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7.5 Strategies for Improved Use of Summary Statistics

© 2025 Princeton University Press, Princeton

© 2025 Princeton University Press, Princeton

Kapitel in diesem Buch

  1. Frontmatter i
  2. CONTENTS vii
  3. PREFACE ix
  4. ACKNOWLEDGMENTS xiii
  5. 1 Mathematical and Biological Introduction
  6. 1.1 Interpreting Population-Genetic Statistics 1
  7. 1.2 A Note about Assumed Background 3
  8. 1.3 Definitions 3
  9. 1.4 Standard Inequalities 7
  10. 1.5 Genetic Diversity and Genetic Homogeneity 8
  11. 1.6 Genetic Differentiation and Genetic Similarity 13
  12. 1.7 Do Statistics “Depend” on Allele Frequencies? 18
  13. 1.8 Exercises 19
  14. 2 Homozygosity and the Most Frequent Allele
  15. 2.1 Arbitrarily Many Distinct Alleles 21
  16. 2.2 A Fixed Value I for the Number of Distinct Alleles 33
  17. 2.3 Example from Human Populations 40
  18. 2.4 Implications 43
  19. 2.5 Exercises 46
  20. 3 Variations on Homozygosity: JA, JB, and JC
  21. 3.1 Boundson JC/JB in Terms of JA 48
  22. 3.2 Example from Drosophila 58
  23. 3.3 Implications 60
  24. 3.4 Exercises 61
  25. 4 The ith Most Frequent Allele
  26. 4.1 Lower Bound on J in Terms of pi 62
  27. 4.2 Upper Bound on J in Terms of pi 66
  28. 4.3 Lower and Upper Bounds on pi in Terms of J 75
  29. 4.4 Example from Human Populations 85
  30. 4.5 Implications 86
  31. 4.6 Exercises 90
  32. 5 α-homozygosity
  33. 5.1 Convexity Inequalities 92
  34. 5.2 Arbitrarily Many Distinct Alleles 94
  35. 5.3 A Fixed Value I for the Number of Distinct Alleles 99
  36. 5.4 Example from Human Populations 103
  37. 5.5 Implications 105
  38. 5.6 Exercises 106
  39. 6 Estimated Homozygosity
  40. 6.1 Samples 108
  41. 6.2 Number of Distinct Alleles Constrained by Sample Size 111
  42. 6.3 A Fixed Value I for the Number of Distinct Alleles 117
  43. 6.4 Example from Human Populations 119
  44. 6.5 Implications 120
  45. 6.6 Exercises 122
  46. 7 Conclusions
  47. 7.1 Summary of Mathematical Results 124
  48. 7.2 Summary of Mathematical Methods 126
  49. 7.3 Benefits of the Mathematical Bounds Approach 128
  50. 7.4 The Continuing Importance of Summary Statistics 137
  51. 7.5 Strategies for Improved Use of Summary Statistics 138
  52. NOTATION 141
  53. SOLUTIONS TO EXERCISES 145
  54. BIBLIOGRAPHY 155
  55. AUTHOR INDEX 163
  56. SUBJECT INDEX 167
Mathematical Properties of Population-Genetic Statistics
Ein Kapitel aus dem Buch Mathematical Properties of Population-Genetic Statistics
https://doi.org/10.1515/9780691261577-042

Kapitel in diesem Buch

  1. Frontmatter i
  2. CONTENTS vii
  3. PREFACE ix
  4. ACKNOWLEDGMENTS xiii
  5. 1 Mathematical and Biological Introduction
  6. 1.1 Interpreting Population-Genetic Statistics 1
  7. 1.2 A Note about Assumed Background 3
  8. 1.3 Definitions 3
  9. 1.4 Standard Inequalities 7
  10. 1.5 Genetic Diversity and Genetic Homogeneity 8
  11. 1.6 Genetic Differentiation and Genetic Similarity 13
  12. 1.7 Do Statistics “Depend” on Allele Frequencies? 18
  13. 1.8 Exercises 19
  14. 2 Homozygosity and the Most Frequent Allele
  15. 2.1 Arbitrarily Many Distinct Alleles 21
  16. 2.2 A Fixed Value I for the Number of Distinct Alleles 33
  17. 2.3 Example from Human Populations 40
  18. 2.4 Implications 43
  19. 2.5 Exercises 46
  20. 3 Variations on Homozygosity: JA, JB, and JC
  21. 3.1 Boundson JC/JB in Terms of JA 48
  22. 3.2 Example from Drosophila 58
  23. 3.3 Implications 60
  24. 3.4 Exercises 61
  25. 4 The ith Most Frequent Allele
  26. 4.1 Lower Bound on J in Terms of pi 62
  27. 4.2 Upper Bound on J in Terms of pi 66
  28. 4.3 Lower and Upper Bounds on pi in Terms of J 75
  29. 4.4 Example from Human Populations 85
  30. 4.5 Implications 86
  31. 4.6 Exercises 90
  32. 5 α-homozygosity
  33. 5.1 Convexity Inequalities 92
  34. 5.2 Arbitrarily Many Distinct Alleles 94
  35. 5.3 A Fixed Value I for the Number of Distinct Alleles 99
  36. 5.4 Example from Human Populations 103
  37. 5.5 Implications 105
  38. 5.6 Exercises 106
  39. 6 Estimated Homozygosity
  40. 6.1 Samples 108
  41. 6.2 Number of Distinct Alleles Constrained by Sample Size 111
  42. 6.3 A Fixed Value I for the Number of Distinct Alleles 117
  43. 6.4 Example from Human Populations 119
  44. 6.5 Implications 120
  45. 6.6 Exercises 122
  46. 7 Conclusions
  47. 7.1 Summary of Mathematical Results 124
  48. 7.2 Summary of Mathematical Methods 126
  49. 7.3 Benefits of the Mathematical Bounds Approach 128
  50. 7.4 The Continuing Importance of Summary Statistics 137
  51. 7.5 Strategies for Improved Use of Summary Statistics 138
  52. NOTATION 141
  53. SOLUTIONS TO EXERCISES 145
  54. BIBLIOGRAPHY 155
  55. AUTHOR INDEX 163
  56. SUBJECT INDEX 167
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