1. Where we are Where we are going
Statistical analyses by number of sample groups
 1 - Single sample Z or t tests
 2 – T tests (dependent, independent)
3 or more ?
Analysis of variance
Between subjects (independent)
Equal N (Easy)
Unequal N (Harder – structural)
Within subjects (dependent – structural)
Two-way (independent - structural)

2. Analysis of Variance Equal N (Easy)
Used with 3 or more groups
Extends logic of independent groups t-test
Some additional things to think about

3. The alternative hypothesis in ANOVA is always
- The population means are different (at least one mean is different from another)
ASSUMPTIONS of ANOVA:
equal variances (required for pooling)
normality (required for test distribution)
The null hypothesis in ANOVA is always:
This implies that any combination of means are also equal

4. **** Null hypothesis is tested by comparing two estimates of the population variance (σ2 ):
(MSB) between-group estimate of (σ2 )
AFFECTED by whether the null is true
(MSW) within-group estimate of (σ2 )
UNAFFECTED by whether the null is true

5. ANOVA: Null Hypothesis Is TRUE
Score Distributions
Sample 3 Scores
Sample 2 Scores
Sample 1 Scores
Mean Distributions, N=9
Sample 3 Mean
Sample 2 Mean
Sample 1 Mean

6. ANOVA: Null Hypothesis Is False
Score Distributions
Mean Distributions (N=9)

7. F ratio = ------- = About 1
-- when the null hypothesis is true
(MSB)
F ratio = = About 1
(MSw)
-- when the null hypothesis is not true
F ratio = = Much greater than 1

8. Let:
K = # of groups (treatments)
J = a particular group 1…K
NJ = # of people in group J _
Xj = Sample mean for group J
NG = Total (grand) number of people
XG = The grand mean
I = Individual
= Variance of the means S2

9. Between Groups variance
Is an estimate of Variance of the means σ2 σ2
Because the variance of the means is the
n variance of the variable divided by sample
size = σ2 S2
(n) estimates the population variance (σ2 ) S2
(n) = MSB IF groups are same size (n1=n2=n3…)

10. Between Groups variance
If Null is true, then these are just random samples S2
(n) estimates σ2 just as well as any random samples would
If Null is false, then these are not just random samples S2
(n) = MSB will be higher than the populations variance because the means are farther away from each other than would be expected by chance

11. Within Groups variance
If Null is true, then these are just random samples
So we can poll the variances just as we did with the independent samples t-test
S12 + S22 + S32 +… Sk2 = k S2
pooled
MSW = S2
pooled
MSB
F =
MSW

12. F Ratio
Critical values of f depend on dfW and dfB
Look up in table

18. Post-hoc tests
ANOVA tells us there is some difference, but it does not tell us which groups are different from each other
ANOVA is like a shotgun – firing many pellets at many different hypothesis like
u1=u2
u2=u3
(u1+u2)/2=u3

19. Post-hoc tests - Tukey
Tests all the pairwise comparisons – does not test complex hypotheses (such as (u1+u2)/2=u3)
u1=u2
u1=u3
u1=u4
u2=u3
u2=u4
u3=u4

20. Apriori tests Also referred to as “planned comparisons”
“planned contrast”
A rifle instead of a shotgun. Used to test a specific hypothesis that is a subset of all hypotheses. For example, with 3 groups – if you wanted to test if group 3 was different from the other two groups, then you would test the following:
(u1+u2)/2=u3

21. But WHAT IF the sample sizes are not the same?

23. 2 2 .

25. F Ratio
Critical values of f depend on dfW and dfB
Look up in table

32. Anova Effect size _ _ Xmax - Xmin dM = --------------
_ _
Xmax - Xmin
dM =
How far apart the means are divided by the standard deviation -
Similar to the effect size for independent t-test (mean difference/stdev)
Small Medium Large S
pooled =