This week

• Techniques
  • Path analysis (esp. path mediation models)
• Functions
  • sem( ) from lavaan
• Reading
  • lavaan tutorial: http://lavaan.ugent.be/tutorial/tutorial.pdf (section 13)
  • Path Analysis chapter (on Learn)

Learning outcomes

• Know how to specify, estimate, and interpret path analysis models in R
• Have a sense of the range of different models that can be fit using path analysis
• Know how to test, interpret and report path mediation models in particular

What is path analysis?

• Links several regression models together
• Tests the set of regression models as a whole
• Useful for situations where there are multiple outcome variables in sequence or parallel
• Models the relations between observed variables (i.e., does not involve latent variables)
• Common example: path mediation model

Mediation

• Is when a predictor X, has an effect on an outcome Y, via a mediating variable M
• The mediator transmits the effect of X to Y
• Examples of mediation hypotheses:
  • Conscientiousness (X) affects health (Y) via health behaviours (M)
  • Conduct problems (X) increase the risk of depression (Y) via peer problems (M)
  • Attitudes to smoking (X) predict intentions to smoke (M) which in turn predicts smoking behaviour (Y)
  • An intervention (X) to reduce youth crime (Y) works by increasing youth self-contol (M)

Visualising a mediation model

• In a SEM diagram we can represent mediation as:

Mediation… not to be confused with moderation

• Mediation is commonly confused with moderation
• Moderation is when a moderator z modifies the effect of X on Y
  • e.g., the effect of X on Y is stronger at higher levels of Z
• Also known as an interaction between X and Z
• Examples of moderation could be:
  • An intervention (X) works better to reduce bullying (Y) at older ages (Z) of school pupil
  • The relation between stress (X) and depression (Y) is lower for those scoring higher on spirituality (Z)

Direct and indirect effects in mediation

• We seldom hypothesise that a mediator completely explains the relation between X and Y
• More commonly, we expect both indirect effects and direct effects of X on Y
  • The indirect effects of X on Y are those transmitted via the mediator
  • The direct effect of X on Y is the remaining effect of X on Y

Visualing direct and indirect effects in mediation

Testing mediation

• Traditionally, mediation was tested using a series of separate regression models:
  1. Y~X
     
  2. Y~X+M
     
  3. M~X

Traditional methods of testing mediation

• The three regression models:

  1. Y~X
     
  2. Y~X+M
     
  3. M~X

• Model 1 estimates the overall effect of X on Y

• Model 2 estimates the partial effects of X and M on Y

• Model 3 estimates the effect of X on M

• If the following conditions were met, mediation was assumed to hold:

  • The effect of X on Y (eq.1) is significant
  • The effect of X on M (eq.3) is significant
  • The effect of X on Y becomes reduced when M is added into the model (eq.2)

Limitations of traditional methods of testing mediation

• Low power
• Very cumbersome for multiple mediators, predictors, or outcomes
• You don’t get an estimate of the magnitude of the indirect effect
• Much better way: path mediation model

BREAK 1

• Time for a pause
• Quiz question:
  • Which of these hypotheses is a mediation hypothesis?
    • 1. Vocabulary development in childhood follows a non-linear trajecrtory
    • 2. The effects of conscientiousness on academic achievement are stronger at low levels of cognitive ability
    • 3. Poverty affects child behaviour problems through increasing parental stress
    • 4. Earlier pubertal onset increases the risk of antisocial behaviour only in girls and not boys

WELCOME BACK 1

• Welcome back!
• The answer to the quiz question is…
  • Which of these hypotheses is a mediation hypothesis?
    • 1. Vocabulary development in childhood follows a non-linear trajecrtory
    • 2. The effects of conscientiousness on academic achievement are stronger at low levels of cognitive ability
    • 3. Poverty affects child behaviour problems through increasing parental stress
    • 4. Earlier pubertal onset increases the risk of antisocial behaviour only in girls and not boys

Testing a path mediation model in lavaan

• Specification
  • Create a lavaan syntax object
• Estimation
  • Estimate the model using e.g., maximum likelihood estimation
• Evaluation/interpretation
  • Inspect the model to judge how good it is
  • Interpret the parameter estimates

Example

• Does peer rejection mediate the association between aggression and depression?

