UTMUNIVERSITI TEKNOLOGI MALAYSIAPensyarah

PM Dr Zaleha IsmailAhli Kumpulan

Hamisan Salim Mohd Zain Husin Dzulkifle Ismail

Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA)A N O V A s a tu p ro s e d u r y a n g d ig u n a k a n u n tu k m e lih a t p e rb e z a a n a n ta ra d u a a ta u le b ih p o p u la s i. (In g a t: d a la m u jia n s ta tis tik k ita g u n a k a n s ta tis tik s a m p e l u n tu k m e m b u a t ru m u s a n m e n g e n a i p o p u la s i). P e r s o a la n : A d a k a h p e rb e z a a n d i a n ta ra d u a a ta u le b ih m in a d a la h d is e b a b k a n o le h tre a tm e n t a ta u c h a n c e ?

C o n to h k e a d a a n d i m a n a A N O V A d ig u n a k a n a d a la h s e p e rti b e rik u t: P o p u la s i A P o p u la s i B P o p u la s i C

1 = ?

2 = ?

3 = ?

S e le p a s tre a tm e n t P e rb e z a a n (a fte r tr e a tm e n t D iffe r e n c e ) Sam pel 1 Sam pel 2

Sam pel 3

X

1

X

2

X

3

S o a la n k a jia n :1 . A d a k a h te rd a p a t p e rb e z a a n s ig n ific a n t d i a n ta ra m in d i a ta s ? 2 . J ik a a d a , y a n g m a n a k a h y a n g b e rb e z a ?

P e rb e z a a n a d a la h d is e b a b k a n h a n y a o le h c h a n c e ? (s a m p lin g e rro r a ta u ra la t p e rs a m p e la n ) a ta u a d a k a h d is e b a b k a n o le h k e s a n tre a tm e n t? M e n g a p a tid a k b u a t u jia n t s e c a ra b e ru la n g ? U jia n - t te rh a d p a d a d u a s a m p e l s a h a ja . R a la t b e rta m b a h A N O V A b o le h g u n a k a n u n tu k m e m b a n d in g b a n y a k b ila n g a n s a m p e l. In i m e m b o le h k a n k ita u n tu k m e re k a b e n tu k k a jia n y a n g a k a n m e lib a tk a n b e b e ra p a n ila i (ta h a p ). U jia n t h a n y a b o le h d ig u n a k a n u n tu k m e m b a n d in g d u a m in s a h a ja .

U jia n H ip o te s is m e n g g u n a k a n A N O V A : C o n to h : S e o ra n g a h li p s ik o lo g i in g in m e lih a t k e s a n s u h u te rh a d a p k e m a m p u a n u n tu k b e la ja r. B e lia u te la h m e le ta k k a n p e la ja r d a la m tig a k u m p u la n m a s in g -m a s in g d a la m b ilik y a n g b e rs u h u 5 0 0, 6 0 0 d a n 7 0 0. A d a k a h s u h u m e m b e ri k e s a n ? D a la m s ta tis tik , k ita m e m b a n d in g d i a n ta ra d u a k e m u n g k in a n : H ip o te s is n o l (H 0), y a n g m e n g a ta k a n b a h a w a s u h u tid a k m e m p u n y a i k e s a n d a n H ip o te ts is a lte rn a tif (H a) y a n g m e n g a ta k a n b a h a w a s u h u a k a n m e m b e ri k e s a n . H ip o te s is n o l: (H 0) :u 1 = u 2 = u 3, d a la m m a n a u m e w a k ili m in p e n c a p a ia n (m is a ln y a ) b a g i s e tia p p o p u la s i d a la m s u h u y a n g te la h d ite ta p k a n .

H ip o te s is a lte rn a tif, a k a n m e n y a ta k a n b a h a w a s e k u ra n g -k u r a n g n y a s a tu m in d a rip a d a p o p u la s i b e rb e z a . Y a k n i: H a: s e k u r a n g -k u r a n g n y a s a tu m in b e r b e z a . A d a b a n y a k k e m u n g k in a n y a n g b o le h b e rla k u . K ita h a d k a n k e p a d a m e lih a t s e k u ra n g k u ra n g n y a s a tu p e rb e z a a n d a h u lu d a n m e lih a t p e rb e z a a n y a n g m a n a k a h s ig n ifik a n d a la m k o n te k s p o s t-h o c A N O V A k e m u d ia n .

