Análisis Factorial Confirmatorio y Confiabilidad en R

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# Análisis Factorial Confirmatorio y Confiabilidad en <code>R</code>
## 3er CIEP - Usil
### Brian N. Peña-Calero y Arnold Tafur-Mendoza
### Avances en Medición Psicológica (AMP) - UNMSM
### 12/10/2019

---

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# .font150[**Temario**]

---

# Temario

1. Modelos Factoriales
 * Definición y utilidad 
 * Conceptos: cargas factoriales, errores, variables observables, variables latentes. 
1. Análisis Factorial Confirmatorio
 * Diferencias y usos entre el análisis factorial exploratorio y confirmatorio
 * Concepto y utilidad
 * Términos básicos: Estimación restringidas (correlación de errores), liberadas, fijas 
 * Modelos AFC conocidos: Primer orden (ortogonales y oblicuos), segundo orden, bifactor
 * Procedimientos de análisis: Especificación, Identificación, Estimación, evaluación y re-especificación
 * Ejemplificación con caso
1. Fiabilidad por consistencia interna
 * Problemática de Alfa y la tau-equivalencia
 * Ventajas de la fiabilidad basado en modelos factoriales
 * Procedimiento en software

---
class: inverse, center, middle

# .font120[**Modelos Factoriales**]

---

# Modelos Factoriales

## Definición y Utilidad
## Conceptos
* ### Cargas Factoriales
* ### Errrores
* ### Variables Observables
* ### Variables Latentes

---
class: inverse, center, middle

# .font120[**Análisis Factorial Confirmatorio**]
---

# Análisis Factorial Confirmatorio
* ### Diferencias y usos entre el análisis factorial exploratorio y confirmatorio
* ### Concepto y utilidad
* ### Términos básicos: Estimación restringidas (correlación de errores), liberadas, fijas
* ### Modelos AFC conocidos: Primer orden (ortogonales y oblicuos), segundo orden, bifactor

---
# Procedimientos de análisis
<br>
## 1. Especificación 
## 2. Identificación
## 3. Estimación
## 4. Evaluación 
## 5. Re-especificación

---
# Procedimientos de análisis

## 1. Especificación 
.left-plot[
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]

.right-code[<br> `lavaan` automatiza la asignación de los errores pertenecientes a cada variable observada (ítem). <br>

```r
model <- " # Modelo de Medición
          `Variable Latente` =~ Item01 + Item02 + Item03
         "
```
Se crea (`<-`) el objeto `model` donde se **especifica**:
* Cuántas y como se llaman las variables latentes
* Cuántos ítems y a donde pertenecen

.center[.font100[*¡Veamos como sería en un modelo más complejo!*]]
]

---
# Procedimientos de análisis

## 1. Especificación 
.left-plot[
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]

.right-code[El símbolo .font150[`↔`] hace referencia a la ***covarianza*** entre los factores.

```r
model <- " # Modelo de Medición
          VL1 =~ Item01 + Item02 + Item03
          VL2 =~ Item04 + Item05 + Item06
          # VL1 ~~ VL2
         "
```
No es necesario especificar `VL1 ~~ VL2`. `lavaan` supone que siempre trabajamos con modelos oblicuos.

**Nota:** *Más adelante veremos modelos ortogonales.*
]

---
# Procedimientos de análisis

## 1. Especificación

Hay más cosas que podemos especificar en nuestros modelos de medición:

--
* Igualar las cargas factoriales de alguno o varios ítems

```r
VL1 =~ a*Item01 + a*Item02 + a*Item03
```

--
* Especificar modelos ortogonales

```r
VL1 ~~ 0*VL2
```

--
* Realizar correlación entre errores (covarianza de varianza específica de los ítems)

```r
# Correlación de errores
  Item01 ~~ Item02 
```

---
# Procedimientos de análisis

## 1. Especificación 
.left-plot[
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]

.right-code[¡ A resolverlo!]

