PRIMJENJENA STATISTIKA

class: center, middle, inverse, title-slide

# PRIMJENJENA STATISTIKA
## Predavanje 10: ANOVA
### 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

- Pregled podataka
- ANOVA
- ANOVA u R
- Efekt veličine
- Post hoc testovi
- Pretpostavke jednostrane ANOVA-e
- Homogenost varijance
- Normalnost distribucije

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.hi[**Pregled podataka**]

```r
load(file.path("../Podatci/clinicaltrial.Rdata")) # Učitaj podatke
str(clin.trial)     # Pregledaj podatke  
```

```
#> 'data.frame':	18 obs. of  3 variables:
#>  $ drug     : Factor w/ 3 levels "placebo","anxifree",..: 1 1 1 2 2 2 3 3 3 1 ...
#>  $ therapy  : Factor w/ 2 levels "no.therapy","CBT": 1 1 1 1 1 1 1 1 1 2 ...
#>  $ mood.gain: num  0.5 0.3 0.1 0.6 0.4 0.2 1.4 1.7 1.3 0.6 ...
```

```r
head(clin.trial, 8)   # Pregledaj podatke
```

```
#>       drug    therapy mood.gain
#> 1  placebo no.therapy       0.5
#> 2  placebo no.therapy       0.3
#> 3  placebo no.therapy       0.1
#> 4 anxifree no.therapy       0.6
#> 5 anxifree no.therapy       0.4
#> 6 anxifree no.therapy       0.2
#> 7 joyzepam no.therapy       1.4
#> 8 joyzepam no.therapy       1.7
```

---

.hi[**Deskriptivna statistika**]

```r
summary(clin.trial)
```

```
#>        drug         therapy    mood.gain     
#>  placebo :6   no.therapy:9   Min.   :0.1000  
#>  anxifree:6   CBT       :9   1st Qu.:0.4250  
#>  joyzepam:6                  Median :0.8500  
#>                              Mean   :0.8833  
#>                              3rd Qu.:1.3000  
#>                              Max.   :1.8000
```

.pull-left[

```r
# Tabuliraj podatke
xtabs( ~ drug, clin.trial )
```

```
#> drug
#>  placebo anxifree joyzepam 
#>        6        6        6
```
]

.pull-right[

```r
# Tabuliraj podatke
xtabs( ~ therapy, clin.trial )
```

```
#> therapy
#> no.therapy        CBT 
#>          9          9
```
]

---

.hi[**Deskriptivna statistika**]

```r
# Agregiraj podatke po primjenjenom ljeku
aggregate( mood.gain ~ drug, clin.trial, mean ) # Prosjek
```

```
#>       drug mood.gain
#> 1  placebo 0.4500000
#> 2 anxifree 0.7166667
#> 3 joyzepam 1.4833333
```

```r
aggregate( mood.gain ~ drug, clin.trial, sd )   # Standardna devijacija
```

```
#>       drug mood.gain
#> 1  placebo 0.2810694
#> 2 anxifree 0.3920034
#> 3 joyzepam 0.2136976
```
---

.hi[**Vizualizacija**]

```r
library(gplots) # Učitaj paket
plotmeans(  formula = mood.gain ~ drug,   # Porast raspoloženja vs. ljek
             data = clin.trial,           # Podatci
             xlab = "Primjenjeni ljek",  
             ylab = "Rast raspoloženja",          
             n.label = FALSE              # Ne prikazuj veličinu uzorka
 )
```

---
layout:false
class: middle, inverse
# ANOVA 
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Usporedba više prosjeka!)
---

layout:true
# ANOVA
---

.hi[**Hipoteze kod ANOVA-e**]

`$$\begin{array}{rcl}
H_0 &:& \mbox{točno je } \mu_P = \mu_A = \mu_J
\end{array}$$`

`$$\begin{array}{rcl}
H_1 &:& \mbox{nije točno } \mu_P = \mu_A = \mu_J
\end{array}$$`

<br>
<br>
.hi[**Izračun varijance kod ANOVA-e**]

`$$\mbox{Var}(Y) = \frac{1}{N} \sum_{k=1}^G \sum_{i=1}^{N_k} \left(Y_{ik} - \bar{Y} \right)^2$$`
---

