PRIMJENJENA STATISTIKA

class: center, middle, inverse, title-slide

# PRIMJENJENA STATISTIKA
## Predavanje 9: Usporedba prosjeka
### Luka Sikic, PhD
### Fakultet hrvatskih studija | <a href="https://github.com/BrbanMiro/WebStatistika">Github PS</a>

---

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# PREGLED PREDAVANJA

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# PREGLED PREDAVANJA
---
<br>

## CILJEVI

- z-test
- t-test
- t-test u nezavisnim uzorcima
- t-test u zavisnim uzorcima
- Jednostrani testovi
- Izvođenje t-testova u R
- Efekt veličine
- Provjera normalnosti distribucije

---

layout:false
class: middle, inverse
# z test 
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Zagrijavanje za t test!)
---

layout:true
# z test

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```r
# Učitaj podatke
load( file.path( "../Podatci/zeppo.Rdata" )) 
print( grades ) # Pregledaj podatke
```

```
#>  [1] 50 60 60 64 66 66 67 69 70 74 76 76 77 79 79 79 81 82 82 89
```

```r
# Izračunaj prosjek
mean(grades)
```

```
#> [1] 72.3
```

.hi[**Postavi hipoteze**]

- Da li su ocjene studenata sociologije više od prosjeka na fakultetu?

`$$\begin{array}{ll}
H_0: & \mu = 67.5 \\
H_1: & \mu \neq 67.5
\end{array}$$`
---

.hi[Prikaži hipoteze grafički]

.footnote[[*]Grafički prikaz nulte i alternativne hipoteze pod pretpostvkom jednostranog z- testa. Nulta i alternativna hipoteza pretpostavljaju da populacija (podatci) prati standarnu distribuciju i da je standardna devijacija poznata (sigma_0). Pod nultom hipotezom je prosjek populacije mu jednak apriori definiranoj vrijednosti mu_0. Pod alternativnom hipotezom prosjek populacije nije jednak tako definiranoj vrijednosti, mu neq mu_0.]

---

.hi[Prikaži podatke grafički]

.footnote[[*]Puna linija predstavlja teoretsku distribuciju pod nultom hipotezom iz koje su "generirane" ocjene studenata sociologije.]

---

.hi[Konstruiraj testnu statistiku]

`$$\bar{X} - \mu_0$$`

`$$X \sim \mbox{Normal}(\mu_0,\sigma^2)$$`

`$$\mbox{SE}({\bar{X}}) = \frac{\sigma}{\sqrt{N}}$$`

`$$\bar{X} \sim \mbox{Normal}(\mu_0,\mbox{SE}({\bar{X}}))$$`

---

`$$z_{\bar{X}} = \frac{\bar{X} - \mu_0}{\mbox{SE}({\bar{X}})}$$`
-

`$$z_{\bar{X}} =  \frac{\bar{X} - \mu_0}{\sigma / \sqrt{N}}$$`

`$$z_{\bar{X}} \sim \mbox{Normal}(0,1)$$`
---

.hi[**Prikaz kritičkih garnica za dvostrani test**]

.footnote[[*]Kritičke regije za dvostrani z-test.]

---

.hi[**Prikaz kritičkih garnica za jednostrani test**]

.footnote[[*]Kritičke regije za jednostrani z-test]

---

.hi[**Izvedi test u R (korak po korak)**]

```r
# Definiraj varijablu sa prosjekom ocjena u uzorku
sample.mean <- mean( grades )
print( sample.mean ) # Pogledaj podatke
```

```
#> [1] 72.3
```

```r
# Definiraj pretpostavljeni prosjek populacije
mu.null <- 67.5
# Definiraj pretpostavljenu standardnu devijaciju populacije
sd.true <- 9.5
```

```r
# Definiraj veličinu uzorka
N <- length( grades )
print( N ) # Pogledaj podatke
```

```
#> [1] 20
```

---

.hi[**Izvedi test u R (korak po korak)**]

```r
# Definiraj standardnu pogrešku sampling distribucije prosjeka (uzorka)
sem.true <- sd.true / sqrt(N)
print(sem.true) # Pogledaj podatke
```

```
#> [1] 2.124265
```

```r
# Spremi testnu statistiku u varijablu 
z.score <- (sample.mean - mu.null) / sem.true
print( z.score ) # Pogledaj podatke
```

```
#> [1] 2.259606
```

---

.hi[**Izračun pripadajuće p-vrijednosti testa**]

```r
# Vjerojatnost u gornjem dijelu distribucije
upper.area <- pnorm( q = z.score, lower.tail = FALSE )
print( upper.area ) # Pogledaj podatke
```

