1. Always be mindful of the kindness and not the faults of others.

2. One-way Anova: Inferences about More than Two Population Means
What is Anova?
One-Way Anova; F tests
Pairwise comparisons: Bonferroni procedure

3. Analysis of Variance & One Factor Designs
Y= DEPENDENT VARIABLE
(“yield”)
(“response variable”)
(“quality indicator”)
X = INDEPENDENT VARIABLE
(A possibly influential FACTOR)

4. = Many other factors (possibly, some we’re unaware of)
OBJECTIVE: To determine the impact of X on Y
Mathematical Model:
Y = f (x, ) , where  = (impact of) all factors other than X
Ex: Y = Forced expiratory volume in one second (liters)
X = Medical center
(John Hopkins, Rancho Los Amigos, St. Louis)
= Many other factors (possibly, some we’re unaware of)

5. Statistical Model • Yij “LEVEL” OF Center Yij = + j + ij
(Brand is, of course, represented as “categorical”)
“LEVEL” OF Center
• • •  •  •  • • • C 1 2 • n
Y11 Y12 • • • • • • •Y1c
Yij = + j + ij
i = 1, , nj
j = 1, , C
Y21 •
YnI •
Yij
Ync
•   •  •   •    •   •    •    • 

6. Let mj = AVERAGE associated with jth level of X
Where
= OVERALL AVERAGE
j = index for FACTOR (center) LEVEL
i= index for “replication”
j = Differential effect (response) associated with jth level of X
and ij = “noise” or “error” associated with the (particular) (i,j)th data value.
Let mj = AVERAGE associated with jth level of X
 tj = mj – m and m = AVERAGE of mj .

7. •••• Y1, Y2, etc., are Column Means 1 2 3 C Y11 Y12 • • • • • •Y1c Y21
YRI
••••
YRc
•  •  •  •  •  •  •  •  • 
Y 1
Y 2
•  •  •  
(Y j)
•  •    
Y c
Y1, Y2, etc., are Column Means
              

8. Y • = Y j /C = “GRAND MEAN”
(assuming same # data points in each column)
(otherwise, Y • = mean of all the data) c
j=1

9. These estimates are based on Gauss’ (1796) PRINCIPLE OF LEAST SQUARES
MODEL: Yij =  + j + ij
Y• estimates 
Yj - Y • estimatesj (= mj – m)
(for all j)
These estimates are based on Gauss’ (1796)
PRINCIPLE OF LEAST SQUARES
and on COMMON SENSE

10. MODEL: Yij =  + j + ij
If you insert the estimates into the MODEL,
(1) Yij = Y • + (Yj - Y • ) + ij.
<
it follows that our estimate of ij is
(2) ij = Yij - Yj
<

11. { { { Then, Yij = Y• + (Yj - Y• ) + ( Yij - Yj)
or, (Yij - Y• ) = (Yj - Y•) + (Yij - Yj ) { { {
(3)
TOTAL
VARIABILITY
in Y
Variability
in Y
associated
with X
Variability
in Y
associated
with all other factors + =

12. SUM OF SQUARES BETWEEN COLUMNS SUM OF SQUARES WITHIN COLUMNS
If you square both sides of (3), and double sum both sides (over i and j), you get, [after some unpleasant algebra, but lots of terms which “cancel”] {{
C nj C
C nj
(Yij - Y• )2 =  nj(Yj - Y•)2 + (Yij - Yj)2 {
j=1 i=1
j=1
j=1 i=1
TSS
TOTAL SUM OF SQUARES (
SSB
SUM OF SQUARES BETWEEN COLUMNS = +
SSW (SSE)
SUM OF SQUARES WITHIN COLUMNS ( ( ( ( (

13. ANOVA TABLE SSB SSB C - 1 = MSB C - 1 SSW = MSW SSW N - C N-C
SOURCE OF VARIABILITY
SSQ DF
Mean
square
(M.S.)
Between Columns (due to center)
SSB
SSB
C - 1 =
MSB
C - 1
Within Columns (due to other factors)
SSW =
MSW
SSW
N - C
N-C
TOTAL TSS N - 1

14. ANOVA TABLE Source of Variability df SSQ M.S. CENTER 1.583 2 0.791=
0.791
ERROR
14.480 57 =
0.254
TOTAL
= 60 -1

15. > 1 , < 1 , We can show: E ( MSB ) = 2 + VCOL E ( MSW ) = 2
This suggests that
There’s some evidence of non-zero VCOL, or “level of X affects Y”
if MSB
> 1 ,
MSW
if MSB
No evidence that VCOL > 0, or that “level of X affects Y”
< 1 ,
MSW

16. With HO: Level of X has no impact on Y
HI: Level of X does have impact on Y,
We need
MSB
> > 1
MSW
to reject HO.

17. More Formally,
HO: 1 = 2 = • • • c = 0
HI: not all j = 0 OR
(All column means are equal)
HO: 1 = 2 = • • • • c
HI: not all j are EQUAL

18.  The distribution of MSB = “Fcalc” , is MSW
The F - distribution with (dfB, dfw)
degrees of freedom 
Assuming
HO true.
Ca = Table Value

19. In our problem: ANOVA TABLE Source of Variability df SSQ M.S. Fcalc
CENTER
1.583 2 =
0.791
3.12=0.791/0.254
ERROR
14.480 57 =
0.254
TOTAL
= 60 -1

20. F table: Table A-5
 = .05
C0.5 = 3.15
Fcal =3.12
(2, 57 DF)

21. Hence, at  =. 05, Do Not Reject Ho , i. e
Hence, at  = .05, Do Not Reject Ho , i.e., Conclude that centers don’t differ significantly on FEV1 at 5% level. P-value is .052, so it is significant at 6% level

22. Multiple Comparison Procedures
Once we reject H0: ==...c in favor of H1: NOT all ’s are equal, we don’t yet know the way in which they’re not all equal, but simply that they’re not all the same. If there are 4 columns, are all 4 ’s different? Are 3 the same and one different? If so, which one? etc.

23. Overall Type I Error Rate
We set up “” as the significance level for a hypothesis test. Suppose we test 3 independent hypotheses, each at = .05; each test has type I error (rej H0 when it’s true) of However,
P(at least one type I error in the 3 tests)
= 1-P( accept all ) = 1 - (.95)3  .14
3, given true

24. Pairwise Comparisons Bonferroni Correction:
Do a series of pairwise t-tests, each with specified  value divided by # of comparisons.
Pairwise Comparisons

25. MINITAB INPUT center fev1 1 3.23 1 3.47 1 1.86 1 2.47 . . 3 2.85
1 3.23
1 3.47
1 1.86
1 2.47
. .
3 2.85
3 2.43
3 3.20
3 3.53

26. ONE FACTOR ANOVA (MINITAB)
MINITAB: STAT>>ANOVA>>ONE-WAY
Click for comparisons

28. Minitab Outputs Fisher 98.3% Individual Confidence Intervals
All Pairwise Comparisons among Levels of center
Simultaneous confidence level = 95.58%
center = 1 subtracted from:
center Lower Center Upper
( * )
( * )
center = 2 subtracted from:
center Lower Center Upper
( * )