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Reed A. Cartwright posted Entry 120 on April 8, 2004 08:00 PM.

Trackback URL: http://www.pandasthumb.org/cgi-bin/mt/mt-tb.fcgi/119

In the first installment of EvoMath, I derived the Hardy-Weinberg Principle and discussed its significance to biology. In the second installment I will demonstrate how to test if a population deviates from Hardy-Weinberg equilibrium.

#### Recap:

A population is considered to be in Hardy-Weinberg equilibrium if the allele and genotype frequencies are as follows.

Genotype | Frequency |
---|---|

AA | |

Aa | |

aa |

Allele | Frequency |
---|---|

A | |

a |

#### Test Procedure:

A goodness-of-fit test can be used to determine if a population is significantly different from the expections of Hardy-Weinberg equilibrium. If we have a series of genotype counts from a population, then we can compare these counts to the ones predicted by the Hardy-Weinberg model. We conclude that the population is not in Hardy-Weinberg equilibrium if the probability that the counts were drawn under the Hardy-Weinberg model is too small for the deviations to be considered due to random chance. The significance level that is typically used is , i.e. the genotype counts have less than a one in twenty chance of being caused by a population in Hardy-Weinberg equilibrium.

In order to calculate this probability, we will use a test statistic, , which was devised in 1900 by Karl Pearson and has a well characterized distribution. If are the set of observed counts, and are the set of expected counts, then

This test statistic has a “chi-square” distribution with degrees of freedom. Since we are testing Hardy-Weinberg equilibrium with two alleles, (rationale not shown). Furthermore, it can be shown that if then . Therefore, if we will reject the null model and conclude that there is significant statistical support that the population is not in Hardy-Weinberg equilibrium.

#### Example 1:

Consider the following samples from a population.

Genotype | Count |
---|---|

AA | 30 |

Aa | 55 |

aa | 15 |

Allele | Frequency |
---|---|

A | 0.575 |

a | 0.425 |

Calculate the value.

Genotype | Observed | Expected | (O-E)^{2}/E |
---|---|---|---|

AA | 30 | 33 | 0.27 |

Aa | 55 | 49 | 0.73 |

aa | 25 | 18 | 0.50 |

Total | 100 | 100 | 1.50 |

Since , we conclude that the genotype frequencies in this population are not significantly different than what would be expected if the population is in Hardy-Weinberg equilibrium.

#### Example 2:

Race and Sanger (1975) determined the blood groups of 1000 Britons as follows (from Hartl and Clarke 1997).

Genotype | Observed | Expected |
---|---|---|

MM | 298 | 294.3 |

MN | 489 | 496.4 |

NN | 213 | 209.3 |

This results in . As in the previous example, the measured genotype frequencies are not significantly different from the expectations of Hardy-Weinberg equilibrium.

#### Example 3:

Matthijis et al. (1998) surveyed a group of 54 people suffering from Jaeken syndrome (from Freeman and Herron 2004).

Genotype | Observed | Expected |
---|---|---|

OO | 11 | 19.44 |

OR | 43 | 25.92 |

RR | 0 | 8.64 |

This results in . Unlike the previous two examples, the measured genotype frequencies are significantly different from the expectations of Hardy-Weinberg equilibrium. This indicates that one or more of the Hardy-Weinberg conditions are being violated; although, it does not tell us which ones.

#### Conclusion:

Although to derive the Hardy-Weinberg principle, we assumed that the size of the population was infinite, these statistical tests demonstrate that finite populations can approximately exist in Hardy-Weinberg equilibrium.

#### References

- Freeman S and Herron JC (2004) Evolutionary Analysis 3
^{rd}ed. Pearson Education, Inc (Upper Saddle River, NJ) - Hartl DL and Clarke AG (1997) Principles of Population Genetics 3
^{rd}ed. Sinauer Associates, Inc (Sutherland, MA) - Matthijis GE et al. (1998) Lack of homozygotes for the most frequent disease allele in carbohydrate-deficient-glycoprotein syndrome type 1A. American Journal of Human Genetics 62: 542-550
- Race RR and Sanger R (1975) Blood Groups in Man 6
^{th}ed. JB Lippincott, Philadelphia

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