Lecture 6: Confirmatory Factor Analysis

class: center, middle, inverse, title-slide

.title[
# Lecture 6: Confirmatory Factor Analysis
]
.author[
### Dr Nemanja Vaci
]
.institute[
### University of Sheffield
]
.date[
### 2026-03-25
]

---

## Press record

---

## Intended learning goals 
Motivate utilisation of path and CFA models; Argue how they relate to other models that we covered 
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.

---

## Dimension reduction (PCA)

Principal Component Analysis (PCA)
- Data reduction technique (not a latent variable model)
- Components are weighted combinations of observed variables
- Maximises total variance explained
- No distinction between common and unique variance
- Used when the goal is summarisation, not theory testing

---

## Latent space of measures (EFA)

Exploratory Factor Analysis (EFA)
- Identifies latent structure underlying observed variables
- Models common variance (separates error/unique variance)
- No strong a priori assumptions about factor structure
- All loadings are estimated (typically rotated for interpretation)
- Used in early stages of scale development
---

## Latent space of measures (CFA)

Confirmatory Factor Analysis (CFA)
- Tests a theory-driven latent structure
- Specifies which variables load on which factors (others fixed to zero)
- Directly evaluates how well the model reproduces the covariance matrix
- Provides formal model fit indices (e.g., CFI, RMSEA)
- Used for validation, comparison of competing models, and measurement testing

---

## Exploratory factor analysis (EFA)

Multivariate statistical procedure (Spearman): understanding and accounting for variation and covariation among of set of observed variables by postulating __latent__ structures (factors)

Factor: unobservable variable that influences more than one observed measure and accounts for their intercorrelation

If we partial out latent construct then intercorrelations would be zero

Factor analysis decomposes variance: __a) common variance (communality)__ and __b) unique variance__

???
Thourough example of EFA in R: https://psu-psychology.github.io/psy-597-SEM/06_factor_models/factor_models.html#overview
---

## EFA versus CFA

Reproduce observer relationships between measured variables with smaller number of latent factors

EFA is data-driven approach: weak or no assumptions on a number of latent dimensions and factor loadings (relations between indicators and factors)

CFA is theory-driven approach: strong assumptions for both things

EFA is used earlier in the process of questionnaire development and construct validation

---

## Factor model

---

## Factor or measurement model

Is linear regression where the main predictor is latent or unobserved:

`$$y_1=\tau_1+\lambda_1*\eta+\epsilon_1$$` 
`$y_1=\tau_1+\lambda_1*\eta+\epsilon_1$` 
`$y_2=\tau_2+\lambda_2*\eta+\epsilon_2$` 
`$y_3=\tau_3+\lambda_3*\eta+\epsilon_3$`

`$\tau$` - the item intercepts or means 
`$\lambda$` - factor loadings - regression coefficients 
`$\epsilon$` - error variances and covariances 
`$\eta$` - the latent predictor of the items 
`$\psi$` - factor variances and covariances 
---

## Exploratory factor model

---

## Confirmatory factor model

---

## Defining latent variables

LVs have no natural scale. We can multiply all loadings by a constant and divide the latent variable by the same constant without changing model fit. We must define the scale (identify the model) before estimation:

1. Marker variable: single factor loading constraint to 1 
2. Standardized latent variables: setting variance of variable to 1 (Z-score) 
3. Effects-coding: constraints that all of the loadings to one LV average 1.0 or that their sum is equal to number of indicators

???
https://www.researchgate.net/publication/255606342_A_Non-arbitrary_Method_of_Identifying_and_Scaling_Latent_Variables_in_SEM_and_MACS_Models
---
## Indicator variable and Standardizing LVs

- Fix one factor loading to 1
- The latent variable is expressed in the units of that observed variable
- All other loadings are relative to the marker variable

---

## Indicator variable and Standardizing LVs

- Fix latent variance to 1
- Latent variable is on a standardised (Z-score) scale
- Loadings can be interpreted like standardised regression coefficients

---

## Effect coding

- Constrain loadings such that: mean of all loadings is 1 or sum is equal to number of indicators
- Latent variable is scaled to the average of its indicators
- Uses all indicators to define the scale (more stable)

---
## Identification of the CFA

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

``` r
Matrix<-cov(vars)
Matrix[upper.tri(Matrix)]<-NA
knitr::kable(Matrix, format = 'html', digits=2) %>%
 kable_styling(font_size = 9)
```

