In a series of papers, Griffing (1967, 1976, 1981a,b) showed theoretically that social effects on trait values alter response to genetic selection, and demonstrated that response depends strongly on relatedness among interacting individuals and on the level of selection. Results of Griffing, however, do not fit easily in common theoretical frameworks for response to selection, and have largely been overlooked. Griffing (1977) summarized his results into a more common theoretical framework, but because it was published in a proceedings was also largely overlooked. Bijma et al. (2007a) rediscovered those results independently 30 years later, but more importantly addressed the importance of those finds.

#### Response to selection

In livestock genetic improvement, response to genetic selection is commonly expressed as the product of the intensity of selection, *ι*, the accuracy of selection, *ρ*, and the genetic standard deviation, *σ*_{G} (e.g., Bourdon 2000),

- (6)

The selection intensity expresses the selection differential, *S*, in standard deviation units, *t* = S/σ. The accuracy is the correlation between the value of the selection criterion (*SC*) and the additive genetic merit for the trait value within individuals,

- (7)

When a measurement taken of an individual’s phenotype is imprecise and subject to measurement error (replicated measurements on the same individual vary from one another), it reduces the correlation between the *SC* (the value of the trait that determines whether a breeder includes or excludes a specific individual) and genetic merit, the underlying genetic basis of the individual’s phenotype. Furthermore, when environmental effects on the phenotype are large, information from an individual’s relatives can be a better predictor of that individual’s genetic merit than any measurement made on its own phenotype. (The terms ‘accuracy’ and ‘genetic merit’ are not found in standard population genetic theory because of the direct mapping of genotype onto phenotype. They are necessary in animal breeding where the mapping can be much more complicated and must be understood through measurement and experimentation.)

In classical quantitative genetic theory, *σ*_{G} is the additive genetic standard deviation in the trait value (Falconer and Mackay 1996). With IGEs, however, *σ*_{G} has a different interpretation, which will be discussed below. Equation 6 applies to any selection strategy and inheritance model. It equals the first term of Price’s Theorem (Price 1970), and represents the change in trait value due to change in allele frequency, keeping average effects of alleles constant for all elements of the inheritance model as in eqn 1.

Equation 6 nicely separates response into three clearly distinct components; a scale-free measure of the strength of selection, *ι;* a scale-free measure of how accurately the SC resembles an individual’s true genetic merit for the trait, *ρ;* and a measure of the magnitude of the heritable differences in the population that can be utilized by genetic selection, *σ*_{G}.

In the following, we first consider the impact of social interactions on *σ*_{G}, and subsequently the effects of multilevel selection and information from kin on the accuracy of selection. (The intensity of selection depends on the ecology or on the breeding design, and will not be considered any further here.)

#### Trait model and heritable variance with IGE

The inheritance of traits affected by social interactions differs from that of classical traits, because trait values are determined in part by heritable effects originating from other individuals (Willham 1963; Griffing 1967; Moore et al. 1997). As a model, consider a population where social interactions occur within groups, each consisting of *n* individuals (note that above, in the population genetic formulation, we assumed groups were large, unlike here where group size is explicit; see also Wade 1980)*.* In this model, individual trait values are the sum of a direct effect due to the focal individual, and the indirect effects due to each of its *n*-1 group members. Both direct and indirect effects can be decomposed into an additive genetic (*i.e.*, heritable) component, *A*, and a remaining nonheritable component, *E*. The trait value of individual *i*, therefore, equals (Griffing 1967).

- (8)

where *A*_{D,i} is the direct genetic effect (DGE) of focal individual *i*, A_{S ,j} the IGE of each of its group members *j*, and *E*_{D,i} and *E*_{S, j} are the corresponding nonheritable terms. From eqn 8, it follows that response to genetic selection equals the change in mean DGE plus group size minus one times the change in mean IGE (Griffing 1967),

- (9)

Equation 9 indicates that an individual’s total genetic merit, representing its heritable impact on the mean trait value of the population, may be defined as (Bijma et al. 2007a)

- (10)

so that response to selection equals change in mean *G*-value, . The *G*_{i} is the sum of all heritable effects of *i* on trait values of individuals, of which *A*_{D,i} surfaces in the focal individual itself, and (*n*-1)*A*_{S,i} in its group members.

