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| Many data sets consist of individuals from different populations, in the cases of structured populations,
| | #REDIRECT [[Genotype_Likelihood_based_Inbreeding_Coefficient]] |
| this usually result in an increased number of homozygotes.
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| The inbreeding coefficient FIC is a measure of deviation away from the Hardy Weinberg Equilibrium.
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| A value of 0 implies no deviation, a negative value implies an excess of heterozygotes and a positive value implies an excess of homozygotes.
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| The following equation gives the estimate of F where the observed genotypes are available.
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| <math>
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| \begin{align}
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| F_{IC} & = 1 - \frac{O(Het)}{E(Het)} \\
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| & = 1 - \frac{\text{No. observed HETs}}{E(Het|\textbf{p)}} \\
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| & = 1 - \frac{\text{No. observed HETs}}{\sum_{i=1}^{n}{\sum_j{P(Het_j|\textbf{p})}}} \\
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| \end{align}
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| </math>
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|
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| The following equation gives the estimate of F where genotype likelihoods are available.
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| <math>
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| \begin{align}
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| F_{IC} & = 1 - \frac{O(Het)}{E(Het)} \\
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| & = 1 - \frac{E(Het|R_i, \textbf{p})}{E(Het|\textbf{p})} \\
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| & = 1 - \frac{\sum_{i=1}^{n}{\sum_j{P(Het_j|R_i , \textbf{p})}}} {\sum_{i=1}^{n}{\sum_{j}{P(Het_j|\textbf{p})}}} \\
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| & = 1 - \frac{\sum_{i=1}^{n}{\sum_j{\frac{P(R_i|Het_j,\textbf{p})P(Het_j|\textbf{p})}{\sum_{(k,l)}{P(R_i|G_{(k,l)},\textbf{p})P(G_{(k,l)}|\textbf{p})}}}}}
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| {\sum_{i=1}^{n}{\sum_j{P(Het_j|\textbf{p})}}} \\
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| \end{align}
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| </math>
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| where:
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| <math>
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| \begin{align}
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| P(G_{(k,l)}|\textbf{g}) & = & g_{(k,l)}
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| \end{align}
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| </math>
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| <math>
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| P(G_{(k,l)}|\textbf{p}) =
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| \begin{cases}
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| p_k^2, & \text{if }k=l \\
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| 2p_kp_l, & \text{if }k \ne l
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| \end{cases}
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| </math>
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