<table class="table" style="font-size: 9px; margin-left: auto; margin-right: auto;">
 <thead>
 <tr>
 <th style="text-align:left;"> </th>
 <th style="text-align:right;"> TimeOnTummy </th>
 <th style="text-align:right;"> PreciseLegMoves </th>
 <th style="text-align:right;"> PreciseHandMoves </th>
 <th style="text-align:right;"> Babbling </th>
 <th style="text-align:right;"> Screeching </th>
 <th style="text-align:right;"> VocalImitation </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> TimeOnTummy </td>
 <td style="text-align:right;"> 24.80 </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;"> NA </td>
 </tr>
 <tr>
 <td style="text-align:left;"> PreciseLegMoves </td>
 <td style="text-align:right;"> 8.82 </td>
 <td style="text-align:right;"> 22.82 </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;"> PreciseHandMoves </td>
 <td style="text-align:right;"> 10.85 </td>
 <td style="text-align:right;"> 9.27 </td>
 <td style="text-align:right;"> 24.27 </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;"> Babbling </td>
 <td style="text-align:right;"> 1.35 </td>
 <td style="text-align:right;"> 4.04 </td>
 <td style="text-align:right;"> 3.52 </td>
 <td style="text-align:right;"> 24.69 </td>
 <td style="text-align:right;"> NA </td>
 <td style="text-align:right;"> NA </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Screeching </td>
 <td style="text-align:right;"> 0.59 </td>
 <td style="text-align:right;"> 3.12 </td>
 <td style="text-align:right;"> 1.72 </td>
 <td style="text-align:right;"> 8.08 </td>
 <td style="text-align:right;"> 21.16 </td>
 <td style="text-align:right;"> NA </td>
 </tr>
 <tr>
 <td style="text-align:left;"> VocalImitation </td>
 <td style="text-align:right;"> 2.71 </td>
 <td style="text-align:right;"> 4.46 </td>
 <td style="text-align:right;"> 4.33 </td>
 <td style="text-align:right;"> 11.31 </td>
 <td style="text-align:right;"> 4.81 </td>
 <td style="text-align:right;"> 25.01 </td>
 </tr>
</tbody>
</table>

---

## Theory and previous results

Previous work in this area found that two __congeneric__ latent factors explain covariances of our six indicators: motoric and verbal latent component

---
## Estimated number of parameters

<img src="parameters.png" width="50%" style="display: block; margin: auto;" />
--
Loadings `$(\lambda)$`: 4 parameters 
Residual variances `$(\epsilon)$` : 6 parameters 
Factor variances and covariances `$(\psi)$` : 3 parameters 
With intercepts: + 6 
---

## Coding of our model

``` r
#install.packages('lavaan')
require(lavaan)
model1<-'
motor =~ TimeOnTummy + PreciseLegMoves + PreciseHandMoves
verbal =~ Babbling + Screeching + VocalImitation
'

fit1<-cfa(model1, data=Babies)
```

pre[class] {
 max-height: 70px;
}
</style>

## Results of the model

``` r
summary(fit1)
```

---

## Results: visual

---

## Interpretation of the coefficients: factor loadings

- When unstandardised and loaded on a single factor, loadings are unstandardised regression coefficients. They represent the expected change in the observed variable for a 1-unit increase in the latent variable

- When loaded on multiple factors, loadings represent partial regression coefficients (check Lecture 1, slide 11). Each loading reflects the relationship between the indicator and one latent variable while controlling for the other latent variables.

- When standardised and loaded on a single factor (congeneric model), loadings approximate the correlation between the indicator and the latent variable

- When standardised and loaded on multiple factors, loadings are standardised partial regression coefficients (beta weights)

---

## Results of the model

``` r
summary(fit1, standardized=TRUE)
```

```
## lavaan 0.6-21 ended normally after 74 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        13
## 
##   Number of observations                           100
## 
## Model Test User Model:
##                                                       
##   Test statistic                                 3.376
##   Degrees of freedom                                 8
##   P-value (Chi-square)                           0.909
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   motor =~                                                              
##     TimeOnTummy       1.000                               3.086    0.623
##     PreciseLegMovs    0.910    0.240    3.791    0.000    2.807    0.591
##     PreciseHandMvs    1.099    0.293    3.746    0.000    3.392    0.692
##   verbal =~                                                             
##     Babbling          1.000                               3.954    0.800
##     Screeching        0.494    0.182    2.718    0.007    1.952    0.426
##     VocalImitation    0.716    0.246    2.906    0.004    2.832    0.569
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   motor ~~                                                              
##     verbal            3.433    1.901    1.806    0.071    0.281    0.281
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .TimeOnTummy      15.031    3.181    4.725    0.000   15.031    0.612
##    .PreciseLegMovs   14.709    2.867    5.130    0.000   14.709    0.651
##    .PreciseHandMvs   12.515    3.338    3.749    0.000   12.515    0.521
##    .Babbling          8.805    5.149    1.710    0.087    8.805    0.360
##    .Screeching       17.139    2.748    6.236    0.000   17.139    0.818
##    .VocalImitation   16.742    3.514    4.764    0.000   16.742    0.676
##     motor             9.523    3.625    2.627    0.009    1.000    1.000
##     verbal           15.635    5.947    2.629    0.009    1.000    1.000
```

---

## Scaling LVs: variance = 1

``` r
model2<-'
motor =~ NA*TimeOnTummy + PreciseLegMoves + PreciseHandMoves
verbal =~ NA*Babbling + Screeching + VocalImitation
motor ~~ 1*motor
verbal ~~ 1*verbal
'

fit2<-cfa(model2, data=Babies)
summary(fit2, standardized=TRUE)
```

```
## lavaan 0.6-21 ended normally after 33 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        13
## 
##   Number of observations                           100
## 
## Model Test User Model:
##                                                       
##   Test statistic                                 3.376
##   Degrees of freedom                                 8
##   P-value (Chi-square)                           0.909
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   motor =~                                                              
##     TimeOnTummy       3.086    0.587    5.254    0.000    3.086    0.623
##     PreciseLegMovs    2.807    0.557    5.042    0.000    2.807    0.591
##     PreciseHandMvs    3.392    0.597    5.680    0.000    3.392    0.692
##   verbal =~                                                             
##     Babbling          3.954    0.752    5.259    0.000    3.954    0.800
##     Screeching        1.952    0.548    3.560    0.000    1.952    0.426
##     VocalImitation    2.832    0.646    4.381    0.000    2.832    0.569
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   motor ~~                                                              
##     verbal            0.281    0.139    2.028    0.043    0.281    0.281
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     motor             1.000                               1.000    1.000
##     verbal            1.000                               1.000    1.000
##    .TimeOnTummy      15.031    3.181    4.725    0.000   15.031    0.612
##    .PreciseLegMovs   14.709    2.867    5.130    0.000   14.709    0.651
##    .PreciseHandMvs   12.515    3.338    3.749    0.000   12.515    0.521
##    .Babbling          8.805    5.149    1.710    0.087    8.805    0.360
##    .Screeching       17.139    2.748    6.236    0.000   17.139    0.818
##    .VocalImitation   16.742    3.514    4.764    0.000   16.742    0.676
```

