PRIMJENJENA STATISTIKA

class: center, middle, inverse, title-slide

# PRIMJENJENA STATISTIKA
## Predavanje 8: Kategorijalna analiza
### Luka Sikic, PhD
### Fakultet hrvatskih studija | <a href="https://github.com/BrbanMiro/WebStatistika">Github PS</a>

---

---
class: inverse, middle
# PREGLED PREDAVANJA

---
layout: true
# PREGLED PREDAVANJA
---
<br>

## CILJEVI

- Goodness-of-fit test
- Test nezavisnosti
- Korekcija kontinuiranosti
- Efekt veličine
- Pretpostavke testa
- Izvođenje chi-sq testova u R
- Fisherov test za male uzorke
- McNemarov test zavisnih uzoraka

---

layout:false
class: middle, inverse
# GOF TEST 
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Jedan od najstarijih statističkih testova!)
---

layout:true
# GOF TEST

---

.hi[**Pregled podataka za analizu**]

```
#>      id choice_1 choice_2
#> 1 subj1   spades    clubs
#> 2 subj2 diamonds    clubs
#> 3 subj3   hearts    clubs
#> 4 subj4   spades    clubs
```

```r
str(cards) # Pregledaj podatke
```

```
#> 'data.frame':	200 obs. of  3 variables:
#>  $ id      : Factor w/ 200 levels "subj1","subj10",..: 1 112 124 135 146 157 168 179 190 2 ...
#>  $ choice_1: Factor w/ 4 levels "clubs","diamonds",..: 4 2 3 4 3 1 3 2 4 2 ...
#>  $ choice_2: Factor w/ 4 levels "clubs","diamonds",..: 1 1 1 1 4 3 2 1 1 4 ...
```
.footnote[[*] Test je osmislio Person (1900), a doradio Fisher (1922).]
---

.hi[**Tabulirani pregled podataka**]

```r
podatci <- table(cards$choice_1) # Tabuliraj prvu varijablu
podatci # Pregledaj tabuliranu varijablu
```

```
#> 
#>    clubs diamonds   hearts   spades 
#>       35       51       64       50
```

.hi[Matematički zapis]

$$
O = (O_1, O_2, O_3, O_4)
$$
.hi[Matematički zapis rezultata]

$$
O = (35, 51, 64, 50)
$$
---

.hi[**Intuitivniji prikaz podataka**]
<br>
<br>
<br>
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> naziv </th>
   <th style="text-align:center;"> indeks i </th>
   <th style="text-align:center;"> simbol </th>
   <th style="text-align:center;"> R naredba </th>
   <th style="text-align:center;"> vrijednost </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> clubs `$\clubsuit$` </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> O_1 </td>
   <td style="text-align:center;"> observed[1] </td>
   <td style="text-align:center;"> 35 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> diamonds `$\diamondsuit$` </td>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;"> O_2 </td>
   <td style="text-align:center;"> observed[2] </td>
   <td style="text-align:center;"> 51 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> hearts `$\heartsuit$` </td>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;"> O_3 </td>
   <td style="text-align:center;"> observed[3] </td>
   <td style="text-align:center;"> 64 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> spades `$\spadesuit$` </td>
   <td style="text-align:center;"> 4 </td>
   <td style="text-align:center;"> O_4 </td>
   <td style="text-align:center;"> observed[4] </td>
   <td style="text-align:center;"> 50 </td>
  </tr>
</tbody>
</table>

---

.hi[Formuliranje nulte hipoteze]

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Hipoteza </th>
   <th style="text-align:left;"> Formalni.Zapis </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> `$H_0$` : Sve četiri karte su izabrane sa jednakom vjerojatnošću </td>
   <td style="text-align:left;"> H_0: P = (.25, .25, .25, .25) </td>
  </tr>
</tbody>
</table>

.hi[Formuliranje alternativne hipoteze]

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Nulta.Hipoteza </th>
   <th style="text-align:left;"> Alternativna.Hipoteza </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> H_0 : Sve četiri karte su izabrane sa jednakom vjerojatnošću </td>
   <td style="text-align:left;"> H_1 : Barem jedan izbor nije sa vjerojatnošću 0.25 </td>
  </tr>
</tbody>
</table>

.hi[Formalni zapis]

---

.hi[**Formiraj vjerojatnosti u R**]

```r
# Pripiši vjerojatnosti varijabli
vjerojatnosti <- c(clubs = .25, diamonds = .25, hearts = .25, spades = .25) 
vjerojatnosti # Pogledaj varijablu
```

```
#>    clubs diamonds   hearts   spades 
#>     0.25     0.25     0.25     0.25
```