The data

library(psych)

## 
## Attaching package: 'psych'

## The following object is masked from 'package:lavaan':
## 
##     cor2cov

describe(agg.data2)

##     vars   n  mean   sd median trimmed  mad   min  max range  skew kurtosis
## Dep    1 500 -0.02 1.06   0.01   -0.03 1.10 -3.10 3.33  6.43  0.06    -0.02
## PR     2 500 -0.04 1.01  -0.06   -0.05 1.01 -3.28 2.80  6.09  0.05     0.03
## Agg    3 500  0.00 0.99  -0.03    0.01 0.91 -3.88 2.56  6.44 -0.18     0.23
##       se
## Dep 0.05
## PR  0.05
## Agg 0.04

#PR = peer rejection, Agg= aggression, Dep= depression

Mediation example

• Does peer rejection mediate the association between aggression and depression?

#Create the model syntax

model1<-'Dep~PR      # Depression predicted by peer rejection
        Dep~Agg     # Depression predicted by aggression (the direct effect)
        PR~Agg      # Peer rejection predicted by aggression'     


#estimate the model 

model1.est<-sem(model1, data=agg.data2)

The model output

summary(model1.est, fit.measures=T)

## lavaan 0.6-5 ended normally after 12 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                          5
##                                                       
##   Number of observations                           500
##                                                       
## Model Test User Model:
##                                                       
##   Test statistic                                 0.000
##   Degrees of freedom                                 0
## 
## Model Test Baseline Model:
## 
##   Test statistic                               221.854
##   Degrees of freedom                                 3
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    1.000
##   Tucker-Lewis Index (TLI)                       1.000
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -1343.461
##   Loglikelihood unrestricted model (H1)      -1343.461
##                                                       
##   Akaike (AIC)                                2696.923
##   Bayesian (BIC)                              2717.996
##   Sample-size adjusted Bayesian (BIC)         2702.125
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.000
##   90 Percent confidence interval - lower         0.000
##   90 Percent confidence interval - upper         0.000
##   P-value RMSEA <= 0.05                             NA
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.000
## 
## Parameter Estimates:
## 
##   Information                                 Expected
##   Information saturated (h1) model          Structured
##   Standard errors                             Standard
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   Dep ~                                               
##     PR                0.319    0.046    6.925    0.000
##     Agg               0.250    0.047    5.294    0.000
##   PR ~                                                
##     Agg               0.444    0.041   10.760    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .Dep               0.887    0.056   15.811    0.000
##    .PR                0.834    0.053   15.811    0.000

Things to note from the model output

• All three regressions paths are statistically significant
• The model is just-identified
  • The degrees of freedom are equal to 0
  • The model fit cannot be tested
  • The model fit statistics (TLI, CFI, RMSEA, SRMR) all suggest perfect fit but this is meaningless

Visualising the model using semPaths()

• We can use semPaths() from the semPlot package to help us visualise the model
  • Shows the parameter estimates within an SEM diagram

library(semPlot)

## Registered S3 methods overwritten by 'huge':
##   method    from   
##   plot.sim  BDgraph
##   print.sim BDgraph

semPaths(model1.est, what='est')

Calculating the indirect effects

• To calculate the indirect effect of X on Y in path mediation, we need to create some new parameters
• The indirect effect of X on Y via M is:
  • \(a*b\)
  • \(a\) = the regression coefficient for M~X
  • \(b\) = the regression coefficient for Y~M

Calculating indirect effects in lavaan

• To calculate the indirect effect of X on Y in lavaan we:
  • Use parameter labels ‘a’ and ‘b’ to label the relevant paths
    • a is for the effect of X on M
    • b is for the effect of M on Y
  • Use the ‘:=’ operator to create a new parameter ‘ind’
    • ‘ind’ represents our indirect effect

  model1<-'Dep~b*PR     # Add b label here     
           Dep~Agg     
           PR~a*Agg     # Add a label here    
           
ind:=a*b                # create a new parameter ind which is the product of a and b'    

