Proceedings, Western Section, American Society of Animal Science Vol. 58, 2007 RETROSPECTIVE ANALYSIS OF SELECTION APPLIED TO A RATIO M. D. MacNeil USDA Agricultural Research Service, Miles City, Montana 59301 ABSTRACT: Use of ratios to adjust one correlated trait for another is fairly commonplace. However, there are statistical arguments that restrict the appropriate use of ratios to certain circumstances. The ratio of a calf’s weaning weight to that of its dam has been used as an indicator of cow efficiency and an evaluation criterion. Objectives of this study were to retrospectively assess selection applied when bulls were selected based on the ratio of their weaning weight to the coincident weight of their dam and predict correlated responses in these weights to that selection. The variance of the ratio tended to increase (P = 0.11) as weaning weight increased, but was independent of cow weight. The observed selection differential for weaning weight was independent of the selection differential for the ratio (P > 0.10). However, the selection differential for cow weight was inversely related to the selection differential for the ratio (P = 0.01). The emphasis on cow weight relative to that given weaning weight ranged from 3% to 109% (SD = 31%) and the variance of the index in retrospect fluctuated more than 5- fold. It follows that the correlated responses of weaning weight and cow weight to selection for their ratio would be variable. These results contraindicate use of the ratio of calf weaning weight to cow weight either as an evaluation criterion for beef cows or as a selection criterion upon which to choose bulls or heifers. By inference, the use of ratios of traits to evaluate animals or as selection criteria is generally likely to be inappropriate. Key words: Breeding aims, Efficiency, Phenotypic selection, Selection index Introduction Use of ratios to adjust one correlated trait for another is fairly commonplace. Examples include weaning weight:cow weight to indicate cow efficiency (Kress et al., 1995; MacNeil, 2005), longissimus muscle area:carcass weight to indicate muscularity (Thomas et al., 2002), weight:height to indicate body condition (Brown et al., 2000; Turconi et al., 2006), and feed:gain to indicate efficiency of growth (Tedeschi et al., 2006). Concerns about insidious effects of ratios were first noted by Pearson (1897) and have been elaborated on numerous times subsequently. Use of a ratio for purposes of comparing subjects assumes three conditions: 1) the relationship between numerator and denominator is linear; 2) the intercept of the regression of numerator on denominator is the origin; and 3) the variance of the ratio increases with increasing values of the numerator and denominator (Weil, 1962). These conditions are rarely tested. When ratios are used to evaluate subjects for the purpose of selection among them, the ratio is often also implicitly assumed to be normally distributed. Yet, a ratio of two normally distributed variables is not normally distributed (Fieller, 1932; Hinkley, 1969). Despite these theoretical issues with the use of ratios, they continue to be commonly used to evaluate animals and as selection criteria. The objective of this study was to conduct a retrospective analysis of selection applied to calf weaning weight and cow weight when sires were selected based on the ratio of these traits. Materials and Methods Data for this study were compiled from MacNeil (2005). Calves were produced in 1989 to 2000 from sires born in 1987 to 1998 and selected on the ratio of their age- of-dam adjusted 200-d weight (WW) to the coincident mature equivalent weight of their dam (CW). Virtually all selection pressure was applied to males and most females were exposed to breeding as yearlings. With the following exceptions, four yearling bulls were selected and used each year. Nine yearling bulls were used in the 1989 and 1990 breeding seasons and five yearling bulls were used in 1992 and 1996 to 1999 breeding seasons. In the present study, attention is focused on the sire selection differentials and predicted responses in the component traits. Individual selection differentials were computed within year based on the adjusted phenotypes. Means and average selection differentials for sires, by birth year, are shown in Table 1. Table 1. Adjusted weaning weight (WW, kg) and mature equivalent cow weight (CW, kg) means and sire selection differentials for WW, CW and their ratio. Mean ♂ Selection Differential Birth Year WW CW WW CW Ratio 1987 246 486 19.7 -2.3 6.3 1988 226 472 23.3 -1.7 8.0 1989 240 499 47.7 25.3 8.9 1990 239 498 30.2 -19.9 8.4 1991 244 515 42.9 24.5 5.5 1992 250 526 47.2 3.2 8.6 1993 260 544 37.9 -25.6 10.7 1994 236 518 37.2 -40.0 11.6 1995 242 546 39.0 8.9 7.5 1996 229 511 31.7 -10.5 7.9 1997 227 508 22.6 -57.7 9.9 1998 240 517 39.0 -7.3 7.8 85