ABSTRACTThe reduction in additive genetic variance due to selection is investigated when index selection using family records is practised. A population of infinite size with no accumulation of inbreeding, an infinitesimal model and discrete generations are assumed. After several generations of selection, the additive genetic variance and the rate of response to selection reach an asymptote. A prediction of the asymptotic rate of response is considered to be more appropriate for comparing response from alternative breeding programmes and for comparing predicted and realized response than the response following the first generation of selection that is classically used. Algorithms to calculate asymptotic response rate are presented for selection based on indices which include some or all of the records of an individual, its full- and half-sibs and its parental estimated breeding values. An index using all this information is used to predict response when selection is based on breeding values estimated by using a Best Linear Unbiased Prediction (BLUP) animal model, and predictions agree well with simulation results. The predictions are extended to multiple trait selection.Asymptotic responses are compared with one-generation responses for a variety of alternative breeding schemes differing in population structure, selection intensity and heritability of the trait. Asymptotic responses can be up to one-quarter less than one-generation responses, the difference increasing with selection intensity and accuracy of the index. Between family variance is reduced considerably by selection, perhaps to less than half its original value, so selection indices which do not account for this tend to place too much emphasis on family information. Asymptotic rates of response to selection, using indices including family information for traits not measurable on the individuals available for selection, such as sex limited or post-slaughter traits, are found to be as much as two-fifths less than their expected one-generation responses. Despite this, the ranking of the breeding schemes is not greatly altered when compared by one-generation rather than asymptotic responses, so the one-generation prediction is usually likely to be adequate for determining optimum breeding structure.