732A35
1
This method was introduced at the last lecture.
Cell means model
𝑌𝑖𝑗𝑘 = 𝜇𝑖𝑗 + 𝜀𝑖𝑗𝑘
Factor effects model
𝑌𝑖𝑗𝑘 = 𝜇∙∙ + 𝛼𝑖 + 𝛽𝑗 + 𝛼𝛽
𝑖𝑗
+ 𝜀𝑖𝑗𝑘
732A35
2
Method of least squares or maximum likelihood is
used to minimize:
𝑄=
𝑌𝑖𝑗𝑘 − 𝜇𝑖𝑗
𝑖
𝑗
2
𝑘
or
𝑄=
𝑌𝑖𝑗𝑘 − 𝜇∙∙ − 𝛼𝑖 − 𝛽𝑗 − 𝛼𝛽
𝑖
𝑗
2
𝑖𝑗
𝑘
732A35
3
𝑆𝑆𝑇𝑂 =
𝑌𝑖𝑗𝑘 − 𝑌∙∙∙
𝑖
𝑗
𝑘
𝑆𝑆𝑇𝑅 = 𝑛
𝑌𝑖𝑗∙ − 𝑌∙∙∙
𝑖
𝑆𝑆𝐸 =
𝑗
𝑘
2
𝑗
𝑌𝑖𝑗𝑘 − 𝑌𝑖𝑗∙
𝑖
2
2
2
𝑒𝑖𝑗𝑘
=
𝑖
𝑗
732A35
𝑘
4
SSTR can be partitioned:
𝑆𝑆𝐴 = 𝑛𝑏
𝑌𝑖∙∙ − 𝑌∙∙∙
2
𝑖
𝑆𝑆𝐵 = 𝑛𝑎
𝑌∙𝑗∙ − 𝑌∙∙∙
2
𝑗
𝑆𝑆𝐴𝐵 = 𝑛
𝑌𝑖𝑗∙ − 𝑌𝑖∙∙ − 𝑌∙𝑗∙ + 𝑌∙∙∙
𝑖
2
𝑗
732A35
5
As usual, the sum of squares are divided by their
degrees of freedom:
𝑆𝑆𝐴
𝑀𝑆𝐴 =
𝑎−1
𝑆𝑆𝐵
𝑀𝑆𝐵 =
𝑏−1
𝑆𝑆𝐴𝐵
𝑀𝑆𝐴𝐵 =
𝑎−1 𝑏−1
𝑆𝑆𝐸
𝑀𝑆𝐸 =
𝑎𝑏 𝑛 − 1
732A35
6
First of all the means should be visualized in a
treatment means plot (Interaction plot in Minitab).
732A35
7
To test if interactions are important, a F-test is
calculated.
𝐻0 : 𝑎𝑙𝑙 𝛼𝛽
𝑖𝑗
𝐻𝑎 : 𝑛𝑜𝑡 𝑎𝑙𝑙 𝛼𝛽
𝐹∗
=0
𝑖𝑗
=0
𝑀𝑆𝐴𝐵
=
𝑀𝑆𝐸
𝐼𝑓 𝐹 ∗ > 𝐹 1 − 𝛼; 𝑎 − 1 𝑏 − 1 , 𝑛 − 1 𝑎𝑏
732A35
→
𝑅𝑒𝑗𝑒𝑐𝑡 𝐻0
8
If the interactions aren’t important, test for factor
effects.
Factor A
Factor B
𝐻0 : 𝑎𝑙𝑙 𝛼𝑖 = 0
𝐻0 : 𝑎𝑙𝑙 𝛽𝑗 = 0
𝐻𝑎 : 𝑛𝑜𝑡 𝑎𝑙𝑙 𝛼𝑖 = 0
𝑀𝑆𝐴
∗
𝐹 =
𝑀𝑆𝐸
Critical value with a-1 and
n-1(ab) degrees of freedom.
𝐻𝑎 : 𝑛𝑜𝑡 𝑎𝑙𝑙 𝛽𝑗 = 0
𝑀𝑆𝐵
∗
𝐹 =
𝑀𝑆𝐸
Critical value with b-1 and
n-1(ab) degrees of freedom.
732A35
9
The methods on the following slides are used when
the interaction is considered unimportant.
Point estimators are:
𝜇𝑖∙ = 𝑌𝑖∙∙∙
𝑎𝑛𝑑
𝜇∙𝑗 = 𝑌∙𝑗∙
Unbiased estimators of the variances:
𝑀𝑆𝐸
𝑀𝑆𝐸
2
2
𝑠 𝑌𝑖∙∙ =
𝑎𝑛𝑑 𝑠 𝑌∙𝑗∙ =
𝑏𝑛
𝑎𝑛
Confidence interval is created with the t distribution:
𝑌 ± 𝑡 1 − 𝛼 2 ; 𝑛 − 1 𝑎𝑏 ∗ 𝑠 𝑌
732A35
10
Tukey is used when all pairwise comparisons are to
be made:
𝐷 = 𝑌𝑖∙∙ − 𝑌𝑖 ′∙∙
1
2𝑀𝑆𝐸
𝐷±
𝑞 1 − 𝛼; 𝑎, 𝑛 − 1 𝑎𝑏 ∗
=
𝑏𝑛
2
𝐷±𝑇∗𝑠 𝐷
𝐷 = 𝑌∙𝑗∙ − 𝑌∙𝑗 ′∙
1
2𝑀𝑆𝐸
𝐷±
𝑞 1 − 𝛼; 𝑏, 𝑛 − 1 𝑎𝑏 ∗
𝑎𝑛
2
732A35
11
If only a few pairwise comparisons is to be made, the
Bonferroni method usually is the best.
Instead of T, B should be used.
𝐵 = 𝑡 1 − 𝛼 2𝑔 ; 𝑛 − 1 𝑎𝑏
Where g is the number of comparisons.
732A35
12
Contrasts can also be obtained, with either the
Scheffé procedure or Bonferroni (see p. 852).
Be aware that the standard deviation and the
multiplicator is not the same for the different
factors.
732A35
13
When the interactions are important, only the
treatment (cell) means should be analyzed.
I only show Tukey at the slides.
𝑌𝑖𝑗∙ − 𝑌𝑖 ′𝑗 ′∙
1
2𝑀𝑆𝐸
±
𝑞 1 − 𝛼; 𝑎𝑏, 𝑛 − 1 𝑎𝑏 ∗
𝑛
2
=𝐷±𝑇∗𝑠 𝐷
732A35
14
On the upcoming (last) lecture we will discuss:
• Two-way ANOVA with only one case per treatment
(ch 20)
• Randomized complete block designs
(ch 21)
• Analysis of Covariance
(ch 22)
732A35
15

Chapter 19

Start reading chapter 20, 21, 22
732A35
16