.hi[**Praktični primjer**]

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> ime </th>
   <th style="text-align:left;"> osoba (p) </th>
   <th style="text-align:left;"> grupa </th>
   <th style="text-align:left;"> broj grupe (k) </th>
   <th style="text-align:left;"> index grupe (i) </th>
   <th style="text-align:left;"> raspolozenje (Y_ik ili Y_p) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Ann </td>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:left;"> cool </td>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:left;"> 20 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Ben </td>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:left;"> cool </td>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:left;"> 55 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cat </td>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:left;"> cool </td>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:left;"> 21 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Dan </td>
   <td style="text-align:left;"> 4 </td>
   <td style="text-align:left;"> uncool </td>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:left;"> 91 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Egg </td>
   <td style="text-align:left;"> 5 </td>
   <td style="text-align:left;"> uncool </td>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:left;"> 22 </td>
  </tr>
</tbody>
</table>

.hi[**Usporedi sa formulom za varijancu**]

`$$\mbox{Var}(Y) = \frac{1}{N} \sum_{p=1}^N  \left(Y_{p} - \bar{Y} \right)^2$$`
---

.hi[**Od varijance do sume kvadrata odstupanja**]

`$$\mbox{SS}_{tot} = \sum_{k=1}^G \sum_{i=1}^{N_k} \left(Y_{ik} - \bar{Y} \right)^2$$`

.hi[**Suma kvadrata odstupanja unutar grupe**]

`$$\mbox{SS}_w = \sum_{k=1}^G \sum_{i=1}^{N_k} \left( Y_{ik} - \bar{Y}_k \right)^2$$`

.hi[**Suma kvadrata odstupanja među grupama**]

`$$\begin{array}{rcl}
\mbox{SS}_{b} &=& \sum_{k=1}^G \sum_{i=1}^{N_k} \left( \bar{Y}_k - \bar{Y} \right)^2
 \\
&=& \sum_{k=1}^G N_k \left( \bar{Y}_k - \bar{Y} \right)^2
\end{array}$$`

.hi[**Ukupna varijacija**]

`$$\mbox{SS}_w  + \mbox{SS}_{b} = \mbox{SS}_{tot}$$`

---

.hi[**Varijacija među grupama**]

<div class="figure" style="text-align: center">
<img src="10_ANOVA_files/figure-html/anovavara-1.svg" alt="Prikaz varijacije između grupa."  />
<p class="caption">Prikaz varijacije između grupa.</p>
</div>

---

.hi[**Varijacija unutar grupa**]

<div class="figure" style="text-align: center">
<img src="10_ANOVA_files/figure-html/anovavarb-1.svg" alt="Prikaz varijacije unutar grupa."  />
<p class="caption">Prikaz varijacije unutar grupa.</p>
</div>

---

.hi[**Od sume kvadrata odstupanja do F-testa**]

- Stupnjevi slobode

`$$\begin{array}{lcl}
\mbox{df}_b &=& G-1 \\
\mbox{df}_w &=& N-G \\
\end{array}$$`

- Kvadrati odstupanja od prosjeka

`$$\begin{array}{lcl}
\mbox{MS}_b &=& \displaystyle\frac{\mbox{SS}_b }{ \mbox{df}_b} \\
\mbox{MS}_w &=& \displaystyle\frac{\mbox{SS}_w }{ \mbox{df}_w} 
\end{array}$$`

- F-statistika

`$$F = \frac{\mbox{MS}_b }{ \mbox{MS}_w }$$`
---

.hi[**Sažeti prikaz u tablici**]

---

.hi[**Izračun ANOVA-e korak po korak**]

- Pregled podataka u tablici

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Grupa (k) </th>
   <th style="text-align:left;"> Ishod (Y_ik) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> placebo </td>
   <td style="text-align:left;"> 0.5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> placebo </td>
   <td style="text-align:left;"> 0.3 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> placebo </td>
   <td style="text-align:left;"> 0.1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> anxifree </td>
   <td style="text-align:left;"> 0.6 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> anxifree </td>
   <td style="text-align:left;"> 0.4 </td>
  </tr>
</tbody>
</table>

