Lecture 5: SEM (Path models)

Lecture 5: SEM (Path models)
Dr Nemanja Vaci
University of Sheffield
2024-03-15
1

Press record2

Intended learning outcomes

Motivate utilisation of path and CFA models; Argue how they connect to other models that we covered at the course.

Calculate number of free parameters and degrees of freedom of the proposed model.

Build a model in R statistical environment, estimate, and interpret the coefficients.

Criticise, modify, compare, and evaluate the fit of the proposed models.

Structural equation modelling (SEM)

General framework that uses various models to test relationships among variables

Other terms: covariance structure analysis, covariance structure modelling, causal modelling

Sewell Wright - "mathematical tool for drawing causal conclusions from a combination of of observational data and theoretical assumptions"

Waves:

Causal modelling through path models
Latent structures - factor analysis
Structural causal models

SEM is a general modelling framework that is composed of measurement model and the structural model.

Judea Pearl - The Causal Foundations of Structural Equation Modeling

Measurement model focuses on the estimation of latent or composite variables
Structural model focuses on the estimation of relations between manifest and/or latent variables in the model (path model)

Terminology:

Manifest variables: observed/collected variables

Latent variables: infered measures - hypothetical constructs

Indicator variables: measures used to infer the latent concepts

Endogenous variables: dependent outcomes

Exogenous variables: predictors

Focus on covariance structure instead of mean

Structural part of the model (path analysis)

Model that test relationship between set of variables, often arranged in some sort of structural form.

A common focus of the path model is the estimation of mediation between X and Y.

First step: Specification of the model

Previous findings show that development of cognitive abilities in people depends on a range of factors in infancy and early childhood. General mental/cognitive abilities (e.g. reading or drawing), varied nutrition, physical exercises, and social engagement have shown to influence the level of cognitive abilities. Based on some of these studies, researchers postulate that social engagement is mediating factor between the behavioural factors and development of cognitive abilities.

Representation of our hypothetical assumptions in the form of the structural equation model

Can model be estimated?

Total Number of the parameters that we can estimate: $\frac{variables*(variables+1)}{2}$

Number of observations

Matrix<-cov(Babies[,c('Nutrition','PhyExer','GMA','SocialBeh','CognitiveAb')])
Matrix[upper.tri(Matrix)]<-NA
knitr::kable(Matrix, format = 'html')

	Nutrition	PhyExer	GMA	SocialBeh	CognitiveAb
Nutrition	45.6689837	NA	NA	NA	NA
PhyExer	-10.1006752	2652.9074	NA	NA	NA
GMA	0.5641485	-249.3049	2478.2889	NA	NA
SocialBeh	-11.6168733	3417.8681	-506.1066	9988.898	NA
CognitiveAb	210.6731970	48916.6339	1254.2100	94358.621	1125746

How many parameters are we estimating (path model)?

How many degrees of freedom do we have without the model?

How many parameters are we estimating (path model)?

How many degrees of freedom do we have without the model?

Number of observations (total number of parameters) = 15
Empty model = variances and covariances
Degrees of freedom (df) = 15 - 8 = 7

Most of the time (CFA model or other software): Degree of freedom for null model = $(\frac{variables*(variables+1)}{2}) - variables$

Matrix<-cov(Babies[,c('Nutrition','PhyExer','GMA','SocialBeh','CognitiveAb')])
Matrix[upper.tri(Matrix)]<-NA
Matrix[lower.tri(Matrix)]<-NA
knitr::kable(Matrix, format = 'html')

	Nutrition	PhyExer	GMA	SocialBeh	CognitiveAb
Nutrition	45.66898	NA	NA	NA	NA
PhyExer	NA	2652.907	NA	NA	NA
GMA	NA	NA	2478.289	NA	NA
SocialBeh	NA	NA	NA	9988.898	NA
CognitiveAb	NA	NA	NA	NA	1125746

How many parameters (our model)?

Free parameters = variances + covariances + regression pathways = 14

Second step: model identificationUnder-indentified: more free parameters than total possible parameters  
 
Just-identified: equal number of free parameters and total possible parameters  

Over-identified: fewer free parameters than total possible parameters  
 
Parameters can either be: free, fixed or constrained  
12

Third step: estimation of the model

modelAbility<-'
SocialBeh~Nutrition+PhyExer+GMA
CognitiveAb~SocialBeh+Nutrition+GMA
'

Third step: estimation of the model

modelAbility<-'
SocialBeh~Nutrition+PhyExer+GMA
CognitiveAb~SocialBeh+Nutrition+GMA
'

fit1<-sem(modelAbility, data=Babies)
summary(fit1)

