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Example 10.3 Falconer & Mackay, chapter 10

Example 10.3 Falconer & Mackay, chapter 10. Sanja Franic VU University Amsterdam 2011. How to translate observational data into estimates of h 2 ?. How to translate observational data into estimates of h 2 ? 2 examples: offspring – parent regression sib analysis.

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Example 10.3 Falconer & Mackay, chapter 10

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  1. Example 10.3Falconer & Mackay, chapter 10 Sanja Franic VU University Amsterdam 2011

  2. How to translate observational data into estimates of h2?

  3. How to translate observational data into estimates of h2? • 2 examples: • offspring – parent regression • sib analysis

  4. How to translate observational data into estimates of h2? • 2 examples: • offspring – parent regression • sib analysis • In both examples we assume random mating and absence of selection on parents.

  5. How to translate observational data into estimates of h2? • 2 examples: • offspring – parent regression • sib analysis • In both examples we assume random mating and absence of selection on parents. • Offspring – parent regression • straightforward

  6. How to translate observational data into estimates of h2? • 2 examples: • offspring – parent regression • sib analysis • In both examples we assume random mating and absence of selection on parents. • Offspring – parent regression • straightforward mean offspring values mid-parent values

  7. How to translate observational data into estimates of h2? • 2 examples: • offspring – parent regression • sib analysis • In both examples we assume random mating and absence of selection on parents. • Offspring – parent regression • straightforward mean offspring values mid-parent values

  8. How to translate observational data into estimates of h2? • 2 examples: • offspring – parent regression • sib analysis • In both examples we assume random mating and absence of selection on parents. • Offspring – parent regression • straightforward bOP = h2 mean offspring values mid-parent values

  9. But: Unequal variances in the sexes pose a complication. • Regression of offspring on parent(s) assumes that covoffspring-mid parent = ½VA (last chapter). • If variances are unequal across sexes, this is not the case; h2 must then be calculated for each sex separately. • Offspring – parent regression • straightforward bOP = h2 mean offspring values mid-parent values

  10. In example 10.3, heritability was estimated by offspring-parent regression. The male and female variances were unequal, so the regression coefficients were calculated separately for each sex of offspring and of parent. Subsequently, these regression coefficients were adjusted by a ratio of male-to-female (or female-to-male) standard deviations, to correct for the difference in the variances across the sexes. Let’s see why this is done. • Offspring – parent regression • straightforward bOP = h2 mean offspring values mid-parent values

  11. Phenotypic values measured in offspring and in parent: bOP = h2 mean offspring values mid-parent values

  12. Phenotypic values measured in offspring and in parent: bOP = h2 mean offspring values mid-parent values

  13. Phenotypic values measured in offspring and in parent: bOP = h2 mean offspring values mid-parent values

  14. Phenotypic values measured in offspring and in parent: bOP = h2 mean offspring values mid-parent values

  15. Phenotypic values measured in offspring and in parent: bOP = h2 mean offspring values mid-parent values

  16. Phenotypic values measured in offspring and in parent: bOP = h2 mean offspring values mid-parent values

  17. Phenotypic values measured in offspring and in parent: bOP = h2 mean offspring values mid-parent values

  18. Phenotypic values measured in offspring and in parent: bOP = h2 mean offspring values mid-parent values

  19. Phenotypic values measured in offspring and in parent: sF/sM= 1.7 / 2.4 = .708 sM/sF = 2.4 / 1.7 = 1.412 bOP = h2 mean offspring values mid-parent values

  20. Phenotypic values measured in offspring and in parent: sF/sM= 1.7 / 2.4 = .708 sM/sF = 2.4 / 1.7 = 1.412 Now we regress offspring on parent: O = b0 + bOPP + ε e bOP = h2 mean offspring values mid-parent values VP bOP P O

  21. Phenotypic values measured in offspring and in parent: sF/sM= 1.7 / 2.4 = .708 sM/sF = 2.4 / 1.7 = 1.412 Now we regress offspring on parent: O = b0 + bOPP + ε covOP = bOP * VP e bOP = h2 mean offspring values mid-parent values VP bOP P O

  22. Phenotypic values measured in offspring and in parent: sF/sM= 1.7 / 2.4 = .708 sM/sF = 2.4 / 1.7 = 1.412 Now we regress offspring on parent: O = b0 + bOPP + ε covOP = bOP * VP bOP = covOP / VP e bOP = h2 mean offspring values mid-parent values VP bOP P O