.hi[**Testna statistika**]

`$$E_i = N \times P_i$$`
.hi[**Formiraj očekivane vjerojatnosti u R**]

```r
N <- 200  # Veličina uzorka
ocekivano <- N * vjerojatnosti # Očekivane frekvencije
ocekivano# Pogledaj podatke
```

```
#>    clubs diamonds   hearts   spades 
#>       50       50       50       50
```

---

.hi[**Prikaži sve podatke**]

<br>
<br>
<table>
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:left;">  </th>
   <th style="text-align:right;"> club </th>
   <th style="text-align:right;"> dijamant </th>
   <th style="text-align:right;"> hertz </th>
   <th style="text-align:right;"> spade </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Očekivana frekvencija </td>
   <td style="text-align:left;"> `$E_i$` </td>
   <td style="text-align:right;"> 50 </td>
   <td style="text-align:right;"> 50 </td>
   <td style="text-align:right;"> 50 </td>
   <td style="text-align:right;"> 50 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Podatci </td>
   <td style="text-align:left;"> `$O_i$` </td>
   <td style="text-align:right;"> 35 </td>
   <td style="text-align:right;"> 51 </td>
   <td style="text-align:right;"> 64 </td>
   <td style="text-align:right;"> 50 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Razlika </td>
   <td style="text-align:left;"> `$O_i - E_i$` </td>
   <td style="text-align:right;"> -15 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 14 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>

---

.hi[**Postupak izračuna u R (korak po korak)**]

```r
# Izračunaj razliku između podataka i očekivanih frekvencija
podatci - ocekivano 
```

```
#> 
#>    clubs diamonds   hearts   spades 
#>      -15        1       14        0
```

```r
# Kvadriraj razlike kako bi se dobili veći brojevi
(podatci - ocekivano)^2
```

```
#> 
#>    clubs diamonds   hearts   spades 
#>      225        1      196        0
```

```r
# Podijeli kvadrirane razlike 
(podatci - ocekivano)^2 / ocekivano
```

```
#> 
#>    clubs diamonds   hearts   spades 
#>     4.50     0.02     3.92     0.00
```

```r
# Zbroji podatke za sve karte
sum( (podatci - ocekivano)^2 / ocekivano )
```

```
#> [1] 8.44
```

---

.hi[Matematički zapis]
<br>
<br>
<br>
`$$X^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i}$$`

.hi[Sampling distribucija GOF statistike]
<br>
<br>
<br>
`$$O_i \sim \mbox{Binomial}(P_i, N)$$`

---

.hi[Stupnjevi slobode]

.footnote[[*]Chi-square distribucija za različite stupnjeve slobode.]

---

.hi[Grafički prikaz]

.footnote[[*]Prikaz kritičke regije za odbacivanje nulte hipoteze.]

---

.hi[**Testiranje u R**]

```r
# Izračunaj 95i centil chi-sq distribucije
qchisq(p = .95, df = 3)
```

```
#> [1] 7.814728
```

```r
# Izračunaj p-vrijednost za podatke iz radnog primjera
pchisq(q = 8.44, df = 3, lower.tail = FALSE)
```

```
#> [1] 0.03774185
```

```r
# Alternativna varijanta izračuna
1-pchisq(q = 8.44, df = 3)
```

```
#> [1] 0.03774185
```

---
.tiny[

```r
# Provedi test u R pomoću funkcije
lsr::goodnessOfFitTest( cards$choice_1 )
```

```
#> 
#>      Chi-square test against specified probabilities
#> 
#> Data variable:   cards$choice_1 
#> 
#> Hypotheses: 
#>    null:        true probabilities are as specified
#>    alternative: true probabilities differ from those specified
#> 
#> Descriptives: 
#>          observed freq. expected freq. specified prob.
#> clubs                35             50            0.25
#> diamonds             51             50            0.25
#> hearts               64             50            0.25
#> spades               50             50            0.25
#> 
#> Test results: 
#>    X-squared statistic:  8.44 
#>    degrees of freedom:  3 
#>    p-value:  0.038
```
]
---

.hi[**Test sa drugim vjerojatnostima**]
<br>
<br>
<br>

```r
# Formuliraj druge vjerojatnosti
nullProbs <- c(clubs = .2, diamonds = .3, hearts = .3, spades = .2)
nullProbs # Pregledaj varijablu
```

```
#>    clubs diamonds   hearts   spades 
#>      0.2      0.3      0.3      0.2
```

