Incorporation of economic values into the component traits
of a ratio: Feed efficiency
C. Y. Lin* and S. E. Aggrey†‡1
*Guelph Food Research Centre, Agriculture and Agri-Food Canada, 93 Stone Road West, Guelph, Ontario,
Canada N1G 5C9; †NutriGenomics Laboratory, Department of Poultry Science, University of Georgia,
Athens 30602-2772; and ‡Institute of Bioinformatics, University of Georgia, Athens 30602-7229
ABSTRACT Direct selection on a ratio (R) of 2 traits
(x1/x2) does not have a mechanism to accommodate
the relative economic values (a1 and a2) between x1 and
x2 because selection criteria x1/x2 and a1x1/a2x2 rank
animals in the same order. This study presented a procedure to incorporate the economic weights into ratio
traits through linear transformation. The partial derivatives of a nonlinear profit function evaluated at the
means were widely taken as economic weights in the
literature. This study showed that the economic weights
derived in this manner were erroneous because they
actually contain a mixture of actual economic weights
and transformation effects. The ratios 1/2 and 2/4 are
considered equal by selection on R, but are treated differently by the linear index. In addition, this study presented a unified approach to compare 4 different selection strategies for genetic improvement of ratio traits:
linear index (I), selection on the ratio (R), selection on
difference between x1 and x2 (D), and selection on x1
alone. This study considered 3 levels of heritability each
for variables x1 and x2 (h12 and h22 ), 2 levels of genetic
correlations (γG), 2 ratios of means (µ1/µ2), and 4 ratios of phenotypic variances (σx21 /σx22 ), giving a total of
96 scenarios. Linear index I was the most efficient of
the 4 criteria compared in all 96 scenarios studied. The
superiority of index I over R, D, and selection on x1
alone are particularly remarkable when x1 and x2 have
a large difference in heritability and are highly correlated. Selection on x1 alone is an economically viable
alternative to criterion I or R for the improvement of
ratio traits particularly when x1 is more heritable than
x2 and when x2 is costly to measure. Selection on D is
more efficient than direct selection on R or selection on
x1 alone when x1 is less heritable than x2 and the difference between µ1 and µ2 is small.
Key words: ratio, economic weight, linear transformation, relative efficiency
2013 Poultry Science 92:916–922
http://dx.doi.org/10.3382/ps.2012-02688
INTRODUCTION
and its component traits by taking the logarithm of the
ratio. Although the ratios differ in original units and in
logs, the ranking of the animals remained unchanged.
Lin (1980) developed a linear index to approximate the
genetic value of feed efficiency ratio (weight gain/feed)
and showed that the linear index was more effective
than selection on BW and restricted selection index for
the improvement of feed efficiency ratio. Both simulation (Gunsett 1984; Famula, 1990) and experimental
studies (Campo and Rodriguez, 1990) indicated that
the linear index proposed by Lin (1980) was more efficient than direct selection on ratio. The advantage of
linear index decreased as the correlation between the
