STATISTIČKI ALATI: USPOREDBA DVA PROSJEKA
Hrvatski studiji
dr.sc. Luka Šikić
16 prosinac, 2019
CILJEVI PREDAVANJA
- 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
Z-test
- Pitanje: Da li podatci pripadju hipotetičkoj distribuciji?
- Uvoz podataka
## [1] 50 60 60 64 66 66 67 69 70 74 76 76 77 79 79 79 81 82 82 89
## [1] 72.3
- Postavi hipoteze
- Da li su ocjene studenata sociologije više od prosjeka na fakultetu?
\[
\begin{array}{ll}
H_0: & \mu = 67.5 \\
H_1: & \mu \neq 67.5
\end{array}
\]
- Prikaži hipoteze grafički
- Konstruiraj testnu statistiku
\[
\bar{X} - \mu_0
\]
\[
X \sim \mbox{Normal}(\mu_0,\sigma^2)
\]
\[
\mbox{SE}({\bar{X}}) = \frac{\sigma}{\sqrt{N}}
\]
\[
\bar{X} \sim \mbox{Normal}(\mu_0,\mbox{SE}({\bar{X}}))
\]
\[
z_{\bar{X}} = \frac{\bar{X} - \mu_0}{\mbox{SE}({\bar{X}})}
\]
\[
z_{\bar{X}} = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{N}}
\]
\[
z_{\bar{X}} \sim \mbox{Normal}(0,1)
\]
- Prikaz kritičkih garnica za jednostrani i dvostrani test
## [1] 72.3
## [1] 20
## [1] 2.124265
## [1] 2.259606
- Izračun pripadajuće p-vrijednosti testa
## [1] 0.01192287
## [1] 0.01192287
## [1] 0.02384574
- Normalnost distribucije
- Nezavisnost podataka u uzorku
- Poznata standardna devijacija
T-test
- grafički prikaži nultu i alternativnu hipotezu kod \(t\)-testa
- Što ako ne znamo standardnu devijaciju populacije?
## [1] 9.520615
- distribucija \(t\)-statistike
\[
t = \frac{\bar{X} - \mu}{\hat{\sigma}/\sqrt{N} }
\] - Prikaži distribuciju grafički
##
## 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
\(t(19) = 2.25\), \(p<.05\), CI\(_{95} = [67.8, 76.8]\)
- Normalnost distribucije
- Nezavisnost
T-test ZA NEZAVISNE UZORKE (Student)
## '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 ...
## grade tutor
## 1 65 Anastasia
## 2 72 Bernadette
## 3 66 Bernadette
## 4 74 Anastasia
## 5 73 Anastasia
## 6 71 Bernadette
| Anastasia |
74.53 |
9.00 |
15 |
| Bernadette |
69.06 |
5.77 |
18 |
- Pregledaj histogram ocjena
- Postavi nultu i alternativnu hipotezu
\[
\begin{array}{ll}
H_0: & \mu_1 = \mu_2 \\
H_1: & \mu_1 \neq \mu_2
\end{array}
\]
\[
t = \frac{\bar{X}_1 - \bar{X}_2}{\mbox{SE}}
\]
- Grafički prikaz \(t\)-distribucije
\[
\mu_1 - \mu_2 = 0
\]
\[
\bar{X}_1 - \bar{X}_2
\]
- 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
\[
\hat\sigma_p = \sqrt{\frac{w_1 {\hat\sigma_1}^2 + w_2 {\hat\sigma_2}^2}{w_1 + w_2}}
\]
- Izračunaj devijaciju od grupnog prosjeka
\[
X_{ik} - \bar{X}_k
\]
- Zbroji za sve opservacije
\[
\frac{\sum_{ik} \left( X_{ik} - \bar{X}_k \right)^2}{N}
\] 3. Izvrši korekciju(nazivnik)
\[
\hat\sigma^2_p = \frac{\sum_{ik} \left( X_{ik} - \bar{X}_k \right)^2}{N -2}
\]
- 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
\[
t = \frac{\bar{X}_1 - \bar{X}_2}{\mbox{SE}({\bar{X}_1 - \bar{X}_2})}
\]
## grade tutor
## 1 65 Anastasia
## 2 72 Bernadette
## 3 66 Bernadette
## 4 74 Anastasia
## 5 73 Anastasia
## 6 71 Bernadette
##
## 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
- Sažeti statistički zapis rezultata testa
\(t(31) = 2.1\), \(p<.05\), \(CI_{95} = [0.2, 10.8]\), \(d = .74\)
- Normalnost distribucije
- Nezavisnost
- Homogenost varijance
T-test ZA NEZAVISNE UZORKE (Welch)
\[
t = \frac{\bar{X}_1 - \bar{X}_2}{\mbox{SE}({\bar{X}_1 - \bar{X}_2})}
\]
- Standardna greška statistike
\[
\mbox{SE}({\bar{X}_1 - \bar{X}_2}) = \sqrt{ \frac{{\hat{\sigma}_1}^2}{N_1} + \frac{{\hat{\sigma}_2}^2}{N_2} }
\]
- Izračun stupnjeva slobode
\[
\mbox{df} = \frac{ ({\hat{\sigma}_1}^2 / N_1 + {\hat{\sigma}_2}^2 / N_2)^2 }{ ({\hat{\sigma}_1}^2 / N_1)^2 / (N_1 -1 ) + ({\hat{\sigma}_2}^2 / N_2)^2 / (N_2 -1 ) }
\]
