Transcript Inferences About Two or More Means

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
 H 0: m 1 = m 2 = m 3
 Pairwise H0: compare each pair of
means
 H 0: m 1 = m 2
 H 0: m 1 = m 3
 H 0: m 2 = m 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 a = .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
H 0: m 1 = m 2 = m 3
 Approximate experimentwise error

m1 = m2
H0: m1 = m3
H0: m2 = m3
H0:
experimentwise
a = .05
a = .05
a = .05
a  .15
ANOVA Notation
0 hrs
10
8
8
6
32
Test scores
4 hrs
14
16
18
16
64
8 hrs
22
14
16
20
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:

XG
X


ij
nG

SXij = sum of all Xi in all groups = 168

nG = n3 + n2 + n3 = 12 ~
Logic of ANOVA
Assume all groups from same
population
2
 with same m and s
 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 s2
 Between groups
 variance of means
 Within groups
 pooled variance of all individual
scores
2
 s 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
MS B  s (n)
2
X

variance of the group means
 find deviations from grand mean
s
2
X

X


j
 XG
k 1

2
Mean Square Within Groups
also MSW: Within Groups Variance
 Pooled variance
 pool variances of all groups
2
 similar to s pooled for t test

s


2
pooled
s s s

k
2
1
2
2
2
3
formula for equal n only
different formula for unequal n ~
F ratio
F test
 Compare the 2 point estimates of s2

MS B
F
MSW
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
SS B   ( X j  X G )
2
df B  k 1
SSW   ( X ij  X j )
2
dfW  nG  k
2
dfT  nG  1
SST   ( X ij  X G )
Partitioning Sums of Squares

Finding Mean Squares
 MS = variance
SS B
MS B 
df B
SSW
MSW 
dfW
Partitioning Sums of Squares

Calculating observed value of F
Fobs
MS B

MSW
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