  23. Phenotypic values measured in offspring and in parent: sF/sM= 1.7 / 2.4 = .708 sM/sF = 2.4 / 1.7 = 1.412 Now we regress offspring on parent: O = b0 + bOPP + ε covOP = bOP * VP bOP = covOP / VP = ½VA / VP e bOP = h2 mean offspring values mid-parent values VP bOP P O

  24. Phenotypic values measured in offspring and in parent: sF/sM= 1.7 / 2.4 = .708 sM/sF = 2.4 / 1.7 = 1.412 Now we regress offspring on parent: O = b0 + bOPP + ε covOP = bOP * VP bOP = covOP / VP = ½VA / VP = ½ h2 e bOP = h2 mean offspring values mid-parent values VP bOP P O

  25. Phenotypic values measured in offspring and in parent: sF/sM= 1.7 / 2.4 = .708 sM/sF = 2.4 / 1.7 = 1.412 Now we regress offspring on parent: O = b0 + bOPP + ε covOP = bOP * VP bOP = covOP / VP = ½VA / VP = ½ h2 h2 = 2bOP e bOP = h2 mean offspring values mid-parent values VP bOP P O

  26. Phenotypic values measured in offspring and in parent: sF/sM= 1.7 / 2.4 = .708 sM/sF = 2.4 / 1.7 = 1.412 Now we regress offspring on parent: O = b0 + bOPP + ε covOP = bOP * VP bOP = covOP / VP = ½VA / VP = ½ h2 h2 = 2bOP Using the actual data, we perform the regression (separately for the sexes, as the variances differ), and obtain estimates of bOP: e bOP = h2 mean offspring values mid-parent values VP bOP P O

  27. Phenotypic values measured in offspring and in parent: sF/sM= 1.7 / 2.4 = .708 sM/sF = 2.4 / 1.7 = 1.412 Now we regress offspring on parent: O = b0 + bOPP + ε covOP = bOP * VP bOP = covOP / VP = ½VA / VP = ½ h2 h2 = 2bOP Using the actual data, we perform the regression (separately for the sexes, as the variances differ), and obtain estimates of bOP: e bOP = h2 mean offspring values mid-parent values VP bOP P O

  28. e VP bOP P O

  29. bOP = covOP / VP e VP bOP P O

  30. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm e VP bOP P O

  31. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm sPm = sM e VP bOP P O

  32. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm sPm = sM e VP bOmPm = covOmPm / sMsM rOmPm = covOmPm / sMsM → bOmPm = rOmPm bOP P O

  33. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf sPm = sM e VP bOmPm = covOmPm / sMsM rOmPm = covOmPm / sMsM → bOmPm = rOmPm bOP P O

  34. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf sPm = sM e VP bOP P O

  35. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf sPm = sM e VP bOmPf = covOmPf / sFsF rOmPf = covOmPf / sMsF → bOmPm≠rOmPm bOP P O

  36. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm sPm = sM e VP bOmPf = covOmPf / sFsF rOmPf = covOmPf / sMsF → bOmPm≠rOmPm bOP P O

  37. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm The ‘correction’ works like this: sPm = sM e VP bOmPf = covOmPf / sFsF rOmPf = covOmPf / sMsF → bOmPm≠rOmPm bOP P O

  38. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm The ‘correction’ works like this: bOmPf = covOmPf / sPfsPf sPm = sM e VP bOmPf = covOmPf / sFsF rOmPf = covOmPf / sMsF → bOmPm≠rOmPm bOP P O

  39. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm The ‘correction’ works like this: bOmPf = covOmPf / sPfsPf bOmPf* sPf /sPm sPm = sM e VP bOmPf = covOmPf / sFsF rOmPf = covOmPf / sMsF → bOmPm≠rOmPm bOP P O

  40. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm The ‘correction’ works like this: bOmPf = covOmPf / sPfsPf bOmPf* sPf /sPm = covOmPf / sPfsPf * sPf /sPm sPm = sM e VP bOmPf = covOmPf / sFsF rOmPf = covOmPf / sMsF → bOmPm≠rOmPm bOP P O

  41. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm The ‘correction’ works like this: bOmPf = covOmPf / sPfsPf bOmPf* sPf /sPm = covOmPf / sPfsPf * sPf /sPm = covOmPfsPf / sPfsPf sPm = covOmPf / sPfsPm sPm = sM e VP bOmPf = covOmPf / sFsF rOmPf = covOmPf / sMsF → bOmPm≠rOmPm bOP P O