--
.right-code[

```r
model <- " # Modelo de Medición
           VL1 =~ Item01 + Item02 + Item03
           VL2 =~ Item04 + Item05 + Item06
           
           # Correlación de errores
           Item04 ~~ Item05
         "
```
La varianza específica del `Item04` y del `Item05` son los que se relacionan de alguna manera. Existe algo que distinto a `VL2` que está explicando el comportamiento de las puntuaciones de esos ítems.
]

---
# Procedimientos de análisis

## 2. Identificación

Este procedimiento hace referencia a la suficiencia de información para el análisis. Dependiendo de la cantidad de información que tengamos y que solicitemos, podremos encontrar una solución satisfactoria.
Por ejemplo:

<box-block> 
X + Y  = 20 <br>
2X + Y = 28 
</box-block>

¿Cuánto vale `X` y cuánto vale `Y`?

--
<br> `X = 8`
<br> `Y = 12`

.center[.font120[*¿Había suficiente información?*]]

---
# Procedimientos de análisis

## 2. Identificación

Imaginemos ahora la siguiente situación:

<box-block> 
 X + Y  = 25 <br>
2X + 2Y = 50
</box-block>

¿Cuánto vale `X` y cuánto vale `Y`?

-- 
Las soluciones pueden ser infinitas
<br> `X = 10; Y = 15`
<br> `X = 8; Y = 17`
<br> `X = 15; Y = 10`

.center[.font120[*¡No tenemos suficiente información!*]]

---
# Procedimientos de análisis

## 2. Identificación

En el caso de un `AFC` la suficiencia de información hace referencia a la cantidad de correlaciones que existe en una matriz de las variables a analizar (en nuestros casos, los ítems). Esto se contrasta con la información solicitada (`Especificaciòn`).

```
## 
## Correlation method: 'pearson'
## Missing treated using: 'pairwise.complete.obs'
```

```
##   rowname   x1   x2   x3
## 1      x1 1.00          
## 2      x2  .30 1.00     
## 3      x3  .44  .34 1.00
```

.center[.font110[¿Cuántas correlaciones tenemos?]]

--
.center[`6`]

---
# Procedimientos de análisis

## 2. Identificación

Veamos otro ejemplo con una cantidad de ítems mayor

```
## 
## Correlation method: 'pearson'
## Missing treated using: 'pairwise.complete.obs'
```

```
##   rowname   x1   x2   x3   x4   x5
## 1      x1 1.00                    
## 2      x2  .30 1.00               
## 3      x3  .44  .34 1.00          
## 4      x4  .37  .15  .16 1.00     
## 5      x5  .29  .14  .08  .73 1.00
```

.center[.font110[¿Cuántas correlaciones tenemos?]]

--
.center[`15`]

---
# Procedimientos de análisis

## 2. Identificación

¡Último ejemplo! *Nota: Encuentren la regla*

.left-code[

```
## 
## Correlation method: 'pearson'
## Missing treated using: 'pairwise.complete.obs'
```

```
##   rowname   x1   x2   x3   x4   x5   x6   x7   x8
## 1      x1 1.00                                   
## 2      x2  .30 1.00                              
## 3      x3  .44  .34 1.00                         
## 4      x4  .37  .15  .16 1.00                    
## 5      x5  .29  .14  .08  .73 1.00               
## 6      x6  .36  .19  .20  .70  .72 1.00          
## 7      x7  .07 -.08  .07  .17  .10  .12 1.00     
## 8      x8  .22  .09  .19  .11  .14  .15  .49 1.00
```
]

.right-code[
.center[.font110[¿Cuántas correlaciones tenemos?]]
<br> .center[`36`]
<br>
.center[$$Información= \frac{n(n+1)}{2}$$]
]

---
# Procedimientos de análisis

## 2. Identificación
.left-plot[
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]

.right-code[¿Cuánta información estamos solicitando aquí? 
En total, solicitamos calcular las `cargas factoriales` (*varianza común*) y el `error de medición` (*varianza específica*):
<br> Cargas factoriales: `λ1, λ2, λ3`
<br> Error de medición: `a, b, c`

.center[*Entonces estamos solicitando 6 informaciones*]

