Considerations on measures of precision and connectedness in mixed linear models of genetic evaluation

Trois criteres d'appreciation de la connexion et de la precision des evaluations genetiques sont etudies et compares. Le premier critere est la variance d'erreur de prediction (PEV), le second mesure la diminution de la PEV quand les effets fixes sont connus (indice de connexion ou IC), et le troisieme est un critere de precision de l'evaluation, exprime par le coefficient de determination generalise (CD). Ces criteres sont presentes a l'aide d'exemples simples bases sur un modele animal. Ils se distinguent par le choix de la matrice etudiee (CD versus PEV, IC), la prise en compte de la seule structure des donnees (IC versus PEV, CD), la presence d'une matrice ou d'un modele de reference (PEV versus IC, CD), et la maniere de prendre en compte les relations de parente entre animaux (CD versus PEV). On montre comment IC favorise les situations ou l'information apportee par les donnees est faible. Un nouvel indice de connexion, s'attachant egalement a la seule structure des donnees, est propose, palliant cet inconvenient. L'interet d'IC et de CD est etudie sur un exemple de modele « troupeau-pere», ou les troupeaux sont de taille fixee, les peres servent dans un seul troupeau, a l'exception d'un pere de reference assurant les liaisons genetiques entre troupeaux. CD permet d'optimiser le plan d'experience par un compromis entre connexion et information contenue dans les donnees, alors que l'utilisation d'IC aboutit au choix d'un plan ou les peres utilises dans un seul troupeau ont un seul veau par troupeau. Si CD et PEV sont equivalents pour des animaux non apparentes, PEV privilegie les forts apparentements, qui diminuent la variance d'erreur de prediction. Mais les parentes diminuent egalement la variabilite genetique, ce que prend en compte CD. Ainsi, on montre, sur un modele animal strictement aleatoire avec meme apparentement entre animaux, comment PEV peut conduire au choix d'un plan minimisant le progres genetique. On retrouve dans ce cas simple la formule classique du progres genetique, ou le CD generalise joue le meme role que le CD individuel d'un indice de selection. CD, compromis entre structure et quantite de donnees, d'une part, et variance d'erreur de prediction et variabilite genetique, d'autre part, est une methode de choix pour l'analyse de la qualite d'une evaluation genetique.


INTRODUCTION
The problem of precision and especially of disconnectedness in BLUP genetic evaluation, is becoming increasingly important in animal breeding. Since the work of Petersen (1978) and Foulley et al (1984Foulley et al ( , 1990, three papers have addressed this subject , Kennedi and Trus (1993), and Laloe (1993).
In the context of genetic evaluation, disconnectedness is not clearly defined. Sometimes, it is the lack of genetic ties between levels of fixed effects, and other times it is defined as the inestimability of contrasts between levels of genetic effects.
Both definitions are somewhat incoherent, since, as Foulley et al (1992) wrote &dquo;From a theoretical point of view, complete disconnectedness among random effects can never occur&dquo;. These authors introduced the concept of &dquo;level (or degree) of disconnectedness&dquo; by relating the prediction error variance (PEV) of the genetic effects to the PEV under a reduced model excluding the fixed effects. They suggested a global measure of connectedness among levels of a factor. Kennedy and Trus (1993) suggested the PEV of differences in predicted genetic values between candidates for selection as the most appropriate measure of connectedness. Lalo6 (1993) introduced the concept of generalized coefficient of determination (CD), the CD of a linear combination of genetic values, and suggested a new definition of disconnectedness among random effects: a design is disconnected for a random factor if the generalized CD of a contrast between its levels is null. Some global measures of the precision of an evaluation or of a set of evaluated animals were suggested.
The aim of this paper is to compare the three methods, theoretically and with some numerical examples based on animal models and sire models.

MODELS, NOTATION AND CRITERIA
Consider a mixed model with one random factor (and the residual effect) where y is the performance vector of dimension n, b the fixed effect vector, X the pertinent incidence matrix, u the random effect vector, Z the corresponding incidence matrix and e the residual vector.
where A is the numerator relationship matrix, and the scalars U2 and u d are the additive and residual variance components, respectively. BLUP (best linear unbiased predictor) of u, denoted u, is the solution of (Z'MZ + !A-1)u = Z'My, where A = o, e 2/ 0 ,2 a, and M = I -X(X'X)-X' is a projection matrix orthogonal to the vector subspace spanned by the columns of X: MX = 0. The joint distribution of u and u is multivariate normal, with a null expectation and variance matrix equal to The distributions of ul û and u -u are multivariate normal: N(u, C°°U e ) and !V(0,C!&dquo;!), respectively.
The following is a second model: With this random model, ul-ii -N(u, Cuuo,,2) and u -Û rv N(0, C!uO&dquo;;), with r = (Z'M r Z + ÀA -1 ) -1 and M r = I -1(1'1)-1 1', the projection matrix orthogonal to the vector 1. This model can be considered to exhibit the information provided by the data in order to predict genetic values, without any loss due to the estimation of fixed effects, except the mean.