---
.tiny[

```r
# Provedi test sa drugačije specificiranim vjerojatnostima
goodnessOfFitTest( x = cards$choice_1, p = nullProbs )
```

```
#> 
#>      Chi-square test against specified probabilities
#> 
#> Data variable:   cards$choice_1 
#> 
#> Hypotheses: 
#>    null:        true probabilities are as specified
#>    alternative: true probabilities differ from those specified
#> 
#> Descriptives: 
#>          observed freq. expected freq. specified prob.
#> clubs                35             40             0.2
#> diamonds             51             60             0.3
#> hearts               64             60             0.3
#> spades               50             40             0.2
#> 
#> Test results: 
#>    X-squared statistic:  4.742 
#>    degrees of freedom:  3 
#>    p-value:  0.192
```
]
---

.hi[**Interpretacija rezultata testa**]
<br>
<br>
<br>
<br>
1. Opis podataka i nulte hipoteze.
<br>
<br>

2. Statistički blok.
<br>
<br>

3. Interpretacija rezultata.

---

layout:false
class: middle, inverse
# TEST NEZAVISNOSTI KATEGORIJA
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Povezanost kategoričkih varijabli.)
---

layout:true
# TEST NEZAVISNOSTI KATEGORIJA

---

.hi[**Pregled podataka**]

```r
str(chapek9) # Pregledaj strukturu podataka
```

```
#> 'data.frame':	180 obs. of  2 variables:
#>  $ species: Factor w/ 2 levels "robot","human": 1 2 2 2 1 2 2 1 2 1 ...
#>  $ choice : Factor w/ 3 levels "puppy","flower",..: 2 3 3 3 3 2 3 3 1 2 ...
```

```r
# Pregledaj podatke
head(chapek9)
```

```
#>   species choice
#> 1   robot flower
#> 2   human   data
#> 3   human   data
#> 4   human   data
#> 5   robot   data
#> 6   human flower
```

---

.hi[**Pregled podataka**]

```r
summary(chapek9)
```

```
#>   species      choice   
#>  robot:87   puppy : 28  
#>  human:93   flower: 43  
#>             data  :109
```

.hi[**Tabuliraj podatke**]

```r
chapekFrequencies <- xtabs( ~ choice + species, data = chapek9)
chapekFrequencies # Pregledaj tabulirane podatke 
```

```
#>         species
#> choice   robot human
#>   puppy     13    15
#>   flower    30    13
#>   data      44    65
```

---

.hi[**Tablični prikaz podataka**]
<br>
<br>
<br>
<table>
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:center;"> Robot </th>
   <th style="text-align:center;"> Human </th>
   <th style="text-align:center;"> Ukupno </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Puppy </td>
   <td style="text-align:center;"> 13 </td>
   <td style="text-align:center;"> 15 </td>
   <td style="text-align:center;"> 28 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Flower </td>
   <td style="text-align:center;"> 30 </td>
   <td style="text-align:center;"> 13 </td>
   <td style="text-align:center;"> 43 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Data file </td>
   <td style="text-align:center;"> 44 </td>
   <td style="text-align:center;"> 65 </td>
   <td style="text-align:center;"> 109 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Ukupno </td>
   <td style="text-align:center;"> 87 </td>
   <td style="text-align:center;"> 93 </td>
   <td style="text-align:center;"> 180 </td>
  </tr>
</tbody>
</table>

---

.hi[**Formalni zapis podataka**]
<br>
<br>
<br>
<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:left;"> Robot </th>
   <th style="text-align:left;"> Human </th>
   <th style="text-align:left;"> Ukupno </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Puppy </td>
   <td style="text-align:left;"> O_11 </td>
   <td style="text-align:left;"> O_12 </td>
   <td style="text-align:left;"> R_1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Flower </td>
   <td style="text-align:left;"> O_21 </td>
   <td style="text-align:left;"> O_22 </td>
   <td style="text-align:left;"> R_2 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Data file </td>
   <td style="text-align:left;"> O_31 </td>
   <td style="text-align:left;"> O_32 </td>
   <td style="text-align:left;"> R_3 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Ukupno </td>
   <td style="text-align:left;"> C_1 </td>
   <td style="text-align:left;"> C_2 </td>
   <td style="text-align:left;"> N </td>
  </tr>
</tbody>
</table>