2 component traits increased or as the heritabilities of
both component traits moved toward equality (Gunsett, 1984).
It is worth noting that a ratio traits x1/x2 used as a
selection criterion does not have a mechanism to accommodate differential economic weights between x1 and
x2. This is because selection criteria x1/x2 and a1x1/a2x2
Many economic traits in livestock and poultry production are expressed as a ratio of 2 component traits.
For example, feed efficiency is defined as a ratio of
weight gain to feed intake and economic efficiency is
defined as income over expense. Percentage traits such
as fat and protein yield are also a type of ratio trait.
Pearson (1897) derived formulae to approximate the
variance of a ratio and phenotypic correlation between
2 ratios. His results have been used to approximate the
heritability of a ratio and genetic correlation between 2
ratios (Turner, 1959; Sutherland, 1965; Gunsett, 1984).
Turner (1959) studied the relationships between a ratio
©2013 Poultry Science Association Inc.
Received August 15, 2012.
Accepted October 5, 2012.
1 Corresponding author: [email protected]
916
917
SELECTION FOR RATIO TRAITS
would rank animals exactly in the same order because
the economic ratio a1/a2 is a constant common to all
individuals. The primary purpose of this study was to
demonstrate how to incorporate the relative economic
values of the 2 component traits of a ratio into selection
decisions, and the secondary purpose was to compare
the efficiency of 4 selection criteria for the improvement
of the ratio traits (linear index, direct selection on the
ratio, selection on difference between x1 and x2, and selection on x1 alone) under 96 combinations of different
heritabilities, genetic correlations, ratios of means, and
ratios of variances.
MATERIALS AND METHODS
To establish notations, consider 2 characters, x1
(weight gain) and x2 (feed intake), have a bivariate normal distribution with respective means µ1 and µ2 and
phenotypic and genetic covariance matrices P and G.
Let the phenotypic ratio (i.e., feed efficiency) be R =
x1/x2, the genetic ratio be Rg = g1/g2, and the genetic
and phenotypic correlations between x1 and x2 be γG
and γP, respectively. The objective is to transform a
genetic ratio into a linear scale to construct a linear
index while accommodating the differential economic
values between x1 and x2.
Let economic value per unit of genetic gains for x1
and x2 be a1 and a2, respectively. A genetic ratio, Rg =
g1/g2, can be linearly transformed using Taylor series
expansion (Mood et al., 1987):
∂Rg
∂g1
|µ1 ,µ2 (g1 − µ1 ) +
∂Rg
∂g 2
|µ1 , µ2 (g 2 − µ2 )
+ (termss of higher order) .
[1]
If the (terms of higher order) in the formula are
dropped, the linear approximation becomes
µ
µ
1
Rg ≅ 1 +
(g1 − µ1 ) − 12 (g 2 − µ2 );[2]
µ2 µ2
µ2
where
≅
∂Rg
∂g1
H = a1 (
µ
1
g1 ) − a2 ( 12 g 2 ) = [g1
µ2
µ2
g ′ Va = g ′w,
µ1
µ
1
g1 − 12 g 2 ,[3]
+
µ2 µ2
µ2
|µ1 ,µ2 is the partial derivative of Rg with re-
spect to g1 evaluated at the point (µ1, µ2). The linear
transformation of a ratio always yields a negative transformation weight of −µ1 /µ22 for the denominator trait
(x2). Because µ1/µ2 (the first term on the right-hand
1

