8/12/2019 Comp Formulas a Nova

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PSYCHOLOGY 315 D R. MCFATTER

Computational Formulas for ANOVAOne-Way ANOVA

Let a = # of levels of the independent variable = # of groups N = total # of observations in the experimentn1 = # of observations in group 1, etc.

H0: 1 = 2 = 3 = . . . = a

NOVA analyzes sample variances to draw inferences about population means. Sample variances can always be calculated asSS/df and these sample variances are called mean squares ( MS ):

SS T = 92 + 8 2 + . . . + 1 2 + 5 2 - (80) 2 /15 = 93.333

SS B = (402

+ 252

+ 152

)/5 - 802

/15 = 63.333 SS W = 93.333 - 63.333 = 30.000

An alternative computational approach emphasizing the conceptual basis of ANOVA is given below.

This is the variance of all scores in the experiment = 6.667.

This is the average of the variances within the groups = 2.50.(1.22 2 + 1.87 2 + 1.58 2)/3 = 2.50.

This is n times the variance of the means = 5(6.333) = 31.667.

Multiple Comparisons: t Crit is the critical value from a t -table using the df of the error termfrom the ANOVA table. The error term is always the denominator ofthe F -ratio. Thus, in the above example, the error df would be 12. The

MS Error would be 2.50; n is always the number of observations eachmean youre comparing is based on.

Example . X 1 X 2 X 3 Placebo Drug A Drug B

9 5 28 4 48 5 36 8 19 3 5

Sum 40 25 15 80M 8 5 3 5.333s 1.224745 1.870829 1.581139

ANOVA Summary Table Source SS df MS F pBetween 63.333 2 31.667 12.67 0.0011Within 30.000 12 2.500Total 93.333 14 6.667

SS Total

SS Between SS Within

s SS

df MS 2 = =

( )SS X

X

N Total =

2

2

( ) ( ) ( ) ( )SS

X

n

X

n

X

n

X

N Betweena

a

= + + + 1

2

1

2

2

2

2 2

K

SS SS SS Within Total Between=

df a Between = 1

df N aWithin = F MS

MS Between

Within

=

df N Total = 1

( )$

T Total

X X

N N

MS 22

2

1=

=

$ W a

Within

s s sa

MS 2 12

22 2

=+ + +

=K

$ $

B M Betweenn MS 2 2

= =

LSD t MS

nCrit

Error =

2

8/12/2019 Comp Formulas a Nova

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PSYCHOLOGY 315 D R. MCFATTER

Two-Way Factorial ANOVALet:

a = # of levels of the independent variable Ac = # of levels of the independent variable Cac = # of cells in the experiment

N = total # of observations in the experimentn1 = # of observations in cell 1, etc.

SS T = 537 - 932 /18 = 56.50

SS B = 122 /3 + 21 2 /3 + . . . - 93 2 /18 = 26.50

SS W = 56.50 - 26.50 = 30.00 SS A = 45

2 /9 + 48 2 /9 - 93 2 /18 = 0.50 SS C = 30

2 /6 + 33 2 /6 + 30 2 /6 - 93 2 /18 = 1.00

SS AC = 26.50 - 0.50 - 1.00 = 25.00

2 3 factorial design Crime (C)DV = Years Forgery Swindle Burglary

3 6 2Attractive 4 7 4

5 8 6

4 3 4Unattractive 6 4 6

Attractivenessof Offender

(A)

8 5 8

Table of Totals

Forgery Swindle BurglaryMarginalTotals

Attractive 12 21 12 45

Unattractive 18 12 18 48MarginalTotals 30 33 30 93

Table of Means

Forgery Swindle BurglaryMarginalMeans

Attractive 4 7 4 5

Unattractive 6 4 6 5.33MarginalMeans 5 5.5 5 5.17

ANOVA Summary TableSource SS df MS F pA 0.50 1 0.500 0.20 0.6627C 1.00 2 0.500 0.20 0.8214AC 25.00 2 12.500 5.00 0.0263Within 30.00 12 2.500Total 56.50 17 3.324

Null hypotheses:

A main effect H0: A = U C main effect H0: F = S = B AC interaction H0: ( AF - UF) = ( AS - US ) = ( AB - UB)

or equivalentlyH0: parallel lines in the cell mean plot

( )SS X

X

N Total =

2

2

( ) ( ) ( ) (SS

X

n

X

n

X

n

X

N Betweenac

ac

= + + + 1

2

1

2

2

2

2

K

SS SS SS Within Total Between=

( ) ( )SS

for each row

n for each row

X

N A =

2 2

( ) ( )SS

for each column

n for each column

X

N C =

2 2

SS SS SS SS AC Between A C =

2

3

4

5

6

7

8

Forgery Swindle BurglaryCrime (C)

Y e a r s

AttractiveUnattractive

2345678

Unattractive AttractiveAttractiveness (A)

Y e a r s

ForgerySwindleBurglary

df a A = 1

df cC = 1

df a c AC = ( )( )1 1

df N acWithin =

df N Total

= 1