---
.hi[**Formuliraj nultu hipotezu**]
<br>
<br>
<br>
<table>
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:left;">  </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> H_0: </td>
   <td style="text-align:left;"> Sve od sljedećeg je točno: </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:left;"> P_11 = P_12 (jednaka vjerojatnost izbora puppy) </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:left;"> P_21 = P_22 (jednaka vjerojatnost izbora flower) i </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:left;"> P_31 = P_32 (jednaka vjerojatnost izbora data). </td>
  </tr>
</tbody>
</table>

---

.hi[**Formuliraj testnu statistiku**]

<br>

`$$\hat{P}_i = \frac{R_i}{N}$$` 
<br>

<br>

`$$X^2 = \sum_{i=1}^r\sum_{j=1}^c\frac{({E}_{ij} - O_{ij})^2}{{E}}_{ij}}$$`

---
.hi[**Stupnjevi slobode kod testa nezavisnosti**]
<br>

`$$df = (r-1)(c-1)$$`
<br>

`$$\begin{array}{rcl}
df &=& \mbox{(broj opservacija)} - \mbox{(broj ograničenja)} \\
&=& (rc) - (c + (r-1)) \\
&=& rc - c - r + 1 \\
&=& (r - 1)(c - 1)
\end{array}$$`

<br>

`$$\begin{array}{rcl}
df &=& \mbox{(broj opservacija)} - \mbox{(broj ograničenja)} \\
&=& rc - ( (c-1) + (r-1) + 1) \\
&=& rc - c - r + 1 \\
&=& (r - 1)(c - 1)
\end{array}$$`

---

.hi[**Izvođenje testa u R**]
<br>
<br>

```r
# Tabuliraj podatke
xtabs( formula = ~ choice + species, data = chapek9 )
```

```
#>         species
#> choice   robot human
#>   puppy     13    15
#>   flower    30    13
#>   data      44    65
```

---

.tiny[

```r
# Izvedi test u R
associationTest( formula = ~ choice + species, data = chapek9 )
```

```
#> 
#>      Chi-square test of categorical association
#> 
#> Variables:   choice, species 
#> 
#> Hypotheses: 
#>    null:        variables are independent of one another
#>    alternative: some contingency exists between variables
#> 
#> Observed contingency table:
#>         species
#> choice   robot human
#>   puppy     13    15
#>   flower    30    13
#>   data      44    65
#> 
#> Expected contingency table under the null hypothesis:
#>         species
#> choice   robot human
#>   puppy   13.5  14.5
#>   flower  20.8  22.2
#>   data    52.7  56.3
#> 
#> Test results: 
#>    X-squared statistic:  10.722 
#>    degrees of freedom:  2 
#>    p-value:  0.005 
#> 
#> Other information: 
#>    estimated effect size (Cramer's v):  0.244
```
]

---

.hi[**Korekcija kontinuiranosti**]
<br>
<br>

`$$X^2 = \sum_{i} \frac{(|E_i - O_i| - 0.5)^2}{E_i}$$`

.hi[**Efekt veličine**]
<br>
<br>

`$$\phi = \sqrt{\frac{X^2}{N}}$$`
---

.hi[**Cramer V**]
<br>
<br>
`$$V = \sqrt{\frac{X^2}{N(k-1)}}$$`

.hi[**Izvedi test u R**]
<br>
<br>

```r
cramersV(chapekFrequencies)
```

```
#> [1] 0.244058
```

---

layout:false
class: middle, inverse
# PRETPOSTAVKE TESTA
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Uvjerljivost testa!)
---

layout:true
# PRETPOSTAVKE TESTA

---

.hi[**Radne pretpostavke GOF i testa nezavisnosti**]
<br>
<br>
<br>
<br>
<br>
1. Očekivane frekvencije su dovoljno velike
<br>
<br>
<br>
<br>
2. Podatci su međusobno neovisni
<br>
<br>

---

layout:false
class: middle, inverse
# IZVOĐENJE CHI-SQ TESTOVA U R
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Uvjerljivost testa!)
---

layout:true
# IZVOĐENJE CHI-SQ TESTOVA U R

---

```r
# Pregledaj podatke
podatci
```

```
#> 
#>    clubs diamonds   hearts   spades 
#>       35       51       64       50
```

```r
# Izvedi chi-sq test na standardni način
chisq.test(x = podatci)
```

```
#> 
#> 	Chi-squared test for given probabilities
#> 
#> data:  podatci
#> X-squared = 8.44, df = 3, p-value = 0.03774
```