```
#> [1] 0.01192287
```

```r
# Vjerojatnost u donjem dijelu distribucije
lower.area <- pnorm( q = -z.score, lower.tail = TRUE )
print( lower.area ) # Pogledaj podatke
```

```
#> [1] 0.01192287
```

```r
# Izračunaj p-vrijednost
p.value <- lower.area + upper.area
print( p.value ) # Pogledaj podatke
```

```
#> [1] 0.02384574
```

---
<br>
<br>

.hi[**Pretpostavke testa**]
<br>
<br>
1. Normalnost distribucije
<br>
2. Nezavisnost podataka u uzorku
<br>
3. Poznata standardna devijacija

---
layout:false
class: middle, inverse
# t test 
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Realnija varijanta...)
---

layout:true
# t test

---

.hi[**Grafički prikaži nultu i alternativnu hipotezu kod t testa**]

.footnote[[*]Grafički prikaz nulte i alternativne hipoteze kod t testa. Primijeti sličnosti u usporedbi sa z testom. Pod nultom hipotezom je prosjek populacije mu jednak nekoj apriori specificiranoj vrijednosti mu_0, a pod alternativnom nije tako. Kao kod z testa prtpostavljamo standardnu distribuciju; razlika se odnosi na to da kod t-testa(distribucije) ne pretpostavljamo da je standardna devijacija sigma unaprijed poznata.]

---

.hi[Što ako ne znamo standardnu devijaciju populacije?]

```r
# Provjeri standardnu devijaciju uzorka na našem primjeru
sd( grades )
```

```
#> [1] 9.520615
```

.hi[**Distribucija t statistike**]

`$$t = \frac{\bar{X} - \mu}{\hat{\sigma}/\sqrt{N} }$$`

---

.hi[**Prikaži distribuciju grafički**]

<img src="../Foto/t_dist.png" width="2665" style="display: block; margin: auto;" />
<br>
<br>
<br>
.footnote[[*]t distribucija sa 2 stupnja slobode(l) i 10 stupnjeva slobode(d) i standardna distribucija(prosjek 0, i st_dev 1) prikazana isprekidanom linijom. t distribucija ima deblje repove(viša asimetričnost) od standardne distribucije. Ova razlika je izražena kod malog broja stupnjeva slobode ali zanemariva za više vrijednosti stupnjeva slobode. Za veći broj stupnjeva slobode je t distribucija skoro identična normalnoj distribuciji.]

---

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

.tiny[

```r
library(lsr) # Učitaj paket
# Provedi test
lsr::oneSampleTTest( x=grades, mu=67.5 )
```

```
#> 
#>    One sample t-test 
#> 
#> Data variable:   grades 
#> 
#> Descriptive statistics: 
#>             grades
#>    mean     72.300
#>    std dev.  9.521
#> 
#> Hypotheses: 
#>    null:        population mean equals 67.5 
#>    alternative: population mean not equal to 67.5 
#> 
#> Test results: 
#>    t-statistic:  2.255 
#>    degrees of freedom:  19 
#>    p-value:  0.036 
#> 
#> Other information: 
#>    two-sided 95% confidence interval:  [67.844, 76.756] 
#>    estimated effect size (Cohen's d):  0.504
```
]
---
<br>
<br>

.hi[**Zapis rezultata testa**]

> t(19) = 2.25, p < .05, CI_{95} = [67.8, 76.8]

.hi[**Pretpostavke t testa**]
<br>
<br>

1. Normalnost distribucije
<br>
2. Nezavisnost

---

layout:false
class: middle, inverse
# t test ZA NEZAVISNE UZORKE (Student) 
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Studentovo rješenje...)
---

layout:true
# t test ZA NEZAVISNE UZORKE (Student)
---

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

```
#> 'data.frame':	33 obs. of  2 variables:
#>  $ grade: num  65 72 66 74 73 71 66 76 69 79 ...
#>  $ tutor: Factor w/ 2 levels "Anastasia","Bernadette": 1 2 2 1 1 2 2 2 2 2 ...
```

```r
head( harpo,4 ) # Pregledaj podatke
```

```
#>   grade      tutor
#> 1    65  Anastasia
#> 2    72 Bernadette
#> 3    66 Bernadette
#> 4    74  Anastasia
```

---
<br>
<br>

```r
# Deskriptivna statistika
library(tidyverse)
harpo %>%
  dplyr::group_by(tutor) %>%
  dplyr::summarise(prosjek = mean(grade),
            stDev = sd(grade))
```