 µ2
g 2 ] 
0



0 
 a 
  1 =
µ1  a2  − 2
µ2 
where the matrix
1

 µ2
V = 
0


Incorporation of Economic Weights
into Linear Index for Improvement
of Ratio Traits
Rg = Rg (µ1 , µ2 ) +
side of equation [3]) is a constant and can be dropped,
the linear net merit is

0 


µ1 
− 2
µ2 
and the vector
w 
w =  1  = Va.
w 
 2
µ
1
g1 ) − a2 ( 12 g 2 ) may be rearµ2
µ2
a2 µ1
a1
ranged as H = ( )g1 + (− 2 )g 2 , where the quantities
µ2
µ2
a2 µ1
a1
( ) and (− 2 ) are constants. Therefore, the net merµ2
µ2
The net merit H = a1(
it H (or aggregate genotype) defined in this study is a
linear combination of individual genotypes (g1 and g2)
a µ
a
weighted by their relative weights ( 1 ) and (− 2 2 1 ),
µ2
µ2
which is in agreement with the original definition of H
by Hazel (1943). It should be noted that the diagonal
matrix V is a transformation matrix used to transform
a nonlinear genetic ratio (Rg = g1/g2) to a linear scale.
It follows that
Linear index I: I = b1x 1 + b2x 2 = x ′b;
Linear merit H: H = w1g1 + w 2g 2 = g ′w.
By index theory, minimizing the squared difference
between I and H leads to Pb = Gw with the index
coefficients being b = P−1Gw = P−1GVa.
The above approach led to the net merit H = g′Va =
g′w to accommodate the relative economic values of the
ratio traits. Notably, these partial derivatives evaluated
at the means (w1:w2) are the product of transformation
matrix (V) and a vector of economic values (a). The
Taylor series expansion is based on approximating the
function by a polynomial under the assumption that
the variables x1 and x2 are continuous. Thus, it does
not apply to categorical or discrete variables.
918
Lin and Aggrey
Selection Criteria Compared
Four different selection strategies were used to compare their relative efficiency in terms of genetic improvement of ratio traits:
1) Linear index (I). The linear index with b =
P−1GVa was derived as above.
2) Selection on phenotypic ratio (R). Taylor series
expansion provides a means for linear transformation of a ratio, making it possible to estimate
(co)variance of a ratio in a linear manner. Taylor
series expansion of a phenotypic ratio, R = x1/
x2, leads to
R=
µ1
µ
µ
1
+
x 1 − 12 x 2 = 1 + 1′ Vx,
µ2 µ2
µ2
µ2
where 1′ is a row vector of ones and V is a transformation matrix as defined above. The ratio of
means µ1/µ2 is a constant and can be dropped.
Therefore, a nonlinear ratio (R = x1/x2) is converted into a linear combination of x1 and x2
weighted by transformation factors (R = 1′Vx).
=
Variance
of
R
is
1′ VPV1
σx21 /µ22 − 2µ1σx1x 2 /µ23 + µ12 σx22 /µ24 . The linear transformation makes it possible to approximately
compare the nonlinear with the linear selection
criteria.
3) Selection on difference between 2 component
traits of a ratio (D). The 2 component traits are
weighted by their economic weights. This criterion takes the form of D = a1x 1 − a2x 2 . This is
equivalent to a base index (Brim et al., 1959;
Williams, 1962). When x2 refers to feed intake, a2
has a negative economic value. The absolute value of a2 is used for D = a1x 1 − a2x 2 , which is
equivalent to D = a1x 1 + a2x 2 with a2 being the
original negative economic value.
4) Selection based on weight gain (x1) alone. Single
trait selection is based on weight gain (x1) rather
than feed intake (x2) mainly because the former
is easier to measure and more heritable than the
latter.
General Formula for Predicting Genetic
Responses to Different Selection Criteria
In matrix notation, the genetic responses in x1 and x2
due to selection criterion j can be calculated using the
general formula below:
∆ j = Gb(i/σ j ), [4]
′
where ∆ j = [∆G1 ∆G2 ] , i = selection intensity, and
σj = SD of criterion j.
(1) Selection is on linear index (j = I): I = x′b, where
b = P−1GVa and σI = b ′Pb.
(2) Selection is on the ratio (j = R): R = 1′Vx =
x′b, where b = V1 and σR = b ′Pb = 1′ VPV1.
(3) Selection is on the difference (j = D): D = a1x1
′
− a2x2 = x′b, where b = [a1 −a2 ] and
σD = b ′Pb = a ′Pa.
′
(4) Selection is on x1 alone: x1 = x′b, where b = [1 0 ]
and σx = b ′Pb = σx2 .
1
1
The genetic response in H due to selection criterion
j is
∆H j = bHj (sel. diff.) =
cov(g ′ Va, x ′b)
σ j2
(sel. diff.) =
, [5]
a ′V ′G b(i/σ j ),
where sel. diff. is the selection differential. Because of
∆ j = Gb(i/σ j ), equation [5] reduces to ∆H j = a ′V ′∆ j .
The above procedure unifies the calculation of genetic
responses to different selection criteria in a single computational scheme.
Equivalence of Different Selection Criteria
Under Special Conditions
When h12 = h22 = h2 and γG = γP, the phenotypic and
genetic covariance matrices are
 σ2
γ p σx1 σx 2 
x1

