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A Consideration of the Power of the χ2 Test to Detect Inbreeding Effects in Natural Populations

Richard H. Ward and Charles F. Sing
The American Naturalist
Vol. 104, No. 938 (Jul. - Aug., 1970), pp. 355-365
Stable URL: http://www.jstor.org/stable/2459121
Page Count: 11
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A Consideration of the Power of the χ2 Test to Detect Inbreeding Effects in Natural Populations
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Abstract

The data above indicate that the sample sizes required to reject the null hypothesis, (F = 0), are much larger than hitherto employed, given the range of expected in a natural population (e.g., from .0004 to .009 in nonprimitive human populations). The use of the χ2 statistic without considering the power of the test to estimate F from small samples would seem to be unjustified since the magnitude of inbreeding expected in a natural population would not cause significant χ2 values. On the other hand, should significant χ2 values be obtained, the magnitude of F required is so great that it would be illogical to attribute the deviations from H-W proportions to inbreeding. Thus it would seem that the problem of estimating inbreeding in a natural population is analogous to the problem of estimating selection (Lewontin and Cockerham 1959; Neel and Schull 1968) because sample sizes far in excess of those normally used are required. The material presented here indicates the size of sample required to detect a specified level of F with an appropriate power. Furthermore, because the noncentrality parameter is a function of k, the number of alleles, choice of a system with many codominant alleles will reduce the required sample size. With this information, a test can be devised that will maximize the chances of detecting a given F for the data set available. Thus, in populations with little inbreeding, one might be prepared to sacrifice some of the power of the test (retaining the significance level) for the sake of a manageable sample, while a higher degree of certainty in highly inbred populations can be obtained by raising the power of the test. In conclusion, the commonly employed method of using the χ2 statistic to estimate F in natural populations is unlikely to give significant estimates unless very large samples are used. In actual fact, with the confounding effect of selection, migration, etc., in natural populations, it appears that the successful estimation of F (and other deterministic processes) in such populations will remain an intractable problem until much larger samples are used in conjunction with more sophisticated models of analysis.

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