---

- Dodaj grupni prosjek
<br>
<br>
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Grupa (k) </th>
   <th style="text-align:left;"> Ishod (Y_ik) </th>
   <th style="text-align:left;"> Grupni prosjek (Y_k) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> placebo </td>
   <td style="text-align:left;"> 0.5 </td>
   <td style="text-align:left;"> 0.45 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> placebo </td>
   <td style="text-align:left;"> 0.3 </td>
   <td style="text-align:left;"> 0.45 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> placebo </td>
   <td style="text-align:left;"> 0.1 </td>
   <td style="text-align:left;"> 0.45 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> anxifree </td>
   <td style="text-align:left;"> 0.6 </td>
   <td style="text-align:left;"> 0.72 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> anxifree </td>
   <td style="text-align:left;"> 0.4 </td>
   <td style="text-align:left;"> 0.72 </td>
  </tr>
</tbody>
</table>

---

- Odstupanja od grupnog prosjeka i kvadrati odstupanja

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Grupa (k) </th>
   <th style="text-align:left;"> Ishod (Y_ik) </th>
   <th style="text-align:left;"> Grupni prosjek (Y_k) </th>
   <th style="text-align:left;"> dodstupanje od gr. prosjeka (Y_ik - Y_k) </th>
   <th style="text-align:left;"> Kvadrat odstupanja (Y_ik - Y_k)^2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> placebo </td>
   <td style="text-align:left;"> 0.5 </td>
   <td style="text-align:left;"> 0.45 </td>
   <td style="text-align:left;"> 0.05 </td>
   <td style="text-align:left;"> 0.0025 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> placebo </td>
   <td style="text-align:left;"> 0.3 </td>
   <td style="text-align:left;"> 0.45 </td>
   <td style="text-align:left;"> -0.15 </td>
   <td style="text-align:left;"> 0.0225 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> placebo </td>
   <td style="text-align:left;"> 0.1 </td>
   <td style="text-align:left;"> 0.45 </td>
   <td style="text-align:left;"> -0.35 </td>
   <td style="text-align:left;"> 0.1225 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> anxifree </td>
   <td style="text-align:left;"> 0.6 </td>
   <td style="text-align:left;"> 0.72 </td>
   <td style="text-align:left;"> -0.12 </td>
   <td style="text-align:left;"> 0.0136 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> anxifree </td>
   <td style="text-align:left;"> 0.4 </td>
   <td style="text-align:left;"> 0.72 </td>
   <td style="text-align:left;"> -0.32 </td>
   <td style="text-align:left;"> 0.1003 </td>
  </tr>
</tbody>
</table>

- Izračunaj sumu kvadrata odstupanja unutar grupa

`$$\begin{array}{rcl}
\mbox{SS}_w &=& 0.0025 + 0.0225 + 0.1225 + 0.0136 + 0.1003 \\
&=& 0.2614
\end{array}$$`

---

.hi[**Izračun u R-u korak po korak**]
<br>
<br>

```r
# Definiraj varijable
outcome <- clin.trial$mood.gain
group <- clin.trial$drug
gp.means <- tapply(outcome,group,mean)
gp.means <- gp.means[group]
dev.from.gp.means <- outcome - gp.means
squared.devs <- dev.from.gp.means ^2
# Poveži u Data Frame
Y <- data.frame( group, outcome, gp.means,
                  dev.from.gp.means, squared.devs )
```

---

```r
# Pregled podataka
print(head(Y,6), digits = 2)
```

```
#>      group outcome gp.means dev.from.gp.means squared.devs
#> 1  placebo     0.5     0.45              0.05       0.0025
#> 2  placebo     0.3     0.45             -0.15       0.0225
#> 3  placebo     0.1     0.45             -0.35       0.1225
#> 4 anxifree     0.6     0.72             -0.12       0.0136
#> 5 anxifree     0.4     0.72             -0.32       0.1003
#> 6 anxifree     0.2     0.72             -0.52       0.2669
```

```r
# Izračunaj sumu kvadrata odstupanja
SSw <- sum( squared.devs )
print( SSw )
```

```
#> [1] 1.391667
```
---
<br>

