MULTIVARIJATNE STATISTIČKE METODE

# MULTIVARIJATNE STATISTIČKE METODE
## Predavanje 5: Faktorska analiza
### Luka Šikić, PhD
### Fakultet hrvatskih studija | <a href="https://lusiki.github.io/WebMultiVar/">Github MV</a>

---

# Pregled predavanja

1. [Jednodimenzionalna faktorska analiza (FA)](#uni)

2. [Eksplorativna FA (EFA)](#multi)

3. [Konfirmatorna FA (CFA)](#conf)

---
class: inverse, center, middle
name: uni

# FAKTORSKA ANALIZA

(Općenito)

---

# Karakteristike FA
 
 
 
- Charles Spearman (1904) zaslužan za uvođenje FA
 
- FA ima mnogo sličnosti sa PCA
 
- Cilj FA je reducirati veliki broj (koreliranih) varijabli na mali broj indeksa (faktora)
 
- FA je statistički model (za razliku od PCA koja je statistička metoda)
 
- Ekstenzija (C)FA je modeliranje strukturnih jednadžbi (SEM) 
 
 
.footnote[[*]Izvstan uvodni [turorial](https://stats.idre.ucla.edu/spss/seminars/introduction-to-factor-analysis/a-practical-introduction-to-factor-analysis/).]
---

# Vrste FA

- koristi se u fazi razvoja modela
- pregled faktorske strukture
- opća slika o varijablama
]

- procijeni kostnrukt
- koristi se nakon razvoja mjere

]

.footnote[[*]Ogledni primjer za motivaciju FA na [poveznici](https://pdf.sciencedirectassets.com/280203/1-s2.0-S1877050915X00123/1-s2.0-S1877050915013605/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEGgaCXVzLWVhc3QtMSJGMEQCIEKL%2BgWTeDULrBDdHjLVVYTpTw0x1b3IcmAwY2fj49agAiB%2F6alKu3d8Cd6HSjQBNDTNSPxADKPdUigLTWG0YLbrpSq9Awih%2F%2F%2F%2F%2F%2F%2F%2F%2F%2F8BEAMaDDA1OTAwMzU0Njg2NSIMR5PnGB3hxYeBGMh5KpEDhNllxnviP6RXrFwFD6GCbdR4qA8%2BcDAcuyonvbAcLcYcjusNF1bVoxijJsUrTO7IHkq1%2F6gfTMW%2Fuis0iw0h5BZsVMg8d2xIt4XVDq2uCE1378Si6BN1zAc7%2BTpI1OX7MlwN0RFYXYcItSs0uGptpELpdqCSC5MbpvOHq2dSk32Rr96UfUtgWP5%2BK00neqh8raUZTOICa9qgQ1e2xraj1yH5PwwZjprfpODiFNnxbUX3FyzYw446DDa0o0hp5Nq0ftEFsQXXLt8PREU4DbGF6URs7yhNp3EUW8FsbNaFBW83FUkq5g8EB22KbnfoaMjL%2FGdX75QKy2Wi2IAKOJ0Gbg1dR8hEMfVvG25lzN3YWBHTZP1CDH43ucW01bUD47CoguLZJ3OlB1YX%2F8VirEyyMqS%2FZDTGRX6PVJw3tt5otbaMDOSxofNnEjIbLMz7POE4XcOPUiTzMorx7xZUqn3n8s6D7jtS%2F9rRExcwJ%2B8KGkG8npqhI70AGEthDjpR6Ja4%2FwnPp2LT07McMcYzGcfORrQwyJHhggY67AGoh98fTEoT0vbi3g%2FgJjuNSkhome8unAE873Wvjb0czEg8VUzqUmq2lgHQFbsrRqyly1HXxTi1garhIhwOXMs7OH%2F13G4dbweUnzHfvHGhkmCGPNEPCO%2BuiAzsv25woyDCYEabk1PIWUbmF5Zy2blhh4tvZ2STlB1w6tM6GVYbXVAHPO3m4CrLYfSDdw86vPFkzWAs2lav7u911%2FCJBWFX%2B18kZ0yN7ChfqR2t2M46ZtQWAbG3NLWlrnWMcpq4hwkth0VI84FgisBGO8jRU5L3wdJcNYFUj7K8DczHTrE6XzDJhIBVldLhUpXjAA%3D%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20210322T080825Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYWHRILQQO%2F20210322%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=6d94252743a7ed60e313fbcccbf494512775492c51e14248e59850866d82cdde&hash=ff6722f6278a608c06c40e587140d6976c1cd76177433d4700cc253bb613a370&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S1877050915013605&tid=spdf-2c946717-28f9-49fa-b844-b8b4f083fea2&sid=b1057b8a1956c546f55a7420b46302cb07cbgxrqb&type=client).]

