ANOVA Sesiuneppt

Analiza variaional ANOVA metod de optimizare a proceselor organizaionaleDaniela COSMA

De ce analiz variaional

Existena unor diferene semnificative ntre nivelul de pregtire al studenilor, metodele de predare istrucie sau evaluare ale cadrelor didactice, caracteristicile disciplinelor de nvmnt fac ca strategiile, modelele i metodele de utilizare optima a resursei umane n procesul de nvmnt fundamentate pe tendina central s fie mai puin eficiente sau eficace.

Rezultatele depind i de factori independei cum ar fi caracteristicile psiho-individuale

Rezultatul implementrii unei stategii de promovare, selecie, formare, perfectionare poate fi evideniat nu mumai pritr-un nivel superior al tendinei centale ci mai ales prin creterea omogenitii populaiei investigate.

De la testul T la FAm putea folosi testul T ca ca s comparm dou cte dou valorile medii ale unui numr de caracteristici.

Aceast metod in cazul cercetrii noastre nu e adecvat din urmtoarele motiveE dificil de luat n considerare toate combinaiile posibile.Orice statistic care are n vedere date pariale (ca n cazul considerrii unei combinaii de dou variabile) e mai puin consistent ca cea care folosete totalitatea informaiilor.Unele comparaii vor furniza rezultate eronate pentru c la nivelul eantioanelor ipoteza va fi admis din ntmplare.

De la T (STUDENT)la F(FICHER)Avem nevoie de un test cu caracter global care va identifica diferenele semnificative ntre caracteristicile diferitelor srategii sau impactul factorilor de mediu asupra procesului de implementare.Dac rspunsul la ntrebarea noastr va fi negativ cercetarea ulterioar nu pateu conduce la rezultate consistente..Un asemenea test de semnificaie aplicabil pe un numr mare de variabile e testul F, sau analiza de variaie, sau ANOVA.

Logica metodei ANOVAIpoteza care se testeaz prin metoda ANOVA se refer la existena unor diferene semnificative ntre tendinele centrale (efectul pe care aplicarea unor srategii l are asupra eantioanelor supuse experimentului) i ne ateptm evident ca ipoteza nul s fie admis.Aceast ntrebare dei se refer la medii i gsete rspunsul prin analiza variaiei.Printre alte motive pentru care ne concentrm atenia asupra variaiei este acela c dorim s testm diferena dintre medii.

DOU SURSE ALE VARIABILITIIIn ANOVA, o estimaie a variabilitii ntre grupuri e comparat cu variabilitatea n cadrul grupurilor.Variabilitatea ntre grupuri crora li s-au aplicat tratamente diferite (s-au experimentat diverse strategii de promovare a profesiei, instituiei militare) conduce la diferene intre medii cu caracter aleatoriu i datorit efectului unor factori de mediu dac e semnificativ desigur variabilitatea n cadrul grupurilor se datoreaz caracteristicilor .Diferite ale elementelor din eantion care evident au participat la acelai experiment.

Variaia ntre grupuriExist o variaie mare ntre mediiMari diferene ntre medii nu se datoreaz ntmprii.E dificil de imaginat c toate grupurile sunt eantioane aleatoare ale aceleiai populaii.Ipoteza nul e respins indicnd un efect al tratamentului adic eficiena cel puin unei stategii utilizate.

Variabilitatea n cadrul grupurilorExist o oarecare variaie ntre mediile grupurilor.Totui variaia n cadrul grupurilor e mai mare pentru fiecare din grupuri.Cu ct variaia n cadrul grupurilor e mai mare cu att sigurana unor ipoteze referitoare la populaie se micoreaz.

RAPORTUL F

DOU SURSE DE VARIAIE

RAPORTUL F

RAPORTUL FEstimarea abaterii mediilor de grup fa de media populaieiGrade de libertatesum of squares betweensum of squares withindegrees of freedom withindegrees of freedom between

RAPORTUL Fsum of squares totaldegrees of freedom total

RAPORTUL F : SS intergrup Abateri individualeVolumul populaiei

RAPORTUL F : SSintragrupDispersia la nivelul populaieiNumrul de indivizi din fiecare grup

RAPORTUL F : SS TotalNumr total de subieci.