Siendo 3 ítems que ingresan al modelo. 
<br> <br>
.center[**¿De cuánta información disponemos?**]
]

---
# Procedimientos de análisis

## 2. Identificación
### Notas importantes
* Mientras tengamos más de 3 ítems en nuestro modelo, no tendremos problemas de identificación (*sub-identificación* o *no identificado*). 
* En los casos que tenemos modelos con solo 3 ítems, se puede calcular siempre y cuando no se soliciten mayor información (*apenas identificado*) como por ejemplo, `correlación de errores entre ítems`. Debido a que con 3 ítems tenemos 6 cantidad de informaciones y estaríamos solicitando un total de 7 informaciones.
* No es imposible trabajar con modelos de 2 o 1 ítem (*ítems únicos*), siempre y cuando se empleen algunos artificios. Por ejemplo, igualar cargas factoriales o ingresar información previa sobre algún parámetro (de esta manera no se volverá a calcular).

---
# Procedimientos de análisis

## 3. Estimación

.center[<b>Tabla Resumen de Estimadores</b>]
<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Estimadores </th>
   <th style="text-align:center;"> Estimadores Robustos </th>
   <th style="text-align:center;"> Descripciones </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> ML (Máxima Verosimilitud) </td>
   <td style="text-align:center;"> MLM, MLR, MLMVS, MLMV </td>
   <td style="text-align:center;"> Datos continuos </td>
  </tr>
  <tr>
   <td style="text-align:center;"> ULS (Mínimos cuadrados no ponderados) </td>
   <td style="text-align:center;"> ULSM, ULSMVS, ULSMV </td>
   <td style="text-align:center;"> Variables Categóricas </td>
  </tr>
  <tr>
   <td style="text-align:center;"> WLS (Mínimos cuadrados ponderados) </td>
   <td style="text-align:center;"> - </td>
   <td style="text-align:center;"> Variables Categóricas </td>
  </tr>
  <tr>
   <td style="text-align:center;"> DWLS (Mínimos cuadrados con diagonal ponderada </td>
   <td style="text-align:center;"> WLSM, WLSMVS, WLSMV </td>
   <td style="text-align:center;"> Variables Categóricas. Es el más recomendado en la actualidad </td>
  </tr>
</tbody>
</table>

---
# Procedimientos de análisis

## 3. Estimación

Aclaración sobre los sufijos de las nomenclaturas para los **estimadores robustos**:
* ABC`M`: Trabaja con errores robustos y corrección para chi-cuadrado *Satorra-Bentler*
* ABC`MVS`: Trabaja con errores robustos y corrección para media-varianza y chi-cuadrado *Satterthwaite*
* ABC`MV`: Trabaja con errores robustos y corrección para media-varianza y chi-cuadrado *scale-shifted*

Los estudios de simulación coinciden en que la corrección **scale-shifted** es quien brinda mejores resultados siempre y cuando se tenga una cantidad de datos *suficiente* (n > 250, dependiendo de la cantidad de ítems).
<br>
Para nuestros casos: Los mejores estimadores a utilizar cuando trabajemos con datos contínuos será `MLR` (en caso de no-normalidad) y `WLSMV` (en caso de datos categóricos).

---
# Procedimientos de análisis

## 4. Evaluación

Este procedimiento hace referencia al cálculo y valoración de los ***índices de ajuste*** de los modelos estimados, así como a las cargas factoriales calculadas.
* Estos índices son derivados del test *chi-cuadrado* por lo que a medida que cambie este, el resto de los índices cambiará. 
* A medida que se emplee un estimador diferente (*recordar que hace correcciones a esta prueba*) y/o aumenten o desciendan el número de ítems/factores, los índices de ajuste cambiarán. 
* Por último a medida que se ingresen o resten especificaciones al modelo, los **índices de ajuste** cambiarán.

Este procedimiento nos permitirá tomar la decisión de finalizar el análisis en este punto o ir al siguiente paso ***Re-Especificación***.