---

# Eksplorativna vs. konfirmatorna FA

---

# OGLEDNI PRIMJER

(Pretpostavke za analizu)

---

# Paket i podatci

- koristimo `psych` paket
- za detalje o paketu pogledajte [Personality project](http://personality-project.org/r/psych/)

```r
# učitaj paket
library(psych)
```

- koristimo `gcbs` podatkovni skup (Generic Conspiracist Beliefs Survey)
- provjerite da li i vas prisluškuju na [Open psychometrics](https://openpsychometrics.org/tests/GCBS/) :-)

```r
gcbs <- readRDS("../Podatci/GCBS_data.rds")
```

---

# Pregledaj podatke

```r
str(gcbs)
```

```
## 'data.frame':	2495 obs. of  15 variables:
##  $ Q1 : int  5 5 2 5 5 1 4 5 1 1 ...
##  $ Q2 : int  5 5 4 4 4 1 3 4 1 2 ...
##  $ Q3 : int  3 5 1 1 1 1 3 3 1 1 ...
##  $ Q4 : int  5 5 2 2 4 1 3 3 1 1 ...
##  $ Q5 : int  5 5 2 4 4 1 4 4 1 1 ...
##  $ Q6 : int  5 3 2 5 5 1 3 5 1 5 ...
##  $ Q7 : int  5 5 4 4 4 1 3 5 1 1 ...
##  $ Q8 : int  3 5 2 1 3 1 4 5 1 1 ...
##  $ Q9 : int  4 1 2 4 1 1 2 5 1 1 ...
##  $ Q10: int  5 4 4 5 5 1 3 5 1 4 ...
##  $ Q11: int  5 4 2 5 5 1 3 5 1 1 ...
##  $ Q12: int  5 5 4 5 5 1 2 5 1 1 ...
##  $ Q13: int  3 4 0 1 3 1 2 3 1 1 ...
##  $ Q14: int  5 4 2 4 5 1 3 4 1 1 ...
##  $ Q15: int  5 5 4 5 5 1 4 5 1 5 ...
```

---

# Konstrukti
 
 
- Zavjera vlade (GM)
 
- Prikrivanje vanzemaljaca (ET)
 
- Zle i zavjereničke globalne organizacije (MG)
 
- Osobna dobrobit (PW)
 
- Kontrola informacija (CI)

Za detalje o varijablama pogledajte [Measuring belief in conspiracy theories](https://www.frontiersin.org/articles/10.3389/fpsyg.2013.00279/full).

---

# Varijable i konstrukti

---

# Jednodimenzionalni konstrukt

---

# Jednodimenzionalni konstrukt

```r
EFA_model <- fa(gcbs) # Default je 1 faktor
EFA_model$loadings
```

```
## 
## Loadings:
##     MR1  
## Q1  0.703
## Q2  0.719
## Q3  0.638
## Q4  0.770
## Q5  0.672
## Q6  0.746
## Q7  0.734
## Q8  0.654
## Q9  0.695
## Q10 0.565
## Q11 0.719
## Q12 0.786
## Q13 0.679
## Q14 0.743
## Q15 0.574
## 
##                  MR1
## SS loadings    7.267
## Proportion Var 0.484
```
]
]

```r
fa.diagram(EFA_model)
```

]

---
class: inverse, center, middle
name: multi

# EKSPLORATIVNA (E)FA

(Bez teorije...)