## lavaan 0.6.15 ended normally after 1 iteration
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                         8
## 
##   Number of observations                           100
## 
## Model Test User Model:
##                                                       
##   Test statistic                               215.236
##   Degrees of freedom                                 1
##   P-value (Chi-square)                           0.000
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Regressions:
##                    Estimate    Std.Err   z-value  P(>|z|)
##   SocialBeh ~                                            
##     Nutrition           0.030     1.105    0.027    0.978
##     PhyExer             1.281     0.146    8.796    0.000
##     GMA                -0.075     0.151   -0.500    0.617
##   CognitiveAb ~                                          
##     SocialBeh           9.579     0.469   20.428    0.000
##     Nutrition           7.019     6.899    1.017    0.309
##     GMA                 2.461     0.941    2.614    0.009
## 
## Variances:
##                    Estimate    Std.Err   z-value  P(>|z|)
##    .SocialBeh        5515.809   780.053    7.071    0.000
##    .CognitiveAb    215129.001 30423.835    7.071    0.000

Step four: model evaluation

Chi-square test: measure of how well model-implied covariance matrix fits data covariance

We would prefer not to reject the null hypothesis in this case

Assumptions:
Multivariate normality
N is sufficiently large (150+)
Parameters are not at boundary or invalid (e.g. variance of zero)

With the large samples it is sensitive to small misfits
Nonormality induces bias

Other fit indices

summary(fit1, fit.measures=TRUE)

## lavaan 0.6.15 ended normally after 1 iteration
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                         8
## 
##   Number of observations                           100
## 
## Model Test User Model:
##                                                       
##   Test statistic                               215.236
##   Degrees of freedom                                 1
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                               438.108
##   Degrees of freedom                                 7
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.503
##   Tucker-Lewis Index (TLI)                      -2.479
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -1328.506
##   Loglikelihood unrestricted model (H1)      -1220.888
##                                                       
##   Akaike (AIC)                                2673.012
##   Bayesian (BIC)                              2693.853
##   Sample-size adjusted Bayesian (SABIC)       2668.587
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          1.464
##   90 Percent confidence interval - lower         1.303
##   90 Percent confidence interval - upper         1.632
##   P-value H_0: RMSEA <= 0.050                    0.000
##   P-value H_0: RMSEA >= 0.080                    1.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.080
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Regressions:
##                    Estimate    Std.Err   z-value  P(>|z|)
##   SocialBeh ~                                            
##     Nutrition           0.030     1.105    0.027    0.978
##     PhyExer             1.281     0.146    8.796    0.000
##     GMA                -0.075     0.151   -0.500    0.617
##   CognitiveAb ~                                          
##     SocialBeh           9.579     0.469   20.428    0.000
##     Nutrition           7.019     6.899    1.017    0.309
##     GMA                 2.461     0.941    2.614    0.009
## 
## Variances:
##                    Estimate    Std.Err   z-value  P(>|z|)
##    .SocialBeh        5515.809   780.053    7.071    0.000
##    .CognitiveAb    215129.001 30423.835    7.071    0.000

Other fit indices

TLI: fit of .95 indicates that the fitted model improves the fit by 95% relative to the null mode, works OK with smaller sample sizes

CFI: Same as TLI, but not very sensitive to sample size

RMSEA: difference between the residuals of the sample covariance matrix and hypothesized model. If we have different scales it is hard to interpret, then we can check standardised root mean square residual (SRMR)

Model modification

Add/take out theoretical pathways:

modelAbility2<-'
SocialBeh~Nutrition+PhyExer+GMA
CognitiveAb~SocialBeh+Nutrition+GMA+PhyExer
'
fit2<-sem(modelAbility2, data=Babies)
summary(fit2, fit.measures=TRUE)

## lavaan 0.6.15 ended normally after 1 iteration
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                         9
## 
##   Number of observations                           100
## 
## Model Test User Model:
##                                                       
##   Test statistic                                 0.000
##   Degrees of freedom                                 0
## 
## Model Test Baseline Model:
## 
##   Test statistic                               438.108
##   Degrees of freedom                                 7
##   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)              -1220.888
##   Loglikelihood unrestricted model (H1)      -1220.888
##                                                       
##   Akaike (AIC)                                2459.776
##   Bayesian (BIC)                              2483.222
##   Sample-size adjusted Bayesian (SABIC)       2454.798
## 
## 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 H_0: RMSEA <= 0.050                       NA
##   P-value H_0: RMSEA >= 0.080                       NA
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.000
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Regressions:
##                    Estimate   Std.Err  z-value  P(>|z|)
##   SocialBeh ~                                          
##     Nutrition          0.030    1.105    0.027    0.978
##     PhyExer            1.281    0.146    8.796    0.000
##     GMA               -0.075    0.151   -0.500    0.617
##   CognitiveAb ~                                        
##     SocialBeh          5.701    0.213   26.781    0.000
##     Nutrition          8.548    2.352    3.634    0.000
##     GMA                2.814    0.321    8.764    0.000
##     PhyExer           11.390    0.413   27.577    0.000
## 
## Variances:
##                    Estimate   Std.Err  z-value  P(>|z|)
##    .SocialBeh       5515.809  780.053    7.071    0.000
##    .CognitiveAb    24999.990 3535.532    7.071    0.000