Criteria
Three criteria are proposed to judge the quality of the prediction of a contrast, ie, a linear combination of the breeding values x'u, where x is a vector whose elements sum to 0: -PEV(x) (Kennedy and Trus, 1993). Comparisons between animals that are poorly connected would have higher prediction error than those that are well connected.
This method is denoted PEV.
-IC(x), the connectedness index , ie, the relative decrease in PEV when fixed effects are exactly known or do not exist (reduced model). It varies between 0 and 1, and is close to 1 when the animals are well connected. This method is denoted IC.
-CD(x), the generalized CD (Lalo6, 1993), which corresponds to the square of the correlation between the predicted and the true difference of genetic values. This method is denoted CD.

AN ANIMAL MODEL EXAMPLE
The examples from Kennedy and Trus (1993) are used to illustrate the three measures. Consider an animal model for which there are two management unit effects that are estimated from the data jointly with the genetic values of four animals. All animals have single records. The first two animals (u i and u 2 ) are in unit 1, and the last two (u 3 and u 4 ) are in unit 2. Heritability equals 0.5 and 0 & d q u o ; ; = or = 1 (A = 1). Two cases are considered: (i) the animals are unrelated, and (ii) animals are unrelated within management unit, but each animal has a full sib in the other management unit; ( Ul , U3 ) and ( U2 , U4 ) are full-sib pairs. Obviously, there are no genetic ties between management units in case (i), and the corresponding design is genetically disconnected. Four contrasts between animals are considered: animals within a management unit (u l -U2 ), animals from different management units (u lu 3 and u 2 -u 3 ) and genetic levels of the units (u i + u 2 -u 3 -U4 )-For each contrast, the above three criteria were calculated, and their values are presented in table I. Some comments about these values allow the identification of following problems.
First, IC could not detect any lack of genetic links between units. Its value was 0.5 in case (i) (unrelated animals) for U1 + U2u 3 -u 4 . Kennedy and Trus (1993) showed that PEV could detect lack of genetic links between units by a covariance of 0 between the BLUE (best linear unbiased estimator) of these units.
Second, disconnectedness was detected by CD, which delivered null CD for the unit comparison, whatever the case, ie, even if the units were genetically linked. Here, the design was such that a difference of genetic levels between units could not be predicted: Ul + u 2 -u 3 -u 4 was always null, whatever the data, as proven in Appendix 1. This concept of connectedness is not equivalent to the lack of genetic links between management units, but to the lack of information provided by the data (var(x'ulû) = var(x'u)). However, PEV showed that the genetic levels of the units were more likely to be the same in case (ii) than in case (i), due to the genetic links between units in case (ii): PEV = 4 in case (i) and PEV = 2 in case (ii).
Finally, the two methods (PEV and CD) accounted for relationships between animals in different ways. Genetic links between units increased the CD of U2 -U3 (unrelated animals of different units), 0.45 (case (ii)) vs 0.25 (case (i)), but the CD of u lu 3 (related animals of different units) decreased, 0.17 (case (ii)) vs 0.25 (case (i)). PEV decreased in both cases. This decrease was higher for related animals, 0.83 (case (ii)) vs 1.5 (case (i)) than for unrelated ones, 1.1 (case (ii)) vs 1.5 (case (ii)). The two methods give, therefore, contradictory results. Indeed, the more the animals were related, the lower the genetic variability of their comparison; PEV(x) decreased, but so did x'Ax. The variance of x'u was proportional to x' Ax -APEV(x). If the relative decrease of PEV(x) were smaller than the relative decrease of x'Ax, the variance of x'u would decrease, and hence the probability that high differences between animals could be exhibited by the evaluation. For instance, in case (i) (unrelated animals), PEV(x) = 1.5 and x'Ax = 2, while in case (ii) (related animals), PEV(x) = 0.83 and x'Ax = 1. The decrease of PEV(x) did not compensate for the loss of genetic variability, and CD(x) went from 0.25 (case (i)) to 0.17 (case (ii)).