---

# Procedura provedbe EFA
 
 
- Provjeri faktorabilnost
 
 
- Izvuci faktore
 
 
- Izaberi odgovarajući broj faktora
 
 
- Rotiraj faktore
 
 
- Interpretiraj rezultate

---

# Podatci

- koristimo podatke [Synthetic Aperture Personality Assessment](https://www.sapa-project.org/survey/start.php) (SAPA)
- 2.800 osoba
- 25 pitanja
.tiny[

```r
data(bfi)
str(bfi)
```

```
## 'data.frame':	2800 obs. of  28 variables:
##  $ A1       : int  2 2 5 4 2 6 2 4 4 2 ...
##  $ A2       : int  4 4 4 4 3 6 5 3 3 5 ...
##  $ A3       : int  3 5 5 6 3 5 5 1 6 6 ...
##  $ A4       : int  4 2 4 5 4 6 3 5 3 6 ...
##  $ A5       : int  4 5 4 5 5 5 5 1 3 5 ...
##  $ C1       : int  2 5 4 4 4 6 5 3 6 6 ...
##  $ C2       : int  3 4 5 4 4 6 4 2 6 5 ...
##  $ C3       : int  3 4 4 3 5 6 4 4 3 6 ...
##  $ C4       : int  4 3 2 5 3 1 2 2 4 2 ...
##  $ C5       : int  4 4 5 5 2 3 3 4 5 1 ...
##  $ E1       : int  3 1 2 5 2 2 4 3 5 2 ...
##  $ E2       : int  3 1 4 3 2 1 3 6 3 2 ...
##  $ E3       : int  3 6 4 4 5 6 4 4 NA 4 ...
##  $ E4       : int  4 4 4 4 4 5 5 2 4 5 ...
##  $ E5       : int  4 3 5 4 5 6 5 1 3 5 ...
##  $ N1       : int  3 3 4 2 2 3 1 6 5 5 ...
##  $ N2       : int  4 3 5 5 3 5 2 3 5 5 ...
##  $ N3       : int  2 3 4 2 4 2 2 2 2 5 ...
##  $ N4       : int  2 5 2 4 4 2 1 6 3 2 ...
##  $ N5       : int  3 5 3 1 3 3 1 4 3 4 ...
##  $ O1       : int  3 4 4 3 3 4 5 3 6 5 ...
##  $ O2       : int  6 2 2 3 3 3 2 2 6 1 ...
##  $ O3       : int  3 4 5 4 4 5 5 4 6 5 ...
##  $ O4       : int  4 3 5 3 3 6 6 5 6 5 ...
##  $ O5       : int  3 3 2 5 3 1 1 3 1 2 ...
##  $ gender   : int  1 2 2 2 1 2 1 1 1 2 ...
##  $ education: int  NA NA NA NA NA 3 NA 2 1 NA ...
##  $ age      : int  16 18 17 17 17 21 18 19 19 17 ...
```
]

---

# Podatci

```r
head(bfi)
```

```
##       A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 E1 E2 E3 E4 E5 N1 N2 N3 N4 N5 O1 O2 O3 O4
## 61617  2  4  3  4  4  2  3  3  4  4  3  3  3  4  4  3  4  2  2  3  3  6  3  4
## 61618  2  4  5  2  5  5  4  4  3  4  1  1  6  4  3  3  3  3  5  5  4  2  4  3
## 61620  5  4  5  4  4  4  5  4  2  5  2  4  4  4  5  4  5  4  2  3  4  2  5  5
## 61621  4  4  6  5  5  4  4  3  5  5  5  3  4  4  4  2  5  2  4  1  3  3  4  3
## 61622  2  3  3  4  5  4  4  5  3  2  2  2  5  4  5  2  3  4  4  3  3  3  4  3
## 61623  6  6  5  6  5  6  6  6  1  3  2  1  6  5  6  3  5  2  2  3  4  3  5  6
##       O5 gender education age
## 61617  3      1        NA  16
## 61618  3      2        NA  18
## 61620  2      2        NA  17
## 61621  5      2        NA  17
## 61622  3      1        NA  17
## 61623  1      2         3  21
```

---

# Podatci

```r
names(bfi)
```

```
##  [1] "A1"        "A2"        "A3"        "A4"        "A5"        "C1"       
##  [7] "C2"        "C3"        "C4"        "C5"        "E1"        "E2"       
## [13] "E3"        "E4"        "E5"        "N1"        "N2"        "N3"       
## [19] "N4"        "N5"        "O1"        "O2"        "O3"        "O4"       
## [25] "O5"        "gender"    "education" "age"
```

---

# Konstrukt

---

# Konstrukt

- konstrukt se ne može direktno mjeriti
- primjeri: samoodređenost, sposobnost razmišljanja, politička pripadnost, ekstrovertiranost 
 
 
<img src="../Foto/konstrukt_FA.png" width="450px" style="display: block; margin: auto;" />

---

# ...bez teorije

<img src="../Foto/noTheory_FA.png" width="550px" style="display: block; margin: auto;" />
---

# Faktorabilnost
 
 
.pull-left[
- `Barlett sphericity` test (BS)

```r
bfi_hetcor <- polycor::hetcor(bfi[,1:25]) # korelacije
bfi_c <- bfi_hetcor$correlations # kor- matrica
bfi_faktorablinost <- cortest.bartlett(bfi_c) # Barlett test
bfi_faktorablinost
```

```
## $chisq
## [1] 671.9842
## 
## $p.value
## [1] 1.497658e-30
## 
## $df
## [1] 300
```

]

```r
KMO(bfi_c) # KMO test
```

```
## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = bfi_c)
## Overall MSA =  0.85
## MSA for each item = 
##   A1   A2   A3   A4   A5   C1   C2   C3   C4   C5   E1   E2   E3   E4   E5   N1 
## 0.75 0.84 0.87 0.88 0.90 0.84 0.80 0.85 0.83 0.86 0.84 0.88 0.90 0.88 0.89 0.78 
##   N2   N3   N4   N5   O1   O2   O3   O4   O5 
## 0.78 0.86 0.89 0.86 0.86 0.78 0.84 0.77 0.76
```

]