---

## Scaling LVs: effect coding

``` r
model3<-'
motor =~ NA*TimeOnTummy+a*TimeOnTummy + b*PreciseLegMoves + c*PreciseHandMoves
verbal =~ NA*Babbling+a1*Babbling + b1*Screeching + c1*VocalImitation
a+b+c==3
a1+b1+c1==3
'

fit3<-cfa(model3, data=Babies)
summary(fit3, standardized=TRUE)
```

```
## lavaan 0.6-21 ended normally after 58 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        15
##   Number of equality constraints                     2
## 
##   Number of observations                           100
## 
## Model Test User Model:
##                                                       
##   Test statistic                                 3.376
##   Degrees of freedom                                 8
##   P-value (Chi-square)                           0.909
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   motor =~                                                              
##     TimOnTmmy  (a)    0.997    0.153    6.524    0.000    3.086    0.623
##     PrcsLgMvs  (b)    0.907    0.146    6.227    0.000    2.807    0.591
##     PrcsHndMv  (c)    1.096    0.162    6.770    0.000    3.392    0.692
##   verbal =~                                                             
##     Babbling  (a1)    1.358    0.236    5.753    0.000    3.954    0.800
##     Screechng (b1)    0.670    0.158    4.236    0.000    1.952    0.426
##     VoclImttn (c1)    0.972    0.189    5.140    0.000    2.832    0.569
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   motor ~~                                                              
##     verbal            2.536    1.341    1.891    0.059    0.281    0.281
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .TimeOnTummy      15.031    3.181    4.725    0.000   15.031    0.612
##    .PreciseLegMovs   14.709    2.867    5.130    0.000   14.709    0.651
##    .PreciseHandMvs   12.515    3.338    3.749    0.000   12.515    0.521
##    .Babbling          8.805    5.149    1.710    0.087    8.805    0.360
##    .Screeching       17.139    2.748    6.236    0.000   17.139    0.818
##    .VocalImitation   16.742    3.514    4.764    0.000   16.742    0.676
##     motor             9.581    2.070    4.628    0.000    1.000    1.000
##     verbal            8.483    1.896    4.475    0.000    1.000    1.000
```

---

## Adding intercepts

``` r
model3<-'
motor =~ TimeOnTummy + PreciseLegMoves + PreciseHandMoves
verbal =~ Babbling + Screeching + VocalImitation
TimeOnTummy ~ 1
PreciseLegMoves ~ 1
PreciseHandMoves ~ 1
Babbling ~ 1
Screeching ~ 1 
VocalImitation ~ 1'
fit3<-cfa(model3, data=Babies)
summary(fit3, standardized=TRUE, fit.measures=T)
```

```
## lavaan 0.6-21 ended normally after 74 iterations
## 
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 19
## 
## Number of observations 100
## 
## Model Test User Model:
## 
## Test statistic 3.376
## Degrees of freedom 8
## P-value (Chi-square) 0.909
## 
## Model Test Baseline Model:
## 
## Test statistic 88.357
## Degrees of freedom 15
## P-value 0.000
## 
## User Model versus Baseline Model:
## 
## Comparative Fit Index (CFI) 1.000
## Tucker-Lewis Index (TLI) 1.118
## 
## Loglikelihood and Information Criteria:
## 
## Loglikelihood user model (H0) -1756.127
## Loglikelihood unrestricted model (H1) -1754.439
## 
## Akaike (AIC) 3550.253
## Bayesian (BIC) 3599.751
## Sample-size adjusted Bayesian (SABIC) 3539.744
## 
## Root Mean Square Error of Approximation:
## 
## RMSEA 0.000
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.047
## P-value H_0: RMSEA <= 0.050 0.954
## P-value H_0: RMSEA >= 0.080 0.015
## 
## Standardized Root Mean Square Residual:
## 
## SRMR 0.034
## 
## Parameter Estimates:
## 
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
## 
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## motor =~ 
## TimeOnTummy 1.000 3.086 0.623
## PreciseLegMovs 0.910 0.240 3.791 0.000 2.807 0.591
## PreciseHandMvs 1.099 0.293 3.746 0.000 3.392 0.692
## verbal =~ 
## Babbling 1.000 3.954 0.800
## Screeching 0.494 0.182 2.718 0.007 1.952 0.426
## VocalImitation 0.716 0.246 2.906 0.004 2.832 0.569
## 
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## motor ~~ 
## verbal 3.433 1.901 1.806 0.071 0.281 0.281
## 
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .TimeOnTummy 31.423 0.496 63.415 0.000 31.423 6.341
## .PreciseLegMovs 30.600 0.475 64.380 0.000 30.600 6.438
## .PreciseHandMvs 29.799 0.490 60.798 0.000 29.799 6.080
## .Babbling 29.946 0.494 60.573 0.000 29.946 6.057
## .Screeching 29.785 0.458 65.078 0.000 29.785 6.508
## .VocalImitation 30.409 0.498 61.110 0.000 30.409 6.111
## 
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .TimeOnTummy 15.031 3.181 4.725 0.000 15.031 0.612
## .PreciseLegMovs 14.709 2.867 5.130 0.000 14.709 0.651
## .PreciseHandMvs 12.515 3.338 3.749 0.000 12.515 0.521
## .Babbling 8.805 5.149 1.710 0.087 8.805 0.360
## .Screeching 17.139 2.748 6.236 0.000 17.139 0.818
## .VocalImitation 16.742 3.514 4.764 0.000 16.742 0.676
## motor 9.523 3.625 2.627 0.009 1.000 1.000
## verbal 15.635 5.947 2.629 0.009 1.000 1.000
```

