# APLIKOVANá ANALYZA DAT PRO KI by dffhrtcv3

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```									    ANALYSIS OF VARIANCE (ANOVA)

test of equality of population means for 3 and more samples
H0 :   1 = 2 = 3 = … = k
H1: AT LEAST between two population means is
a difference

1
ANOVA

We want to evaluate the infuence of fertilisation on
height growth of seedlings

H0: fertilisation
has no
influence           without                         strong
middle          fertilisation
fertilisation   fertilisation

H1: fertilisation has
significant
influence
2
3
ANOVA

FERTILISATION
strong   middle  none
x
x

x3
2
1

3
2
1
ANOVA

x3
x                   x
1                   2

without             middle                      strong
fertilisation       fertilisation               fertilisation
H0 :    1 =    2 = 3             are random and in population
statistically insignificant

4
ANOVA
x3

x
2

x
1

without              middle                        strong
fertilisation        fertilisation                 fertilisation

H1:    1  2  3           are significant and in population
statistically significant
5
ANOVA – parts of variance

TOTAL VARIANCE
(VARIANCE OF ALL EXPERIMENT)

KNOWN SOURCES OF VARIABILITY
(part of total variability explanable by known factor (effect)
and is expressed by the difference among sample means

UNKNOWN SOURCES OF VARIABILITY
(part of total variability explanable by random variations of values
within samples and source of variation is unknown)

6
ANALÝZA ROZPTYLU (ANOVA)

1         2          3
Variability within samples is SMALL in comparison
with variability between samples  GOOD
DIFFERENTIATION OF MEANS

Variability within samples is
LARGE in comparison
with variability between samples 
BAD DIFFERENTIATION OF
7           12   3   MEANS
ANOVA

2 SAMPLES            3 AND MORE SAMPLES
t – test             ANOVA
paired t – test      ANOVA with repeteated
measurements
Mann-Whitney         Kruskal-Wallis test

8
ANOVA - conditions

all samples are independent
all samples come from populations with normal
distributions
all samples come from populations with equal
variances

9
ANOVA – checking of conditions

independence
sample B

sample B
sample A                                    sample A
samples A and B are dependent              samples A and B are independent

normality – test of normality
equality of variances –
Cochran test – for equal sample size,
10                          Barttlet test – for unequal sample size
ANOVA –model

Model of 1-F ANOVA:

yij      =          +        i            +    ij

EFFECT                ERROR
MEASURED       MEAN        change of sample
VALUE                      mean caused by
factor

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ANOVA

12
1-F ANOVA - result table

Degrees    Mean square   Test
Source of
variability   Sum of squares      of      (variance)    criterion
freedom

between
samples

within
samples
(residual)

Total

13                        F > F,k-1,N-k H1
1-F ANOVA – next step?

H0 accepted       STOP

DATA    ANOVA
multiply
H0 rejected   comparison
tests

14
ANOVA – multiply comparisons

H0: A = B , (A  B)             H1: A  B

Comparisons are made for possible combinations of samples.

15
ANOVA – multiply comparisons

Fisher
Tuckey
Scheffe          x1   x2    x3
and many others …

16
TuckeyHSD test

H0: A = B , (A  B) H1: A  B,
xA  xB
q
SE
MR                         MR       1   1 
SE                        SE 
2

n  n 
n                                  A    B

For q > q; N-k; k; (quantil of studentized range) we reject
H0 and difference of means of samples A and B is
significant

17
Scheffe test

H0: A = B , (A  B) H1: A  B,

xA  xB                             1   1 
S                                      n  n 
SE  M R        
SE                               A    B

If S > S   k  1  F ;k 1; N k then there is significant
difference between population means A and B.

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2-F ANOVA - model

y ij = μ + αi + β j +  αiβ j  + εij
yij measured value influenced by i-th level of factor A a j-th
level of factor B
   average common value of yij (common for all samples)
i effect of level Ai
j effect of level Bi
α i β j interaction between factors (optional , there are models
with interaction or without interaction)
ij random error N(0,2)
19
2-F ANOVA - interaction

Study is dealing with the impact of various doses of
nitrogen (N) and phospforus (P) on yield of crop. Both
elements have two levels – N (40, 60), P(10,20). Results
of first three experiments:

Exper.         N           P          Yield
T1            60          10         145
T2            40          10         125
20
T3            40          20         160
2-F ANOVA - interaction
expectation

What yield we
obtain when for N =
measured
60 we increase dose
of P to 20?

Pokus    N       P    Výnos
T1      60      10    145
T2      40      10    125
T3      40      20    160
21    T4      60      20     ?          we expect 180
2-F ANOVA - interaction

Pokus     N        P    Výnos
T1       60       10    145
T2       40       10    125
T3       40       20    160
Result of 4th experiment:    T4       60       20    130
expected

skutečnost
measured

22
2-F ANOVA - interaction

Paralel lines – effects of
factors are additive
(factors are independent)

Crossed lines– – effects of
factors are not additive -
there is interaction
between factors

Interaction is present if effect of one factor is not the same
when levels of the second factors are changed.
So factors are not independent but response on one factor
23   depends on levels of other factors.

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