We can compare the models

lavTestLRT(fit1,fit2)

## 
## Chi-Squared Difference Test
## 
##      Df    AIC    BIC  Chisq Chisq diff  RMSEA Df diff Pr(>Chisq)    
## fit2  0 2459.8 2483.2   0.00                                         
## fit1  1 2673.0 2693.8 215.24     215.24 1.4637       1  < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Or check modification indices

modindices(fit1, sort=TRUE)

##            lhs op         rhs     mi    epc sepc.lv sepc.all sepc.nox
## 15   SocialBeh  ~ CognitiveAb 88.379 -0.228  -0.228   -2.420   -2.420
## 16 CognitiveAb  ~     PhyExer 88.379 11.390  11.390    0.553    0.011
## 22     PhyExer  ~ CognitiveAb 82.143  0.128   0.128    2.635    2.635
## 26         GMA  ~ CognitiveAb  1.601  0.025   0.025    0.529    0.529
## 18   Nutrition  ~ CognitiveAb  1.002  0.007   0.007    1.114    1.114
## 21     PhyExer  ~   SocialBeh  0.000  0.000   0.000    0.000    0.000
## 20   Nutrition  ~         GMA  0.000  0.000   0.000    0.000    0.000
## 19   Nutrition  ~     PhyExer  0.000  0.000   0.000    0.000    0.000
## 24     PhyExer  ~         GMA  0.000  0.000   0.000    0.000    0.000
## 28         GMA  ~     PhyExer  0.000  0.000   0.000    0.000    0.000
## 23     PhyExer  ~   Nutrition  0.000  0.000   0.000    0.000    0.000
## 25         GMA  ~   SocialBeh  0.000  0.000   0.000    0.000    0.000
## 17   Nutrition  ~   SocialBeh  0.000  0.000   0.000    0.000    0.000

Direct and indirect

Direct effect (c): subgroups/cases that differ by one unit on X, but are equal on M are estimated to differ by c units on Y.

Indirect effect:
a) X -> M: cases that differ by one unit in X are estimated to differ by a units on M
b) M -> Y: cases that differ by one unit in M, but are equal on X, are estimated to differ by b units on Y

The indirect effect of X on Y through M is a product of a and b. The two cases that differ by one unit on X are estimated to differ by ab units on Y as a result of the effect of X on M which affects Y.

Direct and indirect

modelAbilityPath<-'
SocialBeh~Nutrition+a*PhyExer+GMA
CognitiveAb~b*SocialBeh+c*PhyExer+GMA
indirect := a*b
direct := c
total := indirect + direct
'
fitPath<-sem(modelAbilityPath, data=Babies)
summary(fitPath)

## lavaan 0.6.15 ended normally after 1 iteration
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                         8
## 
##   Number of observations                           100
## 
## Model Test User Model:
##                                                       
##   Test statistic                                12.401
##   Degrees of freedom                                 1
##   P-value (Chi-square)                           0.000
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Regressions:
##                    Estimate   Std.Err  z-value  P(>|z|)
##   SocialBeh ~                                          
##     Nutrition          0.030    1.105    0.027    0.978
##     PhyExer    (a)     1.281    0.146    8.796    0.000
##     GMA               -0.075    0.151   -0.500    0.617
##   CognitiveAb ~                                        
##     SocialBeh  (b)     5.704    0.227   25.180    0.000
##     PhyExer    (c)    11.355    0.439   25.846    0.000
##     GMA                2.813    0.342    8.233    0.000
## 
## Variances:
##                    Estimate   Std.Err  z-value  P(>|z|)
##    .SocialBeh       5515.809  780.053    7.071    0.000
##    .CognitiveAb    28300.616 4002.312    7.071    0.000
## 
## Defined Parameters:
##                    Estimate   Std.Err  z-value  P(>|z|)
##     indirect           7.308    0.880    8.304    0.000
##     direct            11.355    0.439   25.846    0.000
##     total             18.664    0.894   20.879    0.000

Interaction between the predictors can be included similar to the linear regression model by using (:) sign.

modelAbilityInteraction<-
SocialBeh~Nutrition+PhyExer+GMA+PhyExer:GMA
CognitiveAb~SocialBeh+Nutrition+GMA

Prerequisites

Theory: Strong theoretical assumptions that could be used to draw causal assumptions that could be tested using the data and specification of the model

Data: large samples, N:p rule - 20:1, more data usually better estimates.