---

# Ekstrakcija faktora
 
 
- `minres` : minimum residual [default] (slightly modi,ed methods: ols , wls , gls )
 
- `mle` : Maximum Likelihood Estimation (MLE)
 
- `paf` : Principal Axes Factor (PAF) extraction
 
- `minchi` : minimum sample size weighted chi square
 
- `minrank` : minimum rank
 
- `alpha` : alpha factoring

---

# `minres` metoda

```r
# EFA sa 3 faktora
f_bfi_minres <- fa(bfi_c,nfactors = 3,rotate = "none")
# Sortiraj komunalnosti
f_bfi_minres_common <- sort(f_bfi_minres$communality,decreasing = TRUE)
# Pregled
head(data.frame(f_bfi_minres_common),8)
```

```
##    f_bfi_minres_common
## N1           0.5671549
## N3           0.5611354
## N2           0.5567788
## E4           0.4892165
## C4           0.4494178
## N4           0.4491858
## A5           0.4389367
## E2           0.4208485
```
---

# `minres` metoda

```r
# Sortiraj jedinstvenosti
f_bfi_minres_unique <- sort(f_bfi_minres$uniqueness,decreasing = TRUE)
# Pregled
head(data.frame(f_bfi_minres_unique),10)
```

```
##    f_bfi_minres_unique
## A1           0.9383857
## O5           0.9194360
## O4           0.9180685
## O2           0.9033127
## O1           0.8399469
## A4           0.8087487
## O3           0.7693042
## C3           0.7636974
## E1           0.7349999
## N5           0.7161468
```

---

# `MLE` metoda

```r
# MLE ekstrakcija
f_bfi_mle <- fa(bfi_c, nfactors = 3, fm ="ml")
```

```
## Loading required namespace: GPArotation
```

```r
# Sortiraj komunalnost
f_bfi_mle_common <- sort(f_bfi_mle$communality,decreasing = TRUE)
# Pregled
head(data.frame(f_bfi_mle_common),10)
```

```
##    f_bfi_mle_common
## N1        0.6602592
## N2        0.6427504
## N3        0.5241467
## E4        0.4912137
## C4        0.4756779
## A5        0.4348396
## E2        0.4194426
## C2        0.4188591
## E3        0.4055702
## N4        0.3923124
```

---

# Koliko faktora zadržati?
 
- Kaiser-Guttman test
- Scree test
- Parallel analiza

```r
fa.parallel(bfi_c, n.obs = 200,fa = "fa", fm = "minres") # minres metoda 
```

```
## Parallel analysis suggests that the number of factors =  5  and the number of components =  NA
```

---

# Koliko faktora zadržati?

```r
fa.parallel(bfi_c, n.obs = 200,fa = "fa", fm = "mle") # MLE metoda
```

```
## Parallel analysis suggests that the number of factors =  5  and the number of components =  NA
```

---

# Rotacija faktora
 
 
- Rotacija se radi zbog lakše interpretacije

---

# Rotacija faktora
 
 
 
<img src="../Foto/roto_fa.png" width="700px" style="display: block; margin: auto;" />

---

# Varimax rotacija

```r
f_bfi_varimax <- fa(bfi_c,fm = "minres",nfactors = 5,rotate = "varimax")
```
 
<img src="../Foto/varimax_fa.png" width="450px" style="display: block; margin: auto;" />

---

# Interpretacija

```r
fa.diagram(f_bfi_varimax)
```

<img src="05_FA_files/figure-html/unnamed-chunk-27-1.png" style="display: block; margin: auto;" />
---

# Interpretacija

```r
print(f_bfi_varimax$loadings, cut=0)
```

```
## 
## Loadings:
##    MR2    MR1    MR3    MR5    MR4   
## A1  0.111  0.040  0.023 -0.428 -0.078
## A2  0.030  0.214  0.139  0.627  0.062
## A3  0.009  0.318  0.109  0.651  0.056
## A4 -0.066  0.205  0.231  0.436 -0.113
## A5 -0.122  0.393  0.088  0.537  0.067
## C1  0.010  0.070  0.546  0.039  0.210
## C2  0.090  0.033  0.649  0.103  0.115
## C3 -0.031  0.024  0.557  0.112 -0.005
## C4  0.240 -0.065 -0.634 -0.037 -0.108
## C5  0.290 -0.176 -0.562 -0.048  0.037
## E1  0.043 -0.575  0.033 -0.105 -0.059
## E2  0.245 -0.679 -0.102 -0.113 -0.042
## E3  0.024  0.537  0.083  0.258  0.281
## E4 -0.116  0.647  0.102  0.306 -0.073
## E5  0.036  0.504  0.313  0.090  0.214
## N1  0.787  0.079 -0.046 -0.216 -0.085
## N2  0.754  0.027 -0.031 -0.194 -0.010
## N3  0.732 -0.061 -0.067 -0.028 -0.004
## N4  0.591 -0.345 -0.179  0.006  0.075
## N5  0.538 -0.161 -0.037  0.101 -0.150
## O1 -0.002  0.213  0.115  0.062  0.505
## O2  0.176  0.005 -0.100  0.082 -0.469
## O3  0.027  0.311  0.077  0.127  0.596
## O4  0.221 -0.191 -0.022  0.155  0.369
## O5  0.085 -0.005 -0.063 -0.010 -0.534
## 
##                  MR2   MR1   MR3   MR5   MR4
## SS loadings    2.710 2.473 2.041 1.844 1.522
## Proportion Var 0.108 0.099 0.082 0.074 0.061
## Cumulative Var 0.108 0.207 0.289 0.363 0.424
```
]

