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Table 1 Relationship between \(\gamma\) and the REML estimators obtained from data with QTL plus markers (\(\hat{\gamma }_{\mathrm{QM}}\)) and markers only (\(\hat{\gamma }_{\mathrm{M}}\)), assuming \(\hbox {q}\) fixed and finite, \(\hbox {m}\) very large, \(\hbox {m} > \hbox {q}\), and \(\hbox {q}+\hbox {m}>>> \hbox {n}\)

From: Understanding the potential bias of variance components estimators when using genomic models

Scenario

Population

MAF

QTL/markers

\({\lim _{\mathrm{n}\rightarrow \infty }}\)

\({\mathbb {E}(\hat{\gamma }_{\mathrm{QM}})}\)

\({\mathbb {E}(\hat{\gamma }_{\mathrm{M}})}\)

1

1 generation\(^{*}\)

\(\hbox {f}_{\mathrm{MAF}_{_{\mathrm{QTL}}}}=\hbox {f}_{\mathrm{MAF}_{_{\mathrm{makers}}}}\)

Complete LE

\(\gamma\)

0

2

1 generation\(^{*}\)

\(\hbox {f}_{\mathrm{MAF}_{_{\mathrm{QTL}}}}\ne \hbox {f}_{\mathrm{MAF}_{\mathrm{makers}}}\)

Complete LE

\(\gamma\)

0

3

1 generation\(^{*}\)

\(\hbox {f}_{\mathrm{MAF}_{_{\mathrm{QTL}}}}=\hbox {f}_{\mathrm{MAF}_{\mathrm{makers}}}\)

LD

\(\gamma\)

\(<<< \mathbb {E}(\hat{\gamma }_{\mathrm{QM}})\)

4

2 generations

\(\hbox {f}_{\mathrm{MAF}_{_{\mathrm{QTL}}}}=\hbox {f}_{\mathrm{MAF}_{\mathrm{makers}}}\)

LD

\(\gamma\)

\(<< \mathbb {E}(\hat{\gamma }_{\mathrm{QM}})\)

5

10 generations

\(\hbox {f}_{\mathrm{MAF}_{_{\mathrm{QTL}}}}=\hbox {f}_{\mathrm{MAF}_{\mathrm{makers}}}\)

LD

\(\gamma\)

\(< \mathbb {E}(\hat{\gamma }_{\mathrm{QM}})^{\dagger }\)

6

1 generation\(^{*}\)

\(\hbox {f}_{\mathrm{MAF}_{_{\mathrm{QTL}}}}\ne \hbox {f}_{\mathrm{MAF}_{\mathrm{makers}}}\)

LD

\(<<< \gamma\)

\(<<< \mathbb {E}(\hat{\gamma }_{\mathrm{QM}})\)

7

2 generations

\(\hbox {f}_{\mathrm{MAF}_{_{\mathrm{QTL}}}}\ne \hbox {f}_{\mathrm{MAF}_{\mathrm{makers}}}\)

LD

\(<< \gamma\)

\(<< \mathbb {E}(\hat{\gamma }_{\mathrm{QM}})\)

8

10 generations

\(\hbox {f}_{\mathrm{MAF}_{_{\mathrm{QTL}}}}\ne \hbox {f}_{\mathrm{MAF}_{\mathrm{makers}}}\)

LD

\(< \gamma ^{\dagger }\)

\(< \mathbb {E}(\hat{\gamma }_{\mathrm{QM}})^{\dagger }\)

  1. *Completely unrelated individuals
  2. \(^{\dagger }\lim _{\mathrm{g}\uparrow }\mathrm{h}^{2}_{\mathrm{M}}=\lim _{\mathrm{g}\uparrow }\mathrm{h}^{2}_{\mathrm{QM}}=\mathrm{h}^{2}\) for a large number \(\hbox {g}\) of generations (strongly related individuals)