ONE-WAY ANOVA The traditional approach to analyzing half-sib
ANOVA & sib analysis
description
Transcript of ANOVA & sib analysis
![Page 1: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/1.jpg)
ANOVA & sib analysis
![Page 2: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/2.jpg)
ANOVA & sib analysis• basics of ANOVA - revision• application to sib analysis
• intraclass correlation coefficient
![Page 3: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/3.jpg)
- analysis of variance (ANOVA) is a way of comparing the ratio of systematic variance to unsystematic variance in a study
![Page 4: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/4.jpg)
- analysis of variance (ANOVA) is a way of comparing the ratio of systematic variance to unsystematic variance in a study
ANOVA as regression
![Page 5: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/5.jpg)
- analysis of variance (ANOVA) is a way of comparing the ratio of systematic variance to unsystematic variance in a study
ANOVA as regression
- research question: does exposure to content of Falconer & Mackay (1996) increase knowledge of quantitative genetics?
![Page 6: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/6.jpg)
- analysis of variance (ANOVA) is a way of comparing the ratio of systematic variance to unsystematic variance in a study
ANOVA as regression
- research question: does exposure to content of Falconer & Mackay (1996) increase knowledge of quantitative genetics?
Person ConditionNothing (N) Lectures (L) Lectures +
book(LB)
person 1
0 4 10
person 2
1 7 9
person 3
1 6 8
person 4
2 3 11
person 5
1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
![Page 7: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/7.jpg)
- analysis of variance (ANOVA) is a way of comparing the ratio of systematic variance to unsystematic variance in a study
ANOVA as regression
- research question: does exposure to content of Falconer & Mackay (1996) increase knowledge of quantitative genetics?
Person ConditionNothing (N) Lectures (L) Lectures +
book(LB)
person 1
0 4 10
person 2
1 7 9
person 3
1 6 8
person 4
2 3 11
person 5
1 5 7
μN = 1 μL = 5 μLB = 9 μ = 52 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
person
scor
e
![Page 8: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/8.jpg)
- analysis of variance (ANOVA) is a way of comparing the ratio of systematic variance to unsystematic variance in a study
ANOVA as regression
- research question: does exposure to content of Falconer & Mackay (1996) increase knowledge of quantitative genetics?
outcomeij = model + errorij
Person ConditionNothing (N) Lectures (L) Lectures +
book(LB)
person 1
0 4 10
person 2
1 7 9
person 3
1 6 8
person 4
2 3 11
person 5
1 5 7
μN = 1 μL = 5 μLB = 9 μ = 52 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
person
scor
e
![Page 9: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/9.jpg)
Dummy coding:
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Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Dummy coding:
![Page 11: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/11.jpg)
outcomeij = model + errorij
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Dummy coding:
![Page 12: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/12.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij i = 1, … N, N = number of people per
condition = 5 j = 1, … M, M = number of conditions = 3
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Dummy coding:
![Page 13: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/13.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij i = 1, … N, N = number of people per
condition = 5 j = 1, … M, M = number of conditions = 3
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Dummy coding:
![Page 14: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/14.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij i = 1, … N, N = number of people per
condition = 5 j = 1, … M, M = number of conditions = 3
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Dummy coding:
![Page 15: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/15.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij i = 1, … N, N = number of people per
condition = 5 j = 1, … M, M = number of conditions = 3
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
= b0 + εi1
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Dummy coding:
![Page 16: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/16.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij i = 1, … N, N = number of people per
condition = 5 j = 1, … M, M = number of conditions = 3
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
= b0 + εi1
knowledgei2 = b0 + b1*dummy12 + b2*dummy22 + εi2
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Dummy coding:
![Page 17: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/17.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij i = 1, … N, N = number of people per
condition = 5 j = 1, … M, M = number of conditions = 3
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
= b0 + εi1
knowledgei2 = b0 + b1*dummy12 + b2*dummy22 + εi2
= b0 + b1*1 + b2*0 + εi2
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Dummy coding:
![Page 18: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/18.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij i = 1, … N, N = number of people per