OVERALL INDICES
The best model was different according to the contrasts; when CD was used, we chose case (ii) for considering the contrasts u l -v, 2 and u 2u 3 , but case (i) was the best for the contrast Ul -U 3 -It could be interesting to extend these procedures, defined here for a specific contrast, to a global measure of precision of an evaluation. An overall criterion could be useful when optimizing a design or comparing the precisions of different evaluations. Such overall criteria are derived on the basis of the means of quadratic ratios. As shown in Appendix 2, the ratio of the quadratic forms x'Bx/x'Cx is related to the generalized eigenvalue problem [B -pjc]cj = 0, and two global means of these ratios of quadratic forms are the geometric and the arithmetic means of the corresponding eigenvalues / t i .

Overall connectedness index
The ratio of quadratic forms here is x'cg!x/x'c!!x. The overall index suggested by Foulley et al (1992) is the geometric mean of the eigenvalues of r Ci u _ !C&dquo;&dquo;]c, = 0 or This index is suggested, using the Kullback information (Kullback, 1983) between the joint density of the maximum likelihood estimator of b and u -u and the product of their marginal densities that would prevail if the design were orthonormal in b and u. All the indices of connectedness (IC and IC(x)) are strictly positive and fi 1. The null value never occurs when dealing with random factors, because the random effects are always estimable and the rank of both matrices equals n (eg, Foulley et al, 1990). An IC(x) equal to 1 demonstrates that x'(u-u) is orthogonal to the fixed effects and, for the global IC, that u -u is orthogonal to the fixed effects.
Application of the overall connectedness index among sires in a reference sire system based on planned artificial inseminations with link bulls has already been undertaken in France (Foulley et al, 1990;Hanocq et al, 1992;Lalo6 et al, 1992).

Criteria of precision
Here, we devote our attention to the CDs of the contrasts between genetic values, which could be summarized in the (n -1) greatest eigenvalues u i of the generalized eigenvalue problem (Lalo!, 1993): Some properties of the solutions, written in ascending order, are briefly given here. The pjs are located between 0 and 1: p 2 K CD(x) ! !n; /-L 1 is always null, and the associated eigenvector c i is proportional to A-1 1; the other eigenvectors correspond to contrasts, since (cf, Appendix 2 [A2.12]): c'Ac i = 0 for i > 1 « l' A -1 A Ci = 0 = 1'c i , ie, the definition of a contrast; CD( C i) = / -Li ' Eigenvalues and eigenvectors for case (ii) are reported in table II. It could be verified that eigenvectors corresponding to a null eigenvalue are respectively C1 , proportional to A-1 1, and c 2 , which corresponds to the genetic level comparison of the units. The other eigenvectors correspond to contrasts. Moreover, any contrast x'u can be written as a linear combination of the c i s (i ranging from 2 to n) (cf, Appendix 2 (A2.15!).
From Appendix 2 [A2.6], the CD of any contrast is a weighted mean of the eigenvalues of !7!: Two overall indices of precision can be computed: These criteria have been used to validate the rule of publication of French beef bull genetic values from field data evaluation (Lalo6 and M6nissier, 1995). PEV Kennedy and Trus (1993) did not suggest any overall criterion of precision. By analogy, use of det(C°u)1!! is suggested.
The values of the different criteria are reported in table III. Null values of p 2 showed that both designs were disconnected. P1 was the same for both cases, as IC and det(C°u)1!! favored the design where animals are related.

CONCEPT OF (DIS)CONNECTEDNESS AND RANDOMNESS OF GENETIC EFFECTS
Disconnectedness, as defined in the linear fixed model context (y = Xb-!-e) (use of a generalized inverse of X'X Q e as the variance matrix of BLUE (b)b, occurrence of non-estimable contrasts, 'all or none' characteristic), never occurs when dealing with a random factor. Var(u -u) = C' u oe 2 is always positive definite. However, AC uu is upwardly bound by A, in the sense that, whatever x, AxC uu x <1 x'Ax.
If the PEV of a contrast x'u reaches the upper bound x'Ax, CD(x) = 0 and: Equation [13] implies that x'u does not follow a normal distribution, but a point-mass distribution at 0: P(x'u = 0) = 1. In that sense, disconnectedness for a random factor is an 'all or none' characteristic concerning the distribution of the predictors in the same way as for a fixed factor. If a fixed factor is disconnected, ie, if a contrast between its levels is not estimable, then the CD of a contrast between its levels is null when it is treated as random. Thus the following definition of disconnectedness for random factors is proposed: a random factor is disconnected when at least one contrast between its levels has a null CD. With this definition, the status of a factor with respect to connectedness does not depend on the fixed or random nature of this factor. Connectedness leads to the same consequences in terms of the decrease of a matrix rank or probability laws in both random and fixed cases. Because IC and PEV deal with C uu instead of A-!C°u, they cannot exhibit this kind of disconnectedness for a random factor. As shown below, IC is devoted to the orthogonality between random and fixed factors and can detect perfectly connected contrasts or designs, but not disconnected ones.