---

## Indices of global model fit

``` r
summary(fit1, fit.measures=TRUE)
```

```
## lavaan 0.6-21 ended normally after 74 iterations
## 
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 13
## 
## Number of observations 100
## 
## Model Test User Model:
## 
## Test statistic 3.376
## Degrees of freedom 8
## P-value (Chi-square) 0.909
## 
## Model Test Baseline Model:
## 
## Test statistic 88.357
## Degrees of freedom 15
## P-value 0.000
## 
## User Model versus Baseline Model:
## 
## Comparative Fit Index (CFI) 1.000
## Tucker-Lewis Index (TLI) 1.118
## 
## Loglikelihood and Information Criteria:
## 
## Loglikelihood user model (H0) -1756.127
## Loglikelihood unrestricted model (H1) -1754.439
## 
## Akaike (AIC) 3538.253
## Bayesian (BIC) 3572.120
## Sample-size adjusted Bayesian (SABIC) 3531.063
## 
## Root Mean Square Error of Approximation:
## 
## RMSEA 0.000
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.047
## P-value H_0: RMSEA <= 0.050 0.954
## P-value H_0: RMSEA >= 0.080 0.015
## 
## Standardized Root Mean Square Residual:
## 
## SRMR 0.038
## 
## Parameter Estimates:
## 
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
## 
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## motor =~ 
## TimeOnTummy 1.000 
## PreciseLegMovs 0.910 0.240 3.791 0.000
## PreciseHandMvs 1.099 0.293 3.746 0.000
## verbal =~ 
## Babbling 1.000 
## Screeching 0.494 0.182 2.718 0.007
## VocalImitation 0.716 0.246 2.906 0.004
## 
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## motor ~~ 
## verbal 3.433 1.901 1.806 0.071
## 
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .TimeOnTummy 15.031 3.181 4.725 0.000
## .PreciseLegMovs 14.709 2.867 5.130 0.000
## .PreciseHandMvs 12.515 3.338 3.749 0.000
## .Babbling 8.805 5.149 1.710 0.087
## .Screeching 17.139 2.748 6.236 0.000
## .VocalImitation 16.742 3.514 4.764 0.000
## motor 9.523 3.625 2.627 0.009
## verbal 15.635 5.947 2.629 0.009
```

???
Measurement error in CFA: https://rdrr.io/cran/semTools/man/reliability.html
---
class: inverse, middle, center
# Structural equation model
---

## SEM, finally

---

## Estimation of SEM

``` r
model4<-'
#CFA model
motor =~ TimeOnTummy + PreciseLegMoves + PreciseHandMoves
verbal =~ Babbling + Screeching + VocalImitation

#Path model
motor ~ Age + Weight
verbal ~ Age + Weight
'

fit4<-sem(model4, data=Babies)
```
---
## Structural equation model: Results

``` r
summary(fit4, standardized=TRUE)
```

```
## lavaan 0.6-21 ended normally after 81 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        17
## 
##   Number of observations                           100
## 
## Model Test User Model:
##                                                       
##   Test statistic                                13.018
##   Degrees of freedom                                16
##   P-value (Chi-square)                           0.671
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Latent Variables:
##                    Estimate    Std.Err  z-value  P(>|z|)   Std.lv    Std.all
##   motor =~                                                                  
##     TimeOnTummy         1.000                                 3.064    0.618
##     PreciseLegMovs      0.919    0.242    3.803    0.000      2.816    0.592
##     PreciseHandMvs      1.111    0.295    3.765    0.000      3.403    0.694
##   verbal =~                                                                 
##     Babbling            1.000                                 3.498    0.708
##     Screeching          0.583    0.189    3.089    0.002      2.040    0.446
##     VocalImitation      0.899    0.263    3.422    0.001      3.144    0.632
## 
## Regressions:
##                    Estimate    Std.Err  z-value  P(>|z|)   Std.lv    Std.all
##   motor ~                                                                   
##     Age                -0.016    0.045   -0.355    0.723     -0.005   -0.043
##     Weight              0.000    0.001    0.085    0.932      0.000    0.010
##   verbal ~                                                                  
##     Age                -0.041    0.051   -0.803    0.422     -0.012   -0.097
##     Weight              0.002    0.001    2.108    0.035      0.001    0.263
## 
## Covariances:
##                    Estimate    Std.Err  z-value  P(>|z|)   Std.lv    Std.all
##  .motor ~~                                                                  
##    .verbal              3.292    1.738    1.894    0.058      0.320    0.320
## 
## Variances:
##                    Estimate    Std.Err  z-value  P(>|z|)   Std.lv    Std.all
##    .TimeOnTummy        15.164    3.159    4.800    0.000     15.164    0.618
##    .PreciseLegMovs     14.662    2.862    5.122    0.000     14.662    0.649
##    .PreciseHandMvs     12.440    3.327    3.740    0.000     12.440    0.518
##    .Babbling           12.204    3.740    3.263    0.001     12.204    0.499
##    .Screeching         16.784    2.717    6.177    0.000     16.784    0.801
##    .VocalImitation     14.880    3.440    4.326    0.000     14.880    0.601
##    .motor               9.372    3.577    2.620    0.009      0.998    0.998
##    .verbal             11.299    4.205    2.687    0.007      0.923    0.923
```