```
#> # A tibble: 2 x 3
#>   tutor      prosjek stDev
#>   <fct>        <dbl> <dbl>
#> 1 Anastasia     74.5  9.00
#> 2 Bernadette    69.1  5.77
```

---

.hi[**Histogram ocjena**]

.footnote[[*]Histogram prikazuje distribuciju ocjena u Anastasijnom razredu.]

---

.hi[**Histogram ocjena**]

.footnote[[*]Histogram prikazuje distribuciju ocjena u Bernadettinom razredu.]

---

.hi[**Postavi nultu i alternativnu hipotezu**]

`$$\begin{array}{ll}
H_0: & \mu_1 = \mu_2  \\
H_1: & \mu_1 \neq \mu_2
\end{array}$$`

.hi[**Prikaži test grafički**]

.tiny[
<img src="../Foto/t_promjena.png" width="2667" style="display: block; margin: auto;" />
]
---

.hi[**Testna statistika**]

`$$t = \frac{\bar{X}_1 - \bar{X}_2}{\mbox{SE}}$$`

.hi[**Grafički prikaz t distribucije**]

.footnote[[*] Grafički prikaz nulte i alternativne hipoteze kod Studentovog t testa. Nulta hipoteza pretpostavlja da obje grupe imaju jednak prosjek mu_1 i mu_2 dok su ti prosjeci pod alternativnom hipotezom različiti. Primijeti pretpostavku da su populacijske distribucije normalne i da imaju jednaku standardnu devijaciju.]

---

.hi[**Procjena udružene standardne devijacije**]

- Definiraj pondere

`$$\begin{array}{rcl}
w_1 &=& N_1 - 1\\
w_2 &=& N_2 - 1
\end{array}$$`

- Udružena procjena varijance

`$$\hat\sigma^2_p = \frac{w_1 {\hat\sigma_1}^2 + w_2 {\hat\sigma_2}^2}{w_1 + w_2}$$`

- Udružena procjena standardne devijacije

`$$\hat\sigma_p = \sqrt{\frac{w_1 {\hat\sigma_1}^2 + w_2 {\hat\sigma_2}^2}{w_1 + w_2}}$$`

---

.hi[**Drugačiji način**]

- Izračunaj devijaciju od grupnog prosjeka

`$$X_{ik} - \bar{X}_k$$`

- Zbroji za sve opservacije

`$$\frac{\sum_{ik} \left( X_{ik} - \bar{X}_k \right)^2}{N}$$`
- Izvrši korekciju (nazivnik)

`$$\hat\sigma^2_p = \frac{\sum_{ik} \left( X_{ik} - \bar{X}_k \right)^2}{N -2}$$`

---

.hi[**Drugačiji način**]
<br>
- Izračunaj standardnu pogrešku razlike prosjeka

`$$\mbox{SE}({\bar{X}_1 - \bar{X}_2}) = \hat\sigma \sqrt{\frac{1}{N_1} + \frac{1}{N_2}}$$`

- t statistika u našem testu

`$$t = \frac{\bar{X}_1 - \bar{X}_2}{\mbox{SE}({\bar{X}_1 - \bar{X}_2})}$$`
---

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

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

```
#>   grade      tutor
#> 1    65  Anastasia
#> 2    72 Bernadette
#> 3    66 Bernadette
#> 4    74  Anastasia
#> 5    73  Anastasia
#> 6    71 Bernadette
```

---

.hi[**Izvođenje testa u R**]

.tiny[

```r
# Izvedi test
independentSamplesTTest( 
      formula = grade ~ tutor,  # Formula za zavisnu i nezavisnu varijablu
      data = harpo,             # Podatci
      var.equal = TRUE          # Pretpostavka jednakih varijanci
  )
```

```
#> 
#>    Student's independent samples t-test 
#> 
#> Outcome variable:   grade 
#> Grouping variable:  tutor 
#> 
#> Descriptive statistics: 
#>             Anastasia Bernadette
#>    mean        74.533     69.056
#>    std dev.     8.999      5.775
#> 
#> Hypotheses: 
#>    null:        population means equal for both groups
#>    alternative: different population means in each group
#> 
#> Test results: 
#>    t-statistic:  2.115 
#>    degrees of freedom:  31 
#>    p-value:  0.043 
#> 
#> Other information: 
#>    two-sided 95% confidence interval:  [0.197, 10.759] 
#>    estimated effect size (Cohen's d):  0.74
```
]

---

.hi[**Sažeti statistički zapis rezultata testa**]
<br>
<br>
> t(31) = 2.1, p<.05, CI_{95} = [0.2, 10.8], d = .74

.hi[**Pretpostavke testa**]
<br>
<br>
1. Normalnost distribucije
<br>
2. Nezavisnost
<br>
3. Homogenost varijance

---
layout:false
class: middle, inverse
# t test ZA NEZAVISNE UZORKE (Welch) 
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Welch-eva varijanta...)
---

layout:true
# t test ZA NEZAVISNE UZORKE (Student)
---

.hi[Testna statistika]

.hi[Standardna greška statistike]

.hi[Izračun stupnjeva slobode]

---

.hi[**Prikaži nultu i alternativnu hipotezu grafički**]

.footnote[[*]Grafički prikaz nulte i alternativne hipoteze za Welch t test. Kao kod studentovog t testa pretpostavljamo normalnu distribuciju ali valja primijetiti da pod alternativnom hipotezom više ne zahtijevamo da oba uzorka imaju jednake varijance.]

---

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

.tiny[

