1. Analysis of Variance Compare k Quant Populations k ≥ 2
Case I: Case II: + + + + + + + + + + + + + + + + + + + + + + + +

2. k – Treatment Groups k ≥ 2 nj = # obs in the jth group
X11 X12 X13 …. X1k
X21 X22 X23 …. X2k N = ∑nj ….
Treatment Xn11 Xn22 Xn33 …. Xnkk
Total T1 T2 T3 …. Tk Xj = ∑Xij/nj
Mean X1 X2 X3 …. Xk
Model: Xij = µj + eij Random Error - E(eij) = 0 V(eij) = σ2
Overall Mean µ = ∑µj/k All Means Equal:
Treatment Effect – τj = (µj - µ) Means Differ:

3. Error – eij = (Xij - µj) Grand Mean –
X is estimator for µ Xj is estimator for µj
Consider the Total Variation of X:
Total Variation = Within Group Between Groups
Total = Error Treatments
Total = Residual Model
TSS = SSE SST

4. Analysis of Variance (ANOVA) Table
Source df SS MS Expected
Treatments k MST = SST/k-1 σ2 + V(τ)
Error N-k MSE = SSE/N-k σ2
Total N-1
F Ratio - F = MST/MSE Means Equal : F = 1 Means Differ : F >> 1
H0: All Means Equal
HA: Not All Equal
R: F > Fα(k-1,N-k) F = MST/MSE

5. Example 1:
1 2 3

6. Machine Formulas:
Example 2:
1 2
16 24
20 21
18 22
14 25

8. Multiple Comparisons - Determine which Means Differ
A) Group Mean Interval Estimate 1) 2) 3)

9. B) Difference of Two Means
(µ2 - µ1)
(µ1 - µ3)
(µ2 - µ3)

10. d - e ≤ D ≤ d + e Note: If d > e then A Diff; If d ≤ e then No Diff
Fisher LSD Method (Least Significant Difference)
Groups Contrast LSD Decision

11. Tukey HSD Method (Honestly Significant Difference)
Groups Contrast HSD Decision

12. Example 3: Unequal Group Size 21 26

13. Multiple Comparisons
LSD Method:
Groups d LSD Decision
Tukey - HSD Method:
Groups d HSD Decision

14. 2-Way ANOVA - Complete Randomized Block Design
k Column Groups (Treatments)
b Row Groups (Blocks)
1 2 3 … k Sum Ave
1 X11 X12 X13 … X1k B1 R1
2 X21 X22 X23 … X2k B2 R2
b Xb1 Xb2 Xb3 … Xbk Bk Rk
Sum T1 T2 T3 … Tk
Ave C1 C2 C3 … Ck X

15. Partition the Total Variation of X
Total = Error Blocks Treatments
TSS = SSE SSB SST

16. 2-Way ANOVA Table:
Source df Sum of Squares Mean Square
Treatments k MST = SST/k-1
Blocks b MSB = SSB/b-1
Error N-k-b MSE = SSE/N-k-b+1
Total N-1
There are Two Hypothesis Tests:
1) Equal Column (Treatment) Means FTreat = MST/MSE
2) Equal Row (Block) Means FBlock = MSB/MSE

17. Example 4: 2-Way Example
Package
Ad A B C TV
News

18. Multiple Comparisons:

19. Machine Formulas:

20. Example 5: 2-Way is an Extension of Matched Pair Data
Child Before After A B C
D 89 93
E 85 89

21. Multiple Comparisons:

22. Example 6: Interaction Effect
Temp A B C
Hot
Warm
Cold

23. Multiple Comparisons:

24. Example 7: Interaction Example
Child Placebo Ritalin
Hyper
Normal