## An example: body weight in female mice

The analysis involves 719 offspring from 74 sires and 192 dams, each with one litter. The offspring were spread over 4 generations, and the analysis is performed as a nested ANOVA with the genetic analysis nested within generations. An additional complication is that the design was unbalanced, i.e., unequal numbers of progeny were measured in each sibship. As a result the degrees of freedom don't work out to be quite as simple as what I showed you.8 The results are summarized in Table 6.

Table 6: Quantitative genetic analysis of the inheritance of body weight in female mice (from Falconer and Mackay, pp. 169-170.)
 Composition of Source d.f. Mean square mean square Among sires 70 17.10 Among dams 118 10.79 (within sires) Within progenies 527 2.19

Using the expressions for the composition of the mean square we obtain

Thus,

Why didn't I give a definite number for after my big spiel above about how we can estimate it from a full-sib crossing design? Two reasons. First, if you plug the estimates for and into the formula above for you get , which is clearly impossible since has to be less than and has to be greater than zero. It's a variance. Second, the experimental design confounds two sources of resemblance among full siblings: (1) genetic covariance and (2) environmental covariance. The full-sib families were all raised by the same mother in the same pen. Hence, we don't know to what extent their resemblance is due to a common natal environment.9If we assume , we can estimate the amount of variance accounted for by exposure to a common natal environment, , and by environmental variation within sibships, .10 Similarly, if we assume , then and . In any case, we can estimate the narrow sense heritability as

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