A n a lis is A N O V AC o n to h : K a ta k a n la h k ita m e n d a p a t d a ta s e p e rti b e rik u t:

Sampel 1 Keadaan A (500F)

Sampel 2 Keadaan B (600F)

Sampel 3 Keaadaan C (700F)

1,4,5,6,4

9,8,7,10,6

3,5,7,7,8

X

1

= 4

X

2

= 8

X

3

= 6

S k o r y a n g d ip e ro le h i tid a k la h s a m a , ia itu b e ru b a h . T u g a s k ita ia la h m e n g u k u r ju m la h k e b o le h u b a h a n n y a d a n m e n e ra n g k a n d a rip a d a m a n a ia te rb it.

K o m p o n e n k e b o le h u b a h a n (v a r ia b ility ) T e rd a p a t d u a s u m b e r u ta m a k e b o le h u b a h a n . 1 ) P e rb e z a a n a n ta ra s a m p e l (b e tw e e n s a m p le v a r ia b ility ) 2 ) P e rb e z a a n d a la m s e tia p s a m p e l ( w ith in s a m p le v a r ia b ility ). M e m b u a t a n a lis is ju m la h k e b o le h u b a h a n (v a ria b ility ) d a la m k e d u a k o m p o n e n in i a d a la h k u n c i d a la m A N O V A .

K e b o le h u b a h a n a n ta r a tr e a tm e n t (V a r ia b ility b e tw e e n tr e a tm e n t) P e rb e z a a n (k e b o le h u b a h a n ) a n ta ra m in s a m p e l d is e b a b k a n o le h tig a k e m u n g k in a n : a ) K e s a n tr e a tm e n t. (tr e a tm e n t e ffe c t) b ) P e rb e z a a n in d iv id u (in d iv id u a l d iffe r e n c e ) c ) R a la t k a jia n (e x p e r im e n ta l e r r o r ) P e b e z a a n (k e b o le h u b a h a n ) d a la m s a m p e l m u n g k in d is e b a b k a n o le h : P e rb e z a a n in d iv id u (In d iv id u a l d iffe r e n c e ) a ) R a la t k a jia n (e x p e r im e n ta l e r r o r ) J u m la h k e b o le h u b a h a n (to ta l v a r ia b ility )

K e b o l e h u b a h a n a n t a r a t r e a t m e n t i) K e s a n tre a tm e n t ii) P e rb e z a a n in d iv id u iii) R a la t k a jia n

K e b o le h u b a h a n d a la m tre a tm e n t i) p e rb e z a a n in d iv id u ii) R a la t k a jia n

N is b a h F a d a la h d ita k rifk a n s e b a g a i:k e b o le h u b a h a n a n ta ra " tre a tm e n t" k e b o le h u b a h a n d a la m s a m p e l k e s a n " tre a tm e n t" + p e rb e z a a n in d iv id u + ra la t = P e rb e z a a n in d iv id u + R a la t

N is b a h F =

N ila i p e n y e b u t k a d a n g k a la d i k a ta k a n s e b a g a i e rro r te rm . C a ta ta n : 1 ) D a la m n is b a h F , p e rb e z a a n a n ta ra d u a s e b u ta n (p e n g a ta s d a n p e n y e b u t) a d a la h k e p a d a k e s a n tre a tm e n t E rro r te rm m e m b e ri p e n g u k u ra n te rh a d a p v a ria n s y a n g d is e b a b k a n o le h chance. J ik a H 0 a d a la h b e n a r, ia itu tid a k a d a tre a tm e n t e ffe c t . N ila i F ia la h 1 . 2 ) J ik a H 0 a d a la h p a ls u , ia itu a d a k e s a n tre a tm e n t , m a k a n is b a h F a k a n m e le b e h i 1 .