condition = 5 j = 1, … M, M = number of conditions = 3
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
= b0 + εi1
knowledgei2 = b0 + b1*dummy12 + b2*dummy22 + εi2
= b0 + b1*1 + b2*0 + εi2
= b0 + b1 + εi2
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Dummy coding:
![Page 19: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/19.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij i = 1, … N, N = number of people per
condition = 5 j = 1, … M, M = number of conditions = 3
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
= b0 + εi1
knowledgei2 = b0 + b1*dummy12 + b2*dummy22 + εi2
= b0 + b1*1 + b2*0 + εi2
= b0 + b1 + εi2
knowledgei3 = b0 + b1*dummy13 + b2*dummy23 + εi3
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Dummy coding:
![Page 20: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/20.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij i = 1, … N, N = number of people per
condition = 5 j = 1, … M, M = number of conditions = 3
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
= b0 + εi1
knowledgei2 = b0 + b1*dummy12 + b2*dummy22 + εi2
= b0 + b1*1 + b2*0 + εi2
= b0 + b1 + εi2
knowledgei3 = b0 + b1*dummy13 + b2*dummy23 + εi3
= b0 + b1*0 + b2*1 + εi3
= b0 + b2 + εi3
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Dummy coding:
![Page 21: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/21.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
= b0 + εi1
knowledgei2 = b0 + b1*dummy12 + b2*dummy22 + εi2
= b0 + b1*1 + b2*0 + εi2
= b0 + b1 + εi2
knowledgei3 = b0 + b1*dummy13 + b2*dummy23 + εi3
= b0 + b1*0 + b2*1 + εi3
= b0 + b2 + εi3
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Therefore:
Dummy coding:
![Page 22: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/22.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
= b0 + εi1
knowledgei2 = b0 + b1*dummy12 + b2*dummy22 + εi2
= b0 + b1*1 + b2*0 + εi2
= b0 + b1 + εi2
knowledgei3 = b0 + b1*dummy13 + b2*dummy23 + εi3
= b0 + b1*0 + b2*1 + εi3
= b0 + b2 + εi3
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Therefore:
→ μcondition1 = b0 b0 is the mean of condition 1 (N)
Dummy coding:
![Page 23: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/23.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
= b0 + εi1
knowledgei2 = b0 + b1*dummy12 + b2*dummy22 + εi2
= b0 + b1*1 + b2*0 + εi2
= b0 + b1 + εi2
knowledgei3 = b0 + b1*dummy13 + b2*dummy23 + εi3
= b0 + b1*0 + b2*1 + εi3
= b0 + b2 + εi3
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Therefore:
→ μcondition1 = b0 b0 is the mean of condition 1 (N)
→ μcondition2 = b0 + b1 = μcondition1 + b1
μcondition2 - μcondition1 = b1
Dummy coding:
![Page 24: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/24.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
= b0 + εi1
knowledgei2 = b0 + b1*dummy12 + b2*dummy22 + εi2
= b0 + b1*1 + b2*0 + εi2
= b0 + b1 + εi2
knowledgei3 = b0 + b1*dummy13 + b2*dummy23 + εi3
= b0 + b1*0 + b2*1 + εi3
= b0 + b2 + εi3
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Therefore:
→ μcondition1 = b0 b0 is the mean of condition 1 (N)
→ μcondition2 = b0 + b1 = μcondition1 + b1 b1 is the difference in means of
μcondition2 - μcondition1 = b1 condition 1 (N) and condition 2 (L)
Dummy coding:
![Page 25: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/25.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
= b0 + εi1
knowledgei2 = b0 + b1*dummy12 + b2*dummy22 + εi2
= b0 + b1*1 + b2*0 + εi2
= b0 + b1 + εi2
knowledgei3 = b0 + b1*dummy13 + b2*dummy23 + εi3
= b0 + b1*0 + b2*1 + εi3
= b0 + b2 + εi3
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Therefore:
→ μcondition1 = b0 b0 is the mean of condition 1 (N)
→ μcondition2 = b0 + b1 = μcondition1 + b1 b1 is the difference in means of
μcondition2 - μcondition1 = b1 condition 1 (N) and condition 2 (L)
→ μcondition3 = b0 + b2 = μcondition1 + b2
μcondition3 - μcondition1 = b2
Dummy coding:
![Page 26: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/26.jpg)
outcomeij = model + errorij
knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εij
knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1
= b0 + b1*0 + b2*0 + εi1
= b0 + εi1
knowledgei2 = b0 + b1*dummy12 + b2*dummy22 + εi2
= b0 + b1*1 + b2*0 + εi2
= b0 + b1 + εi2
knowledgei3 = b0 + b1*dummy13 + b2*dummy23 + εi3
= b0 + b1*0 + b2*1 + εi3
= b0 + b2 + εi3
Condition Dummy variableDummy1
(lec)Dummy2 (lecbook)
Nothing (N) 0 0Lectures (L) 1 0Lectures + book (LB)
0 1
Therefore:
→ μcondition1 = b0 b0 is the mean of condition 1 (N)
→ μcondition2 = b0 + b1 = μcondition1 + b1 b1 is the difference in means of
μcondition2 - μcondition1 = b1 condition 1 (N) and condition 2 (L)
→ μcondition3 = b0 + b2 = μcondition1 + b2 b2 is the difference in means of
μcondition3 - μcondition1 = b2 condition 1 (N) and condition 3 (LB)
Dummy coding:
![Page 27: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/27.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
![Page 28: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/28.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
b0
b1
b2
![Page 29: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/29.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
![Page 30: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/30.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
