1. Analysis of Variance
ANOVA

2. Partitioning Variance Framework
Let’s say I collected a random sample of
100 Women
100 Men
--> measured height

4. TOTAL Variance

5. Total Variance = Variance due to group differences +
Variance due to differences within groups

6. Total

7. Among Group
Within Group

8. Ratio of Among to Within
t value for the t-test

10. t = Among / Within = Small / Big = SMALL
Among Group
Within Group

11. t = Among / Within = Big / Small = BIG
Among Group
Within Group

12. Analysis of Variance: ANOVA
Comparing 3 or more means

13. Hypotheses
Ho: A = B
HA: A  B

15. Hypotheses
Ho: A = B = C ; all the means are equal
HA: A  B = C or
A = B  C
--> at least one mean is different or
A  C = B
A  B  C

16. t-test
t-test
t-test

17. Each of the t-tests has a 0
Each of the t-tests has a 0.05 chance (5%) of falsely rejecting the null hypothesis.
So what is our TOTAL chance of falsely rejecting the hypothesis that the means are equal?
t-test

18. --> a function of how many comparisons we do using a t-test.
# samples
being compared 2 3 4 5 10 .
# of
comparisons 1 6 45
Probability of
type I error (=0.05)
0.05
0.14
0.26
0.40
0.9

19. Variable 1 2 3
Group
3 Sources of variation

20. Variable 1 2 3
Group
1) Deviation of individual observations from
the Grand Mean,

21. Variable 1 2 3
Group
2) Deviation of individual observations from
their Group Mean,

22. Variable
Group2 mean lies right
on the grand mean in this instance, not always the case. 1 2 3
Group
3) Deviation of the Group Means from
the Grand Mean,

23. Variable 1 2 3
Group
Total Variation (1 above) in the data can be attributed to contributions made by variation of individual observations about their group mean (2 above) and variation of the groups about the grand mean (3 above).

24. Variable 1 2 3
Group
To test whether group means are equal or not, we look at the ratio of variation due to group deviation from the grand mean to the variation of individual observations from their group mean.
-> We usually refer to these as among and within group sources of variation

25. Variable 1 2 3
Group
The first step in ANOVA is to calculate all of the pertinent sums of squares
Total
Among Group
Within Group

26. Variable 1 2 3
Group
The second step in ANOVA is to calculate all of the mean squares
Total - don’t calculate, is equal to variance for all observations
Among Group -> Among Group SS / Among Group DF
Within Group -> Within Group SS/ Within Group DF

27. Variable 1 2 3
Group
Degrees of Freedom for ANOVA
Total -> total # of observations - 1 n1+n2+n3-1; eg above, 12-1=11
Among Group -> Number of groups -1 = k-1; eg above 3-1=2
Within Group -> Total # of observations - number of groups, N-k eg above 12-3=9

28. SSTotal = SSAmong + SSWithin
Variable 1 2 3
Group
Aside
SSTotal = SSAmong + SSWithin
DFTotal = DFAmong + DFWithin

29. Variable 1 2 3
Group
MSAmong = SSAmong / DFAmong
MSWithinr = SSWithin / DFWithin
F = MSAmong / MSWithin

30. Variable 1 2 3
Group
MSAmong --> Big
MSWithin--> Small
F --> BIG Reject Ho

31. Variable 1 2 3
Group
MSAmong --> Small
MSWithin--> Big
F --> SMALL Do NOT Reject Ho

32. Yields of corn under 4 fertilizer treatments
Control K+N K+P N+P
Mean

35. p

36. ANOVA - testing for differences among means for various groups
--> a couple of ways that we could have come up with our groups
1) we can set them
2) we can determine them randomly

37. Fixed Effect ANOVA Model I
For example, let’s say that there are 26 different medications available to control blood pressure.
Perhaps LHSC uses 4 of the: D, G, M, and P
--> we want to test the effects of these specific drugs on blood pressure.
--> assign patients to each drug randomly and monitor effect on blood pressure
--> do ANOVA to test for differences among means
Fixed Effect ANOVA
Model I

38. Random Effect ANOVA Model II
Alternatively, we may be interested in variability among BP drugs in general
--> select 4 at random from all 26 available
--> do ANOVA to test for differences among means
Random Effect ANOVA
Model II

39. ANOVA, the next level
So far
--> looked at the effect of a single category on the response in some continuous variable
smoking/non-smoking on neonatal birth weight
various fertilizers on crop yield
etc.

