• Regression based models
  
  

• Moderation
  • Theoretical applications
  • Examples
    
    
• Mediation
  • Theoretical applications
  • Examples

• General linear model \(Y = b_0 + b_1 * X + e\)

where:

• \(Y\) is the dependent variable (DV)
• \(X\) is the independent variable (IV)
• \(b_0\) is the intercept
• \(b_1\) is the slope of the relationship between \(X\) and \(Y\)
  • Referred to as the beta coefficient or just beta
  • Standardize beta coefficient (\(\beta\))
• \(e\) is error

• \(Y_{LENGTH} = 7.4225 + 9.7636 * X_{DOSE}\)

• We can expand this model by including other predictors
  • eg., \(X_1, X_2, ... X_p\)
  • and corresponding betas \(b_1, b_2, ... b_p\)
    
    

• With more than one predictor, we interpret each beta as the effect of \(X_p\) on \(Y\) while controlling for all other variables in the model
  
  

• For this talk, assume all DV's are continuous (unless otherwise stated)

• Third variable analyses
  • An extension of the general linear model
    
    
• Used for testing specific hypotheses about the relationship between predictors and outcomes

• Seminal article discussing statistical properties, application, and nomenclature
  
  

• Moderator
  • "… a moderator is a qualitative (e.g., sex, race,class) or quantitative (e.g., level of reward) variable that affects the direction and/or strength of the relation between an independent or predictor variable and a dependent or criterion variable." (p. 1174)
    
    
• Mediator
  • "… a given variable may be said to function as a mediator to the extent that it accounts for the relation between the predictor and the criterion." (p. 1176)
    
    

• Although these two analyses are often brought up together, they are two separate types of analyses testing different hypotheses
  • Don't mix them up

• Used to explore the heterogeneity of results
  
  

• Answers the question "For whom does this relationship exist?" or “does this relationship exist differently in different groups and, if so, how?”
  • Moderation is often interchangeable with the term "interaction" or "effect modification"
    
    
• More common in medical research/practice than mediation
  • Drug interactions
    
    

• Identifying particular at-risk populations or groups
  
  

• Identifying people for whom an intervention is particularly effective
  
  

• Finding effective combinations of intervention components
  • Multiphase Optimization Strategy (MOST)

• Interaction with categorical predictor

• Continuous predictor interacting with itself (curvilinear relationship)

• For example age

• Continuous predictor interacting with itself (curvilinear relationship)

• The model looks like this \(Y = b_0 + b_1PRED + b_2MOD + b_3PRED*MOD\)
  • For polynomials \(Y = b_0 + b_1PRED + b_2PRED^2\)

• Cohen, Cohen, West & Aiken (2013)
  • Applied multiple regression/correlation analysis for the behavioral sciences

• Used to examine the causal (usually sequential) relationship among variables
  
  

• Answers the question "How does this relationship work?"

• Difficult to display the relationship between data, but we can graphically display the model

• From Shanlee's work

• The model displayed here is actually two equations:
  • \(M = aX\)
  • \(Y = bM + cX\)
    
    
• The mediated effect is \(a * b\)
  • We can get the standard error and p value to test for significance

• Model is tested via Structural Equation Modeling (SEM)

• A variable mediates the relationship between an IV and a DV if the product of \(a\) (the effect of the IV on the mediator) and \(b\) (the effect of the mediator on the DV controlling for the IV) is significant
  • Often if \(a\) is significant and \(b\) is significant this will be true
    
    
• Mediation is often associated with causal inference
  • The model implies that the IV causes the mediator (and the DV) and the mediator causes the DV
    
    
• Only methods can produce causality (not analyses)
  • Temporal precedence
  • Randomization

• Specifying a mediation model is largely done using theory
  
  

• In general, mediation is not an exploratory analysis (but moderation can be)
  • Models that violate temporal precedence can fit the data better
    
    

• "… mediation is not a thoughtless routine exercise that can be reduced down to a series of steps. Rather, it requires a detailed knowledge of the process under investigation and a careful and thoughtful analysis of data." (Kenny, 2008)

• Mackinnon (2008)
  • Introduction to statistical mediation analysis

• Moderation
  • For whom does this relationship exist (and to what degree)
  • Single regression model where two (or more) variables are multiplied together
    
    
• Mediation
  • How does this relationship work
  • Multiple regression models where two (or more) coefficients are multiplied together
    
    

• Common issues
  • Requires large sample sizes
  • Need theory to interpret
  • Parsimonious models are easier to interpret
    
    
• Many technical issues we didn't go over
  • It's important that these issues are addressed
  • Violations of assumptions can lead to erroneous conclusions