P e n g ir a a n d a la m A N O V A (A N O V A C a lc u la tio n s ) S tru k tu r p e n g ira a n A N O V A S S ju m la h J u m la h d f

S S a n ta ra

S S d a la m

d f a n ta ra

d f d a la m

R u m u s a n p e n g ira a n a k h ir A N O V A ia la h p e n g ira a n n is b a h -F , d a n m e n g a n d o n g i d u a v a ria n s , F = v a ria n s a n ta ra " tre a tm e n t" (s a m p e l) (v a ria n c e b e tw e e n tre a tm e n t) v a ria n s d a la m " tre a tm e n t" (s a m p e l) (v a ria n c e w ith in tre a tm e n t)

C a ta ta n : in g a t k e m b a li ru m u s a n u n tu k S S d a n v a ria n s s a m p e l.

D a ri c o n to h d i a ta s

Sampel 1 Keadaan A 1,4,5,6,4X1

Sampel 2 Keadaan B 9,8,7,10,6X2

Sampel 3 Keadaan C 3,5,7,7,8X3

= 4

= 8

= 6

n1=5 X = T1= 20= T

n2 =5 X = T2= 40= X

n3 =5 X = T3 = 30

N ila i G

d ib e ri o le h G

A n a lis is h a s ilta m b a h k u a s a d u a (S S ) (a n a ly s is o f th e s u m m a tio n o f th e s u m o f s q u a r e s ) A N O V A m e m e rlu k a n k ira a n ju m la h k e b o le h u b a h a n d a n k e m u d ia n d ib a h a g ik a n k e p a d a d u a k o m p o n e n : a n ta ra tre a tm e n t d a n d a la m tre a tm e n t. 1 ) J u m la h h a s ilta m b a h k u a s a d u a , S S In i a d a la h ju m la h S S b a g i s e m u a n ila i X .ju m la h

SS =

(X - X ) =2

X

2

( X )2 N

J ik a g u n a k a n G , m a k a SSju m la h

=

X

2

G 2 N

M e n g g u n a k a n ru m u s a n d a la m c o n to h 902 = 620 540 = 80 15

SS

ju m la h

= 620

2 . H a s il ta m b a h k u a s a d u a u n tu k d a la m s a m p e l (th e s u m o f s q u a r e s fo r s a m p le ) K e b o le h u b a h a n d a la m s a m p e l ia la h ju m la h S S b a g i s e tia p s a m p e l. S S d a la m = S S s e tia p tre a tm e n t U n tu k c o n to h d i a ta s , SSd a la m

= 14 + 10 +16 = 40

3 . H a s il ta m b a h k u a s a d u a a n ta r a tr e a tm e n t . (S u m o f s q u a r e s b e tw e e n s a m p le ) R u m u s a n h a s il ta m b a h k u a s a d u a a n ta ra s a m p e l d ib e ri o le h : SS =

a n ta ra

T

G 2 n N

2

202 402 302 902 = + + = 80 + 320 +180 540 = 40 5 5 5 15C a ta ta n : S S d a la m + S Sa n ta ra

= SS

ju m la h

D a r ja h k e b e b a s a n d a la m A N O V A (d e g re e o f fre e d o m fo r A N O V A ) 1 ) J u m l a h d a r j a h k e b e b a s a n , d f ju m la h . d f ju m la h = N - 1 , N i a l a h j u m l a h s e m u a s a m p e l . D a l a m c o n t o h , d f ju m la h = 1 5 - 1 = 1 4 2 ) D a r j a h k e b e b a s a n a n t a r a s a m p e l , d f a n ta ra d f a n ta ra = k - 1 , k i a l a h b i l a n g a n k u m p u l a n . D a l a m c o n t o h , d f a n ta ra = 3 - 1 = 2 3 ) D a r j a h k e b e b a s a n d a l a m s a m p e l , d f d a la m d f d a la m = ( n - 1 ) d a l a m c o n t o h d f d a la m = 4 + 4 + 4 = 1 2 .

P e n g ir a a n V a r ia n s d a n n is b a h F (c a lc u la tio n o f v a r ia n c e a n d F r a tio ) P e n g ira a n v a ria n s a d a la h d ib e ri o le h

v a r ia n s =

SS SS = n 1 df

D a la m A N O V A v a ria n s d iru ju k s e b a g a i m in k u a s a d u a (m e a n s q u a re ) a ta u M S . Ia itu :

M S =D a la m c o n to h d i a ta s , M S danM Sd a la m

SS df

a n ta ra

=

S S a n ta ra 40 = = 20 d f a n ta ra 2S S d a la m 40 = = 3 .3 3 3 d f d a la m 12