```r
independentSamplesTTest( 
      formula = grade ~ tutor,  # Formula za zavisnu i nezavisnu varijablu
      data = harpo)          # Podatci
```

```
#> 
#>    Welch's independent samples t-test 
#> 
#> Outcome variable:   grade 
#> Grouping variable:  tutor 
#> 
#> Descriptive statistics: 
#>             Anastasia Bernadette
#>    mean        74.533     69.056
#>    std dev.     8.999      5.775
#> 
#> Hypotheses: 
#>    null:        population means equal for both groups
#>    alternative: different population means in each group
#> 
#> Test results: 
#>    t-statistic:  2.034 
#>    degrees of freedom:  23.025 
#>    p-value:  0.054 
#> 
#> Other information: 
#>    two-sided 95% confidence interval:  [-0.092, 11.048] 
#>    estimated effect size (Cohen's d):  0.724
```
]
---
<br>
.hi[**Pretpostavke testa**]
<br>
<br>
1. Normalnost distribucije
<br>
2. Nezavisnost

---
layout:false
class: middle, inverse
# t test ZA ZAVISNE UZORKE 
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(U zavisnim uzorcima...)
---

layout:true
# t test ZA ZAVISNE UZORKE
---

```r
# Učitaj podatke
load( file.path("../Podatci/chico.Rdata" ))
str(chico)  # Pogledaj podatke   
```

```
#> 'data.frame':	20 obs. of  3 variables:
#>  $ id         : Factor w/ 20 levels "student1","student10",..: 1 12 14 15 16 17 18 19 20 2 ...
#>  $ grade_test1: num  42.9 51.8 71.7 51.6 63.5 58 59.8 50.8 62.5 61.9 ...
#>  $ grade_test2: num  44.6 54 72.3 53.4 63.8 59.3 60.8 51.6 64.3 63.2 ...
```

```r
# Pregledaj podatke
head( chico,4 )
```

```
#>         id grade_test1 grade_test2
#> 1 student1        42.9        44.6
#> 2 student2        51.8        54.0
#> 3 student3        71.7        72.3
#> 4 student4        51.6        53.4
```

---

```r
library( psych ) # Učitaj paket
psych::describe( chico ) # Pregledaj podatke
```

```
#>             vars  n  mean   sd median trimmed  mad  min  max range  skew
#> id*            1 20 10.50 5.92   10.5   10.50 7.41  1.0 20.0  19.0  0.00
#> grade_test1    2 20 56.98 6.62   57.7   56.92 7.71 42.9 71.7  28.8  0.05
#> grade_test2    3 20 58.38 6.41   59.7   58.35 6.45 44.6 72.3  27.7 -0.05
#>             kurtosis   se
#> id*            -1.38 1.32
#> grade_test1    -0.35 1.48
#> grade_test2    -0.39 1.43
```

---

.hi[**Prikaži podatke grafički**]

.footnote[[*]Prosječna ocjena za test 1 i 2, uz prateće 95% intervale pouzdanosti.]

---

.footnote[[*]Dijagram rasipanja za odnos ocjena na prvom i drugom testu.]

---

.hi[Izračunaj razliku u ocjeni]

```r
# Stvori vektor razlika u ocjenama između prvog i drugog testa
chico$improvement <- chico$grade_test2 - chico$grade_test1 
```

```r
# Pregledaj podatke
head( chico,5 )
```

```
#>         id grade_test1 grade_test2 improvement
#> 1 student1        42.9        44.6         1.7
#> 2 student2        51.8        54.0         2.2
#> 3 student3        71.7        72.3         0.6
#> 4 student4        51.6        53.4         1.8
#> 5 student5        63.5        63.8         0.3
```

---

.hi[**Prikaži razlike grafički**]

.footnote[[*]Histogram prikazuje individualna poboljšanja ocjene između prvog i drugog testa. Valja primijetiti da je gotovo cjelokupna distribucija iznad 0: najveći broj studenata je poboljšao rezultat na drugom testu.]

---