---
class: inverse, center, middle

# Procjena kvalitete modela

(Koliko je pouzdana procjena!?)

---

# Podijeli podatke

```r
N <- nrow(bfi)
indices <- seq(1, N)
indices_EFA <- sample(indices, floor((.5*N)))
indices_CFA <- indices[!(indices %in% indices_EFA)]
bfi_EFA <- bfi[indices_EFA, ]
bfi_CFA <- bfi[indices_CFA, ]
```

---

# Podijeli podatke

```r
head(bfi_EFA, 2)
```

```
##       A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 E1 E2 E3 E4 E5 N1 N2 N3 N4 N5 O1 O2 O3 O4
## 64069  2  6  5  3  4  2  3  6  5  5  1  3  4  6  5  3  4  5  1  4  4  3  4  6
## 64495  3  5  4  6  5  5  5  6  1  1  6  6  2  2  4  4  4  2  4  4  5  1  5  6
##       O5 gender education age
## 64069  3      2         4  24
## 64495  2      1         4  28
```

```r
head(bfi_CFA, 2)
```

```
##       A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 E1 E2 E3 E4 E5 N1 N2 N3 N4 N5 O1 O2 O3 O4
## 61617  2  4  3  4  4  2  3  3  4  4  3  3  3  4  4  3  4  2  2  3  3  6  3  4
## 61620  5  4  5  4  4  4  5  4  2  5  2  4  4  4  5  4  5  4  2  3  4  2  5  5
##       O5 gender education age
## 61617  3      1        NA  16
## 61620  2      2        NA  17
```

---

# Korelacijska matrica

```r
bfi_EFA_cor <- cor(bfi_EFA, use = "pairwise.complete.obs")
head(bfi_EFA_cor,4) 
```

```
##            A1         A2         A3         A4         A5         C1         C2
## A1  1.0000000 -0.3352959 -0.2728057 -0.1375883 -0.1514420 0.01739205 0.01060102
## A2 -0.3352959  1.0000000  0.4877745  0.3459091  0.3811325 0.12697746 0.14392010
## A3 -0.2728057  0.4877745  1.0000000  0.3613521  0.4900486 0.12489855 0.13785354
## A4 -0.1375883  0.3459091  0.3613521  1.0000000  0.3235077 0.09467829 0.23279038
##             C3         C4          C5         E1          E2         E3
## A1 -0.01783306  0.1376209  0.04123172  0.1112480  0.09490553 -0.0283627
## A2  0.23100935 -0.1588774 -0.12757642 -0.2207431 -0.22859332  0.2721867
## A3  0.16741553 -0.1454697 -0.16289472 -0.2483077 -0.32009550  0.4171098
## A4  0.13725781 -0.1463744 -0.20863281 -0.1310965 -0.18430445  0.1784194
##             E4          E5          N1          N2          N3          N4
## A1 -0.06212886 -0.03560382  0.15099350  0.12728828  0.10307092  0.02589476
## A2  0.30960792  0.29424172 -0.09966623 -0.07923077 -0.05475419 -0.10669241
## A3  0.39248887  0.26610977 -0.10400972 -0.08546580 -0.05685266 -0.14574324
## A4  0.30785232  0.16574148 -0.08886486 -0.12879214 -0.06764519 -0.13969324
##             N5         O1          O2          O3          O4          O5
## A1  0.01821888 0.02036992  0.09238427 -0.06042714 -0.07379101  0.10281922
## A2  0.02189162 0.12222326  0.02202875  0.15042011  0.08569123 -0.09351295
## A3 -0.03098240 0.12210058 -0.02616394  0.23878685  0.02402204 -0.06915835
## A4  0.03268684 0.06035262  0.04658407  0.07070544 -0.04244752  0.01176520
##        gender   education         age
## A1 -0.1186945 -0.13282060 -0.15790071
## A2  0.1957883 -0.01801807  0.12943842
## A3  0.1592532 -0.01504874  0.07515617
## A4  0.1500163 -0.04727867  0.14161828
```
]