Sums of squares
![Page 31: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/31.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
Sums of squares
SST = Σ(scoreij - μ)2
![Page 32: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/32.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
Sums of squares
SST = Σ(scoreij - μ)2
![Page 33: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/33.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
Sums of squares
SST = Σ(scoreij - μ)2
SSB = ΣNj(μj - μ)2
![Page 34: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/34.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
Sums of squares
SST = Σ(scoreij - μ)2
SSB = ΣNj(μj - μ)2
![Page 35: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/35.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
Sums of squares
SST = Σ(scoreij - μ)2
SSB = ΣNj(μj - μ)2
SSW = Σ(scoreij - μj)2
![Page 36: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/36.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
Sums of squares
SST = Σ(scoreij - μ)2
SSB = ΣNj(μj - μ)2
SSW = Σ(scoreij - μj)2
![Page 37: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/37.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
Sums of squares
SST = Σ(scoreij - μ)2
SSB = ΣNj(μj - μ)2
SSW = Σ(scoreij - μj)2
SST = SSB + SSW
![Page 38: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/38.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
Sums of squares
SST = Σ(scoreij - μ)2
SSB = ΣNj(μj - μ)2
SSW = Σ(scoreij - μj)2
SST = SSB + SSW
Degrees of freedom
dfT = MN - 1dfB = M – 1dfW = M(N – 1)
N = number of people per conditionM = number of conditions
![Page 39: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/39.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
Sums of squares
SST = Σ(scoreij - μ)2
SSB = ΣNj(μj - μ)2
SSW = Σ(scoreij - μj)2
SST = SSB + SSW
Degrees of freedom
dfT = MN - 1dfB = M – 1dfW = M(N – 1)
Mean squares
MST = SST/dfT
MSB = SSB/dfB
MSW = SSW/dfW
N = number of people per conditionM = number of conditions
![Page 40: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/40.jpg)
Person ConditionNothing (N) Lectures (L) Lectures + book
(LB)person 1 0 4 10person 2 1 7 9person 3 1 6 8person 4 2 3 11person 5 1 5 7
μN = 1 μL = 5 μLB = 9 μ = 5
2 4 6 8 10 12 14
02
46
810
Offspring
Phen
otyp
ic v
alue
μN
μL
μLB
μ
Sums of squares
SST = Σ(scoreij - μ)2
SSB = ΣNj(μj - μ)2
SSW = Σ(scoreij - μj)2
SST = SSB + SSW
Degrees of freedom
dfT = MN - 1dfB = M – 1dfW = M(N – 1)
Mean squares
MST = SST/dfT
MSB = SSB/dfB
MSW = SSW/dfW
N = number of people per conditionM = number of conditions
F-ratio
F = MSB/MSW
= MSmodel/MSerror
![Page 41: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/41.jpg)
Sib analysis
![Page 42: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/42.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
- number of males (sires) each mated to number of females (dams)
![Page 43: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/43.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
- number of males (sires) each mated to number of females (dams)
- mating and selection of sires and dams → random
![Page 44: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/44.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
- number of males (sires) each mated to number of females (dams)
- mating and selection of sires and dams → random
- thus: population of full sibs (same father, same mother; same cell in table) and half sibs (same father, different mother; same row in table)
![Page 45: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/45.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
- number of males (sires) each mated to number of females (dams)
- mating and selection of sires and dams → random
- thus: population of full sibs (same father, same mother; same cell in table) and half sibs (same father, different mother; same row in table)
- data: measurements of all offspring
![Page 46: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/46.jpg)
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
0 5 10 15
02
46
810
Offspring
Phen
otyp
ic v
alue
μsire1
μdam1sire1
scoreoffspring1dam1sire1
Sib analysis
- example with 3 sires:
![Page 47: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/47.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
![Page 48: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/48.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component
![Page 49: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/49.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component - component attributable to differences between the progeny of different males
![Page 50: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/50.jpg)
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
0 5 10 15
02
46
810
Offspring
Phen
otyp
ic v
alue
μsire1
Sib analysis
μsire2
μsire3
![Page 51: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/51.jpg)
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
0 5 10 15
02
46
810
Offspring
Phen
otyp
ic v
alue
μsire1
Sib analysis
μsire2
μsire3
![Page 52: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/52.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component
![Page 53: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/53.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component- between-dam, within-sire component