```r
# Pogledaj granice pouzdanosti za varijablu poboljšanja ocjena
ciMean( x = chico$improvement )
```

```
#>           2.5%    97.5%
#> [1,] 0.9508686 1.859131
```

---

.hi[**Tehnički aspekti testa**]

- Definiraj razliku

`$$D_{i} = X_{i1} - X_{i2}$$`

- Formuliraj hipoteze

`$$\begin{array}{ll}
H_0: & \mu_D = 0  \\
H_1: & \mu_D \neq 0
\end{array}$$`

- Testna statistika

`$$t = \frac{\bar{D}}{\mbox{SE}({\bar{D}})}$$`

---

.hi[**Izvođenje testa u R**]

I. način

.tiny[

```r
lsr::oneSampleTTest( chico$improvement, mu=0 )
```

```
#> 
#>    One sample t-test 
#> 
#> Data variable:   chico$improvement 
#> 
#> Descriptive statistics: 
#>             improvement
#>    mean           1.405
#>    std dev.       0.970
#> 
#> Hypotheses: 
#>    null:        population mean equals 0 
#>    alternative: population mean not equal to 0 
#> 
#> Test results: 
#>    t-statistic:  6.475 
#>    degrees of freedom:  19 
#>    p-value:  <.001 
#> 
#> Other information: 
#>    two-sided 95% confidence interval:  [0.951, 1.859] 
#>    estimated effect size (Cohen's d):  1.448
```
]
---

.hi[**Izvođenje testa u R**]

II. način

.tiny[

```r
lsr::pairedSamplesTTest( 
     formula = ~ grade_test2 + grade_test1, # Formula za definiranje zavisnih i nezavisnih varijabli
     data = chico                           # Podatci 
  )
```

```
#> 
#>    Paired samples t-test 
#> 
#> Variables:  grade_test2 , grade_test1 
#> 
#> Descriptive statistics: 
#>             grade_test2 grade_test1 difference
#>    mean          58.385      56.980      1.405
#>    std dev.       6.406       6.616      0.970
#> 
#> Hypotheses: 
#>    null:        population means equal for both measurements
#>    alternative: different population means for each measurement
#> 
#> Test results: 
#>    t-statistic:  6.475 
#>    degrees of freedom:  19 
#>    p-value:  <.001 
#> 
#> Other information: 
#>    two-sided 95% confidence interval:  [0.951, 1.859] 
#>    estimated effect size (Cohen's d):  1.448
```
]
---

.hi[**Izvođenje testa u R**]

II. način

```r
# Prestrukturiraj podatke
chico2 <- wideToLong( chico, within="time" )
head( chico2 ) # Pregledaj podatke
```

```
#>         id improvement  time grade
#> 1 student1         1.7 test1  42.9
#> 2 student2         2.2 test1  51.8
#> 3 student3         0.6 test1  71.7
#> 4 student4         1.8 test1  51.6
#> 5 student5         0.3 test1  63.5
#> 6 student6         1.3 test1  58.0
```

---
II. način

```r
# Sortiraj podatke
chico2 <- sortFrame( chico2, id )
head( chico2 ) # Pregledaj podatke
```

```
#>           id improvement  time grade
#> 1   student1         1.7 test1  42.9
#> 21  student1         1.7 test2  44.6
#> 10 student10         1.3 test1  61.9
#> 30 student10         1.3 test2  63.2
#> 11 student11         1.4 test1  50.4
#> 31 student11         1.4 test2  51.8
```

---

II. način
.tiny[

```r
# Provedi test
lsr::pairedSamplesTTest( 
     formula = grade ~ time,  # Definiraj formulu
     data = chico2,           # Podatci
     id = "id"                # Naziv id
  )
```

```
#> 
#>    Paired samples t-test 
#> 
#> Outcome variable:   grade 
#> Grouping variable:  time 
#> ID variable:        id 
#> 
#> Descriptive statistics: 
#>              test1  test2 difference
#>    mean     56.980 58.385     -1.405
#>    std dev.  6.616  6.406      0.970
#> 
#> Hypotheses: 
#>    null:        population means equal for both measurements
#>    alternative: different population means for each measurement
#> 
#> Test results: 
#>    t-statistic:  -6.475 
#>    degrees of freedom:  19 
#>    p-value:  <.001 
#> 
#> Other information: 
#>    two-sided 95% confidence interval:  [-1.859, -0.951] 
#>    estimated effect size (Cohen's d):  1.448
```
]
---

.hi[**Alternativna specifikacija testa**]
<br>
<br>