From eqn 10, it follows that the genetic variance determining the potential response to selection in the population equals (Bijma et al. 2007a),

- (11)

This result shows that IGEs alter the genetic variance that determines the potential response to selection in the population. In the absence of IGEs, eqn 11 reduces to , which is the classical result. Equation 11 can be generalized to account for other types of social interactions, such as the combination of social and maternal genetic effects (Bijma 2010; Bouwman et al. in press).

Note that eqn 11 defines genetic variance from a response-to-selection perspective. In other words, the as defined in eqn 11 is the genetic variance that is valid for use in eqn 6. It differs from the genetic component of phenotypic variance, *Var*_{G}(*z*). For example, when interacting individuals are unrelated, it follows from eqn 8 that , but eqn (9) implies that . Because is not a component of phenotypic variance, IGEs create a situation where the genetic variance available for response to selection may exceed the phenotypically observed variance, . Hence, part of the genetic variance is hidden. Thus, in theory, with IGEs a response expressed in phenotypic standard deviation units can be substantially greater than in classical quantitative genetic theory as we saw with heritability exceeding 1 in the first section on empirical results. Eaglen and Bijma (2009) discuss the distinction between and *Var*_{G}(*z*) in the context of maternal genetic effects.

#### The effect of information from kin and multilevel selection on accuracy

As shown above, the follows directly from the inheritance model; it does not depend on the mode of selection. Rather, information from kin and multilevel selection affect the accuracy with which selection at the phenotypic level translates into a change in mean *G*-value. Hence, in terms of eqn 6, they affect the value of *ρ*.

First consider individual selection, where individual fitness is determined entirely by individual trait value, *SC*_{i}* = z*_{i}. In this case, accuracy equals the correlation between an individual’s *G*-value and its trait value, *ρ* = *Corr*(*G, z*). Substitution of eqns 8 and 10 shows that accuracy of individual selection equals

- (12)

where *r* denotes relatedness between interacting individuals, which takes values between 0 and 1. (Note, *r* in eqn 12 is identical to *f* in eqns 2–5; the use of *f* is common in population genetics, whereas the use of *r* is common in quantitative genetics and kin selection theory.)

The numerator of this result shows that accuracy can be partitioned into two components. First, a component due to relatedness, , which represents the proportion of selection acting directly on the *G*-values of individuals, and which is always positive. Second, a component due to the complement of relatedness, , which may take negative values when direct effects and IGEs are negatively correlated. The second component explains why individual selection without kin information can yield a response opposite to the direction of selection, which was first shown theoretically by Griffing (1967), and observed empirically by Craig and Muir (1996). When this component is negative, individual selection alone increases competition so much that the net response becomes negative as we discussed earlier. Equation 12 shows that relatedness among interacting individuals shifts selection away from a potentially negative term toward the genetic variance in trait value. In other words, relatedness causes utilization of the available genetic variance, thereby avoiding response in the ‘wrong’ direction.

Second, consider multilevel selection when group members are unrelated. Only individual and only group selection represent the extremes of a continuous scale of multi-level selection [see Bijma and Wade (2008) or van Dyken (2010) for a treatment of soft-selection in this context]. A selection model allowing for a continuous degree of multilevel selection is given by (Bijma et al. 2007a)

- (13)

where the sum is taken over the *n*-1 group members of the focal individual, and *g* represents the degree of between-group selection. A *g* = 0 yields *SC*_{i}*= z*_{i}, indicating selection solely on individual trait value. A *g* = 1 yields , the sum being taken over all *n* group members including the focal individual, so that all group member have the same *SC*-value and selection occurs fully between groups. Hence, *g* is a measure for the degree of between-group selection, and takes values in the same range as relatedness, *g* [0...1]

