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[
### 2024-03-15
]

---

## 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.

---

## Latent space of measures

Principal Component Analysis (PCA) 
Exploratory Factor Analysis (EFA) 
Confirmatory Factor Analysis (CFA)

???
Differences between PCA and EFA: 
[Link 1](https://stats.stackexchange.com/a/95106) 
[Link 2](https://stats.stackexchange.com/a/288646)
---

## 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 are not measured, however we can still infer them from the observed data. To be able to do so, we need to define their scale:

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

---

## Effect coding

---
## 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 unstandardized and loaded on a single factor, then unstandardized regression coefficients. Model predicted difference in the predictor (measured variable) between groups that differ in 1-unit on the LVs

- When loaded on multiple factors, then regression coefficients become contingent on other factors (check Lecture 1, slide 11)

- When standardized and loaded on a single factor (congeneric structure), then standardized loadings are estimated correlations between indicators and LVs

- When standardized and loaded on a multiple factors, then same as the second option only standardized (beta weights)

---

## Results of the model

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

```
## lavaan 0.6.15 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 0.6.15 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 0.6.15 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
## 
## Constraints:
##                                                |Slack|
##     a+b+c - (3)                                  0.000
##     a1+b1+c1 - (3)                               0.000
```

---

## 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 0.6.15 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
## motor 0.000 0.000 0.000
## verbal 0.000 0.000 0.000
## 
## 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 0.6.15 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 0.6.15 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 0.6.15 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
##     motor             0.000                           
##     verbal            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
##     motor             0.000                           
##     verbal            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 0.6.15 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
##     motor             0.000                           
##     verbal            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
##     motor             0.000                           
##     verbal            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 0.6.15 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
##     motor             0.000                           
##     verbal            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 0.6.15 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
##     motor             0.000                           
##     verbal            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