=

N is b a h F ia la h F =

M S M S

a n ta ra d a la m

D a la m c o n to h d i a ta s , N ila i F ia la h ,

F =

20 = 6 3 .3 3 3

B ia s a n y a k e p u tu s a n p e n g ira a n d ile ta k k a n d a la m ja d u a l A N O V A s e p e r ti b e rik u t

Sumber(source) Antara Sampel

SS 40

df 2

MS 20 F= 6.00

Dalam Sampel

40

12

3.333

Jumlah

80

14

T abu ran =F C a ta ta n : 1 ) N ila i F a d a la h s e n tia s a p o s itif k e ra n a n ila i v a ria n s a d a la h s e n tia s a p o s itif. 2 ) B ila H 0 a d a la h b e n a r, p e n g a ta s d a n p e n y e b u t a k a n m e n g u k u r v a ria n s y a n g s a m a d a n n ila i F a k a n h a m p ir s a m a d e n g a n 1 .

B e n tu k ta b u ra n F a d a la h s e p e rti b e rik u t:

J a d u a l n ila i F d a p a t d ilih a t d a la m ja d u a l D e n g a n m e n e ta p n ila i , k ita a k a n d a p a t m e lih a t n ila i k ritik a l u n tu k m e n e n tu k a n r a n ta u p e n o la k a n y a n g b e r k a ita n , B a g i c o n to h d ia ta s d e n g a n m e n g a m b il n ila i = 0 .0 1 F0 .0 1 ; 2 ,1 2

= 6 .9 3

D a la m c o n to h n ila i F = 6 , tid a k te rle ta k d a la m k a w a s a n p e n o la k a n , m a k a k ita g a g a l m e n o la k H 0

JALAN KERJA ANOVAContoh soalan : Bina jadual ANOVA bagi data berikut.

**Sebelum menyelesaikan langkah-langkah jawapan tentukan dulu: 1) T bagi setiap kumpulan. 2) Darjah kebebasan/Degree of freedom ( df ). Apakah keputusan mengenai perbezaan di antara kumpulan boleh dibuat oleh penyelidik? Uji pada tahap = 0.01 (perhatikan bahawa bilangan sampel tidak sama dalam ketiga kumpulan) LANGKAH 1:-NYATAKAN HIPOTESIS HIPOTESIS NOL ( Ho ) : 1 = 2= 3 HIPOTESIS ALTERNATIF ( Ha ) : Salah satu min berbeza.

LANGKAH 2 :-TENTUKAN KRITERIA Diberi = 0.01 Cari nilai dfantara : 3 (I,II,III kumpulan ) - 1 kumpulan (II) : 3 1= 2 : 2 Cari nilai dfdalam : = ( 10-1 ) + ( 9-1 ) + ( 7-1 ) = 9 + 8 + 6 = 23 Cari nilai dfjumlah : [26= (10+9+7) ] [1(kumpulan tengah)] = 26 - 1 = 25 Tuliskan kesimpulan berdasarkan dfantara, dfdalam ;

LANGKAH 3- LAKUKAN KIRAAN2 2

Cari : a)SS antara

formulanya = T - G __ __ n N2 2

b)SS jumlah formulanya = x - G __ N2 2

c)SS dalam formulanya

= x (x)

Penyelesaian : a) SS antara2 2

formulanya = T - G __ __ n N2 2 2

2

= 109 + 131 + 110 ___ ___ ___ 10 9 7 = 4823.45 = 111.92

-

( 109+131+110) ______________ 26 4711.53

2

2

b)SSjumlah

= x - G __ N2

= 5384 - ( 109 + 131 + 110 ) _____________ 26 = 5384 - 4711.5 = 672.5

Ssjumlah - ssantara2 2

c) SS dalam = x (x) = 672.5 - 111.92 = 559.9 dibundarkan kepada terhampir = 560

ms antara = SSantara ------------dfantara = 111.92 _____ 2 = 55.96

ms dalam = Ssdalam ----------dfdalam = 560 ___ 23 = 24.35

Bina Jadual ANOVA

msantara F -----------msdalam df 2 ms 55.96 F 55.96 ------- = 2.298 24.35

Sumber antara

SS 111.92

dalam jumlah

560 672.5

23 25

24.35

F=2.298