![Page 54: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/54.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component- between-dam, within-sire component - component attributable to differences between progeny of females mated to same male
![Page 55: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/55.jpg)
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
0 5 10 15
02
46
810
Offspring
Phen
otyp
ic v
alue
μsire1
Sib analysis
μsire2
μsire3
![Page 56: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/56.jpg)
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
0 5 10 15
02
46
810
Offspring
Phen
otyp
ic v
alue
μsire1
Sib analysis
μsire2
μsire3
![Page 57: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/57.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component- between-dam, within-sire component
![Page 58: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/58.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component- between-dam, within-sire component- within-progeny component
![Page 59: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/59.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component- between-dam, within-sire component- within-progeny component - component attributable to differences between offspring of the same female
![Page 60: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/60.jpg)
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
0 5 10 15
02
46
810
Offspring
Phen
otyp
ic v
alue
μsire1
Sib analysis
μsire2
μsire3
![Page 61: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/61.jpg)
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
0 5 10 15
02
46
810
Offspring
Phen
otyp
ic v
alue
μsire1
Sib analysis
μsire2
μsire3
![Page 62: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/62.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component- between-dam, within-sire component- within-progeny component
![Page 63: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/63.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component - between-dam, within-sire component - within-progeny component
![Page 64: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/64.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component (σ2S)
- between-dam, within-sire component (σ2D)
- within-progeny component (σ2W)
![Page 65: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/65.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
σ2T = σ2
S + σ2D +
σ2W
- between-sire component (σ2S)
- between-dam, within-sire component (σ2D)
- within-progeny component (σ2W)
![Page 66: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/66.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component (σ2S) = variance between means of half-sib families = covHS = ¼VA
- between-dam, within-sire component (σ2D)
- within-progeny component (σ2W)
σ2T = σ2
S + σ2D +
σ2W
![Page 67: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/67.jpg)
0 5 10 15
02
46
810
Offspring
Phen
otyp
ic v
alue
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
![Page 68: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/68.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component (σ2S) = variance between means of half-sib families = covHS = ¼VA
- between-dam, within-sire component (σ2D)
- within-progeny component (σ2W)
σ2T = σ2
S + σ2D +
σ2W
![Page 69: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/69.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component (σ2S) = variance between means of half-sib families = covHS = ¼VA
- between-dam, within-sire component (σ2D)
- within-progeny component (σ2W) = total variance minus variance between groups = VP – covFS = ½VA +
¾VD + VEw
σ2T = σ2
S + σ2D +
σ2W
![Page 70: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/70.jpg)
Sib analysis
Sire 1
Sire 2
Sire 3
Sire 4
Sire 5
Sire 6
Sire 7
Sire 8
ANOVA:
Partitioning the phenotypic variance (VP):
- between-sire component (σ2S) = variance between means of half-sib families = covHS = ¼VA
- between-dam, within-sire component (σ2D) = σ2
T - σ2S - σ2
W = covFS – covHS = ¼VA + ¼VD + VEc
- within-progeny component (σ2W) = total variance minus variance between groups = VP – covFS = ½VA +
¾VD + VEw
σ2T = σ2
S + σ2D +
σ2W
![Page 71: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/71.jpg)
Question:
Why is any between group variance component equal to the covariance of the members of the groups?
![Page 72: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/72.jpg)
Question:
Why is any between group variance component equal to the covariance of the members of the groups?
Conceptually:
If all offspring in a group have relatively high values, the mean value for that group will also be relatively
high. Conversely, when all members of a group have relatively low values, the mean for that group will be
relatively low.
![Page 73: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/73.jpg)
Question:
Why is any between group variance component equal to the covariance of the members of the groups?
Conceptually:
If all offspring in a group have relatively high values, the mean value for that group will also be relatively
high. Conversely, when all members of a group have relatively low values, the mean for that group will be
relatively low.
