Analysis of Variance:Inferences about 2 or More Means

Chapter 13

Homework: 1, 2, 7, 8, 9

Analysis of Variance

or ANOVA Procedure for testing hypotheses about 2 or

more means simultaneously e.g., amount of sleep effects on test scores

group 1: 0 hrs

group 2: 4 hrs

group 3: 8 hrs ~

ANOVA: Null Hypothesis

Omnibus H0: all possible H0

H0: 1 = 2 = 3

Pairwise H0: compare each pair of means

H0: 1 = 2

H0: 1 = 3

H0: 2 = 3

ANOVA: assume H0 true for all comparisons ~

ANOVA: Alternative Null Hypothesis

Best way to state: the null hypothesis is false at least one of all the possible H0 is

false Does not tell us which one is false

Post hoc tests (Ch 14) ~

Experimentwise Error

Why can’t we just use t tests? Type 1 error: incorrectly rejecting H0 each comparison = .05 but we have multiple comparisons

Experimentwise probability of type 1 error P (1 or more Type 1 errors)

ANOVA: only one H0 ~

Experimentwise Error

H0: 1 = 2 = 3

Approximate experimentwise error H0: 1 = 2 = .05

H0: 1 = 3 = .05

H0: 2 = 3 = .05

experimentwise .15

ANOVA Notation

Test scores

0 hrs 4 hrs 8 hrs

10 14 22

8 16 14

8 18 16

6 16 20

32 64 72

ANOVA Notation columns = groups

jth group j = 2 = 2d column = group 2 (4hrs)

k = total # groups (columns) k = 3

nj = # observations in group j

n3 = # observations in group 3 ~

ANOVA Notation

sj2 = variance of group j

Xi = ith observation in group

X4 = 4th observation in group

Xij = ith observation in group j

X31 = 3d observation in group 1 ~

ANOVA Notation

subscript G = grand refers to all data points in all groups

taken together Grand mean:

G

ijG

n

XX

Xij = sum of all Xi in all groups = 168

nG = n3 + n2 + n3 = 12 ~

Logic of ANOVA

Assume all groups from same population with same and 2

Comparing means are they far enough apart to reject H0? ask same question for ANOVA

MORE THAN 2 MEANS ~

Logic of ANOVA

ANOVA: 2 point estimates of

Between groups variance of means

Within groups pooled variance of all individual

scores s2

pooled ~

Logic of ANOVA

Are differences between groups (means)

bigger than difference between individuals? If is H0 false then distance between groups should be

larger We will work with groups of equal size

n1 = n2 = n3 Unequal n

different formulas same logic & overall method ~

Mean Square Between Groups also called MSB

Mean Square Between Groups

1

2

2

k

XXs

Gj

X

variance of the group means find deviations from grand mean

)(2 nsMSXB

Mean Square Within Groups

also MSW: Within Groups Variance Pooled variance

pool variances of all groups similar to s2 pooled for t test

k

sssspooled

23

22

212

formula for equal n only different formula for unequal n ~

F ratio

F test Compare the 2 point estimates of 2

W

B

MS

MSF

F ratio

If H0 is true then MSB = MSW then F = 1 if means are far apart then MSB > MSW

F > 1

Set criterion to reject H0

determine how much greater than 1 Test statistic: Fobs

compare to FCV Table A.4 (p 478) ~

F ratio: degrees of freedom

Required to determine FCV ~ df for numerator and denominator of F

dfB = (k - 1) (number of groups) - 1

dfW = (nG - k)

df1 + df2 + df3 +.... + dfk ~ ANOVA nondirectional

even though shade only right tail F is always positive ~

TABLE A.4: Critical values of F (a = .05)

Partitioning Sums of Squares

Sums of Squares sum of squared deviations

2)( GjB XXSS

2)( jijW XXSS

1kdfB

kndf GW

2)( GijT XXSS 1 GT ndf

Partitioning Sums of Squares

Finding Mean Squares MS = variance

B

BB df

SSMS

W

WW df

SSMS

Partitioning Sums of Squares

Calculating observed value of F

W

Bobs MS

MSF

ANOVA Summary Table

Output of most computer programs partitioned SS

_________________________________

Source SS df MS F

Between SSB dfB MSB Fobs

Within SSW dfW MSW

Total SST dfT