40. ANOVA also allows us to look at the effects of multiple factors on the response variable of interest
For example, in the blood pressure example above,
--> we could have added the effects of gender as well
2-Factor ANOVA
--> and the effect of race
3-factor ANOVA

41. 2 Factor ANOVA
Let’s say, --> look at the influence of drug and gender on BP
Questions:
What effect does DRUG have?
What effect does GENDER have?
Does DRUG have the same effect on each GENDER?

42. Hypotheses:
Ho: D = G
HA: D  G
Ho: male = female
HA: male  female
Ho: the drugs effect males and females in the same way
HA: the drugs effect males and females in different ways
Main Effects
Interaction

43. G D D G

44. G D D G

45. TOTAL SS
DF = N-1

46. Cells SS
kA* kB - 1 = ab - 1

47. Within Cells (Error) SS
DF = ab(n-1)

48. Factor A SS
DF = a - 1

49. Factor B SS
DF = b - 1

50. Interaction SS = Cells SS - Factor A SS - Factor B SS
DF = Cells DF - Factor A DF - Factor B SS

51. All MS’s are equal to
SS divided by appropriate DF
EG Factor A MS = Factor A SS / Factor A DF

52. ? Source of Variation df SS MS F Total Cells Factor A Factor B A x B
Within Cells ?

53. Various F calculations - depend on what type of ANOVA.
Fixed --> Model I
both factors are fixed
Random --> Model II
both factors are random
Mixed model --> Model III
one factor fixed and one factor random

54. Computation of F - statistics for Model I and III ANOVA

55. Model I - both factors are fixed
An Example
A researcher is investigating the effect of three drug therapies (labeled 1, 2 and 3) on two different disorders (called depressives and schizophrenics)
Model I - both factors are fixed
9 patients with each disorder (18 total) assigned randomly to each drug treatment (3 in each cell)
Drug 1 Drug 2 Drug 3
Sch
Dep

56. Sch
Dep
Drug Drug 2 Drug 3

57. Hypotheses:
Drug
Ho: There is no difference among drugs in Psych score change
HA: There is a difference among drugs in Psych score change
Disorder
Ho: There is no difference among disorders in Psych score change
HA: There is a difference among disorders in Psych score change
Interaction:
Ho: There is no difference in the way the disorders respond to the drugs
HA: There is a difference in the way the disorders respond to the drugs

59. Examine the interaction term first!!!
ANOVA Results
Effect df MS df MS
(F/R) Effect Effect Error Error F p
{1}VAR1 Fixed
{2}VAR2 Fixed
1*2 Fixed
Examine the interaction term first!!!
< 0.05, therefore reject Ho, there is a significant interaction
--> the two disorders respond differently to the two drug treatments
If interaction significant, STOP

61. Hypotheses: Psychiatric Treatment (Factor A)
Ho: There is no difference in months to recovery among Psychiatric treatments
HA: There is a difference in months to recovery among Psychiatric treatments
Physical Therapy Treatment (Factor B)
Ho: There is no difference in months to recovery among Physical Therapy treatments
HA: There is a difference in months to recovery among Physical Therapy treatments
Interaction (AxB)
Ho: Physical Therapy treatments respond the same among the Psychiatric treatments
HA: Physical Therapy treatments respond differently among the Psychiatric treatments

62. Reject Interaction Ho ANOVA Results: MONTHS (2way.sta)
Effect df MS df MS
(F/R) Effect Effect Error Error F p
{1}PSYCHTRT
{2}PYSTHER 1*
Reject Interaction Ho

64. Hypotheses:
Pot Use
Ho: There is no difference in maturity among Pot Usage groups
HA: There is a difference in maturity among Pot Usage groups
Age
Ho: There is no difference in maturity among Age groups
HA: There is a difference in maturity among Age groups
Interaction
Ho: The effect of Age on Maturity is the same among the Usage groups
HA: The effect of Age on Maturity is NOT the same among the Usage groups

65. Do NOT reject Interaction Ho (p = 0.2182 > 0.05)
ANOVA Results: VAR10 (2way.sta)
Effect df MS df MS
(F/R) Effect Effect Error Error F p
Use Fixed
Age Fixed
1*2 Fixed
Do NOT reject Interaction Ho (p = > 0.05)
Reject Use Ho (p = < 0.05)
Reject Age Ho (p= < 0.05)