.hi[**Izračunaj grupna odstupanja od "master" prosjeka**]
<br>
<br>

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Grupa (k) </th>
   <th style="text-align:left;"> Group mean (Y_k) </th>
   <th style="text-align:left;"> Grand mean (Y) </th>
   <th style="text-align:left;"> devijacija (Y_k - Y) </th>
   <th style="text-align:left;"> Kvadrirane devijacije (Y_k - Y)^2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> placebo </td>
   <td style="text-align:left;"> 0.45 </td>
   <td style="text-align:left;"> 0.88 </td>
   <td style="text-align:left;"> -0.43 </td>
   <td style="text-align:left;"> 0.18 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> anxifree </td>
   <td style="text-align:left;"> 0.72 </td>
   <td style="text-align:left;"> 0.88 </td>
   <td style="text-align:left;"> -0.16 </td>
   <td style="text-align:left;"> 0.03 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> joyzepam </td>
   <td style="text-align:left;"> 1.48 </td>
   <td style="text-align:left;"> 0.88 </td>
   <td style="text-align:left;"> 0.60 </td>
   <td style="text-align:left;"> 0.36 </td>
  </tr>
</tbody>
</table>

---

- Pomnoži sa veličinom uzorka

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Grupa (k) </th>
   <th style="text-align:left;"> Kvadrirane devijacije (Y_k - Y)^2 </th>
   <th style="text-align:left;"> Veličina uzorka (N_k) </th>
   <th style="text-align:left;"> Ponderirani kv. odst. N_k*(Y_k - Y)^2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> placebo </td>
   <td style="text-align:left;"> 0.18 </td>
   <td style="text-align:left;"> 6 </td>
   <td style="text-align:left;"> 1.11 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> anxifree </td>
   <td style="text-align:left;"> 0.03 </td>
   <td style="text-align:left;"> 6 </td>
   <td style="text-align:left;"> 0.16 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> joyzepam </td>
   <td style="text-align:left;"> 0.36 </td>
   <td style="text-align:left;"> 6 </td>
   <td style="text-align:left;"> 2.18 </td>
  </tr>
</tbody>
</table>

- Zbroji kvadrate odstupanja između grupa

`$$\begin{array}{rcl}
\mbox{SS}_{b} &=& 1.11 + 0.16 + 2.18 \\
&=& 3.45
\end{array}$$`

---
<br>

.hi[**Provedi proceduru u R**]
<br>

```r
# Definiraj varijable
gp.means <- tapply(outcome,group,mean)
grand.mean <- mean(outcome)
dev.from.grand.mean <- gp.means - grand.mean
squared.devs <- dev.from.grand.mean ^2
gp.sizes <- tapply(outcome,group,length)
wt.squared.devs <- gp.sizes * squared.devs
# Poveži u DF
Y <- data.frame( gp.means, grand.mean, dev.from.grand.mean, 
                  squared.devs, gp.sizes, wt.squared.devs )
```

---

```r
# Prikaži podatke
print(head(Y,6), digits = 2)
```

```
#>          gp.means grand.mean dev.from.grand.mean squared.devs gp.sizes
#> placebo      0.45       0.88               -0.43        0.188        6
#> anxifree     0.72       0.88               -0.17        0.028        6
#> joyzepam     1.48       0.88                0.60        0.360        6
#>          wt.squared.devs
#> placebo             1.13
#> anxifree            0.17
#> joyzepam            2.16
```

```r
# Izračunaj sumu kvadrata odstupanja
SSb <- sum( wt.squared.devs )
print( SSb )
```

```
#> [1] 3.453333
```
---

.hi[**Izračunaj stupnjeve slobode**]
<br>
`$$\begin{array}{lclcl}
\mbox{df}_b &=& G - 1 &=& 2 \\
\mbox{df}_w &=& N - G &=& 15 
\end{array}$$`

.hi[**Izračunaj MS statistike**]
<br>
<br>
`$$\begin{array}{lclclcl}
\mbox{MS}_b &=& \displaystyle\frac{\mbox{SS}_b }{  \mbox{df}_b } &=& \displaystyle\frac{3.45}{ 2}  &=& 1.73
\end{array}$$`