---
# Procedimientos de análisis

## 4. Evaluación

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Índices de ajuste </th>
   <th style="text-align:center;"> Descriptivos </th>
   <th style="text-align:center;"> Puntos de Corte </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> χ² </td>
   <td style="text-align:center;"> Test Chi-Cuadrado (bondad de ajuste) </td>
   <td style="text-align:center;"> No estadísticamente significativo </td>
  </tr>
  <tr>
   <td style="text-align:center;"> χ²/gl </td>
   <td style="text-align:center;"> Medida de parsimonia </td>
   <td style="text-align:center;"> Menor a 3 o 2 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> CFI (Comparative Fit Index) </td>
   <td style="text-align:center;vertical-align: middle !important;" rowspan="2"> Medida de ajuste independiente o incremental </td>
   <td style="text-align:center;vertical-align: middle !important;" rowspan="2"> ≥ .90 = ajuste adecuado; ≥ .95 = buen ajuste </td>
  </tr>
  <tr>
   <td style="text-align:center;"> TLI (Tucker Lewis Index) o NNFI (Non-Normed Fit Index) </td>
   
   
  </tr>
  <tr>
   <td style="text-align:center;"> RMSEA (Root Mean Square Error of Approximation) </td>
   <td style="text-align:center;"> Evalúa que tan lejos está de un modelo perfecto </td>
   <td style="text-align:center;"> ≥ .10 = ajuste pobre; ≤ .08 = adecuado ajuste, 
                        ≤ .05 = buen ajuste </td>
  </tr>
  <tr>
   <td style="text-align:center;"> SRMR (Standardized Root Mean Residual) </td>
   <td style="text-align:center;"> Evalúa que tan grande es el error de reproducir el modelo </td>
   <td style="text-align:center;"> ≤ .08 = buen ajuste, ≤ .06: ideal </td>
  </tr>
</tbody>
</table>

---
# Procedimientos de análisis

## 5. Re-Especificación

Este procedimiento hace referencia a empezar nuevamente el flujo del análisis, quitar, aumentar algo en el proceso de `Espeficación` que permita tener un modelo factorial idóneo.

Una de las cosas que ayudan en esta etapa es el cálculo de los `índices de modificación`.
En el paquete `lavaan` podremos realizar rápidamente con la sintaxis:

```r
modificationindices()
```

--
<br>
.center[.font150[¿Listos para hacer un ejemplo?]]

---

# Ejemplificación de caso

Para este primer ejemplo usaremos la BD de Holzinger & Swineford (1939). Se tienen 9 pruebas que se estructuran en 3 factores latentes: Visual, Textual y Velocidad.

```r
lavaan::HolzingerSwineford1939
```
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class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>id<\/th>\n      <th>sex<\/th>\n      <th>ageyr<\/th>\n      <th>agemo<\/th>\n      <th>school<\/th>\n      <th>grade<\/th>\n      <th>x1<\/th>\n      <th>x2<\/th>\n      <th>x3<\/th>\n      <th>x4<\/th>\n      <th>x5<\/th>\n      <th>x6<\/th>\n      <th>x7<\/th>\n      <th>x8<\/th>\n      <th>x9<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"pageLength":4,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4,6,7,8,9,10,11,12,13,14,15]},{"orderable":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[4,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

# Ejemplificación de caso

## 1. Especificación

```r
model <-  " visual  =~ x1 + x2 + x3
            textual =~ x4 + x5 + x6
            speed   =~ x7 + x8 + x9 "
```

## 2. Identificación y Estimación
Almacenamos la estimación en el objeto `fit` sobre el cuál consultaremos para obtener información.