Un exemplu: ANOVA

Jocul 1

Jocul 2

Jocul 3

X1

X2

X2

X2

X3

X2

Eantionul ex grupa mic

12

144

81

6

36

Eantionul de control grupa mic

10

100

7

49

7

49

Eantionul ex grupa mijlocie

11

121

6

36

2

4

Eantionul de control grupa mic

7

49

9

81

3

9

Eantionul ex grupa mare

10

100

4

16

2

4

50

514

35

263

20

102

MEDIA=

10

7

4

_1256058706.unknown

_1256058755.unknown

ANOVA INDEX

A=514+263+102=879;

B=(50+35+20)2/15=735

C=(50)2/5+(35)2/5+(20)2/5=825

ANOVAs index va fi:

Table nr :2

DISPERSION

SS

Df

MS

F

INTERGROUP

90

2

45

10,00

INTRAGROUP

54

12

4,5

SCORE

144

14

An Example: ANOVAIpoteza testat.EXISTA DIFERENE SEMNIFICATIVE NTRE STATEGII?Ipoteza statistic corespunzatoare.

Rezultat Test

F(2,12)=10,00,p

ANOVA cu msuratori repetate

Table nr: 3

S

Primul ex grupa Mmruelor

Primul ex grupa Iepurailor

Al doilea ex grupa Mmruelor

Al doilea ex grupa Iepurailor

x

X2

X

X2

x

X2

x

X2

1

6

36

9

81

12

144

11

121

2

8

64

10

100

14

196

15

225

3

5

25

6

36

10

100

11

121

4

7

49

9

81

9

81

10

100

5

4

16

8

64

10

100

9

81

6

9

81

6

36

11

121

10

100

39

271

48

398

66

742

66

748

M

6,5

8,0

11,0

11,0

Calculm A,B,C values:

A=271+398+748=2159

B=(39+48+66+66)2/24=1998,375

C=[(6+9+12+11)2+(8+10+14+15)2+(5+6+10+11)2+(7+9+9+10)2+(4+8+10+9)2+(9+6+11+10)2]/4+2039,75

D=(392+482+662+662)/6=2089,5

Calculm SS

SS individual=C-B=2039,5-1998,375=41,375

SS true(experiment)=D-B=2089,5-1998,375=91,125

SS residual=(A-B)-(C-B)+(D-B)=(2159-1988,375)-[(2039,75-1988,375)+(2089,5-1998,375)]=28,125.

SS total = A-B = 2159-1998,375=160,625.

SS total = SS individual + SS experiment + SS residual

Calculam gradele de libertate:

df individual =n-1=6-1=5.

df experimental =k-1=4-1=3.

df residual =(k-1)(n-1)=(6-1)(4-1)=15.

df total = N-1=24-1=23.

calculame(MS):

MS individual=SS individual / df individual = 41,375/5=8,275.

MS experimental=SS experiment / df experimental=91,125/3=30,375.

MS residual=SS residual / df residual = 28,125/15=1,875

calculam F pentru ANOVA cu msurtori repetate:

F=MS experimental/MS residual;

F=30,375/1,875=16,2.