---

```r
# Izvedi chi-sq test na standardni način; drugačije definirane vjerojatnosti
chisq.test(x = podatci, p = c(.2,.3,.3,.2))
```

```
#> 
#> 	Chi-squared test for given probabilities
#> 
#> data:  podatci
#> X-squared = 4.7417, df = 3, p-value = 0.1917
```

---

```r
# Test sa drugim podatcima
# Pregledaj podatke
chapekFrequencies
```

```
#>         species
#> choice   robot human
#>   puppy     13    15
#>   flower    30    13
#>   data      44    65
```

```r
# Izvedi chi-sq test
chisq.test(chapekFrequencies)
```

```
#> 
#> 	Pearson's Chi-squared test
#> 
#> data:  chapekFrequencies
#> X-squared = 10.722, df = 2, p-value = 0.004697
```

---

layout:false
class: middle, inverse
# FISHEROV TEST
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Premalo podataka...)
---

layout:true
# FISHEROV TEST

---
.hi[**Pregledaj podatke**]

```r
head(trial,5) # prvih pet redova
```

```
#>   happy on.fire
#> 1  TRUE   FALSE
#> 2  TRUE   FALSE
#> 3 FALSE   FALSE
#> 4 FALSE    TRUE
#> 5  TRUE   FALSE
```

```r
str(trial) # Struktura podataka
```

```
#> 'data.frame':	16 obs. of  2 variables:
#>  $ happy  : logi  TRUE TRUE FALSE FALSE TRUE TRUE ...
#>  $ on.fire: logi  FALSE FALSE FALSE TRUE FALSE FALSE ...
```

---

.hi[**Tabuliraj podatke**]

```r
salem.tabs <- table( trial )
# Prikaži podatke
print( salem.tabs )
```

```
#>        on.fire
#> happy   FALSE TRUE
#>   FALSE     3    3
#>   TRUE     10    0
```
---

.hi[**Provedi test**]

```r
# Izvedi chi-sq test na tabuliranim podatcima
chisq.test( salem.tabs )
```

```
#> 
#> 	Pearson's Chi-squared test with Yates' continuity correction
#> 
#> data:  salem.tabs
#> X-squared = 3.3094, df = 1, p-value = 0.06888
```
---

.hi[**Pregled varijabli u formalnom zapisu**]
<br>
<br>
<br>

<table>
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:left;"> Sretan </th>
   <th style="text-align:left;"> Tužan </th>
   <th style="text-align:left;"> Ukupno </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Zapaljena </td>
   <td style="text-align:left;"> `$O_{11}$` </td>
   <td style="text-align:left;"> `$O_{12}$` </td>
   <td style="text-align:left;"> `$R_{1}$` </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Nije zapaljena </td>
   <td style="text-align:left;"> `$O_{21}$` </td>
   <td style="text-align:left;"> `$O_{22}$` </td>
   <td style="text-align:left;"> `$R_{2}$` </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Ukupno </td>
   <td style="text-align:left;"> `$C_{1}$` </td>
   <td style="text-align:left;"> `$C_{2}$` </td>
   <td style="text-align:left;"> `$N$` </td>
  </tr>
</tbody>
</table>

.hi[**Formalni zapis vjerojatnosti za test**]

`$$P(O_{11}, O_{12}, O_{21}, O_{22} \ | \ R_1, R_2, C_1, C_2)$$`
---

.hi[**Provedi Fisherov test**]

```r
# Izvedi Fisherov test u R
fisher.test(salem.tabs)
```

```
#> 
#> 	Fisher's Exact Test for Count Data
#> 
#> data:  salem.tabs
#> p-value = 0.03571
#> alternative hypothesis: true odds ratio is not equal to 1
#> 95 percent confidence interval:
#>  0.000000 1.202913
#> sample estimates:
#> odds ratio 
#>          0
```

---

layout:false
class: middle, inverse
# McNEMAROV TEST
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Međusobno zavisni podatci...)
---

layout:true
# McNEMAROV TEST

---

.hi[**Pregledaj podatke**]