---
class: inverse, middle, center
# Measurement invariance 
---

## Measurement invariance

Compare our model between the groups: 
 - Configural invarience: Model fitted for each group separately 
 - Metric invariance: restriction of the factor loadings, but intercepts are allowed to vary 
 - Scalar invariance: restriction of the both, factor loadings and intercepts 
 - Strict invariance: restriction on factor loadings, intercepts and residual variances

---

## Configural invariance

``` r
modelMI<-'
motor =~ TimeOnTummy + PreciseLegMoves + PreciseHandMoves
verbal =~ Babbling + Screeching + VocalImitation
'

fitMIC<-cfa(modelMI, data=Babies, group='Gender')
summary(fitMIC)
```

```
## lavaan 0.6-21 ended normally after 141 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        38
## 
##   Number of observations per group:                   
##     Girls                                           48
##     Boys                                            52
## 
## Model Test User Model:
##                                                       
##   Test statistic                                15.880
##   Degrees of freedom                                16
##   P-value (Chi-square)                           0.461
##   Test statistic for each group:
##     Girls                                        6.246
##     Boys                                         9.634
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## 
## Group 1 [Girls]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor =~                                            
##     TimeOnTummy       1.000                           
##     PreciseLegMovs    0.754    0.282    2.677    0.007
##     PreciseHandMvs    0.982    0.345    2.849    0.004
##   verbal =~                                           
##     Babbling          1.000                           
##     Screeching        0.428    0.237    1.809    0.070
##     VocalImitation    0.632    0.313    2.021    0.043
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor ~~                                            
##     verbal            7.298    3.772    1.935    0.053
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TimeOnTummy      31.723    0.769   41.272    0.000
##    .PreciseLegMovs   30.863    0.734   42.020    0.000
##    .PreciseHandMvs   30.204    0.725   41.681    0.000
##    .Babbling         29.693    0.781   37.997    0.000
##    .Screeching       29.281    0.669   43.755    0.000
##    .VocalImitation   30.893    0.750   41.215    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TimeOnTummy      15.328    5.201    2.947    0.003
##    .PreciseLegMovs   18.476    4.570    4.042    0.000
##    .PreciseHandMvs   12.648    4.739    2.669    0.008
##    .Babbling         11.319    8.343    1.357    0.175
##    .Screeching       18.192    4.110    4.426    0.000
##    .VocalImitation   19.781    5.256    3.764    0.000
##     motor            13.031    6.402    2.036    0.042
##     verbal           17.992    9.733    1.849    0.065
## 
## 
## Group 2 [Boys]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor =~                                            
##     TimeOnTummy       1.000                           
##     PreciseLegMovs    1.168    0.448    2.607    0.009
##     PreciseHandMvs    1.106    0.409    2.706    0.007
##   verbal =~                                           
##     Babbling          1.000                           
##     Screeching        0.410    0.238    1.727    0.084
##     VocalImitation    0.562    0.302    1.861    0.063
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor ~~                                            
##     verbal           -0.787    2.007   -0.392    0.695
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TimeOnTummy      31.147    0.634   49.148    0.000
##    .PreciseLegMovs   30.357    0.611   49.675    0.000
##    .PreciseHandMvs   29.426    0.660   44.593    0.000
##    .Babbling         30.179    0.617   48.873    0.000
##    .Screeching       30.251    0.620   48.790    0.000
##    .VocalImitation   29.962    0.655   45.744    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TimeOnTummy      13.783    3.710    3.715    0.000
##    .PreciseLegMovs    9.737    3.951    2.465    0.014
##    .PreciseHandMvs   13.963    4.139    3.373    0.001
##    .Babbling         -0.009    9.707   -0.001    0.999
##    .Screeching       16.649    3.652    4.559    0.000
##    .VocalImitation   16.035    4.395    3.649    0.000
##     motor             7.101    3.990    1.779    0.075
##     verbal           19.837   10.457    1.897    0.058
```

