1. doc.Ing. Zlata Sojková,CSc.
Analysis of Variance
doc.Ing. Zlata Sojková,CSc.

2. doc.Ing. Zlata Sojková,CSc.
In practice it is often necessary to compare a large number of independent random selections in terms of level, we are interested in hypothesis:
for at least one i (i = 1, 2,…m)
for m > 2, when i , i =1, 2, …m are mean values of normally distributed populations with equal variances 2 , t.j. N(, 2)
To verify this hypothesis is used important statistical method called Analysis of variance,
abbreviated ANOVA (resp. AV)
doc.Ing. Zlata Sojková,CSc.

3. doc.Ing. Zlata Sojková,CSc.
In practice is AV used for examination of the impact of one, or more factors (treatments) on the statistical sign.
Factors are labeled A, B,…in AV they will be regarded as qualitative attributes with different variations – levels of factor
Result will be quantitative statistical sign denoted Y
AV is frequently used in the evaluation of biological experiments
The simplest case is AV with single factor called One factor analysis of variance
doc.Ing. Zlata Sojková,CSc.

4. doc.Ing. Zlata Sojková,CSc.
Level of the factor refer to :
 certain amount of quantitative factor,
e.g. Amount of pure nutrients in manure, different income groups of households
Certain kind of qualitative factor, e.g. different types of the same crop, methods of products placing in stores,
AV is a generalization of Student's t-test for independent choices
AV also examines the impact of qualitative factors resulting in a quantitative character -> analyzes the relationships between attributes
doc.Ing. Zlata Sojková,CSc.

5. Scheme of single-factor experiment “balanced attempt”
row sum
row average
Repetition
A 1 2… j… n Yi . yi .
1 y11 y12 y1j y1n Y1. y1.
2 y21 y22 y2j y2n Y2. y2.
… ………..
i yi1 yi2 yij yin Yi. yi.
m ym1 ym2 ymj ymn Ym. ym.
Y.. y..
Levels
of the
factor
Overall
average
Total sum
doc.Ing. Zlata Sojková,CSc.

6. doc.Ing. Zlata Sojková,CSc.
Row sum:
Total sum:
Row average:
Overall average:
doc.Ing. Zlata Sojková,CSc.

7. doc.Ing. Zlata Sojková,CSc.
Model for resulting observed value:
where i = 1, 2,…, m
j = 1,2,…, n
 expected values for all levels of the factor and observed values
i impact of i-th level of the factor A
eij random error, every measurement is biased,
resp. impact of random factors
doc.Ing. Zlata Sojková,CSc.

8. doc.Ing. Zlata Sojková,CSc.or
Then we can formulate null hypothesis:
Ho : 1 = 2 =… i = m = 0
-> effects of all levels of factor A are zero, insignificant, against the alternative hypothesis
H1: i  0 for at least one i (i = 1,2…m)
effect i at least one i – level of the factor is significant,
=> significantly different from zero
doc.Ing. Zlata Sojková,CSc.

9. doc.Ing. Zlata Sojková,CSc.
Estimates of parameters are sample characteristics: :
What can be rewrited:
doc.Ing. Zlata Sojková,CSc.

10. Comparison of two experiments with three levels of factor3 1 2 1 2 3
doc.Ing. Zlata Sojková,CSc.

11. Principle of the ANOVA
Essence of the analysis of variance lies in the decomposition of the total variability of the investigated sign. Sr Sc S1
Variability between levels of factor, caused by the action of factor A,
“variability between groups”
Random variability,
residual,
“variability within groups“
Total
variability
doc.Ing. Zlata Sojková,CSc.

12. doc.Ing. Zlata Sojková,CSc.2
Degrees of freedom 3
Mean square (MS)
(1/2) 1
Sum of squares
(SS) 4
F critical
Variability
Variability between groups
m-1
s12 S1
Variability within groups
m.n - m
sr2 Sr
Total variability
N-1=
m .n-1 Sc
doc.Ing. Zlata Sojková,CSc.

13. doc.Ing. Zlata Sojková,CSc.
Test statistics for one factor ANOVA can be written:
F value will be compared with appropriate table value for F-distribution:
F , with (m-1) and (m.n - m) degrees of freedom
doc.Ing. Zlata Sojková,CSc.

14. doc.Ing. Zlata Sojková,CSc.
Decision about test result:
If F vyp  F. ((m-1,(N-m)) We reject H0,
In that case is effect of at least one level of the factor significant, thus average level of the indicator is significantly different from others. => At least one effect i is statistically significantly different from zero. If
F vyp  F
Do not reject Ho F
Acceptance regon Ho
Rejection region H0
doc.Ing. Zlata Sojková,CSc.

15. If null hypothesis is rejected:
We found only that effect of the factor on examined attribute is significant.
It is also necessary to identify levels of the factor, which are significantly different - for this purpose are used tests of contrasts
Test of contrast: Duncan test, Scheffe test, Tuckey test and others…..
doc.Ing. Zlata Sojková,CSc.

16. doc.Ing. Zlata Sojková,CSc.
Terms of use AV:
Samples have normal distribution,
violating of this assumption has significant effect on the results of AV
statistical independence of random errors eij
Identical residual variances
12 = 22 = …. = 2 , t.j. D(eij) = 2
for all i = 1,2…., m, j=1,2, …n
this assumption is more serious and can be verified by Cochran, resp. Bartlett test.
doc.Ing. Zlata Sojková,CSc.

17. Scheme of single-factor experiment “unbalanced attempt”
row sum
Row average
Different number of repetitions
A 1 2… j … ni Yi . yi .
1 y11 y12 y1j n1 Y1. y1.
2 y21 y22 y2j n2 Y2. y2.
… ………..
i yi1 yi2 yij ni Yi. yi.
m ym1 ym2 ymj nm Ym. ym.
Y.. y..
Levels
of the
factor
Overall average
Where
doc.Ing. Zlata Sojková,CSc.