```
pairedSamplesTTest( 
     formula = grade ~ time + (id),
     data = chico2
  )
  
ili  
  
```

```
 pairedSamplesTTest( grade ~ time + (id), chico2 )
 
```

---
layout:false
class: middle, inverse
# JEDNOSTRANI TESTOVI 
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(U slučaju kada znamo smjer odnosa...)
---

layout:true
# JEDNOSTRANI TESTOVI
---

.hi[**Provedi test sa pretpostavkom da je prosjek uzorka veći od pretpostavljenog prosjeka populacije**]

.tiny[

```r
library(psych)
library(lsr)
 # Provedi test
lsr::oneSampleTTest( x = grades,
                mu = 67.5,
                one.sided = "greater" # Gornja granica
                ) 
```

```
#> 
#>    One sample t-test 
#> 
#> Data variable:   grades 
#> 
#> Descriptive statistics: 
#>             grades
#>    mean     72.300
#>    std dev.  9.521
#> 
#> Hypotheses: 
#>    null:        population mean less than or equal to 67.5 
#>    alternative: population mean greater than 67.5 
#> 
#> Test results: 
#>    t-statistic:  2.255 
#>    degrees of freedom:  19 
#>    p-value:  0.018 
#> 
#> Other information: 
#>    one-sided 95% confidence interval:  [68.619, Inf] 
#>    estimated effect size (Cohen's d):  0.504
```
]

---

.tiny[

```r
# Provedi test na drugi način
lsr::independentSamplesTTest( 
    formula = grade ~ tutor, 
    data = harpo, 
    one.sided = "Anastasia"
  )
```

```
#> 
#>    Welch's independent samples t-test 
#> 
#> Outcome variable:   grade 
#> Grouping variable:  tutor 
#> 
#> Descriptive statistics: 
#>             Anastasia Bernadette
#>    mean        74.533     69.056
#>    std dev.     8.999      5.775
#> 
#> Hypotheses: 
#>    null:        population means are equal, or smaller for group 'Anastasia' 
#>    alternative: population mean is larger for group 'Anastasia' 
#> 
#> Test results: 
#>    t-statistic:  2.034 
#>    degrees of freedom:  23.025 
#>    p-value:  0.027 
#> 
#> Other information: 
#>    one-sided 95% confidence interval:  [0.863, Inf] 
#>    estimated effect size (Cohen's d):  0.724
```
]

---

.tiny[

```r
# Provedi test na treći način
pairedSamplesTTest( 
     formula = ~ grade_test2 + grade_test1, 
     data = chico, 
     one.sided = "grade_test2" 
  )
```

```
#> 
#>    Paired samples t-test 
#> 
#> Variables:  grade_test2 , grade_test1 
#> 
#> Descriptive statistics: 
#>             grade_test2 grade_test1 difference
#>    mean          58.385      56.980      1.405
#>    std dev.       6.406       6.616      0.970
#> 
#> Hypotheses: 
#>    null:        population means are equal, or smaller for measurement 'grade_test2' 
#>    alternative: population mean is larger for measurement 'grade_test2' 
#> 
#> Test results: 
#>    t-statistic:  6.475 
#>    degrees of freedom:  19 
#>    p-value:  <.001 
#> 
#> Other information: 
#>    one-sided 95% confidence interval:  [1.03, Inf] 
#>    estimated effect size (Cohen's d):  1.448
```
]

---

<br>

```r
# Alternativne specifikacije testa
> pairedSamplesTTest( 
    formula = grade ~ time, 
    data = chico2, 
    id = "id", 
    one.sided = "test2" 
  )
> pairedSamplesTTest( 
    formula = grade ~ time + (id), 
    data = chico2, 
    one.sided = "test2" 
  )
```

---
layout:false
class: middle, inverse
# STANDARDNI NAČIN PROVOĐENJA t-testa  
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Test u praksi...)
---

layout:true
# STANDARDNI NAČIN PROVOĐENJA t-testa U R
---

```r
# Provedi standardni test usporedbe prosjeka
t.test( x = grades, # Definiraj podatke
        mu = 67.5   # Definiraj prosjek 
        )
```

```
#> 
#> 	One Sample t-test
#> 
#> data:  grades
#> t = 2.2547, df = 19, p-value = 0.03615
#> alternative hypothesis: true mean is not equal to 67.5
#> 95 percent confidence interval:
#>  67.84422 76.75578
#> sample estimates:
#> mean of x 
#>      72.3
```
---

```r
# Provedi test za nezavisne uzorke
t.test( formula = grade ~ tutor, # Definiraj formulu
        data = harpo )           # Definiraj podatke
```

```
#> 
#> 	Welch Two Sample t-test
#> 
#> data:  grade by tutor
#> t = 2.0342, df = 23.025, p-value = 0.05361
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -0.09249349 11.04804904
#> sample estimates:
#>  mean in group Anastasia mean in group Bernadette 
#>                 74.53333                 69.05556
```

---

