1. Analysis of Variance: Inferences about 2 or More Means
Chapter 13
Homework: 1, 2, 7, 8, 9

2. 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 ~

3. ANOVA: Null Hypothesis
Omnibus H0: all possible H0
H0: m1 = m2 = m3
Pairwise H0: compare each pair of means
H0: m1 = m2
H0: m1 = m3
H0: m2 = m3
ANOVA: assume H0 true for all comparisons ~

4. 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) ~

5. 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 ~

6. Experimentwise Error H0: m1 = m2 = m3 Approximate experimentwise error
H0: m1 = m2 a = .05
H0: m1 = m3 a = .05
H0: m2 = m3 a = .05
experimentwise a » .15

7. ANOVA Notation Test scores 0 hrs 4 hrs 8 hrs 10 14 22 8 16 14 8 18 16

8. 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 ~

9. 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 ~

10. ANOVA Notation subscript G = grand
refers to all data points in all groups taken together
Grand mean:
SXij = sum of all Xi in all groups = 168
nG = n3 + n2 + n3 = 12 ~

11. Logic of ANOVA Assume all groups from same population
with same m and s2
Comparing means
are they far enough apart to reject H0?
ask same question for ANOVA
MORE THAN 2 MEANS ~

12. Logic of ANOVA ANOVA: 2 point estimates of s2 Between groups
variance of means
Within groups
pooled variance of all individual scores
s2pooled ~

13. 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 ~

14. Mean Square Between Groups
also called MSB
Mean Square Between Groups
variance of the group means
find deviations from grand mean

15. Mean Square Within Groups
also MSW: Within Groups Variance
Pooled variance
pool variances of all groups
similar to s2 pooled for t test
formula for equal n only
different formula for unequal n ~

16. F ratio
F test
Compare the 2 point estimates of s2

17. 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) ~

18. 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 + df dfk ~
ANOVA nondirectional
even though shade only right tail
F is always positive ~

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

20. Partitioning Sums of Squares
sum of squared deviations

21. Partitioning Sums of Squares
Finding Mean Squares
MS = variance

22. Partitioning Sums of Squares
Calculating observed value of F

23. 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