  42. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm The ‘correction’ works like this: bOmPf = covOmPf / sPfsPf bOmPf* sPf /sPm = covOmPf / sPfsPf * sPf /sPm = covOmPfsPf / sPfsPf sPm = covOmPf / sPfsPm sPm = sM e VP b’OmPf = covOmPf / sMsF rOmPf = covOmPf / sMsF → b’OmPm=rOmPm bOP P O

  43. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm The ‘correction’ works like this: bOmPf = covOmPf / sPfsPf bOmPf* sPf /sPm = covOmPf / sPfsPf * sPf /sPm = covOmPfsPf / sPfsPf sPm = covOmPf / sPfsPm = rOmPf sPm = sM e VP b’OmPf = covOmPf / sMsF rOmPf = covOmPf / sMsF → b’OmPm=rOmPm bOP P O

  44. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm The ‘correction’ works like this: bOmPf = covOmPf / sPfsPf bOmPf* sPf /sPm = covOmPf / sPfsPf * sPf /sPm = covOmPfsPf / sPfsPf sPm = covOmPf / sPfsPm = rOmPf bOfPm* sPm /sPf = covOfPmsPm / sPmsPm sPf = covOfPm / sPmsPf = rOfPm sPm = sM e VP b’OmPf = covOmPf / sMsF rOmPf = covOmPf / sMsF → b’OmPm=rOmPm bOP P O

  45. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm The ‘correction’ works like this: bOmPf = covOmPf / sPfsPf bOmPf* sPf /sPm = covOmPf / sPfsPf * sPf /sPm = covOmPfsPf / sPfsPf sPm = covOmPf / sPfsPm = rOmPf bOfPm* sPm /sPf = covOfPmsPm / sPmsPm sPf = covOfPm / sPmsPf = rOfPm bOmPf * sPf /sPm = .3 * .708 = .2124 sPm = sM e VP b’OmPf = covOmPf / sMsF rOmPf = covOmPf / sMsF → b’OmPm=rOmPm bOP P O

  46. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm The ‘correction’ works like this: bOmPf = covOmPf / sPfsPf bOmPf* sPf /sPm = covOmPf / sPfsPf * sPf /sPm = covOmPfsPf / sPfsPf sPm = covOmPf / sPfsPm = rOmPf bOfPm* sPm /sPf = covOfPmsPm / sPmsPm sPf = covOfPm / sPmsPf = rOfPm bOmPf * sPf /sPm = .3 * .708 = .2124 sPm = sM e VP b’OmPf = covOmPf / sMsF rOmPf = covOmPf / sMsF → b’OmPm=rOmPm bOP P O

  47. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm The ‘correction’ works like this: bOmPf = covOmPf / sPfsPf bOmPf* sPf /sPm = covOmPf / sPfsPf * sPf /sPm = covOmPfsPf / sPfsPf sPm = covOmPf / sPfsPm = rOmPf bOfPm* sPm /sPf = covOfPmsPm / sPmsPm sPf = covOfPm / sPmsPf = rOfPm bOmPf * sPf /sPm = .3 * .708 = .2124 bOfPm * sPm /sPf = .1 * 1.412 = .1412 sPm = sM e VP b’OmPf = covOmPf / sMsF rOmPf = covOmPf / sMsF → b’OmPm=rOmPm bOP P O

  48. bOP = covOP / VP → so for males, bOmPm = covOmPm / VPm = covOmPm / sPmsPm → for females, bOfPf = covOfPf / VPf = covOfPf / sPfsPf → but for males-females, bOmPf = covOmPf / VPf = covOmPf / sPfsPf → and for females-males, bOfPm = covOfPm / VPm = covOmPf / sPmsPm The ‘correction’ works like this: bOmPf = covOmPf / sPfsPf bOmPf* sPf /sPm = covOmPf / sPfsPf * sPf /sPm = covOmPfsPf / sPfsPf sPm = covOmPf / sPfsPm = rOmPf bOfPm* sPm /sPf = covOfPmsPm / sPmsPm sPf = covOfPm / sPmsPf = rOfPm bOmPf * sPf /sPm = .3 * .708 = .2124 bOfPm * sPm /sPf = .1 * 1.412 = .1412 sPm = sM e VP b’OmPf = covOmPf / sMsF rOmPf = covOmPf / sMsF → b’OmPm=rOmPm bOP P O

  49. e VP bOP P O

  50. bOP e h2 = 2bOP VP bOP P O

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