Indirect effects in the output

model1.est<-sem(model1, data=agg.data2)
summary(model1.est)

## lavaan 0.6-5 ended normally after 12 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                          5
##                                                       
##   Number of observations                           500
##                                                       
## Model Test User Model:
##                                                       
##   Test statistic                                 0.000
##   Degrees of freedom                                 0
## 
## Parameter Estimates:
## 
##   Information                                 Expected
##   Information saturated (h1) model          Structured
##   Standard errors                             Standard
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   Dep ~                                               
##     PR         (b)    0.319    0.046    6.925    0.000
##     Agg               0.250    0.047    5.294    0.000
##   PR ~                                                
##     Agg        (a)    0.444    0.041   10.760    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .Dep               0.887    0.056   15.811    0.000
##    .PR                0.834    0.053   15.811    0.000
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     ind               0.142    0.024    5.823    0.000

Statistical significance of the indirect effects

• Default method of assessing the statistical significance of indirect effects assume normal sampling distribution
• May not hold for indirect effects which are the product of regression coefficients
• Instead we can use bootstrapping
  • Provides an estimate of the sampling variance of a coefficient based on the actual data
    • as opposed to a theoretical sampling distribution
  • Resamples with replacement repeatedly from the observed data
  • Calculates the sampling variance based on variation of the coefficient across resamples
    • Number of resamples usually between 1000 and 10000
  • Allows 95% confidence intervals (CIs) to be computed
  • If 95% CI includes 0, the indirect effect is not significant at alpha=.05

Bootstapped CIs for indirect effect in lavaan

  model1<-'Dep~b*PR          
           Dep~Agg     
           PR~a*Agg      
ind:=a*b'    

model1.est<-sem(model1, data=agg.data2, se='bootstrap') #we add the argument se='bootstrap'

Output for bootstrapped CIs for an indirect effect in lavaan

summary(model1.est, ci=T) # we add the argument ci=T to see the confidence intervals in the output

## lavaan 0.6-5 ended normally after 12 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                          5
##                                                       
##   Number of observations                           500
##                                                       
## Model Test User Model:
##                                                       
##   Test statistic                                 0.000
##   Degrees of freedom                                 0
## 
## Parameter Estimates:
## 
##   Standard errors                            Bootstrap
##   Number of requested bootstrap draws             1000
##   Number of successful bootstrap draws            1000
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
##   Dep ~                                                                 
##     PR         (b)    0.319    0.044    7.212    0.000    0.235    0.406
##     Agg               0.250    0.045    5.570    0.000    0.159    0.338
##   PR ~                                                                  
##     Agg        (a)    0.444    0.045    9.821    0.000    0.352    0.533
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
##    .Dep               0.887    0.058   15.379    0.000    0.772    0.999
##    .PR                0.834    0.051   16.312    0.000    0.733    0.933
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
##     ind               0.142    0.024    5.922    0.000    0.097    0.193

Total effects in path mediation

• As well as the direct and indirect effect, it is often of interest to know the total effect of X on Y \[Total = Indirect + Direct \]

Total effects in path mediation

\[Total = a*b + c\]

Total effect in lavaan

  model1<-'Dep~b*PR          
           Dep~c*Agg         # we add the label c for our direct effect    
           PR~a*Agg      
ind:=a*b
total:=a*b+c                 # we add a new parameter for the total effect'    

model1.est<-sem(model1, data=agg.data2, se='bootstrap') #we add the argument se='bootstrap'

Total effect in lavaan output

summary(model1.est, ci=T)