---

## Metric invariance: 1

``` r
modelMI<-'
motor =~ TimeOnTummy + PreciseLegMoves + PreciseHandMoves
verbal =~ Babbling + Screeching + VocalImitation
'

fitMIM<-cfa(modelMI, data=Babies, group='Gender',group.equal='loadings')
summary(fitMIM)
```

```
## lavaan 0.6-21 ended normally after 135 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        38
##   Number of equality constraints                     4
## 
##   Number of observations per group:                   
##     Girls                                           48
##     Boys                                            52
## 
## Model Test User Model:
##                                                       
##   Test statistic                                16.556
##   Degrees of freedom                                20
##   P-value (Chi-square)                           0.682
##   Test statistic for each group:
##     Girls                                        6.606
##     Boys                                         9.951
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## 
## Group 1 [Girls]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor =~                                            
##     TmOnTmm           1.000                           
##     PrcsLgM (.p2.)    0.949    0.247    3.840    0.000
##     PrcsHnM (.p3.)    1.037    0.267    3.880    0.000
##   verbal =~                                           
##     Babblng           1.000                           
##     Scrchng (.p5.)    0.430    0.164    2.619    0.009
##     VclImtt (.p6.)    0.601    0.211    2.854    0.004
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor ~~                                            
##     verbal            7.124    3.399    2.096    0.036
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TimeOnTummy      31.723    0.757   41.929    0.000
##    .PreciseLegMovs   30.863    0.751   41.121    0.000
##    .PreciseHandMvs   30.204    0.722   41.828    0.000
##    .Babbling         29.693    0.782   37.967    0.000
##    .Screeching       29.281    0.671   43.661    0.000
##    .VocalImitation   30.893    0.747   41.353    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TimeOnTummy      16.616    4.612    3.603    0.000
##    .PreciseLegMovs   17.266    4.551    3.794    0.000
##    .PreciseHandMvs   13.341    4.238    3.148    0.002
##    .Babbling         10.817    7.130    1.517    0.129
##    .Screeching       18.163    4.020    4.518    0.000
##    .VocalImitation   20.094    4.867    4.128    0.000
##     motor            10.861    4.723    2.300    0.021
##     verbal           18.541    8.400    2.207    0.027
## 
## 
## Group 2 [Boys]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor =~                                            
##     TmOnTmm           1.000                           
##     PrcsLgM (.p2.)    0.949    0.247    3.840    0.000
##     PrcsHnM (.p3.)    1.037    0.267    3.880    0.000
##   verbal =~                                           
##     Babblng           1.000                           
##     Scrchng (.p5.)    0.430    0.164    2.619    0.009
##     VclImtt (.p6.)    0.601    0.211    2.854    0.004
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor ~~                                            
##     verbal           -0.695    2.209   -0.315    0.753
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TimeOnTummy      31.147    0.643   48.419    0.000
##    .PreciseLegMovs   30.357    0.600   50.563    0.000
##    .PreciseHandMvs   29.426    0.662   44.430    0.000
##    .Babbling         30.179    0.617   48.879    0.000
##    .Screeching       30.251    0.619   48.885    0.000
##    .VocalImitation   29.962    0.657   45.613    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TimeOnTummy      12.951    3.622    3.576    0.000
##    .PreciseLegMovs   11.037    3.171    3.481    0.000
##    .PreciseHandMvs   13.593    3.843    3.537    0.000
##    .Babbling          1.076    6.336    0.170    0.865
##    .Screeching       16.449    3.442    4.779    0.000
##    .VocalImitation   15.667    3.841    4.079    0.000
##     motor             8.566    3.646    2.349    0.019
##     verbal           18.747    7.377    2.541    0.011
```

---

## Metric invariance: 2

``` r
#install.packages('semTools')
require(semTools)
summary(compareFit(fitMIC, fitMIM))
```

```
## ################### Nested Model Comparison #########################
## 
## Chi-Squared Difference Test
## 
##        Df    AIC    BIC  Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
## fitMIC 16 3573.3 3672.3 15.880                                    
## fitMIM 20 3566.0 3654.6 16.556    0.67673     0       4     0.9542
## 
## ####################### Model Fit Indices ###########################
##          chisq df pvalue rmsea    cfi    tli  srmr       aic       bic
## fitMIC 15.880† 16   .461 .000† 1.000† 1.003  .065  3573.335  3672.332 
## fitMIM 16.556  20   .682 .000† 1.000† 1.067† .064† 3566.012† 3654.588†
## 
## ################## Differences in Fit Indices #######################
##                 df rmsea cfi   tli   srmr    aic     bic
## fitMIM - fitMIC  4     0   0 0.064 -0.001 -7.323 -17.744
```