Hence, the greater the correlation, the greater will be the variability among the means of the groups (i.e.,
between-groups variability) as a proportion of the total variability, and the smaller will be the proportion of
total variability inside the groups (i.e., within-groups variability).
![Page 74: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/74.jpg)
Question:
Why is any between group variance component equal to the covariance of the members of the groups?
Conceptually:
If all offspring in a group have relatively high values, the mean value for that group will also be relatively
high. Conversely, when all members of a group have relatively low values, the mean for that group will be
relatively low.
Hence, the greater the correlation, the greater will be the variability among the means of the groups (i.e.,
between-groups variability) as a proportion of the total variability, and the smaller will be the proportion of
total variability inside the groups (i.e., within-groups variability).
Computationally:
We can illustrate this using the intraclass correlation coefficient (ICC).
![Page 75: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/75.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
- not a nested design -> each sire mated to only 1 dam -> families of full sibs
![Page 76: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/76.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
2 4 6 8 10 12
24
68
1012
Offspring
Phen
otyp
ic v
alue
μs1
μs2
μs3
μ
![Page 77: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/77.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
2 4 6 8 10 12
24
68
1012
Offspring
Phen
otyp
ic v
alue
μs1
μs2
μs3
μ
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
![Page 78: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/78.jpg)
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
- this is the variance between the means of 3 groups, or the between-group variance component
2 4 6 8 10 12
24
68
1012
Offspring
Phen
otyp
ic v
alue
μs1
μs2
μs3
μ
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
![Page 79: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/79.jpg)
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
- this is the variance between the means of 3 groups, or the between-group variance component
- how does the magnitude of this variance component relate to the covariance within the groups?
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
2 4 6 8 10 12
24
68
1012
Offspring
Phen
otyp
ic v
alue
μs1
μs2
μs3
μ
![Page 80: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/80.jpg)
Families of sibs
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
- this is the variance between the means of 3 groups, or the between-group variance component
- how does the magnitude of this variance component relate to the covariance within the groups?
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
2 4 6 8 10 12
24
68
1012
Offspring
Phen
otyp
ic v
alue
μs1
μs2
μs3
μ
![Page 81: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/81.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
2 4 6 8 10 12
24
68
1012
Offspring
Phen
otyp
ic v
alue
μs1
μs2
μs3
μ
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
- this is the variance between the means of 3 groups, or the between-group variance component
- how does the magnitude of this variance component relate to the covariance within the groups?
![Page 82: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/82.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
- this is the variance between the means of 3 groups, or the between-group variance component
- how does the magnitude of this variance component relate to the covariance within the groups?
How to summarize the correlations between these 4 variables?
![Page 83: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/83.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
- this is the variance between the means of 3 groups, or the between-group variance component
- how does the magnitude of this variance component relate to the covariance within the groups?
How to summarize the correlations between these 4 variables?
- use Pearson r (bivariately) to obtain a correlation matrix?
![Page 84: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/84.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
- this is the variance between the means of 3 groups, or the between-group variance component
- how does the magnitude of this variance component relate to the covariance within the groups?
How to summarize the correlations between these 4 variables?
- use Pearson r (bivariately) to obtain a correlation matrix?- no, because a) we need a single measure of relationship
![Page 85: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/85.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
- this is the variance between the means of 3 groups, or the between-group variance component
- how does the magnitude of this variance component relate to the covariance within the groups?
How to summarize the correlations between these 4 variables?
- use Pearson r (bivariately) to obtain a correlation matrix?- no, because a) we need a single measure of relationship b) r sensitive to reshuffling data in rows (thus, if we reshuffle data in rows, the row means [μs] and between-group variance component [σ2
s] would stay the same, while r would change)
![Page 86: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/86.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
- this is the variance between the means of 3 groups, or the between-group variance component
- how does the magnitude of this variance component relate to the covariance within the groups?
How to summarize the correlations between these 4 variables?
- use Pearson r (bivariately) to obtain a correlation matrix?- no, because a) we need a single measure of relationship b) r sensitive to reshuffling data in rows (thus, if we reshuffle data in rows, the row means [μs] and between-group variance component [σ2
s] would stay the same, while r would change)
- solution: ICC (intraclass correlation coefficient):
![Page 87: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/87.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
- this is the variance between the means of 3 groups, or the between-group variance component
- how does the magnitude of this variance component relate to the covariance within the groups?
How to summarize the correlations between these 4 variables?