P=

2
 γ p σx σx

σ
x


1
2
2
and
2
 σ
γG σx1 σx 2 
x1

G = h2 
.
 γG σx σx
σx22 

1
2

Then G = h2P because of γG = γP.
Index weights for linear index (I) are b = P−1GVa =
2
h Va where constant h2 can be dropped. If a1 = a2 =
1, then vector b reduces to V1, which is identical to
selection on the ratio (R). This proves that if a1 = a2,
h12 = h22 , and γG = γP, selection on I is equivalent to
selection on R regardless of the choice of matrices G,
P, and V.
When µ1 = µ2 = µ, h12 = h22 , and γG = γP, then Va
reduces to Va = (1/µ)[a1 − a2]′. In this case, index
weights for I is b = P−1GVa = (h2/µ)[a1 − a2]′ = [a1
− a2]′ because h2/µ can be dropped without affecting
the proportionality of b, indicating that selection on I
is identical to selection on difference (D) between x1
and x2 regardless of the choice of G, P, and a.
919
SELECTION FOR RATIO TRAITS
Table 1. Relative efficiency of 4 selection methods for improving ratio trait with γG =
0.31
µ1:µ2 = 1:2
h12
h22
Criterion
0.1
 
 
 
 
0.1
 
 
 
 
0.3
 
 
 
 
0.3
 
 
 
 
0.5
 
 
 
 
0.5
 
 
 
0.3
 
 
 
 
0.5
 
 
 
 
0.1
 
 
 
 
0.5
 
 
 
 
0.1
 
 
 
 
0.3
Index
x1/x2
x1 − x2
x1
 
Index
x1/x2
x1 − x2
x1
 
Index
x1/x2
x1 − x2
x1
 
Index
x1/x2
x1 − x2
x1
 
Index
x1/x2
x1 − x2
x1
 
Index
x1/x2
x1 − x2
x1
 
 
 
σx21 :σx22
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
=
µ1:µ2 = 1:8
1:2
1:4
1:8
1:16
1:2
1:4
1:8
1:16
1.16
1
1.16
0.37
 