---

## Scalar invariance: 1

``` r
modelMI<-'
motor =~ TimeOnTummy + PreciseLegMoves + PreciseHandMoves
verbal =~ Babbling + Screeching + VocalImitation
'

fitMISc<-cfa(modelMI, data=Babies, group='Gender',group.equal=c('loadings','intercepts'))
summary(fitMISc)
```

```
## lavaan 0.6-21 ended normally after 153 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        40
##   Number of equality constraints                    10
## 
##   Number of observations per group:                   
##     Girls                                           48
##     Boys                                            52
## 
## Model Test User Model:
##                                                       
##   Test statistic                                19.394
##   Degrees of freedom                                24
##   P-value (Chi-square)                           0.731
##   Test statistic for each group:
##     Girls                                        8.252
##     Boys                                        11.142
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## 
## Group 1 [Girls]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor =~                                            
##     TmOnTmm           1.000                           
##     PrcsLgM (.p2.)    0.949    0.244    3.880    0.000
##     PrcsHnM (.p3.)    1.045    0.266    3.925    0.000
##   verbal =~                                           
##     Babblng           1.000                           
##     Scrchng (.p5.)    0.410    0.163    2.510    0.012
##     VclImtt (.p6.)    0.560    0.206    2.715    0.007
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor ~~                                            
##     verbal            7.181    3.418    2.101    0.036
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TmOnTmm (.16.)   31.753    0.661   48.007    0.000
##    .PrcsLgM (.17.)   30.920    0.638   48.461    0.000
##    .PrcsHnM (.18.)   30.141    0.661   45.611    0.000
##    .Babblng (.19.)   29.775    0.767   38.801    0.000
##    .Scrchng (.20.)   29.718    0.504   58.930    0.000
##    .VclImtt (.21.)   30.219    0.575   52.515    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TimeOnTummy      16.671    4.605    3.620    0.000
##    .PreciseLegMovs   17.293    4.543    3.807    0.000
##    .PreciseHandMvs   13.265    4.238    3.130    0.002
##    .Babbling          9.888    7.683    1.287    0.198
##    .Screeching       18.494    4.071    4.543    0.000
##    .VocalImitation   20.960    4.956    4.229    0.000
##     motor            10.803    4.679    2.309    0.021
##     verbal           19.542    8.992    2.173    0.030
## 
## 
## Group 2 [Boys]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor =~                                            
##     TmOnTmm           1.000                           
##     PrcsLgM (.p2.)    0.949    0.244    3.880    0.000
##     PrcsHnM (.p3.)    1.045    0.266    3.925    0.000
##   verbal =~                                           
##     Babblng           1.000                           
##     Scrchng (.p5.)    0.410    0.163    2.510    0.012
##     VclImtt (.p6.)    0.560    0.206    2.715    0.007
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor ~~                                            
##     verbal           -0.874    2.208   -0.396    0.692
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TmOnTmm (.16.)   31.753    0.661   48.007    0.000
##    .PrcsLgM (.17.)   30.920    0.638   48.461    0.000
##    .PrcsHnM (.18.)   30.141    0.661   45.611    0.000
##    .Babblng (.19.)   29.775    0.767   38.801    0.000
##    .Scrchng (.20.)   29.718    0.504   58.930    0.000
##    .VclImtt (.21.)   30.219    0.575   52.515    0.000
##     motor            -0.628    0.766   -0.819    0.413
##     verbal            0.405    0.985    0.411    0.681
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TimeOnTummy      12.970    3.609    3.594    0.000
##    .PreciseLegMovs   11.054    3.159    3.500    0.000
##    .PreciseHandMvs   13.564    3.847    3.526    0.000
##    .Babbling         -0.129    7.015   -0.018    0.985
##    .Screeching       16.806    3.500    4.802    0.000
##    .VocalImitation   16.307    3.880    4.203    0.000
##     motor             8.523    3.612    2.359    0.018
##     verbal           19.957    8.027    2.486    0.013
```

---

## Scalar invariance: 2

``` r
summary(compareFit(fitMIM,fitMISc))
```

```
## ################### Nested Model Comparison #########################
## 
## Chi-Squared Difference Test
## 
##         Df    AIC    BIC  Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
## fitMIM  20 3566.0 3654.6 16.556                                    
## fitMISc 24 3560.8 3639.0 19.394     2.8375     0       4     0.5854
## 
## ####################### Model Fit Indices ###########################
##           chisq df pvalue rmsea    cfi    tli  srmr       aic       bic
## fitMIM  16.556† 20   .682 .000† 1.000† 1.067  .064† 3566.012  3654.588 
## fitMISc 19.394  24   .731 .000† 1.000† 1.074† .071  3560.850† 3639.005†
## 
## ################## Differences in Fit Indices #######################
##                  df rmsea cfi   tli  srmr    aic     bic
## fitMISc - fitMIM  4     0   0 0.008 0.007 -5.162 -15.583
```

