefficients[2], 
                  linetype = "-1SD Cal"), size = 1) +
  geom_abline(aes(intercept = modmodel$coefficients[1],
                  slope = modmodel$coefficients[2], 
                  linetype = "Average Cal"), size = 1) +
  geom_abline(aes(intercept = modmodelhigh$coefficients[1],
                  slope = modmodelhigh$coefficients[2], 
                  linetype = "+1SD Cal"), size = 1) +
  scale_linetype_manual(values = c("dotted", "dashed", "solid"),
                        breaks = c("-1SD Cal", "Average Cal", "+1SD Cal"),
                        name = "Simple Slope") +
  cleanup 

Moderation: MeMoBootR

• We can use the MeMoBootR to complete the entire processing, including data screening for us!
• You would enter the raw variables, as the centering is completed for you.

mod_model <- moderation1(y = "Aggression",
                         x = "Vid_Games",
                         m = "CaUnTs",
                         df = master)

Moderation: MeMoBootR

#data screening
head(mod_model$datascreening$fulldata)
#mod_model$datascreening$correl
#mod_model$datascreening$linearity
#mod_model$datascreening$normality
#mod_model$datascreening$homogen

#models
summary(mod_model$model1)
#summary(mod_model$model1low)
#summary(mod_model$model1high)
mod_model$interpretation

#graphs
mod_model$graphslopes

Summary

• In this lecture, we have covered:

  • Mediation
  • Moderation
  • Special packages for these models
  • Follow ups for these models

doomlab/learnSTATS documentation built on June 9, 2022, 12:54 a.m.