```r
# Provedi test za zavisne uzorke
t.test( x = chico$grade_test2,   # Definiraj varijablu
        y = chico$grade_test1,   # Definiraj varijablu
         paired = TRUE           # Zavisni uzorci
 )
```

```
#> 
#> 	Paired t-test
#> 
#> data:  chico$grade_test2 and chico$grade_test1
#> t = 6.4754, df = 19, p-value = 3.321e-06
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  0.9508686 1.8591314
#> sample estimates:
#> mean of the differences 
#>                   1.405
```

---
layout:false
class: middle, inverse
# EFEKT VELIČINE 
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Snaga procijenjenog odnosa...)
---

layout:true
# EFEKT VELIČINE
---

.hi[**Osnovna ideja**]

`$$d = \frac{\mbox{(prosjek 1)} - \mbox{(prosjek 2)}}{\mbox{std dev}}$$`
.hi[**Interpretacija**]

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> `$d$`-vrijednost </th>
   <th style="text-align:left;"> okvirna interpretacija </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> otprilike 0.2 </td>
   <td style="text-align:left;"> mali efekt </td>
  </tr>
  <tr>
   <td style="text-align:left;"> otprilike 0.5 </td>
   <td style="text-align:left;"> umjereni efekt </td>
  </tr>
  <tr>
   <td style="text-align:left;"> otprilike 0.8 </td>
   <td style="text-align:left;"> veliki efekt </td>
  </tr>
</tbody>
</table>

.footnote[[*] Okvirni prikaz za interpretaciju Choenove d statistike. d statistika pokazuje kolika je razlika između dva prosjeka izraženo u standardnim devijacijama.]

---

.hi[**Jedan uzorak**]

`$$d = \frac{\bar{X} - \mu_0}{\hat{\sigma}}$$`

```r
# Provedi Choen test za jedan uzorak
lsr::cohensD( x = grades,    # Podatci
          mu = 67.5     # cUsporedi sa prosjekom od 67.5
 )
```

```
#> [1] 0.5041691
```

```r
# Provedi test "ručno" (bez funkcije)
(mean(grades) - 67.5 ) / sd(grades)
```

```
#> [1] 0.5041691
```

---

.hi[**Efekt veličine Studentovog t testa**]

`$$\delta = \frac{\mu_1 - \mu_2}{\sigma}$$`

`$$d = \frac{\bar{X}_1 - \bar{X}_2}{\hat{\sigma}_p}$$`

```r
# Provedi test u R
lsr::cohensD( formula = grade ~ tutor,   # Definiraj formulu
          data = harpo,             # Podatci 
          method = "pooled"         # Tip testa
)
```

```
#> [1] 0.7395614
```
---

.hi[**Efekt veličine Welchovog t testa**]

`$$\delta^\prime = \frac{\mu_1 - \mu_2}{\sigma^\prime}$$`

`$$\sigma^\prime = \sqrt{\displaystyle{\frac{ {\sigma_1}^2 + {\sigma_2}^2}{2}}}$$`

`$$d = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\displaystyle{\frac{ {\hat\sigma_1}^2 + {\hat\sigma_2}^2}{2}}}}$$`

```r
# Provedi test u R
lsr::cohensD( formula = grade ~ tutor,  # Definiraj formulu
          data = harpo,            # Podatci
          method = "unequal"       # Tip testa
 )
```

```
#> [1] 0.7244995
```
---

.hi[**Efekt veličine kod zavisnih uzoraka**]

`$$d = \frac{\bar{D}}{\hat{\sigma}_D}$$`

```r
# Provedi test u R
lsr::cohensD( x = chico$grade_test2,  # Definiraj prvu varijablu
         y = chico$grade_test1,  # Definiraj drugu varijablu
          method = "paired"      # Izaberi metodu
 )
```

```
#> [1] 1.447952
```

---
layout:false
class: middle, inverse
# PROVJERA NORMALNOSTI DISTRIBUCIJE 
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Jesu li zadovoljene pretpostavke?)
---

layout:true
# PROVJERA NORMALNOSTI DISTRIBUCIJE
---

```
#> Normalno distribuirani podatci 
#> Asimetrija= -0.02936155 
#> Zakrivljenost= -0.06035938 
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  data
#> W = 0.99108, p-value = 0.7515
```

<br>
.footnote[[*]Histogram normalno distribuiranih podataka; prikaz se odnosi na simulaciju 100 opservacija.]

---

.footnote[[*]QQ plot normalno distribuiranih podataka, prikaz se odnosi na simulaciju 100 opservacija.]

---