Table nr: 4

The source of the dispersion

SS

df

MS

F

F, (p

Repartitia F (Fisher Snedecor)Valorile functiei F (F = s12/ s22 ) pentru l1, l2 grade de libertate si =0,05, = 0,01 nivel de semnificatie

l 2

1

l1 = 1

l1 = 2

l1 = 3

l1 = 4

l1 = 5

= 0,05

= 0,01

= 0,05

= 0,01

= 0,05

= 0,01

= 0,05

= 0,01

= 0,05

= 0,01

1

23

161,40

4052,00

199,50

4999,00

215,70

5403,00

224,60

5625,00

230,20

5764,00

2

34

4

18,51

98,49

19,00

99,00

19,16

99,17

19,25

99,25

19,30

99,30

3

10,13

34,12

9,55

30,81

9,28

29,46

9,12

28,71

9,01

28,24

4

5

6

7,71

21,20

6,94

18,00

6,59

16,69

6,39

15,98

6,26

15,52

5

6,61

16,26

5,79

13,27

5,41

12,06

5,19

11,39

5,05

10,97

6

7

8

9

10

11

12

13

14

5,99

13,74

5,14

10,91

4,76

9,78

4,53

9,15

4,39

8,75

7

5,59

12,25

4,74

9,55

4,35

8,45

4,12

7,85

3,97

7,45

8

5,32

11,26

4,46

8,65

4,07

7,59

3,84

7,01

3,69

6,63

9

5,12

10,56

4,26

8,02

3,86

6,99

3,63

6,42

3,48

6,06

10

4,96

10,04

4,10

7,56

3,71

6,55

3,48

5,99

3,33

5,64

11

4,84

9,65

3,98

7,20

3,59

6,22

3,36

5,67

3,20

5,32

12

4,75

9,33

3,88

6,93

3,49

5,95

3,26

5,41

3,11

5,06

13

4,67

9,07

3,80

6,70

3,41

5,74

3,18

5,20

3,02

4,86

14

15

16

17

18

4,60

8,86

3,74

6,51

3,34

5,56

3,11

5,03

2,96

4,69

15

4,54

8,68

3,68

6,36

3,29

5,42

3,06

4,89

2,90

4,56

16

4,49

8,53

3,63

6,23

3,24

5,29

3,01

4,77

2,85

4,44

17

4,45

8,40

3,59

6,11

3,20

5,18

2,96

4,67

2,81

4,34

18

19

20

21

22

23

4,41

8,28

3,55

6,01

3,16

5,09

2,93

4,58

2,77

4,25

19

4,38

8,18

3,52

5,93

3,13

5,01

2,90

4,50

2,74

4,17

20

4,35

8,10

3,49

5,85

3,10

4,94

2,87

4,43

2,71

4,10

21

4,32

8,02

3,47

5,78

3,07

4,87

2,84

4,37

2,68

4,04

22

4,30

7,94

3,44

5,72

3,05

4,82

2,82

4,31

2,66

3,99

23

24

4,28

7,88

3,42

5,66

3,03

4,76

2,80

4,26

2,64

3,94

24

25

26

27

4,26

7,82

3,40

5,61

3,01

4,72

2,78

4,22

2,62

3,90

25

4,24

7,77

3,38

5,57

2,99

4,68

2,76

4,18

2,60

3,86

26

4,22

7,72

3,37

5,53

2,98

4,64

2,74

4,14

2,59

3,82

27

28

29

30

40

4,21

7,68

3,35

5,49

2,96

4,60

2,73

4,11

2,57

3,78

28

4,20

7,64

3,34

5,45

2,95

4,57

2,71

4,07

2,56

3,75

29

4,18

7,60

3,33

5,42

2,93

4,54

2,70

4,04

2,54

3,73

30

60

120

4,17

7,56

3,32

5,39

2,92

4,51

2,69

4,02

2,53

3,70

40

4,08

7,31

3,23

5,18

2,84

4,31

2,61

3,83

2,45

3,51

60

4,00

7,08

3,15

4,98

2,76

4,13

2,52

3,65

2,37

3,34

120

3,92

6,85

3,07

4,79

2,68

3,95

2,45

3,48

2,29

3,17

3,84

6,64

2,99

4,60

2,60

3,78

2,37

3,32

2,21

3,02

The following program was designed to perform ANOVA methods.