`$$\begin{array}{lclclcl}
\mbox{MS}_w &=& \displaystyle\frac{\mbox{SS}_w }{  \mbox{df}_w } &=& \displaystyle\frac{1.39}{15} &=& 0.09
\end{array}$$`
---
<br>

.hi[**Izračunaj F-statistiku**]
<br>
`$$F \ = \ \frac{\mbox{MS}_b }{ \mbox{MS}_w } \ = \ \frac{1.73}{0.09} \ = \ 18.6$$`

<br>
.hi[**Provjeri p-vrijednost**]
<br>

```r
pf( 18.6, df1 = 2, df2 = 15, lower.tail = FALSE)
```

```
#> [1] 8.672727e-05
```

---
<br>
.hi[**Prikaži rezultate u tablici**]
<br>
<br>
<br>
<table>
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:left;"> df </th>
   <th style="text-align:left;"> suma kvadrata </th>
   <th style="text-align:left;"> srednji kvadrati </th>
   <th style="text-align:left;"> F-statistika </th>
   <th style="text-align:left;"> p-vrijednost </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> between groups </td>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:left;"> 3.45 </td>
   <td style="text-align:left;"> 1.73 </td>
   <td style="text-align:left;"> 18.6 </td>
   <td style="text-align:left;"> 8.67 * 10^-5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> within groups </td>
   <td style="text-align:left;"> 15 </td>
   <td style="text-align:left;"> 1.39 </td>
   <td style="text-align:left;"> 0.09 </td>
   <td style="text-align:left;"> - </td>
   <td style="text-align:left;"> - </td>
  </tr>
</tbody>
</table>

---
layout:false
class: middle, inverse
# ANOVA U R
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Četiri načina!)
---

layout:true
# ANOVA U R
---

<br>
.hi[**Opcije**]
<br>
<br>

```r
# Različiti načini za provođenje ANOVA postupka
aov( formula = mood.gain ~ drug, data = clin.trial ) # Puna specifikacija
aov( clin.trial$mood.gain ~ clin.trial$drug )        # Definiranje varijabli
aov( mood.gain ~ drug, clin.trial )                  # Skraćeno;definirani df
```

```r
# Spremi rezultate testa u objekt
my.anova <- aov( mood.gain ~ drug, clin.trial ) 
```

---

```r
# Provjeri klasu objekta
class( my.anova )
```

```
#> [1] "aov" "lm"
```

```r
# Provjeri što sadrži objekt
names( my.anova )
```

```
#>  [1] "coefficients"  "residuals"     "effects"       "rank"         
#>  [5] "fitted.values" "assign"        "qr"            "df.residual"  
#>  [9] "contrasts"     "xlevels"       "call"          "terms"        
#> [13] "model"
```
---

```r
# Prikaži procjenjeni objekt
print( my.anova )
```

```
#> Call:
#>    aov(formula = mood.gain ~ drug, data = clin.trial)
#> 
#> Terms:
#>                     drug Residuals
#> Sum of Squares  3.453333  1.391667
#> Deg. of Freedom        2        15
#> 
#> Residual standard error: 0.3045944
#> Estimated effects may be unbalanced
```
---
<br>
.hi[**Testiranje ANOVA hipoteza**]
<br>
<br>

```r
# Prikaži deskriptivnu statistiku
summary( my.anova )
```

```
#>             Df Sum Sq Mean Sq F value   Pr(>F)    
#> drug         2  3.453  1.7267   18.61 8.65e-05 ***
#> Residuals   15  1.392  0.0928                     
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---
layout:false
class: middle, inverse
# EFEKT VELIČINE
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(Snaga odnosa)
---

layout:true
# EFEKT VELIČINE
---

.hi[**Formula**]

`$$\eta^2 = \frac{\mbox{SS}_b}{\mbox{SS}_{tot}}$$`

.hi[**Izračun**]

`$$\eta^2 = \frac{3.45}{4.84} = 0.71$$`

.hi[**Makni kvadrat**]

`$$\eta= \sqrt{\frac{\mbox{SS}_b}{\mbox{SS}_{tot}}}$$`
---

.hi[**Napravi procjenitelj u R**]
<br>
<br>

```r
SStot <- SSb + SSw          # Ukupna suma kvadrata
eta.squared <- SSb / SStot  # Eta kvadrat
print( eta.squared )        # Prikaži rezultate
```

