• Are "Get angry easily" and "Get irritated easily" locally independent given Neuroticism?
• Are "Don't talk a lot" and "Find it difficult to approach others" locally independent given Extroversion?
• Are "Am indifferent to the feelings of others" and "Inquire about others' well-being" locally independent given Agreeableness?
• Are "Avoid difficult reading material" and "Will not probe deeply into a subject" locally independent given Openness to Experience?
• Are "Do things in a half-way manner" and "Waste my time" locally independent given Conscientiousness?

• Part 1: Markov Random Fields
  • Gaussian Graphical Model
  • Ising Model
  • LASSO regularization
• Part 2: Residual Interaction Modeling
  • Combining factor analysis and network modeling
  • rim demonstration
• Part 3: Longitudinal analysis
  • Single person: GraphicalVAR
  • Multiple persons: mlVAR

• A network is a set of nodes connected by a set of edges
• Nodes represent variables
• Edges can be directed or undirected and represent interactions
• Color and width indicate the strength and sign of an edge (Epskamp et al. 2012)

• A pairwise Markov Random Field (MRF) is an undirected network
• Two nodes are connected if they are not independent conditional on all other nodes.
• More importantly, two nodes are NOT connected if they are independent conditioned on all nodes:
• \(X_i \!\perp\!\!\!\perp X_j \mid \boldsymbol{X}^{-(i,j)} = \boldsymbol{x}^{-(i,j)} \iff (i,j) \not\in E\)
• A node separates two nodes if it on all paths from one node to another
• Assumption: only pairwise effects
• No equivalent models!
  • Clear saturated model is a fully connected network
• Naturally cyclic!

• \(B\) separates \(A\) and \(C\)
• \(A \!\perp\!\!\!\perp C \mid B\)

• Worrying and fatigue separate Insomnia and Concentration

If this model is the generating model, does:

• \(A\) predict \(B\)?
  • Yes!
• \(B\) predict \(A\)?
  • Yes!
• \(A\) predict \(B\) just as well as \(B\) predict \(A\)?
  • Using linear or logistic regression, yes!

# Generate data (binary):
A <- sample(c(0,1), 10000, replace = TRUE)
B <- 1 * (runif(10000) < ifelse(A==1, 0.8, 0.2))

# Predict A from B (logistic bregression):
AonB <- glm(A ~ B, family = "binomial")
coef(AonB)

## (Intercept)           B 
##   -1.369453    2.757481

# Predict B from A (logistic regression):
BonA <- glm(B ~ A, family = "binomial")
coef(BonA)

## (Intercept)           A 
##   -1.363489    2.757481

• The logistic regression parameters are equal!

• \(A\) predicts \(B\) and \(B\) predicts \(A\)

If this model is the generating model, does:

• \(A\) predict \(C\) or vise versa?
  • Yes, they are correlated
• \(A\) predict \(C\) or vise versa when also taking \(B\) into account?
  • No!
• In a multiple (logistic) regression, \(C\) should not predict \(A\) when \(B\) is also taken as predictor

# Generate data (Gaussian):
A <- rnorm(10000)
B <- A + rnorm(10000)
C <- B + 2*rnorm(10000)

# Predict A from C:
AonC <- lm(A ~ C)
summary(AonC)

## 
## Call:
## lm(formula = A ~ C)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.2389 -0.6162  0.0059  0.6132  3.3662 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.003797   0.009093  -0.418    0.676    
## C            0.170163   0.003742  45.469   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9092 on 9998 degrees of freedom
## Multiple R-squared:  0.1713, Adjusted R-squared:  0.1713 
## F-statistic:  2067 on 1 and 9998 DF,  p-value: < 2.2e-16

# Predict A from B and C:
AonBC <- lm(A ~ B + C)
summary(AonBC)

## 
## Call:
## lm(formula = A ~ B + C)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.66254 -0.47831 -0.00956  0.47492  2.53815 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.005440   0.007056  -0.771    0.441    
## B            0.494626   0.006086  81.269   <2e-16 ***
## C            0.003341   0.003556   0.939    0.348    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7056 on 9997 degrees of freedom
## Multiple R-squared:  0.501,  Adjusted R-squared:  0.5009 
## F-statistic:  5019 on 2 and 9997 DF,  p-value: < 2.2e-16

• \(A\) predicts \(B\) better than \(C\) predicts \(B\)
• The relationship between \(A\) and \(C\) is mediated by \(B\)

• A MRF can not represent the exact implied independence relationship of a collider structure
  • Three edges are needed instead of two
• However, exogenous variables are commonly modeled to be correlated anyway