- Prikaži nultu i alternativnu hipotezu grafički
##
## 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
- Normalnost distribucije
- Nezavisnost
T-test ZA ZAVISNE UZORKE
## '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 ...
## 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
## 5 student5 63.5 63.8
## 6 student6 58.0 59.3
## Warning: package 'psych' was built under R version 3.6.1
## 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
- Izračunaj razliku u ocjeni
## 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
## 6 student6 58.0 59.3 1.3
## 2.5% 97.5%
## [1,] 0.9508686 1.859131
- 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
\[
t = \frac{\bar{D}}{\mbox{SE}({\bar{D}})}
\]
\[
t = \frac{\bar{D}}{\hat\sigma_D / \sqrt{N}}
\]
- način
##
## 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
- način
##
## 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
- način
## 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
## 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
##
## 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
- Alternativna specifikacija testa
pairedSamplesTTest(
formula = grade ~ time + (id),
data = chico2
)
pairedSamplesTTest( grade ~ time + (id), chico2 )
JEDNOSTRANI TESTOVI
- Provedi test sa pretpostavkom da je prosjek uzorka veći od pretpostavljenog prosjeka populacije
##
## 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
##
## 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
##
## 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
STANDARDNI NAČIN PROVOĐENJA t-testa U R
##
## 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
##
## 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
##
## 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
(Choen) EFEKT VELIČINE
\[
d = \frac{\mbox{(prosjek 1)} - \mbox{(prosjek 2)}}{\mbox{std dev}}
\]
| otprilike 0.2 |
mali efekt |
| otprilike 0.5 |
umjereni efekt |
| otprilike 0.8 |
veliki efekt |
\[
d = \frac{\bar{X} - \mu_0}{\hat{\sigma}}
\]
## [1] 0.5041691
## [1] 0.5041691
- Efekt veličine Studentovog \(t\)-testa
\[
\delta = \frac{\mu_1 - \mu_2}{\sigma}
\]
\[
d = \frac{\bar{X}_1 - \bar{X}_2}{\hat{\sigma}_p}
\]
## [1] 0.7395614
- Efekt veličine Welchovog \(t\)-testa
\[
\delta^\prime = \frac{\mu_1 - \mu_2}{\sigma^\prime}
\]
\[
\sigma^\prime = \sqrt{\displaystyle{\frac{ {\sigma_1}^2 + {\sigma_2}^2}{2}}}
\]
\[
d = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\displaystyle{\frac{ {\hat\sigma_1}^2 + {\hat\sigma_2}^2}{2}}}}
\]
## [1] 0.7244995
- Efekt veličine kod zavisnih uzoraka
\[
d = \frac{\bar{D}}{\hat{\sigma}_D}
\]
## [1] 1.447952
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
## Podatci sa gamma distribucijom.
## Asimetrija= 1.889475
## Zakrivljenost= 4.4396
##
## Shapiro-Wilk normality test
##
## data: data
## W = 0.81758, p-value = 8.908e-10
## Heavy-Tailed Data
## Asimetrija= -0.05308273
## Zakrivljenost= 7.508765
##
## Shapiro-Wilk normality test
##
## data: data
## W = 0.83892, p-value = 4.718e-09
\[
W = \frac{ \left( \sum_{i = 1}^N a_i X_i \right)^2 }{ \sum_{i = 1}^N (X_i - \bar{X})^2}
\]
##
## Shapiro-Wilk normality test
##
## data: normal.data
## W = 0.98654, p-value = 0.4076
TESTIRANJE NE-NORMALNO DISTRIBUIRANIH PODATAKA
- Wilcoxson test za dva uzorka
## 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
##
## Wilcoxon rank sum test
##
## data: scores by group
## W = 3, p-value = 0.05556
## alternative hypothesis: true location shift is not equal to 0
## [1] 6.4 10.7 11.9 7.3 10.0
## [1] 14.5 10.4 12.9 11.7 13.0
##
## Wilcoxon rank sum test
##
## data: score.A and score.B
## W = 3, p-value = 0.05556
## alternative hypothesis: true location shift is not equal to 0
- Wilcoxson test za jedan uzorak
## 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
##
## Wilcoxon signed rank test
##
## data: happiness$change
## V = 7, p-value = 0.03711
## alternative hypothesis: true location is not equal to 0
##
## Wilcoxon signed rank test
##
## data: happiness$after and happiness$before
## V = 7, p-value = 0.03711
## alternative hypothesis: true location shift is not equal to 0