---

## Strict invariance: 1

``` r
modelMI<-'
motor =~ TimeOnTummy + PreciseLegMoves + PreciseHandMoves
verbal =~ Babbling + Screeching + VocalImitation
'

fitMISt<-cfa(modelMI, data=Babies, group='Gender',group.equal=c('loadings','intercepts','residuals'))
summary(fitMISt)
```

```
## lavaan 0.6-21 ended normally after 107 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        40
##   Number of equality constraints                    16
## 
##   Number of observations per group:                   
##     Girls                                           48
##     Boys                                            52
## 
## Model Test User Model:
##                                                       
##   Test statistic                                25.722
##   Degrees of freedom                                30
##   P-value (Chi-square)                           0.689
##   Test statistic for each group:
##     Girls                                       10.852
##     Boys                                        14.870
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## 
## Group 1 [Girls]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor =~                                            
##     TmOnTmm           1.000                           
##     PrcsLgM (.p2.)    0.916    0.237    3.858    0.000
##     PrcsHnM (.p3.)    1.046    0.270    3.872    0.000
##   verbal =~                                           
##     Babblng           1.000                           
##     Scrchng (.p5.)    0.357    0.152    2.352    0.019
##     VclImtt (.p6.)    0.497    0.194    2.561    0.010
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor ~~                                            
##     verbal            7.237    3.475    2.083    0.037
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TmOnTmm (.16.)   31.754    0.663   47.879    0.000
##    .PrcsLgM (.17.)   30.903    0.624   49.533    0.000
##    .PrcsHnM (.18.)   30.145    0.673   44.815    0.000
##    .Babblng (.19.)   29.707    0.776   38.266    0.000
##    .Scrchng (.20.)   29.700    0.506   58.645    0.000
##    .VclImtt (.21.)   30.290    0.582   52.081    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TmOnTmm (.p7.)   14.722    3.138    4.691    0.000
##    .PrcsLgM (.p8.)   14.342    2.843    5.045    0.000
##    .PrcsHnM (.p9.)   13.256    3.161    4.194    0.000
##    .Babblng (.10.)    1.930    7.776    0.248    0.804
##    .Scrchng (.11.)   18.072    2.752    6.568    0.000
##    .VclImtt (.12.)   19.193    3.335    5.755    0.000
##     motor            11.438    4.800    2.383    0.017
##     verbal           27.035    9.847    2.745    0.006
## 
## 
## Group 2 [Boys]:
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor =~                                            
##     TmOnTmm           1.000                           
##     PrcsLgM (.p2.)    0.916    0.237    3.858    0.000
##     PrcsHnM (.p3.)    1.046    0.270    3.872    0.000
##   verbal =~                                           
##     Babblng           1.000                           
##     Scrchng (.p5.)    0.357    0.152    2.352    0.019
##     VclImtt (.p6.)    0.497    0.194    2.561    0.010
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   motor ~~                                            
##     verbal           -0.554    2.239   -0.248    0.804
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TmOnTmm (.16.)   31.754    0.663   47.879    0.000
##    .PrcsLgM (.17.)   30.903    0.624   49.533    0.000
##    .PrcsHnM (.18.)   30.145    0.673   44.815    0.000
##    .Babblng (.19.)   29.707    0.776   38.266    0.000
##    .Scrchng (.20.)   29.700    0.506   58.645    0.000
##    .VclImtt (.21.)   30.290    0.582   52.081    0.000
##     motor            -0.635    0.772   -0.823    0.411
##     verbal            0.459    0.994    0.462    0.644
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .TmOnTmm (.p7.)   14.722    3.138    4.691    0.000
##    .PrcsLgM (.p8.)   14.342    2.843    5.045    0.000
##    .PrcsHnM (.p9.)   13.256    3.161    4.194    0.000
##    .Babblng (.10.)    1.930    7.776    0.248    0.804
##    .Scrchng (.11.)   18.072    2.752    6.568    0.000
##    .VclImtt (.12.)   19.193    3.335    5.755    0.000
##     motor             8.157    3.558    2.292    0.022
##     verbal           18.233    8.616    2.116    0.034
```

---

## Strict invariance: 2

``` r
summary(compareFit(fitMISc,fitMISt))
```

```
## ################### Nested Model Comparison #########################
## 
## Chi-Squared Difference Test
## 
##         Df    AIC    BIC  Chisq Chisq diff    RMSEA Df diff Pr(>Chisq)
## fitMISc 24 3560.8 3639.0 19.394                                       
## fitMISt 30 3555.2 3617.7 25.722      6.328 0.033067       6     0.3875
## 
## ####################### Model Fit Indices ###########################
##           chisq df pvalue rmsea    cfi    tli  srmr       aic       bic
## fitMISc 19.394† 24   .731 .000† 1.000† 1.074† .071† 3560.850  3639.005 
## fitMISt 25.722  30   .689 .000† 1.000† 1.055  .079  3555.178† 3617.702†
## 
## ################## Differences in Fit Indices #######################
##                   df rmsea cfi    tli  srmr    aic     bic
## fitMISt - fitMISc  6     0   0 -0.019 0.008 -5.672 -21.303
```

---

## Where are the differences

``` r
lavTestScore(fitMISc)
```

```
## $test
## 
## total score test:
## 
##    test    X2 df p.value
## 1 score 3.387 10   0.971
## 
## $uni
## 
## univariate score tests:
## 
##      lhs op   rhs    X2 df p.value
## 1   .p2. == .p25. 0.479  1   0.489
## 2   .p3. == .p26. 0.015  1   0.903
## 3   .p5. == .p28. 0.006  1   0.939
## 4   .p6. == .p29. 0.047  1   0.828
## 5  .p16. == .p39. 0.007  1   0.936
## 6  .p17. == .p40. 0.021  1   0.885
## 7  .p18. == .p41. 0.045  1   0.831
## 8  .p19. == .p42. 0.278  1   0.598
## 9  .p20. == .p43. 0.958  1   0.328
## 10 .p21. == .p44. 1.948  1   0.163
```

---
class: inverse, middle, center
# Practical aspect
---

## Theory and data

Holzinger and Swineford data (1939) - [LINK](https://www.rdocumentation.org/packages/psychTools/versions/2.0.8/topics/holzinger.swineford)

``` r
#install.packages('sem')
require(sem)
data('HS.data')
```

<img src="image7.png" width="55%" style="display: block; margin: auto;" />
---

## Model development

---

## Important aspects: theory

- Understanding differences between Exploratory FA and Confirmatory FA 
- How is linear model defined in the CFA 
- Scaling of the latent variables 
- Interpretation of the coefficients 
- Number of free parameters versus total number of parameters

---

## Important aspects: practice

- Specifying and estimating CFA model 
- Scaling the LVs by using marker variable or by scaling LVs variance to 1 
- Adding intercepts to your CFA model 
- Making a full SEM model

---

## Literature

Confirmatory Factor Analysis for Applied Research by Timothy A. Brown

Chapters 9 of Principles and Practice of Structural Equation Modeling by Rex B. Kline

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

---

## Thank you for your attention