1.31
1
1.27
0.21
 
1.06
1
0.74
0.99
 
1.04
1
1.01
0.59
 
1.08
1
0.71
1.03
 
1.02
1
0.81
0.89
1.12
1
1.12
0.19
 
1.18
1
1.17
0.08
 
1.12
1
0.71
0.99
 
1.03
1
1.02
0.38
 
1.18
1
0.65
1.10
 
1.03
1
0.80
0.80
1.06
1
1.06
0.07
 
1.08
1
1.08
0.01
 
1.16
1
0.74
0.94
 
1.02
1
1.01
0.20
 
1.31
1
0.65
1.16
 
1.04
1
0.85
0.64
1.02
1
1.02
−0.01
 
1.03
1
1.03
−0.04
 
1.16
1
0.82
0.78
 
1.01
1
1.00
0.07
 
1.38
1
0.72
1.13
 
1.02
1
0.91
0.42
1.01
1
0.67
0.96
 
1.06
1
0.84
0.92
 
1.00
1
0.56
0.98
 
1.00
1
0.57
0.98
 
1.00
1
0.57
0.98
 
1.00
1
0.55
0.98
1.04
1
0.70
0.90
 
1.18
1
1.00
0.81
 
1.00
1
0.41
0.98
 
1.00
1
0.52
0.95
 
1.00
1
0.41
0.98
 
1.00
1
0.41
0.98
1.10
1
0.82
0.77
 
1.32
1
1.19
0.63
 
1.01
1
0.30
0.98
 
1.01
1
0.54
0.88
 
1.01
1
0.29
0.98
 
1.00
1
0.33
0.96
1.16
1
0.97
0.59
 
1.38
1
1.29
0.41
 
1.02
1
0.25
0.98
 
1.02
1
0.64
0.76
 
1.03
1
0.22
0.99
 
1.01
1
0.32
0.94
1Selection
on x1/x2 was used as a basis for comparison within scenario. γG = genetic correlation; x1 = trait 1; x2 = trait 2; µ1 = mean of trait 1; µ2
= mean of trait 2; σx21 = variance of trait 1; σx22 variance of trait 2; h2 = heritability.
Numerical Example
This study examined 3 levels of h2 for x1 and x2
= 0.1, 0.3, or 0.5), 2 levels of genetic correlations (γG = 0.3 or 0.8), 2 ratios of means (µ1/µ2 = 1/2
or 1/8), and 4 phenotypic variance ratios (σx21 /σx22 =
1/2, 1/4, 1/8, or 1/16), giving a total of 96 scenarios
(Tables 1 and 2). These scenarios emulate low, moderate, and high h2, low, and high genetic correlations in
combination with varying ratios of both means and
phenotypic variances. To reduce the number of combinations, phenotypic and genetic correlations were set
equal (γG = γP) and x1 and x2 were assumed to have
equal economic values (a1 = a2 = 1). Note that the
transformation matrix V is invariant to the changes in
h2, γG, and γP. Selection intensity (i) was set to be
unity for the purpose of comparison. Genetic responses
in x1, x2, and H due to the 4 selection criteria were
computed according to the general formulae [4] and [5].
The relative efficiency of the 4 selection criteria was
computed within each scenario using direct selection on
the ratio as a basis for comparison. Although a total of
96 scenarios were examined, the general treatment developed in this study permits the comparison of different selection criteria for any desired combination of h2,
γG, µ1/µ2, a1/a2, and σx21 /σx22 .
(h12 , h22
RESULTS AND DISCUSSION
Linear Transformation and Relative
Economic Values
The relative economic values have been derived as
the partial derivative of a nonlinear profit function with
respect to the traits evaluated at the means (Moav and
Hill, 1966; Harris, 1970; Goddard, 1983; Brascamp et.
al. 1985; Itoh and Yamada, 1988). This approach is
incorrect because these partial derivatives evaluated at
the means are shown to be a product of transformation
weights and relative economic values (Va) where the
matrix V results from the transformation of a nonlinear function into a linear scale. As illustrated above,
the partial derivatives of a genetic ratio (Rg = g1/g2)
evaluated at the mean are (1/µ2) for g1 and (−µ1 /µ22 )
for g2. Obviously, the quantities (1/µ2) and (−µ1 /µ22 )
are the results of linear transformation and have nothing to do with the economic values of x1 and x2. Furthermore, when the economic weights are obtained
based on the derivatives of a nonlinear profit function,
the product of Va would vary depending upon how the
profit function is defined. As an example, the functions
x1/x2, x2/x1, and x1x2 would yield different transformation matrices (V), respectively, and thus different sets
920
Lin and Aggrey
Table 2. Relative efficiency of 4 selection methods for improving ratio trait with γG = 0.81
µ1:µ2 = 1:2
h12
h22
Criterion
0.1
 
 
 
 
0.1
 
 
 
 
0.3
 
 
 
 
0.3
 
 
 
 
0.5
 
 
 
 
0.5
 
 
 
0.3
 
 
 
 
0.5
 
 
 
 
0.1
 
 
 
 
0.5
 
 
 
 
0.1
 
 
 
 
0.3
Index
x1/x2
x1 − x2
x1
 
Index
x1/x2
x1 − x2
x1
 
Index
x1/x2
x1 − x2
x1
 
Index
x1/x2
x1 − x2
x1
 
Index
x1/x2
x1 − x2
x1
 
Index
x1/x2
x1 − x2
x1
 
 
 
σx21 :σx22
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
=
µ1:µ2 = 1:8
1:2
1:4
1:8
1:16
1:2
1:4
1:8
1:16
1.41
1
1.39
0.02
 
