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ANOVA & sib analysis

ANOVA & sib analysis. ANOVA & sib analysis. basics of ANOVA - revision application to sib analysis intraclass correlation coefficient. analysis of variance (ANOVA) is a way of comparing the ratio of systematic variance to unsystematic variance in a study .

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ANOVA & sib analysis

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  1. ANOVA & sib analysis

  2. ANOVA & sib analysis • basics of ANOVA - revision • application to sib analysis • intraclass correlation coefficient

  3. analysis of variance (ANOVA) is a way of comparing the ratio of systematic variance to unsystematic variance in a study

  4. analysis of variance (ANOVA) is a way of comparing the ratio of systematic variance to unsystematic variance in a study • ANOVA as regression

  5. 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?

  6. 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?

  7. 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? score person

  8. 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 score person

  9. Dummy coding:

  10. Dummy coding:

  11. Dummy coding: outcomeij = model + errorij

  12. Dummy coding: outcomeij = model + errorij knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εiji = 1, … N, N = number of people per condition = 5 j = 1, … M, M = number of conditions = 3

  13. Dummy coding: outcomeij = model + errorij knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εiji = 1, … N, N = number of people per condition = 5 j = 1, … M, M = number of conditions = 3 knowledgei1 = b0 + b1*dummy11 + b2*dummy21 + εi1

  14. Dummy coding: outcomeij = model + errorij knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εiji = 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

  15. Dummy coding: outcomeij = model + errorij knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εiji = 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

  16. Dummy coding: outcomeij = model + errorij knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εiji = 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

  17. Dummy coding: outcomeij = model + errorij knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εiji = 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

  18. Dummy coding: outcomeij = model + errorij knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εiji = 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

  19. Dummy coding: outcomeij = model + errorij knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εiji = 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

  20. Dummy coding: outcomeij = model + errorij knowledgeij = b0 + b1*dummy1j + b2*dummy2j + εiji = 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

  21. Dummy coding: 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 Therefore:

  22. Dummy coding: 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 Therefore: → μcondition1 = b0b0 is the mean of condition 1 (N)

  23. Dummy coding: 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 Therefore: → μcondition1 = b0b0 is the mean of condition 1 (N) → μcondition2 = b0 + b1 = μcondition1 + b1 μcondition2 - μcondition1= b1

  24. Dummy coding: 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 Therefore: → μcondition1 = b0b0 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)

  25. Dummy coding: 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 Therefore: → μcondition1 = b0b0 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

  26. Dummy coding: 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 Therefore: → μcondition1 = b0b0 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)

  27. μLB μ μL μN

  28. μLB b2 μ μL b1 μN b0

  29. μLB μ μL μN

  30. μLB μ μL μN Sums of squares

  31. μLB μ μL μN Sums of squares SST = Σ(scoreij - μ)2

  32. μLB μ μL μN Sums of squares SST = Σ(scoreij - μ)2

  33. μLB μ μL μN Sums of squares SST = Σ(scoreij - μ)2 SSB = ΣNj(μj - μ)2

  34. μLB μ μL μN Sums of squares SST = Σ(scoreij - μ)2 SSB = ΣNj(μj - μ)2

  35. μLB μ μL μN Sums of squares SST = Σ(scoreij - μ)2 SSB = ΣNj(μj - μ)2 SSW = Σ(scoreij - μj)2

  36. μLB μ μL μN Sums of squares SST = Σ(scoreij - μ)2 SSB = ΣNj(μj - μ)2 SSW = Σ(scoreij - μj)2

  37. μLB μ μL μN Sums of squares SST = Σ(scoreij - μ)2 SSB = ΣNj(μj - μ)2 SSW = Σ(scoreij - μj)2 SST = SSB + SSW

  38. μLB μ μL μN 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 - 1 dfB = M – 1 dfW = M(N – 1) N = number of people per condition M = number of conditions

  39. μLB μ μL μN 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 - 1 dfB = M – 1 dfW = M(N – 1) Mean squares MST = SST/dfT MSB = SSB/dfB MSW = SSW/dfW N = number of people per condition M = number of conditions

  40. μLB μ μL μN 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 - 1 dfB = M – 1 dfW = M(N – 1) Mean squares MST = SST/dfT MSB = SSB/dfB MSW = SSW/dfW F-ratio F = MSB/MSW = MSmodel/MSerror N = number of people per condition M = number of conditions

  41. Sib analysis

  42. Sib analysis • number of males (sires) each mated to number of females (dams)

  43. Sib analysis • number of males (sires) each mated to number of females (dams) • mating and selection of sires and dams→ random

  44. Sib analysis • 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)

  45. Sib analysis • 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

  46. Sib analysis • example with 3 sires: scoreoffspring1dam1sire1 μdam1sire1 μsire1

  47. Sib analysis • ANOVA: • Partitioning the phenotypic variance (VP):

  48. Sib analysis • ANOVA: • Partitioning the phenotypic variance (VP): • between-sire component

  49. Sib analysis • ANOVA: • Partitioning the phenotypic variance (VP): • between-sire component • - component attributable to differences • between the progeny of different males

  50. Sib analysis μsire3 μsire2 μsire1

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