## lavaan 0.6-5 ended normally after 12 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                          5
##                                                       
##   Number of observations                           500
##                                                       
## Model Test User Model:
##                                                       
##   Test statistic                                 0.000
##   Degrees of freedom                                 0
## 
## Parameter Estimates:
## 
##   Standard errors                            Bootstrap
##   Number of requested bootstrap draws             1000
##   Number of successful bootstrap draws            1000
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
##   Dep ~                                                                 
##     PR         (b)    0.319    0.045    7.118    0.000    0.228    0.403
##     Agg        (c)    0.250    0.045    5.505    0.000    0.161    0.340
##   PR ~                                                                  
##     Agg        (a)    0.444    0.044   10.012    0.000    0.359    0.529
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
##    .Dep               0.887    0.060   14.704    0.000    0.771    1.008
##    .PR                0.834    0.052   15.987    0.000    0.735    0.938
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
##     ind               0.142    0.024    5.827    0.000    0.095    0.193
##     total             0.392    0.043    9.204    0.000    0.309    0.475

Why code the total effect in lavaan?

• We could have just added up the coefficients for the direct and indirect effects
• By coding it in lavaan, however, we can assess the statistical significance of the total effect
• Useful because sometimes the direct and indirect effects are not individually significant but the total effect is
  • May be especially relevant in cases where there are many mediators of small effect

Interpreting the total, direct, and indirect effect coefficients

• The total effect can be interpreted as the unit increase in Y expected to occur when X increases by one unit
• The indirect effect can be interpreted as the unit increase in Y expected to occur via M when X increases by one unit
• The direct effect can be interpreted as the unit increase in Y expected to occur with a unit increase in X over and above the increase transmitted by M
  • Note: ‘direct’ effect may not actually be direct - it may be acting via other mediators not included in our model

Standardised parameters

• As with CFA models, standardised parameters can be obtained using:

summary(model1.est, ci=T, standardized=T)

## lavaan 0.6-5 ended normally after 12 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                          5
##                                                       
##   Number of observations                           500
##                                                       
## Model Test User Model:
##                                                       
##   Test statistic                                 0.000
##   Degrees of freedom                                 0
## 
## Parameter Estimates:
## 
##   Standard errors                            Bootstrap
##   Number of requested bootstrap draws             1000
##   Number of successful bootstrap draws            1000
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
##   Dep ~                                                                 
##     PR         (b)    0.319    0.045    7.118    0.000    0.228    0.403
##     Agg        (c)    0.250    0.045    5.505    0.000    0.161    0.340
##   PR ~                                                                  
##     Agg        (a)    0.444    0.044   10.012    0.000    0.359    0.529
##    Std.lv  Std.all
##                   
##     0.319    0.306
##     0.250    0.234
##                   
##     0.444    0.434
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
##    .Dep               0.887    0.060   14.704    0.000    0.771    1.008
##    .PR                0.834    0.052   15.987    0.000    0.735    0.938
##    Std.lv  Std.all
##     0.887    0.790
##     0.834    0.812
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
##     ind               0.142    0.024    5.827    0.000    0.095    0.193
##     total             0.392    0.043    9.204    0.000    0.309    0.475
##    Std.lv  Std.all
##     0.142    0.132
##     0.392    0.366

BREAK 2

• Time for a pause
• Quiz question
  • If the effect of X on M is b=.30 and the effect of M on Y is b=.10, what is the indirect effect of X on Y?
    • 1. b=.40
    • 2. b=.03
    • 3. b=.30
    • 4. b=.10

Welcome back 2

• Welcome back!
• The answer to the quiz question is…
• Quiz question
  • If the effect of X on M is b=.30 and the effect of M on Y is b=.10, what is the indirect effect of X on Y?
    • 1. b=.40
    • 2. b=.03
    • 3. b=.30
    • 4. b=.10

Reporting path mediation models

• Methods
  • The model being tested
  • e.g. ‘Y was regressed on both X and M and M was regressed on X’
  • The estimator used (e.g., maximum likelihood estimation)
  • The method used to test the significance of indirect effects (‘bootstrapped 95% confidence intervals’)
• Results
  • Model fit (for over-identified models)
  • The parameter estimates for the path mediation and their statistical significance
    • Can be useful to present these in a SEM diagram
    • The diagrams from R not considered ‘publication quality’ - draw in powerpoint or similar