- use Pearson r (bivariately) to obtain a correlation matrix?- no, because a) we need a single measure of relationship b) r sensitive to reshuffling data in rows (thus, if we reshuffle data in rows, the row means [μs] and between-group variance component [σ2
s] would stay the same, while r would change)
- solution: ICC (intraclass correlation coefficient): a) a single measure b) insensitive to reshuffling data in rows
![Page 88: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/88.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
ICC = σ2s/(σ2
s + σ2w)
How to summarize the correlations between these 4 variables?
- use Pearson r (bivariately) to obtain a correlation matrix?- no, because a) we need a single measure of relationship b) r sensitive to reshuffling data in rows (thus, if we reshuffle data in rows, the row means [μs] and between-group variance component [σ2
s] would stay the same, while r would change)
- solution: ICC (intraclass correlation coefficient): a) a single measure b) insensitive to reshuffling data in rows
![Page 89: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/89.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
ICC = σ2s/(σ2
s + σ2w)
σ2w = Σ(pij - μSi)2/dfw
How to summarize the correlations between these 4 variables?
- use Pearson r (bivariately) to obtain a correlation matrix?- no, because a) we need a single measure of relationship b) r sensitive to reshuffling data in rows (thus, if we reshuffle data in rows, the row means [μs] and between-group variance component [σ2
s] would stay the same, while r would change)
- solution: ICC (intraclass correlation coefficient): a) a single measure b) insensitive to reshuffling data in rows
![Page 90: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/90.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
ICC = σ2s/(σ2
s + σ2w)
σ2w = Σ(pij - μSi)2/dfw
= [(p11 - μs1)2 + (p12 - μs1)2 + … + (p21 – μs2)2 + … + (p34 – μs3)2]/3(4-1) = 15/9 = 1.67
How to summarize the correlations between these 4 variables?
- use Pearson r (bivariately) to obtain a correlation matrix?- no, because a) we need a single measure of relationship b) r sensitive to reshuffling data in rows (thus, if we reshuffle data in rows, the row means [μs] and between-group variance component [σ2
s] would stay the same, while r would change)
- solution: ICC (intraclass correlation coefficient): a) a single measure b) insensitive to reshuffling data in rows
![Page 91: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/91.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
σ2s = Σ(μSi - μ)2/dfs
= [N1(μs1 - μ)2 + N2(μs2 - μ)2 + N3(μs3 - μ)2]/(3-1) = [4*(2.5 – 6.5)2 + 4*(6.5 – 6.5)2 + 4*(10.5 – 6.5)2]/2 = (4*42 + 4*0 + 4*42)/2 = 64
ICC = σ2s/(σ2
s + σ2w)
= 64/(64+1.67) = 0.97
σ2w = Σ(pij - μSi)2/dfw
= [(p11 - μs1)2 + (p12 - μs1)2 + … + (p21 – μs2)2 + … + (p34 – μs3)2]/3(4-1) = 15/9 = 1.67
How to summarize the correlations between these 4 variables?
- use Pearson r (bivariately) to obtain a correlation matrix?- no, because a) we need a single measure of relationship b) r sensitive to reshuffling data in rows (thus, if we reshuffle data in rows, the row means [μs] and between-group variance component [σ2
s] would stay the same, while r would change)
- solution: ICC (intraclass correlation coefficient): a) a single measure b) insensitive to reshuffling data in rows
![Page 92: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/92.jpg)
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 2 3 4 μs1 = 2.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 9 10 11 12 μs3 = 10.5
μ = 6.5
ICC = 0.97
2 4 6 8 10 12
24
68
1012
Offspring
Phen
otyp
ic v
alue
![Page 93: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/93.jpg)
2 4 6 8 10 12
24
68
1012
Offspring
Phen
otyp
ic v
alue
ICC = 0.10
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 1 9 4 11 μs1 =
Sire 2 7 2 12 8 μs2 =
Sire 3 6 3 10 5 μs3 =
μ =
![Page 94: ANOVA & sib analysis](https://reader035.fdocuments.net/reader035/viewer/2022062812/56816354550346895dd3f9fd/html5/thumbnails/94.jpg)
2 4 6 8 10 12
24
68
10
Offspring
Phen
otyp
ic v
alue
Measures of phenotype
Offspring 1 Offspring 2 Offspring 3 Offspring 4 Mean
Sire 1 3 5 8 10 μs1 = 6.5
Sire 2 5 6 7 8 μs2 = 6.5
Sire 3 2 4 9 11 μs3 = 6.5
μ = 6.5
ICC = 0