1.61
1
1.61
−0.17
 
1.01
1
0.49
0.80
 
1.08
1
0.90
0.44
 
1.00
1
0.53
0.76
 
1.01
1
0.48
0.81
1.14
1
1.09
−0.19
 
1.14
1
1.10
−0.20
 
1.14
1
0.32
0.83
 
1.07
1
0.97
−0.03
 
1.14
1
0.32
0.84
 
1.07
1
0.45
0.67
1.01
1
0.94
−0.27
 
1.00
1
0.92
−0.22
 
1.41
1
0.42
0.83
 
1.01
1
0.94
−0.28
 
1.61
1
0.30
1.11
 
1.08
1
0.70
0.24
1.01
1
0.92
−0.31
 
1.02
1
0.90
−0.25
 
1.24
1
0.76
0.21
 
1.00
1
0.95
−0.40
 
1.93
1
0.51
1.03
 
1.01
1
0.93
−0.35
1.24
1
−0.76
1.08
 
1.50
1
−1.15
1.13
 
1.09
1
0.41
0.94
 
1.04
1
−0.33
1.03
 
1.16
1
0.55
0.92
 
1.03
1
0.21
0.96
1.12
1
−0.76
1.08
 
1.12
1
−0.76
1.08
 
1.07
1
0.10
0.91
 
1.03
1
−0.51
1.03
 
1.12
1
0.21
0.89
 
1.02
1
−0.08
0.95
1.00
1
−0.37
0.97
 
1.21
1
0.30
0.76
 
1.04
1
−0.07
0.88
 
1.01
1
−0.46
0.99
 
1.08
1
0.03
0.85
 
1.01
1
−0.21
0.92
1.24
1
0.43
0.57
 
1.93
1
1.32
0.15
 
1.01
1
−0.14
0.83
 
1.01
1
−0.17
0.85
 
1.02
1
−0.06
0.80
 
1.00
1
−0.23
0.87
1Selection
on x1/x2 was used as a basis for comparison within scenario. γG = genetic correlation; x1 = trait 1; x2 = trait 2; µ1 = mean of trait 1; µ2
= mean of trait 2; σx21 = variance of trait 1; σx22 variance of trait 2; h2 = heritability.
of “economic weights Va” between x1 and x2. It is improper to have the relative economic weights between
x1 and x2 vary with the functions. Therefore, it is inappropriate to use the partial derivatives of a nonlinear
profit function to derive the economic values. This
study demonstrates 2 alternative approaches to incorporate the relative economic values between x1 and x2
to achieve maximum response in net merit. James
(1982) took the partial derivatives of a1x1/a2x2 to derive the relative weights. This is actually equivalent to
taking the partial derivatives of x1/x2 because a1/a2 is
a constant and can be dropped.
Relative Efficiency of Selection Criteria
Tables 1 and 2 show the relative efficiency of the 4
selection criteria compared. Note that a negative relative efficiency denotes a decline in a ratio value (i.e.,
negative gain in net merit) and a relative efficiency of
1.00 is larger than an integer of 1 (base value set for
selection on the ratio) because of truncation to 2 decimal places. Of 96 scenarios compared (Tables 1 and 2),
index selection is always more efficient than direct selection on the ratio. However, when x1 is moderate to
highly heritable (h12 = 0.3 or 0.5) and has a low correlation with x2 (γG = 0.3) in the case of µ1:µ2 = 1:8, the
advantage of linear index over direct selection on ratio
is negligible (Table 1). Generally, the superiority of linear index over direct selection on ratio is greater at a
high correlation between x1 and x2 than at a low correlation (comparison of Tables 1 and 2) and becomes
more apparent when large difference in h2 between x1
and x2 exists (h12 = 0.1 and h22 = 0.5 or vice versa). This
study proved theoretically that criteria I and R are
equivalent when h12 = h22 , a1 = a2, and γG = γP. Therefore, the advantage of I over R is not large when h12
differs slightly from h22 (e.g., h12 = 0.3 and h22 = 0.5 in
Tables 1 and 2). Similarly, Gunsett (1984) reported
that I and R yielded similar response when x1 and x2
are equally heritable. Davis (1987) found little advantage of I over R. This was because x1 and x2 have similar heritability in his study (0.50 for feed intake vs. 0.45
for weight gain).
Selection on difference (D = x1 − x2) is more efficient
than selection on R when there is a small difference in
means between x1 and x2 (e.g., µ1:µ2 = 1:2) and when
x1 has higher heritability than x2 with a low correlation
(Table 1). The efficiency of selection on D is much lower than selection on R when h12 > h22 or when there is a
large difference in means (e.g., µ1:µ2 = 1:8). In the case
of a low correlation and a small difference between u1
and u2, selection on D is as efficient as a linear index
when h12 = 0.1 and h22 = 0.3 or 0.5 (Table 1). However,
