Lecture 5: SEM (Path models)

class: center, middle, inverse, title-slide

.title[
# Lecture 5: SEM (Path models)
]
.author[
### Dr Nemanja Vaci
]
.institute[
### University of Sheffield
]
.date[
### 2024-03-15
]

---

## Press record

---

## 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:
 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.

???
Judea Pearl - [The Causal Foundations of Structural Equation Modeling](https://ftp.cs.ucla.edu/pub/stat_ser/r370.pdf)

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.

.center[
<img src="GeneralExample.png", width = "80%"> 
]

???
.center[
<img src="graphical.png", width = "120%"> 
]
---

## 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.

.center[
<img src="GeneralExample.png", width = "60%"> 
]
???
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: `\(\frac{variables*(variables+1)}{2}\)`

.center[
<img src="GeneralExample.png", width = "60%"> 
]

---
## Number of observations

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

<table>
 <thead>
 <tr>
 <th style="text-align:left;"> </th>
 <th style="text-align:right;"> Nutrition </th>
 <th style="text-align:right;"> PhyExer </th>
 <th style="text-align:right;"> GMA </th>
 <th style="text-align:right;"> SocialBeh </th>
 <th style="text-align:right;"> CognitiveAb </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Nutrition </td>
 <td style="text-align:right;"> 45.6689837 </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 </tr>
 <tr>
 <td style="text-align:left;"> PhyExer </td>
 <td style="text-align:right;"> -10.1006752 </td>
 <td style="text-align:right;"> 2652.9074 </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 </tr>
 <tr>
 <td style="text-align:left;"> GMA </td>
 <td style="text-align:right;"> 0.5641485 </td>
 <td style="text-align:right;"> -249.3049 </td>
 <td style="text-align:right;"> 2478.2889 </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SocialBeh </td>
 <td style="text-align:right;"> -11.6168733 </td>
 <td style="text-align:right;"> 3417.8681 </td>
 <td style="text-align:right;"> -506.1066 </td>
 <td style="text-align:right;"> 9988.898 </td>
 <td style="text-align:right;"> NA </td>
 </tr>
 <tr>
 <td style="text-align:left;"> CognitiveAb </td>
 <td style="text-align:right;"> 210.6731970 </td>
 <td style="text-align:right;"> 48916.6339 </td>
 <td style="text-align:right;"> 1254.2100 </td>
 <td style="text-align:right;"> 94358.621 </td>
 <td style="text-align:right;"> 1125746 </td>
 </tr>
</tbody>
</table>

---
## How many parameters are we estimating (path model)?

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

--
.center[
<img src="ModelParameters.png", width = "60%"> 
]

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 = `\((\frac{variables*(variables+1)}{2}) - variables\)`

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

<table>
 <thead>
 <tr>
 <th style="text-align:left;"> </th>
 <th style="text-align:right;"> Nutrition </th>
 <th style="text-align:right;"> PhyExer </th>
 <th style="text-align:right;"> GMA </th>
 <th style="text-align:right;"> SocialBeh </th>
 <th style="text-align:right;"> CognitiveAb </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Nutrition </td>
 <td style="text-align:right;"> 45.66898 </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 </tr>
 <tr>
 <td style="text-align:left;"> PhyExer </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> 2652.907 </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 </tr>
 <tr>
 <td style="text-align:left;"> GMA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> 2478.289 </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SocialBeh </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> 9988.898 </td>
 <td style="text-align:right;"> NA </td>
 </tr>
 <tr>
 <td style="text-align:left;"> CognitiveAb </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> 1125746 </td>
 </tr>
</tbody>
</table>
---
## How many parameters (our model)?

.center[
<img src="ModelParameters.png", width = "60%"> 
]

Free parameters = variances + covariances + regression pathways = 14
---

## 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

---

## Third step: estimation of the model

pre[class] {
 max-height: 80px;
}
</style>

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

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

---

## 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

```r
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
.center[
<img src="fitInd.png", width = "60%">
]

???
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:

```r
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

```r
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

```r
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

.center[
<img src="simplified.png", width = "60%"> 
]

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

```r
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)
```

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 alternatives

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

---
## PiecewiseSEM
.center[
<img src="PiecewiseSEM.png", width = "50%"> 
]

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

```r
#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: 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

---

## 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

---
## 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 attention