---

# Svojsvene vrijednosti

```r
scree(bfi_EFA_cor, factors = FALSE)
```

---

# Provedi FA

```r
EFA_model <- fa(bfi_EFA, nfactors = 6)
EFA_model
```

```
## Factor Analysis using method = minres
## Call: fa(r = bfi_EFA, nfactors = 6)
## Standardized loadings (pattern matrix) based upon correlation matrix
## MR2 MR1 MR3 MR5 MR4 MR6 h2 u2 com
## A1 0.07 -0.05 0.06 -0.55 0.00 0.37 0.371 0.63 1.9
## A2 -0.01 -0.07 0.10 0.63 0.03 -0.01 0.473 0.53 1.1
## A3 -0.05 -0.18 0.04 0.57 0.09 0.11 0.500 0.50 1.4
## A4 -0.06 -0.08 0.19 0.40 -0.12 0.13 0.283 0.72 2.1
## A5 -0.19 -0.20 0.02 0.44 0.08 0.21 0.458 0.54 2.5
## C1 0.03 0.09 0.51 -0.02 0.18 0.08 0.325 0.68 1.4
## C2 0.09 0.14 0.65 0.02 0.11 0.20 0.492 0.51 1.4
## C3 0.03 0.09 0.54 0.12 -0.04 0.06 0.302 0.70 1.2
## C4 0.07 0.06 -0.65 -0.04 -0.01 0.27 0.543 0.46 1.4
## C5 0.14 0.16 -0.60 0.03 0.10 0.05 0.457 0.54 1.3
## E1 -0.14 0.58 0.10 -0.15 -0.09 0.05 0.382 0.62 1.4
## E2 0.03 0.73 0.00 -0.06 -0.06 0.03 0.590 0.41 1.0
## E3 0.00 -0.32 0.01 0.16 0.37 0.28 0.486 0.51 3.2
## E4 -0.07 -0.54 0.03 0.21 -0.02 0.27 0.546 0.45 1.9
## E5 0.17 -0.41 0.25 0.05 0.26 0.00 0.415 0.59 2.9
## N1 0.84 -0.11 0.01 -0.09 -0.04 -0.02 0.681 0.32 1.1
## N2 0.80 -0.05 0.01 -0.06 0.03 -0.06 0.623 0.38 1.0
## N3 0.68 0.12 -0.04 0.07 0.03 0.08 0.543 0.46 1.1
## N4 0.45 0.39 -0.16 0.12 0.06 0.02 0.491 0.51 2.4
## N5 0.46 0.20 -0.01 0.24 -0.18 0.15 0.396 0.60 2.5
## O1 -0.02 -0.05 0.05 -0.06 0.56 0.09 0.338 0.66 1.1
## O2 0.11 -0.02 -0.08 0.09 -0.41 0.28 0.284 0.72 2.2
## O3 0.01 -0.06 0.02 0.04 0.66 0.05 0.487 0.51 1.0
## O4 0.09 0.38 -0.01 0.18 0.34 0.02 0.246 0.75 2.6
## O5 0.05 -0.05 -0.03 -0.02 -0.52 0.27 0.348 0.65 1.5
## gender 0.19 -0.06 0.13 0.31 -0.18 -0.07 0.150 0.85 3.0
## education -0.01 0.09 0.00 0.08 0.09 -0.21 0.062 0.94 2.1
## age -0.02 0.02 0.07 0.24 -0.01 -0.23 0.103 0.90 2.2
## 
## MR2 MR1 MR3 MR5 MR4 MR6
## SS loadings 2.52 2.19 2.09 1.95 1.78 0.84
## Proportion Var 0.09 0.08 0.07 0.07 0.06 0.03
## Cumulative Var 0.09 0.17 0.24 0.31 0.38 0.41
## Proportion Explained 0.22 0.19 0.18 0.17 0.16 0.07
## Cumulative Proportion 0.22 0.41 0.60 0.77 0.93 1.00
## 
## With factor correlations of 
## MR2 MR1 MR3 MR5 MR4 MR6
## MR2 1.00 0.24 -0.18 -0.06 -0.01 0.16
## MR1 0.24 1.00 -0.22 -0.27 -0.20 -0.12
## MR3 -0.18 -0.22 1.00 0.18 0.24 0.03
## MR5 -0.06 -0.27 0.18 1.00 0.19 0.17
## MR4 -0.01 -0.20 0.24 0.19 1.00 0.04
## MR6 0.16 -0.12 0.03 0.17 0.04 1.00
## 
## Mean item complexity = 1.8
## Test of the hypothesis that 6 factors are sufficient.
## 
## The degrees of freedom for the null model are 378 and the objective function was 7.75 with Chi Square of 10763.45
## The degrees of freedom for the model are 225 and the objective function was 0.66 
## 
## The root mean square of the residuals (RMSR) is 0.03 
## The df corrected root mean square of the residuals is 0.03 
## 
## The harmonic number of observations is 1374 with the empirical chi square 657.2 with prob < 6.4e-44 
## The total number of observations was 1400 with Likelihood Chi Square = 914.57 with prob < 7.3e-84 
## 
## Tucker Lewis Index of factoring reliability = 0.888
## RMSEA index = 0.047 and the 90 % confidence intervals are 0.044 0.05
## BIC = -715.39
## Fit based upon off diagonal values = 0.98
## Measures of factor score adequacy 
## MR2 MR1 MR3 MR5 MR4 MR6
## Correlation of (regression) scores with factors 0.93 0.89 0.89 0.87 0.86 0.76
## Multiple R square of scores with factors 0.86 0.80 0.78 0.76 0.74 0.58
## Minimum correlation of possible factor scores 0.72 0.60 0.57 0.52 0.49 0.17
```
]