selection on D is less effective than a linear index in
SELECTION FOR RATIO TRAITS
almost all other scenarios. When γG = 0.8 and µ1:µ2 =
1:8, selection on D tends to deteriorate the ratio values
as indicated by negative relative efficiency in Table 2.
Selection on x1 alone is more efficient than selection
on R when h2 is 0.5 for x1 and 0.1 for x2 with γG being
0.3 (Table 1) or when x1 is less heritable than x2 coupled with a higher correlation, a larger difference in
means (µ1:µ2 = 1:8) and a smaller difference in variance (σx21 :σx22 = 1:2 or 1:4; Table 2). In these circumstances, selection on x1 alone is superior to selection on
R not simply because it is more efficient, but most important, because it does not involve the costly measurement of x2 (e.g., individual feed intake). Selection on x1
alone is the least efficient of the 4 selection criteria
compared when h2 is low for x1 and high for x2 and the
ratio of µ1 to µ2 is 1:2 (Tables 1 and 2). This is particularly noticeable as shown by the negative relative
efficiency when genetic correlation is high (Table 2).
However, selection on x1 alone is only slightly less efficient than selection on I or R when x1 has a higher
heritability than x2 with a low correlation of 0.3 and
µ1:µ2 = 1:8 (Table 1). In these cases, selection on x1
alone is a viable alternative to selection on I or R in
terms of improving the ratio because there is no need to
go to the trouble of measuring x2.
Essl (1989) and Famula (1990) reported that correlated response to selection was greater in the denominator trait (x2) than in the numerator trait (x1) in spite
of higher heritability for x1. This study found that their
findings were true in their specific situations, but are
not generally true as the correlated responses in x1 and
x2 vary widely depending upon the combination of h2,
γG, µ1/µ2, σx21 /σx22 , and selection criteria applied. Genetic responses in net merit (ΔH) and component traits
(ΔG1 and ΔG2) to each of the 4 selection criteria in the
96 scenarios were not presented due to a large number
of figures.
It is a statistical artifact that denominator trait (x2)
plays a greater part in changing the value of x1/x2 than
the numerator trait (x1) because 1 unit of decrease in
x2 would improve the value of x1/x2 more than does
1 unit of increase in x1. However, selection against x2
to improve feed efficiency is ineffective and impractical for 3 reasons: 1) feed intake (x2) generally has a
lower heritability than weight gain (x1); 2) it is easier
and more accurate to measure individual weight gain
than to determine individual feed intake; and 3) it is
more desirable economically to improve feed efficiency
through increased weight gain than through reduced
feed intake. For example, 2 animals with respective
feed efficiency ratios of 2/4 and1/2 are equally efficient. However, a fast-growing animal with a ratio of
2/4 is more desirable than a slow-growing animal with
a ratio of 1/2 in terms of investments, management,
and carcass yields. In fact, an animal with a feed efficiency ratio of 2/4 is worth more than 2 animals with
a ratio of 1/2 each.
921
Different Characteristics Between Linear
Index and Direct Selection on Ratio
The construction of the linear index takes into account genetic and phenotypic (co)variances, whereas
direct selection on the ratio x1/x2 considers only phenotypic values. This is the main reason why a linear index
is more efficient than direct selection on ratio. A ratio
of 2 normally distributed dependent variables is not
normally distributed (Fieller, 1932). Both phenotypic
and genetic ratios are not linear and do not have bivariate normal distribution. Theoretical prediction of selection progress (a product of heritability and selection
differential) is valid only if a character has a normal
distribution and the regression of its genetic on phenotypic value is linear. Because ratio traits do not satisfy