We are not that interested in significance:

a) Overall behaviour of the model more interesting

b) More data higher probability of significant results (weak effects)

c) Latent models are estimated by anchoring on indicator variables, different estimation can result in different patterns

Problems with SEM and alternativesVariables derived from the normal distribution 
Observations independent 
Large sample size 
24

PiecewiseSEM

Variables are causally dependent if there is an arrow between them
They are causally independent if there are no arrows between them

X1 is causally independent from Y2 conditional on Y1

PiecewiseSEM performs a test of directional separation (d-sep) and asks whether causally independent paths are significant when controlling for variables on which causal process is conditional.

https://jonlefcheck.net/2014/07/06/piecewise-structural-equation-modeling-in-ecological-research/

PiecewiseSEM

#install.packages('piecewiseSEM)
require(piecewiseSEM)
model1<-psem(lm(SocialBeh~Nutrition+PhyExer+GMA, data=Babies),
             lm(CognitiveAb~SocialBeh+Nutrition+GMA, data=Babies))
summary(model1, .progressBar=FALSE)

## 
## Structural Equation Model of model1 
## 
## Call:
##   SocialBeh ~ Nutrition + PhyExer + GMA
##   CognitiveAb ~ SocialBeh + Nutrition + GMA
## 
##     AIC      BIC
##  229.364   255.416
## 
## ---
## Tests of directed separation:
## 
##                Independ.Claim Test.Type DF Crit.Value P.Value    
##   CognitiveAb ~ PhyExer + ...      coef 95    26.8792       0 ***
## 
## Global goodness-of-fit:
## 
##   Fisher's C = 209.364 with P-value = 0 and on 2 degrees of freedom
## 
## ---
## Coefficients:
## 
##      Response Predictor Estimate Std.Error DF Crit.Value P.Value Std.Estimate
##     SocialBeh Nutrition   0.0300    1.1278 96     0.0266  0.9789       0.0020
##     SocialBeh   PhyExer   1.2814    0.1487 96     8.6187  0.0000       0.6604
##     SocialBeh       GMA  -0.0753    0.1538 96    -0.4899  0.6253      -0.0375
##   CognitiveAb SocialBeh   9.5792    0.4786 96    20.0156  0.0000       0.9023
##   CognitiveAb Nutrition   7.0193    7.0413 96     0.9969  0.3213       0.0447
##   CognitiveAb       GMA   2.4607    0.9607 96     2.5614  0.0120       0.1155
##      
##      
##   ***
##      
##   ***
##      
##     *
## 
##   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05
## 
## ---
## Individual R-squared:
## 
##      Response method R.squared
##     SocialBeh   none      0.44
##   CognitiveAb   none      0.81

Important aspects: theoryDifference between moderation and mediation 
Interpretation of the predictors 
Calculation of free parameters and total parameters 
Model identification: three-types of identifications 
Overall fit of the model 
27

Important aspects: practiceBuilding path model: both continous and categorical exogenous variables 
Calculation of the direct and indirect pathways for predictors of interest 
Adding an interaction to path model 
Interpretation of the coefficients 
Getting fit indices of the model 
28

Literature

Chapters 1 to 5 of Principles and Practice of Structural Equation Modeling by Rex B. Kline

Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach by Andrew F. Hayes

Latent Variable Modeling Using R: A Step-by-Step Guide by A. Alexander Beaujean

Thank you for your attention30

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Lecture 5: SEM (Path models)

Dr Nemanja Vaci

University of Sheffield

2024-03-15

Press record

Intended learning outcomes

Structural equation modelling (SEM)

Structural part of the model (path analysis)

First step: Specification of the model

Can model be estimated?

Number of observations

How many parameters are we estimating (path model)?

How many parameters are we estimating (path model)?

How many parameters (our model)?

Second step: model identification

Third step: estimation of the model

Third step: estimation of the model

Step four: model evaluation

Other fit indices

Other fit indices

Model modification

We can compare the models

Or check modification indices

Direct and indirect

Direct and indirect

Prerequisites

Problems with SEM and alternatives

PiecewiseSEM

PiecewiseSEM

Important aspects: theory

Important aspects: practice

Literature

Thank you for your attention

Press record

Help