Enter your data for a Analysis of Variance. For this to make sense you should have several groups of data (at least 3; maximum: 26).Number of groups:

Each group includes a certain number of data items. (Often all the groups have the same number of items, but that is not required.) What is the size (i.e., the number of items) of largest group? (maximum: 99)Size of largest group:

There is no harm is over estimating the group size: blanks will be ignored. You do need to correctly enter the number of groups.

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_1257287741.unknown

Data Entry: ANOVA

Enter in the below set of boxes your data for each group (order makes no difference within a group) and then click on the Calculate Now button. Empty boxes will be ignored.

Partea superioar a machetei

HTMLCONTROL Forms.HTML:Hidden.1

HTMLCONTROL Forms.HTML:Submitbutton.1

HTMLCONTROL Forms.HTML:Reset.1

Data for Group A

A01= A02= A03= A04= A05=

Data for Group B

B01= B02= B03= B04= B05=

Data for Group C

C01= C02= C03= C04= C05=

_1257288052.unknown

_1257288057.unknown

_1257288059.unknown

_1257288061.unknown

_1257288063.unknown

_1257288064.unknown

_1257288062.unknown

_1257288060.unknown

_1257288058.unknown

_1257288054.unknown

_1257288056.unknown

_1257288053.unknown

_1257288048.unknown

_1257288050.unknown

_1257288051.unknown

_1257288049.unknown

_1257288046.unknown

_1257288047.unknown

_1257288045.unknown

ANOVA: Results

The results of a ANOVA statistical test performed at 09:56 on 12-NOV-2007

Source of Sum of d.f. Mean F

Variation Squares Squares

between 90.00 2 45.00 10.00

error 54.00 12 4.500

total 144.0 14

The probability of this result, assuming the null hypothesis, is 0.003

Group A: Number of items= 57.00 10.0 10.0 11.0 12.0

Mean = 10.0 95% confidence interval for Mean: 7.933 thru 12.07 Standard Deviation = 1.87 Hi = 12.0 Low = 7.00 Median = 10.0 Average Absolute Deviation from Median = 1.20

Group B: Number of items= 54.00 6.00 7.00 9.00 9.00


Group C: Number of items= 52.00 2.00 3.00 6.00 7.00


An Example: ANOVACalculate the test statistic.Grand Total: 104

ImaginedRetrospectiveCurrent7491214486463686410100525981121446361112110100T:24146T:40410T:40408

An Example: ANOVA

An Example: ANOVADetermine if your result is significant.Reject H0, 9.61>4.26Interpret your results.There is a significant difference in the ratings of the intensity of unrequited love depending on when (or if) the emotion was felt.ANOVA Summary TableIn the literature, the ANOVA results are often summarized in a table.SourcedfSSMSFBetween Groups242.6721.349.61Within Groups9202.22Total1162.67

After the F TestWhen an F turns out to be significant, we know, with some degree of confidence, that there is a real difference somewhere among our means.But if there are more than two groups, we dont know where that difference is.Post hoc tests have been designed for doing pair-wise comparisons after a significant F is obtained.

An Example: ANOVAA psychologist interested in artistic preference randomly assigns a group of 15 subjects to one of three conditions in which they view a series of unfamiliar abstract paintings. The 5 participants in the famous condition are led to believe that these are each famous paintings. The 5 participants in the critically acclaimed condition are led to believe that these are paintings that are not famous but are highly thought of by a group of professional art critics. The 5 in the control condition are given no special information about the paintings. Does what people are told about paintings make a difference in how well they are liked? Use the .01 level of significance.

FamousCritically AcclaimedNo Information10547165391073843

An Example: ANOVAState the research hypothesis.Does what people are told about paintings make a difference in how well they are liked?State the statistical hypothesis.

An Example: ANOVASet decision rule.

An Example: ANOVAGrand Total: 85

FamousCritically AcclaimedNo Information101005254167491163652539981101007493986441639T:40338T:20100T:25151

An Example: ANOVA

An Example: ANOVADetermine if your result is significant.Retain H0, 4.06

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ANOVA Sesiuneppt