Reporting path mediation models - example of SEM diagram with results

Note. *=significant at p<.05

• Include the key parameter estimates
• Indicate statistically significant paths (e.g. with an ’*’)
• Include a figure note that explains how statistically significant paths (and at what level) are signified

Reporting path mediation models - the indirect effects

• Results

  • The coefficient for the indirect effect and the bootstrapped 95% confidence intervals
  • Common to also report proportion mediation:

  \[\frac{indirect}{total}\]

• However, important to be aware of limitations:

  • Big proportion mediation possible when total effect is small - makes effect seem more impressive
  • Small proportion mediation even when total effect is big - can underplay importance of effect
  • Should be interpreted in context of total effect

• Tricky interpretation if there are a mix of negative and positive effects involved

Extensions of path mediation models

• We can extend our path mediation model in various ways:
  • Several mediators in sequence or parallel
  • Multiple outcomes
  • Multiple predictors
  • Multiple groups (e.g., comparing direct and indirect effects across males and females)
  • Add covariates to adjust for potential confounders

Example: multiple mediation model

model2<-'Dep~b2*Aca  
         Aca~a2*Agg
         
         Dep~b1*PR
         PR~a1*Agg
         
         Dep~c*Agg  # direct effect

ind1:=a1*b1
ind2:=a2*b2
total=a1*b1+a2*b2+c'

Other path analysis models

• Path mediation models are a common application of path models
• But they are just one example
• Anything that can be expressed in terms of regressions between observed variables can be tested as a path model
• Can include ordinal or binary variables
• Can include moderation
• Other common path analysis models include:
  • Autoregressive models for longitudinal data
  • Cross-lagged panel models for longitudinal data

Other path analysis models - AR

• autoregressive models to examine the stability of a construct over time

##creating a lavaan syntax object for an autoregressive model

Autoregressive<-'AggT3~AggT2
                 AggT2~AggT1'

Other path analysis models - CLPM

• cross-lagged panel models to examine the relations between constructs over time
  • autoregressive paths control for previous levels of each construct
  • cross-lagged paths capture the relations between the two constructs

# creating a lavaan syntax object for a CLPM
CLPM<-'AggT3~AggT2+DepT2
       AggT2~AggT1+DepT1
       DepT3~DepT2+AggT2
       DepT2~DepT1+AggT1'

Other path analysis models - CLPM with mediation

• longitudinal mediation models using a cross-lagged panel model

#creating a lavaan syntax object for a longitudinal mediation model

CLPM.med<-'AggT3~AggT2+DepT2+PRT2  
           AggT2~AggT1+DepT1+PRT1
           DepT3~DepT2+AggT2+b*PRT2   # label the effect of M on Y as b
           DepT2~DepT1+AggT1+PRT1     
           PRT3~PRT2+AggT2+DepT2
           PRT2~PRT1+a*AggT1+DepT1    # label the effect of X on M as a

ind:=a*b'

Making model modifications

• As in CFA models, you may want to make some modifications to your initially hypothesised model
  • non-significant paths that you want to trim
  • include some additional paths not initially included
• Remember that this now moves us into exploratory territory where:
  • Model modifications should be substantively as well as statistically justifiable
  • You must be aware of the possibility that you are capitalising on chance
  • You should aim to replicate the modifications in independent data

Cautions regarding path analysis models

• Assumption that the paths represent causal effects is only an assumption
  • Especially if using cross-sectional data
• The parameters are only accurate if the model is correctly specified

Cautions regarding path analysis models - indistinguishable models

Measurement error in path analysis

• Path analysis models use observed variables
• Assumes no measurement error in these variables
• Path coefficients likely to be attenuated due to unmodelled measurement error
• Structural equation models solve this issue
• They are path analysis models where the paths are between latent rather than observed variables
• …more on this next week

Path analysis summary

• Path analysis can be used to fit sets of regression models
  • Common path analysis model is the path mediation model
  • But very flexible - huge range of models that can be tested
• In R, path analysis can be done using the sem( ) function in lavaan
• Need to be aware that we aren’t testing causality but assuming it