1. summary

2. Homoscedasticity

3. Tests for homoscedasticity
𝐻 0 : 𝜎 1 = 𝜎 2
F-test of equality of variances (Hartley's test), 𝐹 𝑛 𝐿 −1, 𝑛 𝑆 −1 𝑛 𝐿 −1, 𝑛 𝑆 −1 = 𝑠 𝐿 2 𝑠 𝑆 2
F-test is extremely sensitive to the departures from the normality.
Alternative test is the Levene's test – performed over absolute values of the deviations from the mean, test statistic distribution: F-distribution

4. Power of the test
A probability that it correctly rejects the null hypothesis (H0) when it is false.
Equivalently, it is the probability of correctly accepting the alternative hypothesis (Ha) when it is true - that is, the ability of a test to detect an effect, if the effect actually exists.
Probability of FP is α
Decision
Reject H0
Retain H0
State of the world
H0 true
Type I error
H0 false
Type II error
Probability of FN is β
power = 1 - β

5. What factors affect the power?
To increase the power of your test, you may do any of the following:
Increase the effect size (the difference between the null and alternative values) to be detected
The reasoning is that any test will have trouble rejecting the null hypothesis if the null hypothesis is only 'slightly' wrong. If the effect size is large, then it is easier to detect and the null hypothesis will be soundly rejected.
Increase the sample size(s) – power analysis
Decrease the variability in the sample(s)
Increase the significance level (α) of the test
The shortcoming of setting a higher α is that Type I errors will be more likely. This may not be desirable.

6. new stuff

7. Effect size
When a difference is statistically significant, it does not necessarily mean that it is big, important or helpful in decision-making. It simply means you can be confident that there is a difference.
For example, you evaluate the effect of sun erruptions on student knowledge (𝑛=2000).
The mean score on the pretest was 84 out of 100. The mean score on the posttest was 83.
Although you find that the difference in scores is statistically significant (because of a large sample size), the difference is very small suggesting that erruptions do not lead to a meaningful decrease in student knowledge.

8. Effect size
To know if an observed difference is not only statistically significant, but also factually important, you have to calculate its effect size.
The effect size in our case is 84 – 83 = 1.
The effect size is transformed on a common scale by standardizing (i.e., the difference is divided by a s.d.).

9. Power analysis
To ensure that your sample size is big enough, you will need to conduct a power analysis.
For any power calculation, you will need to know:
What type of test you plan to use (e.g., independent t-test)
The alpha value (usually 0.05)
The expected effect size
The sample size you are planning to use
Because the effect size can only be calculated after you collect data, you will have to use an estimate for the power analysis.
Cohen suggests that for t-test values of 0.2, 0.5, and 0.8 represent small, medium and large effect sizes respectively.

10. Power analysis in R (paired t-test)
install.packages("pwr")
library(pwr)
pwr.t.test(d=0.8,power=0.8,sig.level=0.05,type="paired",alternative="two.sided")
Paired t test power calculation
n =
d = 0.8
sig.level = 0.05
power = 0.8
alternative = two.sided
NOTE: n is number of *pairs*

11. Check for normality – histogram

12. Check for normality – QQ-plot
qqnorm(rivers)
qqline(rivers)

13. Check for normality – tests
The graphical methods for checking data normality still leave much to your own interpretation. If you show any of these plots to ten different statisticians, you can get ten different answers.
H0: Data follow a normal distribution.
Shapiro-Wilk test
> shapiro.test(rivers)
Shapiro-Wilk normality test
data: rivers
W = , p-value < 2.2e-16

14. p-value < 2.2e-16
log
p-value = 3.945e-05

15. Nonparametric statistics
Small samples from considerably non-normal distributions.
non-parametric tests
No assumption about the shape of the distribution.
No assumption about the parameters of the distribution (thus they are called non-parametric).
Simple to do, however their theory is extremely complicated. Of course, we won't cover it at all.
However, they are less accurate than their parametric counterparts.
So if your data fullfill the assumptions about normality, use paramatric tests (t-test, F-test).

16. Nonparametric tests
If the normality assumption of the t-test is violated, then its nonparametric alternative should be used.
The nonparametric alternative of t-test is Wilcoxon test.
wilcox.test()

17. anova (analýza rozptylu)

18. A problem
You're comparing three brands of beer.

19. A problem You buy four bottles of each brand for the following prices.
What do you think, which of these brands have significantly different prices?
No significant difference between any of these.
Primátor and Kocour
Primátor and Matuška
Kocour and Matuška
Primátor
Kocour
Matuška 15 39 65 12 45 14 48 32 11 60 38

20. t-test
We can do three t-tests to show if there is a significant difference between these brands.
How many t-tests would you need to compare four samples? 6
To compare 10 samples, you need 45 t-tests! This is a lot. We don’t want to do a million t-tests.
But in this lesson you'll learn a simpler method.
Its called Analysis of variance (Analýza rozptylu) – ANOVA.

21. Multiple comparisons problem
If you make two comparisons and assuming that both null hypothesis are true, what is the chance that both comparisons will not be statistically significant (𝛼=0.5)?
0.95×0.95=0.9025
And what is the chance that one or both comparisons will result in a statistically significant conclusion just by chance?
1.0−0.9025= ~ 10%
For N comparisons, this probability is generally 1.00 − 0.95𝑁.
So, for example, for 13 independent tests there is about 50:50 chance of obtaining at least one FP.

