. 2014 Apr 16;30(15):2189–2196. doi: 10.1093/bioinformatics/btu198

Table 1.

Approximation of rare variant sharing probabilities for three second cousins in small populations

Number of founders		200		400
Number of founders		Sample $ϕ^{f}$ ^a	Population $ϕ^{f}$	Sample $ϕ^{f}$	Population $ϕ^{f}$
Analytical approximation
RMSE^b	Absolute	0.0015	0.0026	0.0006	0.0007
RMSE^b	Relative	0.27	0.34	0.24	0.28
Bias^c	Absolute	−0.0009	−0.0012	−0.0002	−0.0002
Bias^c	Relative	−0.18	−0.18	−0.02	0.01
Monte Carlo approximation
RMSE	Absolute	0.0015	0.0026	0.0006	0.0008
RMSE	Relative	0.27	0.35	0.26	0.32
Bias	Absolute	−0.0009	−0.0012	−0.0002	−0.0001
Bias	Relative	−0.17	−0.17	−0.01	0.02

^a $ϕ^{f}$ Mean kinship coefficient among founders from the sample or in the population. ^bRMSE: Root mean square error. If we denote the RV sharing probability at the rth replicate by $β_{r}$ and its approximation by ${\hat{β}}_{r}$ , then the absolute RMSE is equal to $\sqrt{\frac{1}{R} \sum_{r = 1}^{R} {({\hat{β}}_{r} - β_{r})}^{2}}$ , and the relative RMSE is equal to $\sqrt{\frac{1}{R} \sum_{r = 1}^{R} {(({\hat{β}}_{r} - β_{r}) / β_{r})}^{2}}$ where $R = 100$ . ^cThe absolute bias is equal to $\frac{1}{R} \sum_{r = 1}^{R} ({\hat{β}}_{r} - β_{r})$ , and the relative bias is equal to $\frac{1}{R} \sum_{r = 1}^{R} (({\hat{β}}_{r} - β_{r}) / β_{r})$ with $R = 100$ .