*Recordar:* Es importante el almacenar información con `<-`

```r
library(lavaan)
```

```
## This is lavaan 0.6-6
```

```
## lavaan is BETA software! Please report any bugs.
```

```r
fit <- cfa(model = model,
           data  = HolzingerSwineford1939)
```

---

# Ejemplificación de caso

## 4. Evaluación

.scroll-output[

```r
summary(fit)
```

---

# Ejemplificación de caso

## 4. Evaluación

.scroll-output[

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

```
## lavaan 0.6-6 ended normally after 35 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                         21
##                                                       
##   Number of observations                           301
##                                                       
## Model Test User Model:
##                                                       
##   Test statistic                                85.306
##   Degrees of freedom                                24
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                               918.852
##   Degrees of freedom                                36
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.931
##   Tucker-Lewis Index (TLI)                       0.896
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -3737.745
##   Loglikelihood unrestricted model (H1)      -3695.092
##                                                       
##   Akaike (AIC)                                7517.490
##   Bayesian (BIC)                              7595.339
##   Sample-size adjusted Bayesian (BIC)         7528.739
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.092
##   90 Percent confidence interval - lower         0.071
##   90 Percent confidence interval - upper         0.114
##   P-value RMSEA <= 0.05                          0.001
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.065
## 
## 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
##   visual =~                                                             
##     x1                1.000                               0.900    0.772
##     x2                0.554    0.100    5.554    0.000    0.498    0.424
##     x3                0.729    0.109    6.685    0.000    0.656    0.581
##   textual =~                                                            
##     x4                1.000                               0.990    0.852
##     x5                1.113    0.065   17.014    0.000    1.102    0.855
##     x6                0.926    0.055   16.703    0.000    0.917    0.838
##   speed =~                                                              
##     x7                1.000                               0.619    0.570
##     x8                1.180    0.165    7.152    0.000    0.731    0.723
##     x9                1.082    0.151    7.155    0.000    0.670    0.665
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   visual ~~                                                             
##     textual           0.408    0.074    5.552    0.000    0.459    0.459
##     speed             0.262    0.056    4.660    0.000    0.471    0.471
##   textual ~~                                                            
##     speed             0.173    0.049    3.518    0.000    0.283    0.283
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .x1                0.549    0.114    4.833    0.000    0.549    0.404
##    .x2                1.134    0.102   11.146    0.000    1.134    0.821
##    .x3                0.844    0.091    9.317    0.000    0.844    0.662
##    .x4                0.371    0.048    7.779    0.000    0.371    0.275
##    .x5                0.446    0.058    7.642    0.000    0.446    0.269
##    .x6                0.356    0.043    8.277    0.000    0.356    0.298
##    .x7                0.799    0.081    9.823    0.000    0.799    0.676
##    .x8                0.488    0.074    6.573    0.000    0.488    0.477
##    .x9                0.566    0.071    8.003    0.000    0.566    0.558
##     visual            0.809    0.145    5.564    0.000    1.000    1.000
##     textual           0.979    0.112    8.737    0.000    1.000    1.000
##     speed             0.384    0.086    4.451    0.000    1.000    1.000
```
]

---

## 5. Re-Especificación

```r
modindices(fit, sort = TRUE, maximum.number = 10)  
```

```
##        lhs op rhs     mi    epc sepc.lv sepc.all sepc.nox
## 30  visual =~  x9 36.411  0.577   0.519    0.515    0.515
## 76      x7 ~~  x8 34.145  0.536   0.536    0.859    0.859
## 28  visual =~  x7 18.631 -0.422  -0.380   -0.349   -0.349
## 78      x8 ~~  x9 14.946 -0.423  -0.423   -0.805   -0.805
## 33 textual =~  x3  9.151 -0.272  -0.269   -0.238   -0.238
## 55      x2 ~~  x7  8.918 -0.183  -0.183   -0.192   -0.192
## 31 textual =~  x1  8.903  0.350   0.347    0.297    0.297
## 51      x2 ~~  x3  8.532  0.218   0.218    0.223    0.223
## 59      x3 ~~  x5  7.858 -0.130  -0.130   -0.212   -0.212
## 26  visual =~  x5  7.441 -0.210  -0.189   -0.147   -0.147
```

---
class: inverse, center, middle

# .font120[**Fiabilidad por consistencia interna**]

---

# Fiabilidad por consistencia interna
* ### Problemática de Alfa y la tau-equivalencia
* ### Procedimiento en software

---
class: inverse, center, middle

# .font120[**Gracias**]