<br>
<br>
<br>
<table>
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:center;"> Prije </th>
   <th style="text-align:center;"> Poslije </th>
   <th style="text-align:center;"> Ukupno </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Da </td>
   <td style="text-align:center;"> 30 </td>
   <td style="text-align:center;"> 10 </td>
   <td style="text-align:center;"> 40 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Ne </td>
   <td style="text-align:center;"> 70 </td>
   <td style="text-align:center;"> 90 </td>
   <td style="text-align:center;"> 160 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Ukupno </td>
   <td style="text-align:center;"> 100 </td>
   <td style="text-align:center;"> 100 </td>
   <td style="text-align:center;"> 200 </td>
  </tr>
</tbody>
</table>

.footnote[[*]Podatci su međusobno zavisni je ima 200 opservacija, a 100 ljudi]

---

.hi[**Drugačiji prikaz podataka**]
<br>
<br>
<br>
<table>
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:center;"> Prije: Da </th>
   <th style="text-align:center;"> Prije: Ne </th>
   <th style="text-align:center;"> Ukupno </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Poslije: Da </td>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;"> 10 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Poslije: Ne </td>
   <td style="text-align:center;"> 25 </td>
   <td style="text-align:center;"> 65 </td>
   <td style="text-align:center;"> 90 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Ukupno </td>
   <td style="text-align:center;"> 30 </td>
   <td style="text-align:center;"> 70 </td>
   <td style="text-align:center;"> 100 </td>
  </tr>
</tbody>
</table>

---

.hi[**Formalni zapis testa**]

<br>
`$$X^2 = \frac{(|b-c| - 0.5)^2}{b+c}$$`

<br>
<br>
`$$X^2 = \frac{(|O_{12}-O_{21}| - 0.5)^2}{O_{12} + O_{21}}$$`
---

.hi[**Pregled podataka**]

```r
str(agpp)      # Pregled podataka
```

```
#> 'data.frame':	100 obs. of  3 variables:
#>  $ id             : Factor w/ 100 levels "subj.1","subj.10",..: 1 13 24 35 46 57 68 79 90 2 ...
#>  $ response_before: Factor w/ 2 levels "no","yes": 1 2 2 2 1 1 1 1 1 1 ...
#>  $ response_after : Factor w/ 2 levels "no","yes": 2 1 1 1 1 1 1 2 1 1 ...
```

```r
head(agpp,5)     # Pregled podataka
```

```
#>       id response_before response_after
#> 1 subj.1              no            yes
#> 2 subj.2             yes             no
#> 3 subj.3             yes             no
#> 4 subj.4             yes             no
#> 5 subj.5              no             no
```

---

.hi[**Tabuliraj podatke**]

```r
summary(agpp ) # Pregled podataka
```

```
#>         id     response_before response_after
#>  subj.1  : 1   no :70          no :90        
#>  subj.10 : 1   yes:30          yes:10        
#>  subj.100: 1                                 
#>  subj.11 : 1                                 
#>  subj.12 : 1                                 
#>  subj.13 : 1                                 
#>  (Other) :94
```

```r
pravaTablica <- xtabs( ~ response_before + response_after, data = agpp)
print(pravaTablica) # Pregled podataka
```

```
#>                response_after
#> response_before no yes
#>             no  65   5
#>             yes 25   5
```

---
.hi[**Provedi test u R**]
<br>
<br>
<br>

```r
mcnemar.test(pravaTablica)
```

```
#> 
#> 	McNemar's Chi-squared test with continuity correction
#> 
#> data:  pravaTablica
#> McNemar's chi-squared = 12.033, df = 1, p-value = 0.0005226
```

---
.hi[**Razlika među testovima**]

```r
# Tabuliraj podatke
izborKarata <- xtabs( ~ choice_1 + choice_2, data = cards )
izborKarata # Pregled podataka
```

```
#>           choice_2
#> choice_1   clubs diamonds hearts spades
#>   clubs       10        9     10      6
#>   diamonds    20        4     13     14
#>   hearts      20       18      3     23
#>   spades      18       13     15      4
```

---

.hi[**Provedi testove**]

```r
# Provedi standardni chi-sq test
chisq.test(izborKarata)
```

```
#> 
#> 	Pearson's Chi-squared test
#> 
#> data:  izborKarata
#> X-squared = 29.237, df = 9, p-value = 0.0005909
```

```r
# Provedi McNemarov tets
mcnemar.test(izborKarata)
```

```
#> 
#> 	McNemar's Chi-squared test
#> 
#> data:  izborKarata
#> McNemar's chi-squared = 16.033, df = 6, p-value = 0.01358
```

---

layout:false
class: middle, inverse
# HVALA NA PAŽNJI!
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Sljedeće predavanje: Usporedba prosjeka.)