18. doc.Ing. Zlata Sojková,CSc.1
Sum of squares
(SS) 3
Mean square (MS)
(1/2) 4
F- critical 2
Degrees of freedom
Variability
Variability between groups
m-1
s12 S1
Variability within groups
N - m
sr2 Sr
Total variability
N-1 S
doc.Ing. Zlata Sojková,CSc.

19. doc.Ing. Zlata Sojková,CSc.
Two-factor analysis of variance with one observation in each subclass.... TAV
Consider the effect of factor A, which we investigate on the m - levels, i = 1,2, ...., m
Then consider the effect of factor B, which is observed on n - levels , j = 1,2, …, n
On every i-level of factor A and j-level of factor B we have only one observation (repetition) yij
=>We are veryfying two null hypothesis
doc.Ing. Zlata Sojková,CSc.

20. doc.Ing. Zlata Sojková,CSc.
Scheme for Two-factor experiment with one observation in each subclass TAV
row sum
n- levels of factor B B
A … j … n Yi . yi .
1 y11 y12 y1j y1n Y1. Y1.
2 y21 y22 y2j y2n Y2. y2.
… ………..
i yi1 yi2 yij yin Yi. yi.
m ym1 ym2 ymj ymn Ym. ym.
Y.1 Y Y.j ... Y Y.. y.1 y y.j ... y.1 y..
m-levels
of factor A
Row average
Overall average
Column sum
Column average
doc.Ing. Zlata Sojková,CSc.

21. We are verifying the validity of two null hypothesis
We can write model for examined attribute as follows:
We are verifying the validity of two null hypothesis
Hypothesis for factor A:
Ho 1: 1 = 2 =… i = m = 0
t.j. All effects of factor A levels are equal to zero,
thus insignificant, against alternative hypothesis
H11 : i  0 for at least one i (i = 1,2…m)
effect i of at least one i – level of factor A is significant, significantly different from zero
doc.Ing. Zlata Sojková,CSc.

22. doc.Ing. Zlata Sojková,CSc.
Hypothesis for factor B:
Ho 2:  1 =  2 =…  j =  n = 0
=> All effects of factor A levels are equal to zero,
thus insignificant, against alternative hypothesis
H12 :  j  0 for at least one j (j = 1,2…m)
effect  j of at least one j – level of the factor B is significant, significantly different from zero
doc.Ing. Zlata Sojková,CSc.

23. doc.Ing. Zlata Sojková,CSc.2
Degrees of freedom 3
Mean square (MS)
(1/2) 4
F - critical
TAV
Variability 1
Sum of squares
(SS)
Variability between rows S1
m-1
s12
Variability
between columns S2
n-1
s22
Residual
variability Sr
sr2
(m-1)(n-1)
Total
variability Sc
m.n -1
doc.Ing. Zlata Sojková,CSc.

24. Decomposition of the total variability Sc= S1 + S2 + S r
Variability between rows,
effect of factor A
Variability between columns,
Effect of factor B
Residual
variability
Total variability
doc.Ing. Zlata Sojková,CSc.

25. Investigating the relationships between statistical attributes
Investigating the relationship between qualitative attributes, e.g. AB , called measurement of the association
Investigating the relationship between quantitative attributes - regression and correlation analysis
doc.Ing. Zlata Sojková,CSc.

26. Inestigating the association
Based on the association, resp. pivot tables
For testing the existence of  significant relationship between qualitative signs we use
2 - test of independence
Ho: two signs A and B are independent
H1: signsA and B are dependent
Attribute A has m - levels, variations
Attribute B has k - levels , variation
doc.Ing. Zlata Sojková,CSc.

27. Hypotheses formulation
Dependence of the attributes will appear in different frequency
E.g. We examine wheter the size of the package is affected by the size of the family
Ho : Choice of the package size depend on the count of family members
H1 : Choice of package size is affected by the size of the family
The procedure lies in comparing empirical and theoretical frequencies, (how should be empirical frequencies, if the attributes A and B were independent
doc.Ing. Zlata Sojková,CSc.

28. Simultanous frequencies, frequencies of the second order (aibj)
Marginal frequencies
(ai) resp.(bj)
Size of the family
Package size
< Total
(b1) (b2) (b3)
do 100g
(a1) (a1b1) (a1 b2) g
(a2)
250g <
(a3) (a3b3)
Total
Total count of the respondents n
doc.Ing. Zlata Sojková,CSc.

29. Determination of theoretical frequencies
Based on the sentence about independence of the random events A and B:
P(AB) = P(A) . P(B), thus signs A and B are independent, then:
P(aibj) = P(ai) .P(bj)
Estimate based on the relative frequencies:
(aibj)o = (ai) . (bj)  (aibj)o = (ai) .(bj)
n n n n
Theoretical frequencies
doc.Ing. Zlata Sojková,CSc.

30. Calculation of theoretical frequencies (a1b1)o = 70.40/300 = 9,33
Family size
Package size
and < Total
(b1) (b2) (b3)
do 100g
(a1) , g
(a2)
250g <
(a3)
Total
Total count of respondents n
doc.Ing. Zlata Sojková,CSc.

31. Calculation of test criteria and decision:
If 2 calculated  2 for significance  for
degrees of freedom (m-1).(k-1)
 Ho is rejected => signs A and B are dependent
In our case it means, that count of the family members significantly affects choice of the package size. Further, we should measure strength (power) of the dependence.
doc.Ing. Zlata Sojková,CSc.