Combining Multiple Genetic Estimates of <i> N _e </i>

Researchers often use multiple genetic methods to estimate contemporary effective population size (N_e), but few formally combine estimates despite potential benefits for increasing precision. Maximising benefits requires an optimal, inverse-variance weighting scheme. Methods should be estimating the same parameter, which can be appropriate either for estimates using the same method applied to different time periods, or estimates using different methods applied to the same time period. Previous approaches focused on var( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mover><mml:mi>N</mml:mi> <mml:mo>̂</mml:mo></mml:mover> <mml:mi>e</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {\hat{N}}_e $$</mml:annotation></mml:semantics> </mml:math> ) for weighting, but that is problematical because <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mover><mml:mi>N</mml:mi> <mml:mo>̂</mml:mo></mml:mover> <mml:mi>e</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {\hat{N}}_e $$</mml:annotation></mml:semantics> </mml:math> is highly skewed and can be infinitely large. A new approach is described using weights inversely proportional to var(1/ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mover><mml:mi>N</mml:mi> <mml:mo>̂</mml:mo></mml:mover> <mml:mi>e</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {\hat{N}}_e $$</mml:annotation></mml:semantics> </mml:math> ), which is the drift signal that N_e-estimation methods respond to. The distribution of 1/ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mover><mml:mi>N</mml:mi> <mml:mo>̂</mml:mo></mml:mover> <mml:mi>e</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {\hat{N}}_e $$</mml:annotation></mml:semantics> </mml:math> is close to normal even when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mover><mml:mi>N</mml:mi> <mml:mo>̂</mml:mo></mml:mover> <mml:mi>e</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {\hat{N}}_e $$</mml:annotation></mml:semantics> </mml:math> assumes extreme values. Benefits are maximised under three general conditions: estimators have approximately equal variances; they are uncorrelated or have weak positive correlations; individual estimates have low precision (i.e., if data are limited and/or true N_e is large). Analytical and numerical results demonstrate that: (1) existing theory allows robust estimates of var(1/ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mover><mml:mi>N</mml:mi> <mml:mo>̂</mml:mo></mml:mover> <mml:mi>e</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {\hat{N}}_e $$</mml:annotation></mml:semantics> </mml:math> ) for the temporal and LD methods, which provide independent information about N_e-both of which facilitate optimally combining those methods; (2) estimates for the LD and sibship methods are essentially uncorrelated when data are limited but can be strongly positively correlated in genomics-scale datasets. General theory predicting var(1/ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub><mml:mover><mml:mi>N</mml:mi> <mml:mo>̂</mml:mo></mml:mover> <mml:mi>e</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {\hat{N}}_e $$</mml:annotation></mml:semantics> </mml:math> ) for the sibship method is lacking, but values for specific scenarios have been published. New software (ComboNe) is introduced to calculate combined estimates.