---

# Faktorska opterećenja

```r
EFA_model$loadings
```

```
## 
## Loadings:
##           MR2    MR1    MR3    MR5    MR4    MR6   
## A1                             -0.548         0.369
## A2                              0.633              
## A3               -0.179         0.566         0.107
## A4                       0.194  0.397 -0.119  0.131
## A5        -0.190 -0.201         0.438         0.214
## C1                       0.512         0.184       
## C2                0.138  0.654         0.105  0.197
## C3                       0.537  0.123              
## C4                      -0.649                0.267
## C5         0.140  0.163 -0.600         0.102       
## E1        -0.140  0.576        -0.147              
## E2                0.729                            
## E3               -0.318         0.156  0.372  0.281
## E4               -0.537         0.209         0.272
## E5         0.168 -0.406  0.248         0.262       
## N1         0.841 -0.113                            
## N2         0.804                                   
## N3         0.679  0.117                            
## N4         0.455  0.385 -0.164  0.119              
## N5         0.462  0.197         0.237 -0.175  0.154
## O1                                     0.555       
## O2         0.109                      -0.412  0.281
## O3                                     0.662       
## O4                0.376         0.181  0.338       
## O5                                    -0.525  0.266
## gender     0.191         0.131  0.313 -0.177       
## education                                    -0.214
## age                             0.241        -0.230
## 
##                  MR2   MR1   MR3   MR5   MR4   MR6
## SS loadings    2.430 1.948 1.948 1.782 1.689 0.787
## Proportion Var 0.087 0.070 0.070 0.064 0.060 0.028
## Cumulative Var 0.087 0.156 0.226 0.290 0.350 0.378
```
]

---

# Faktorski koeficijenti

```r
head(EFA_model$scores)
```

```
##              MR2        MR1        MR3        MR5        MR4        MR6
## 64069 0.36384774 -0.5520311 -1.1317340  0.5768891 -0.3937724  0.3201839
## 64495 0.07988474  1.6059104  1.2631588 -0.3233387  0.1540410 -0.7019641
## 62795 0.54030891  0.4617641 -1.2126318 -0.1319860 -2.7327064  0.3696351
## 62942 0.23525170  0.1962169  0.7800890  0.6559302 -0.3013955  0.1670891
## 67238 0.06193102  0.1612078 -0.4647004 -1.6962687  0.2489364  0.3774843
## 66264         NA         NA         NA         NA         NA         NA
```

---

# Procjena kvalitete modela (fit)
 
 
##### Apsolutne mjere

- Chi-square test: nesignifikanstnost
- Tucker-Lewis Index (TLI) : > 0.90
- Root Mean Square Error of Approximation (RMSEA): < 0.05

##### Relativne mjere

- Bayesian Information Criterion (BIC)

---

# Procjena kvalitete modela (relativna)

```r
EFA_model <- fa(bfi_EFA, nfactors = 6)
EFA_model
```

```r
bfi_theory <- fa(bfi_EFA, nfactors = 5)
bfi_eigen <- fa(bfi_EFA, nfactors = 6)

bfi_theory$BIC
```

```
## [1] -466.7392
```

```r
bfi_eigen$BIC
```

```
## [1] -715.3859
```

---
class: inverse, center, middle
name: conf

# KONFIRMATORNA FA (CFA)

(Teorija u pozadini...)