22. Multiple comparisons problem
Bennet et al., Journal of Serendipitous and Unexpected Results, 1, 1-5, 2010

23. Correcting for multiple comparisons
Bonferroni correction – the simplest approach is to divide the α value by the number of comparisons N. Then define the particular comparison as statistically significant when its p-value is less than 𝛼/𝑁.
For example, for 100 comparisons reject the null in each if its p-value is less than =
However, this is a bit too conservative, other approaches exist.
> p.adjust()
“There seems no reason to use the unmodified Bonferroni correction because it is dominated by Holm's method”

24. Main idea of ANOVA
To compate three or more samples, we can use the same ideas that underlie t-tests.
In t-test, the general form of t-statistic is
𝑡= 𝑥 1 − 𝑥 2 𝑆𝐸
Similarly, for three or more samples we assess the variability between sample means in numerator and the error (variability within samples) in denominator.
Variability between sample means
Error, variability within samples

25. Variability between sample means

26. Variability within samples

27. ANOVA hypothesis 𝐻 0 : 𝜇 1 = 𝜇 2 = 𝜇 3
𝐻 0 : 𝜇 1 = 𝜇 2 = 𝜇 3
𝐻 1 : at least one pair of samples is significantly different
Follow-up multiple comparison steps – see which means are different from each other.

28. F ratio 𝐹= between−group variability within−group variability
As between-group variability (variabilita mezi skupinami) increases, F-statistic increases and this leans more in favor of the alternative hypothesis that at least one pair of means is significantly different.
As within-group variability (variabilita v rámci skupin) increases, F-statistic decreases and this leans more in favor of the null hypothesis that the means are not siginificantly different.
𝐹= between−group variability within−group variability

29. Beer brands – a boxplot 𝑥 𝑃 𝑥 𝐺 𝑥 𝐾 𝑥 𝑀 13 35 45 48 Primátor Kocour
Matuška 15 39 65 12 45 14 48 32 11 60 38 13 35 45 48
𝑥 𝑃
𝑥 𝐺
𝑥 𝐾
𝑥 𝑀

30. Between-group variability
SS – sum of squares, součet čtverců
MS – mean square, průměrný čtverec
SSB – součet čtverců mezi skupinami
MSB – průměrný čtverec mezi skupinami
𝑥 𝑃 − 𝑥 𝐺 2
𝑥 𝑀 − 𝑥 𝐺 2
𝑆𝑆𝐵= 𝑘 𝑛 𝑘 𝑥 𝑘 − 𝑥 𝐺 2
𝑑 𝑓 𝐵 =𝑘−1
𝑀𝑆𝐵= 𝑘 𝑛 𝐾 𝑥 𝑘 − 𝑥 𝐺 2 𝑘−1
𝑥 𝐾 − 𝑥 𝐺 2 13 35 45 48
𝑥 𝑃
𝑥 𝐺
𝑥 𝐾
𝑥 𝑀

31. Within-group variability
SSW – součet čtverců uvnitř skupin
MSW – průměrný čtverec uvnitř skupin
𝑀𝑆𝑊= 𝑆𝑆𝑊 𝑑 𝑓 𝑊 = 𝑘 𝑥 𝑖 − 𝑥 𝑘 𝑁−𝑘

32. The summary of variabilities
𝑀𝑆𝑊= 𝑆𝑆𝑊 𝑑 𝑓 𝑊 = 𝑘 𝑥 𝑖 − 𝑥 𝑘 𝑁−𝑘
𝑀𝑆𝐵= 𝑆𝑆𝐵 𝑑 𝑓 𝐵 = 𝑘 𝑥 𝑘 − 𝑥 𝐺 𝑘−1
Primátor
Kocour
Matuška 15 39 65 12 45 14 48 32 11 60 38
𝑥 𝑖 ... value of each data point
𝑥 𝑘 ... sample mean
𝑁 ... total number of data points
𝑘 ... number of samples
𝑛 𝐾 ... number of data points in each sample
𝑥 𝐺 ... grand mean

33. 𝐹 𝑑 𝑓 𝐵 , 𝑑 𝑓 𝑊 = 𝑀𝑆𝐵 𝑀𝑆𝑊 𝑑 𝑓 𝐵 =𝑘−1 𝑑 𝑓 𝑊 =𝑁−𝑘
F-ratio
𝐹 𝑑 𝑓 𝐵 , 𝑑 𝑓 𝑊 = 𝑀𝑆𝐵 𝑀𝑆𝑊 𝑑 𝑓 𝐵 =𝑘−1 𝑑 𝑓 𝑊 =𝑁−𝑘

34. F-distribution

35. F distribution

36. Beer prices 𝑀𝑆𝐵= 𝑆𝑆𝐵 𝑑 𝑓 𝐵 =1505.3 𝑀𝑆𝑊= 𝑆𝑆𝑊 𝑑 𝑓 𝑊 =95.78 𝑥 𝑘
𝑀𝑆𝐵= 𝑆𝑆𝐵 𝑑 𝑓 𝐵 =1505.3
Primátor
Kocour
Matuška 15 39 65 12 45 14 48 32 11 60 38 13
𝑀𝑆𝑊= 𝑆𝑆𝑊 𝑑 𝑓 𝑊 =95.78
𝑥 𝑘
𝑥 𝐺 =35.33
𝑆𝑆𝐵=𝑛 𝑘 𝑥 𝑘 − 𝑥 𝐺 2 =3011
𝑑 𝑓 𝐵 =𝑘−1=2
𝑆𝑆𝑊= 𝑘 𝑥 𝑖 − 𝑥 𝑘 2 =862
𝑑 𝑓 𝑊 =𝑁−𝑘=9
𝐹 2,9 = 𝑀𝑆𝐵 𝑀𝑆𝑊 =15.72
𝐹 2,9 ∗ =4.25

37. F9,2
F2,9

38. Beer brands – ANOVA