Substitution of eqns 10 and 13 into eqn 7 shows that the accuracy of multilevel selection in the absence of kin equals

- (14)

Note that this result is strikingly similar to that for individual selection with related individuals (eqn 12); in the numerator, *r* is replaced by *g*. Hence, accuracy is partitioned into a proportion *g* acting directly on the *G*-values of individuals, and a remaining proportion (1-*g*) acting on a potentially negative term. Hence, the effect of between-group selection on the sign of the accuracy is identical to the effect of relatedness. For example, to obtain positive accuracy while is negative, either requires a certain degree of relatedness, or exactly the same degree of between-group selection.

Third, consider combined selection. Substitution of eqn 13 into eqn 7, and accounting for relatedness among individuals, shows that the accuracy of combined kin and multilevel selection equals

- (15)

The first term of the numerator demonstrates that both relatedness and multilevel selection act directly on the genetic variance in trait value, and thus contribute to a positive accuracy. Moreover, as indicated by the term (*n−2*)*gr*, the positive effects of relatedness and multilevel selection on accuracy amplify each other. Hence, if direct and IGE are negatively correlated, then the combination of kin and multilevel selection is a very powerful way to avoid negative response due to increased competition.

Figure 3 illustrates the relationship between *ρ*, *g* and *r* for a correlation of -0.6 between direct and IGE. In this example, a positive accuracy requires for example *r* > 0.11 with *g* = 0, or *g* > 0.11 with *r* = 0, or a combination of *g* and *r* such as *g* = *r* = 0.05. For *g*, *r* > 0.11, accuracy increases less with multilevel selection than with relatedness, because greater *g* yields greater σ_{SC} which limits the increase in accuracy (eqn 14).

Finally, consider selection based on relatives. In this case, the selection candidates are housed individually and they are selected based on the information of relatives kept in family groups (Ellen et al. 2007). Selection based on phenotypes recorded on relatives is very common in livestock genetic improvement, for example because recording the trait requires sacrificing the individual (e.g., carcass meat yield), or the trait is expressed only in females (e.g., egg number or litter size). In the absence of IGEs, the accuracy of selection based on relatives is commonly expressed in terms of relatedness between the candidate and its relatives, *r*, the square root of heritability, *h*, and the intraclass correlation *t* between the relatives. Also with IGEs, the accuracy of selection based on relatives can be expressed in that way, provided that that the relatives are kept in family groups. With IGEs, the accuracy is given by (Ellen et al. 2007)

- (16)

where η = σ_{G}/σ_{TPV} is an analogy of the square root of heritability, *h = σ*_{A}*/σ*_{P}, τ = *r*_{br}η^{2} is an analogy of the intraclass correlation between relatives t = *r*_{br}h^{2}, and *mn* is the number of relatives in *m* groups consisting of *n* individuals each. The *r* denotes relatedness between the candidate and its relatives, whereas *r*_{br} denotes mutual relatedness among the relatives. For example, when selection male candidates based on phenotypes recorded on their half-sib offspring, then *r* = 0.5 and *r*_{br} = 0.25. The *η* and *τ* account for interactions among individuals, and, therefore, depend on the genetic variance in the population and on the total phenotypic value (TPV) contributed by an individual. The TPV is the phenotypic analogy of the genetic merit of an individual (*G*, defined in eqn 10). The TPV represents an individual’s total phenotypic effect on the population mean, which equals its direct phenotypic effect plus *n*-1 its indirect phenotypic effect. Hence, analogous to eqn 11, . (Further details are in Ellen et al. 2007).

In conclusion, response to genetic selection can be partitioned into the strength of selection, the accuracy of selection, and the genetic variance available in the population (eqn 6). Social interactions alter the genetic variance available for response (eqn 11). Kin and multilevel selection shift the accuracy of selection in a positive direction, and therefore increase the utilization of genetic variance by selection. Finally, the effects of kin and multilevel selection on accuracy are strikingly similar (eqn 15).