---

# Razlozi za korištenje
 
 
 
- Eksplicitno specificiran odnos među varijablama
 
 
- Testiranje teorije koja je unaprijed poznata
 
 
- Pristup koji se koristi kod razvoja nove mjere

---

# Skica I (*jedan konstrukt*)

---

# Skica II (*više konstrukata*)
 
 
<img src="assets/8697169.png" width="507" style="display: block; margin: auto;" />

---

# Podatci

```r
hz <- lavaan::HolzingerSwineford1939
hz %>% dplyr::glimpse()
```

```
## Rows: 301
## Columns: 15
## $ id <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, ~
## $ sex <int> 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, ~
## $ ageyr <int> 13, 13, 13, 13, 12, 14, 12, 12, 13, 12, 12, 12, 12, 12, 12, 12,~
## $ agemo <int> 1, 7, 1, 2, 2, 1, 1, 2, 0, 5, 2, 11, 7, 8, 6, 1, 11, 5, 8, 3, 1~
## $ school <fct> Pasteur, Pasteur, Pasteur, Pasteur, Pasteur, Pasteur, Pasteur, ~
## $ grade <int> 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, ~
## $ x1 <dbl> 3.333333, 5.333333, 4.500000, 5.333333, 4.833333, 5.333333, 2.8~
## $ x2 <dbl> 7.75, 5.25, 5.25, 7.75, 4.75, 5.00, 6.00, 6.25, 5.75, 5.25, 5.7~
## $ x3 <dbl> 0.375, 2.125, 1.875, 3.000, 0.875, 2.250, 1.000, 1.875, 1.500, ~
## $ x4 <dbl> 2.333333, 1.666667, 1.000000, 2.666667, 2.666667, 1.000000, 3.3~
## $ x5 <dbl> 5.75, 3.00, 1.75, 4.50, 4.00, 3.00, 6.00, 4.25, 5.75, 5.00, 3.5~
## $ x6 <dbl> 1.2857143, 1.2857143, 0.4285714, 2.4285714, 2.5714286, 0.857142~
## $ x7 <dbl> 3.391304, 3.782609, 3.260870, 3.000000, 3.695652, 4.347826, 4.6~
## $ x8 <dbl> 5.75, 6.25, 3.90, 5.30, 6.30, 6.65, 6.20, 5.15, 4.65, 4.55, 5.7~
## $ x9 <dbl> 6.361111, 7.916667, 4.416667, 4.861111, 5.916667, 7.500000, 4.8~
```
.footnote[[*] Vidi detalje o podatcima na [poveznici](https://rdrr.io/cran/lavaan/man/HolzingerSwineford1939.html).]

---

# Definiraj model

##### Sintaksa za definiranje CFA modela:

```r
hz.model <- '
visual =~ x1 + x2 + x3
writing =~ x4 + x5 + x6
maths =~ x7 + x8 + x9
'
```

.footnote[ [*]Za detalje o lavaan projektu pogledajte [stranicu](https://lavaan.ugent.be/tutorial/syntax1.html).]

---

# Provedi model

```r
hz.fit <- lavaan::cfa(hz.model, data=hz)
lavaan::summary(hz.fit, fit.measures=TRUE)
```

```
## lavaan 0.6-8 ended normally after 35 iterations
## 
## Estimator ML
## Optimization method NLMINB
## Number of model 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|)
## visual =~ 
## x1 1.000 
## x2 0.554 0.100 5.554 0.000
## x3 0.729 0.109 6.685 0.000
## writing =~ 
## x4 1.000 
## x5 1.113 0.065 17.014 0.000
## x6 0.926 0.055 16.703 0.000
## maths =~ 
## x7 1.000 
## x8 1.180 0.165 7.152 0.000
## x9 1.082 0.151 7.155 0.000
## 
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## visual ~~ 
## writing 0.408 0.074 5.552 0.000
## maths 0.262 0.056 4.660 0.000
## writing ~~ 
## maths 0.173 0.049 3.518 0.000
## 
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .x1 0.549 0.114 4.833 0.000
## .x2 1.134 0.102 11.146 0.000
## .x3 0.844 0.091 9.317 0.000
## .x4 0.371 0.048 7.779 0.000
## .x5 0.446 0.058 7.642 0.000
## .x6 0.356 0.043 8.277 0.000
## .x7 0.799 0.081 9.823 0.000
## .x8 0.488 0.074 6.573 0.000
## .x9 0.566 0.071 8.003 0.000
## visual 0.809 0.145 5.564 0.000
## writing 0.979 0.112 8.737 0.000
## maths 0.384 0.086 4.451 0.000
```
]

---

# Prikaži grafički

```r
semPlot::semPaths(hz.fit)
```

---

# Prikaži grafički

```r
semPlot::semPaths(hz.fit, "std")
```

---

# Kvaliteta procjene (Fit)

```r
lavaan::fitmeasures(hz.fit, c('cfi', 'rmsea', 'rmsea.ci.upper', 'bic'))
```

```
##            cfi          rmsea rmsea.ci.upper            bic 
##          0.931          0.092          0.114       7595.339
```

---

# Hvala na pažnji

(Nastavak: Klaster analiza)