```
#> [1] 0.7127623
```

```r
library(lsr)
# Izračunaj pomoću funkcije
lsr::etaSquared( x = my.anova )
```

```
#>         eta.sq eta.sq.part
#> drug 0.7127623   0.7127623
```

---
layout:false
class: middle, inverse
# PRETPOSTAVKE JEDNOSTRANE ANOVA-e
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(Robusnost...)
---

layout:true
# PRETPOSTAVKE JEDNOSTRANE ANOVA-e
---
<br>
<br>

.pull-left[
.hi[**Statistički model ANOVA-e**]
<br>

`$$\begin{array}{lrcl}
H_0: & Y_{ik} &=& \mu + \epsilon_{ik} \\
H_1: & Y_{ik} &=& \mu_k + \epsilon_{ik} 
\end{array}$$`

]

.pull-right[
.hi[**Pretpostavka o rezidualnoj strukturi**]
<br>
`$$\epsilon_{ik} \sim \mbox{Normal}(0, \sigma^2)$$`
]

<br>
.hi[**Pretpostavke**]
<br>
- Normalnost distribucije
- Homogenost varijance
- Nezavisnost

---
layout:false
class: middle, inverse
# HOMOGENOST VARIJANCE
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(Kruške ili jabuke!?)
---

layout:true
# HOMOGENOST VARIJANCE
---

<br>
<br>
.hi[**Provjera homogenosti Leven-ovim testom**]
<br>
`$$Z_{ik} = \left| Y_{ik} - \bar{Y}_k \right|$$`

<br>
<br>
.hi[**Provjera homogenosti Brown-Forsythe testom**]
<br>
`$$Z_{ik} = \left| Y_{ik} - \mbox{median}_k(Y) \right|$$`
---

.hi[**Provedi test u R**]

```r
library( car )  # Učitaj paket
car::leveneTest( my.anova ) # Provedi Levene test
```

```
#> Levene's Test for Homogeneity of Variance (center = median)
#>       Df F value Pr(>F)
#> group  2  1.4672 0.2618
#>       15
```

```r
car::leveneTest( y = my.anova, center = mean ) # Provedi B-F test 
```

```
#> Levene's Test for Homogeneity of Variance (center = mean)
#>       Df F value Pr(>F)
#> group  2  1.4497 0.2657
#>       15
```
---

```r
# Provedi (Levene) test korak po korak
Y <- clin.trial $ mood.gain    # Ishod
G <- clin.trial $ drug         # Grupa
gp.mean <- tapply(Y, G, mean)  # Izračunaj grupne prosjeke
Ybar <- gp.mean[G]             # Izvadi grupne prosjeke vezane uz svaku ops
Z <- abs(Y - Ybar)             # Transformirana varijabla
summary( aov(Z ~ G) )          # Izvrši ANOVA-u 
```

```
#>             Df Sum Sq Mean Sq F value Pr(>F)
#> G            2 0.0616 0.03080    1.45  0.266
#> Residuals   15 0.3187 0.02125
```

---

.hi[**Ukoliko su varijance heterogene:**]

```r
# Welch-ev test za ANOVA-u
oneway.test(mood.gain ~ drug, data = clin.trial)
```

```
#> 
#> 	One-way analysis of means (not assuming equal variances)
#> 
#> data:  mood.gain and drug
#> F = 26.322, num df = 2.0000, denom df = 9.4932, p-value = 0.000134
```

```r
# Usporedi sa rezultatima testa za jednake varijance
oneway.test(mood.gain ~ drug, data = clin.trial, var.equal = TRUE)
```

```
#> 
#> 	One-way analysis of means
#> 
#> data:  mood.gain and drug
#> F = 18.611, num df = 2, denom df = 15, p-value = 8.646e-05
```

---
layout:false
class: middle, inverse
# NORMALNOST DISTRIBUCIJE
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(Vizualizacija i formalni test!)
---

layout:true
#  NORMALNOST DISTRIBUCIJE
---

.hi[**Provjera normalnosti distribucije**]

```r
my.anova.residuals <- residuals( object = my.anova )   # Izvadi reziduale
```