```
#> Podatci sa gamma distribucijom. 
#> Asimetrija= 1.889475 
#> Zakrivljenost= 4.4396 
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  data
#> W = 0.81758, p-value = 8.908e-10
```

.footnote[[*]Histogram 100 opservacija "zakrivljeno" distribuiranih podataka.]

---

.footnote[[*]QQ plot normalno distribuiranih, "zakrivljenih" podataka, prikaz se odnosi na simulaciju 100 opservacija.]

---

```
#> Heavy-Tailed Data 
#> Asimetrija= -0.05308273 
#> Zakrivljenost= 7.508765 
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  data
#> W = 0.83892, p-value = 4.718e-09
```

.footnote[[*]Histogram 100 opservacija u distribuciji koja ima puno mase u repovima.]

---

.footnote[[*]Histogram 100 opservacija u distribuciji koja ima puno mase u repovima.]

---

```r
normal.data <- rnorm( n = 100 )  # Stvori 100 normalno distribuiranih brojeva
par(mfrow=c(1,2))                # Postavi grid za prikaz grafikona
hist( x = normal.data )          # Napravi histogram
qqnorm( y = normal.data )        # Napravi QQ grafikon
```

```r
par(mfrow=c(1,1))                # Resetiraj grid za prikaz grafikona
```

---

.hi[**Shapiro-Wilk test**]

`$$W = \frac{ \left( \sum_{i = 1}^N a_i X_i \right)^2 }{ \sum_{i = 1}^N (X_i - \bar{X})^2}$$`

```r
# Provedi test na nizu normalno distribuiranih podataka
shapiro.test( x = normal.data )
```

```
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  normal.data
#> W = 0.98654, p-value = 0.4076
```

---
layout:false
class: middle, inverse
# NE-STANDARDNA DISTRIBUCIJA  
<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
(Kada nisu zadovoljene pretpostavke...)
---

layout:true
# NE-STANDARDNA DISTRIBUCIJA
---

.hi[**Wilcoxson test za dva uzorka**]

```r
load(file.path("../Podatci/awesome.Rdata")) # Uvezi podatke
print( awesome ) # Prikaži podatke
```

```
#>    scores group
#> 1     6.4     A
#> 2    10.7     A
#> 3    11.9     A
#> 4     7.3     A
#> 5    10.0     A
#> 6    14.5     B
#> 7    10.4     B
#> 8    12.9     B
#> 9    11.7     B
#> 10   13.0     B
```
---

```r
# Provedi Wicoxonov test
wilcox.test( formula = scores ~ group, data = awesome)
```

```
#> 
#> 	Wilcoxon rank sum exact test
#> 
#> data:  scores by group
#> W = 3, p-value = 0.05556
#> alternative hypothesis: true location shift is not equal to 0
```

---

```r
# Uvezi drugačije organizirane podatke
load( file.path("../Podatci/awesome2.Rdata" ))
score.A
```

```
#> [1]  6.4 10.7 11.9  7.3 10.0
```

```r
score.B
```

```
#> [1] 14.5 10.4 12.9 11.7 13.0
```

```r
# Provedi Wicoxonov test
wilcox.test( x = score.A, y = score.B )
```

```
#> 
#> 	Wilcoxon rank sum exact test
#> 
#> data:  score.A and score.B
#> W = 3, p-value = 0.05556
#> alternative hypothesis: true location shift is not equal to 0
```

---

.hi[**Wilcoxson test za jedan uzorak**]

```r
# Učitaj podatke
load( file.path("../podatci/happy.Rdata" ))
print( happiness ) # Prikaži podatke
```

```
#>    before after change
#> 1      30     6    -24
#> 2      43    29    -14
#> 3      21    11    -10
#> 4      24    31      7
#> 5      23    17     -6
#> 6      40     2    -38
#> 7      29    31      2
#> 8      56    21    -35
#> 9      38     8    -30
#> 10     16    21      5
```
---

```r
# Wilcoxconov test za jedan uzorak
wilcox.test( x = happiness$change,
              mu = 0
)
```

```
#> 
#> 	Wilcoxon signed rank exact test
#> 
#> data:  happiness$change
#> V = 7, p-value = 0.03711
#> alternative hypothesis: true location is not equal to 0
```
---

```r
# Wilcoxonov test za zavisne uzorke
wilcox.test( x = happiness$after,
             y = happiness$before,
              paired = TRUE 
)
```

```
#> 
#> 	Wilcoxon signed rank exact test
#> 
#> data:  happiness$after and happiness$before
#> V = 7, p-value = 0.03711
#> alternative hypothesis: true location shift is not equal to 0
```

---

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 više od dva prosjeka.)