these 2 underlying assumptions, the realized response
to selection on ratio does not agree with expected response (Kennedy, 1984; Gunsett, 1987; Campo and Rodriguez, 1990), indicating that the heritability estimate
of a ratio is not accurate for predicting genetic response
in a ratio. In spite of the serious drawbacks associated
with the use of a ratio, animal scientists remain keenly
interested in measuring biological or economic efficiency in terms of ratios. Transformation of a nonlinear
ratio to a linear scale for index construction meets the
assumption of both linearity and normality because a
linear combination of 2 normally distributed variables
is linear and is normally distributed.
Two animals with equal feed efficiency ratios, say 1/2
and 2/4, respectively present a selection problem as to
which one should be selected based on a ratio. This
dilemma does not exist for index selection because
these 2 animals with the same ratio values do not have
the same index values on a linear scale. For example,
when µ1:µ2 = 1:2, σx21 :σx22 = 1:4, h12 = 0.3, h22 = 0.1, and
γG = 0.3 (Table 1), the linear index computed for this
scenario was I = 0.1442x1 − 0.0336x2. The animal with
a ratio of 1/2 (x1 = 1 and x2 = 2) has an index value of
0.077 compared with 0.154 for the other with a ratio of
2/4 (x1 = 2 and x2 = 4). Obviously, index selection will
favor the animal with a ratio of 2/4 over the other with
a ratio of 1/2, although both animals have the same
ratio values. This selection decision by linear index is
correct because an efficiency ratio of 2/4 is preferable
to that of 1/2 as explained in the preceding section. In
contrast, when the economic efficiency is defined as a
ratio of net income to input cost, there would be no
preference between economic efficiencies of 2/4 and
1/2.
Tables 1 and 2 showed that for a given combination
of h2, γG, and σx21 /σx22 , the ratio of the means (µ1/µ2) is
an important factor in determining the relative efficiency of selection criteria. Most studies on genetic improvement of ratio traits failed to recognize this crucial
factor. Because the proportional change between µ1
and µ2 (e.g., change µ1/µ2 = 1/2 to µ1/µ2 = 2/4) or
922
Lin and Aggrey
between σx21 and σx22 (e.g., change σx21 /σx22 from 1/4 to
2/8) does not alter the relative efficiency of the selection criteria, the general procedure presented herein allows for comparing the relative efficiency of different
selection criteria for any desired combination of the parameters without the need of knowing the absolute values of µ1, µ2, σx21 , and σx22 .
Implications
Transformation of a nonlinear ratio x1/x2 to a linear
scale for index construction not only meets the required
assumptions of linearity and normality for the estimation of genetic parameters and the prediction of genetic
responses but also maximizes the genetic responses in
ratio traits. A procedure was developed for incorporating economic weights into the component traits of a
ratio such as feed efficiency. The relative efficiency of
different selection strategies varies widely, depending
upon the combination of heritabilities, genetic correlation, the ratio of the means, and the ratio of phenotypic
variances between the 2 component traits. Experimental or simulation results reported in the literature covered only a small part of the whole picture. This study
provides a general procedure for comparing different
selection strategies for the improvement of a ratio trait,
taking into account the relative economic weights of
the component traits of a ratio for any combination of
genetic and phenotypic parameters.
ACKNOWLEDGMENTS
This work was supported by USDA National Research Initiative grant 2009-35205-05208 and Georgia
Food Industry Partnership grant 10.26KR696-110.
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