```r
hist( x = my.anova.residuals )           # Napravi histogram reziduala
```

---

```r
qqnorm( y = my.anova.residuals )         # Napravi QQ plot
```

```r
shapiro.test( x = my.anova.residuals )   # Provedi Shapiro-Wilk test
```

```
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  my.anova.residuals
#> W = 0.96019, p-value = 0.6053
```

---

.hi[**Kruskal-Wallis test**]

- Izračunaj prosječni rank vezan uz svaku grupu
<br>
`$$\bar{R}_k = \frac{1}{N_K} \sum_{i} R_{ik}$$`

- Izračunaj rank grand prosjeka
<br>
`$$\bar{R} = \frac{1}{N} \sum_{i} \sum_{k} R_{ik}$$`

- Izračunaj sumu kvadrata ukupnih rank-odstupanja
<br>

`$$\mbox{RSS}_{tot} = \sum_k \sum_i ( R_{ik} - \bar{R} )^2$$`

---

.hi[**Kruskal-Wallis test**]

- Izračunaj sumu kvadrata rank-odstupanja među grupama

`$$\begin{array}{rcl}
\mbox{RSS}_{b} &=& \sum_k \sum_i ( \bar{R}_k  - \bar{R} )^2 \\
&=& \sum_k N_k ( \bar{R}_k  - \bar{R} )^2 
\end{array}$$`

- Izračunaj Kruskal-Wallis statistiku

`$$K = (N - 1) \times \frac{\mbox{RSS}_b}{\mbox{RSS}_{tot}}$$`

- Drugi način za izračun K-W statistike

`$$K = \frac{12}{N(N-1)} \sum_k N_k {\bar{R}_k}^2    -  3(N+1)$$`
---

.hi[**Kruskal-Wallis test**]

- Prikaži rangirane podatke

```r
f <- table( clin.trial$mood.gain )   # Tabuliraj i spremi u varijablu
print(f)                             # Prikaži
```

```
#> 
#> 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.9 1.1 1.2 1.3 1.4 1.7 1.8 
#>   1   1   2   1   1   2   1   1   1   1   2   2   1   1
```

- Izvrši K-W test u R

```r
# Prvi način
kruskal.test(mood.gain ~ drug, data = clin.trial)
```

```
#> 
#> 	Kruskal-Wallis rank sum test
#> 
#> data:  mood.gain by drug
#> Kruskal-Wallis chi-squared = 12.076, df = 2, p-value = 0.002386
```

---

.hi[**Kruskal-Wallis test**]

```r
# Drugi način
kruskal.test(x = clin.trial$mood.gain, g = clin.trial$drug)
```

```
#> 
#> 	Kruskal-Wallis rank sum test
#> 
#> data:  clin.trial$mood.gain and clin.trial$drug
#> Kruskal-Wallis chi-squared = 12.076, df = 2, p-value = 0.002386
```
<br>

```r
# Treći način
mood.gain <- list( placebo, joyzepam, anxifree ) # Spremi serije u listu
kruskal.test( x = mood.gain )                    # Provedi test
```

---
layout:false
class: middle, inverse
# ANOVA vs STUDENTOV t-test
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Postoji li razlika?!)
---

layout:true
# ANOVA vs t-test
---
<br>

```r
# Provedi ANOVA test
summary( aov( mood.gain ~ therapy, data = clin.trial ))
```

```
#>             Df Sum Sq Mean Sq F value Pr(>F)
#> therapy      1  0.467  0.4672   1.708   0.21
#> Residuals   16  4.378  0.2736
```

---

```r
# Provedi studentov t-test
t.test( mood.gain ~ therapy, data = clin.trial, var.equal = TRUE )
```

```
#> 
#> 	Two Sample t-test
#> 
#> data:  mood.gain by therapy
#> t = -1.3068, df = 16, p-value = 0.2098
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -0.8449518  0.2005073
#> sample estimates:
#> mean in group no.therapy        mean in group CBT 
#>                0.7222222                1.0444444
```

```r
# Kvadriraj za F-statistiku
1.3068 ^ 2
```

```
#> [1] 1.707726
```

---
layout:false
class: middle, inverse
# Hvala na pažnji!
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Sljedeće predavanje: Linearna regresija)