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Lecture 5: SEM (Path models)

Dr Nemanja Vaci

University of Sheffield

2024-03-15

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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.

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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:

  1. Causal modelling through path models
  2. Latent structures - factor analysis
  3. Structural causal models


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

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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.


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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.


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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: variables(variables+1)2



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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
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How many parameters are we estimating (path model)?

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

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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

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Most of the time (CFA model or other software): Degree of freedom for null model = (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

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Second step: model identification

  1. Under-indentified: more free parameters than total possible parameters

  2. Just-identified: equal number of free parameters and total possible parameters

  3. Over-identified: fewer free parameters than total possible parameters


    Parameters can either be: free, fixed or constrained
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Third step: estimation of the model

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

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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
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Other fit indices

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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
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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
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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
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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.

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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
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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

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Problems with SEM and alternatives

  1. Variables derived from the normal distribution
  2. Observations independent
  3. Large sample size
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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.

25

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
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Important aspects: theory

  • Difference 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: practice

  • Building 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
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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

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Thank you for your attention

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