BOUNDARIES AND RELATIVE EVOLUTION OF CRITERIA
Lower boundary of the index of connectedness Since C' u is positive definite, IC(x) is never null and the index of connectedness never reaches the null value. It is interesting to characterize the lower boundary of this index, and how it varies.
Consider a contrast x'u, and denote the generalized coefficient of determination of x'u obtained with model [2] as CD r (x). CD r (x) can be considered as the amount of information provided by data, independent of the design. A formula relating IC(x), CD(x) and CD r (x) could be derived from [4] and [5]: IC(x) has a minimal value when x is disconnected in the complete model [1] (CD(x) = 0) and is equal to 1 -CD r (x), by applying [14]. Thus, the index of connectedness of a disconnected contrast increases as the amount of data decreases, contrary to the assumption of IC accounting only for the design.
The connectedness index of a contrast x'u is then located in the interval [1 -CDr(x),1!. Particularly, when CD r (x) = 0, IC(x) = 1. This case occurs, for instance, when considering a contrast between a sire and a dam known only by their common progeny. Their predicted genetic values will always be equal whatever the performances. Thus, the question of whether there is any assortative mating cannot be answered. IC(x), however, is always equal to 1 and these animals would be declared as perfectly connected and then comparable.
The same kind of result can be found again when working with a design as a whole; consider a nested, balanced 'herd/sire' model, with t progeny per sire, h herds and n different sires per herd. This design is clearly disconnected.
Some values of p l and IC in relation to t are indicated in table IV, where h and n are equal to 5 and 2, respectively. Heritability equals 0.2. Though all these designs are disconnected, IC varies from 0.980 (t = 1) to 0 (t = oo). The greater the amount of data, the lower IC. The design where t = 1 seemed to be very well connected, the index of connectedness can not exhibit any disconnectedness and favors designs with low precision. The variation of this index for similar disconnected situations makes it unreliable for use.
Another index of connectedness is proposed, in order to study the causes of low precision of an evaluation. This low precision could be caused by a lack of information provided by the data or the design structure. It would be interesting to determine the main cause of this low precision. This would allow the precisions obtained in both reduced and complete models to be compared, on the basis of the matrices A-C&dquo;&dquo; and A-C r in order to avoid the above-described drawback of IC. This new index is denoted ø(x) for a contrast x'u and is equal to CD(x)/CD r (x) or to the ratio of quadratic forms x'(A-C°°)x/x'(A-Cr°)x. ø(x) is located between 0 (disconnectedness) and 1 (no impact of the fixed effects), whatever CD r (x). The overall indices of connectedness are: where p lr and p 2r are the overall criteria of precision P1 and p 2 obtained with the reduced model, respectively.
In the above sire model example, </J 2 = 0, revealing again that the design is disconnected. It can be shown in this example that <P 1 = (n -1)h/(nh -1), ie, the proportion of connected contrasts among all the contrasts. It does not depend on the heritability or the amount of information provided by the data, ie, the number of progeny per sire. For the situations reported in table IV, the values of <P 1 and 4> 2 are constant, and equal to 0.556 and 0, respectively, as the value of IC varies from 0 to 0.980.
These new indices obviously have the same limitations as the original one (they only take into account the impact of the fixed effects, orthogonality is favored) and can not be the only criterion used to judge a design. They could be used, however, to see if a low value of a CD is caused by a small amount of data or by a poor design, and also to evaluate the global loss of information due to the design.
Upper boundary of the index of connectedness: complete connectedness Consider a completely connected design, ie, one whose overall index of connectedness is 1. Then, for any x, x' ÀC!ux = X'!C°&dquo;X. Since both matrices are positive definite, Cu' = C uu and, consequently, Z'MZ = Z'M r Z. It can be seen that the condition of complete connectedness is independent of the relationship matrix. This equality characterizes a design where, in a fixed effects model context, u is orthogonal to all other effects (except the mean). This kind of orthogonal design must be complete with proportional frequencies (Coursol, 1980;Mukhopadhyay, 1983).
All the levels of the random factor must then be identically distributed among all levels of all the fixed factors. For instance, for a sire model, the following equality must be satisfied for any sire and any level of factors included in the model: where n oo is the total number of progeny, n 2o the sire i number, n Oj ( k ) the number of the level j of the kth fixed factor, and n ij ( k ) the sire i number in the level j of the kth fixed factor.

Boundaries of the criteria of precision
The CD of a contrast is the square of correlation between x'u and x i i, which varies between 0 and 1. A value of zero indicates that the data does not provide any information about the comparison: var(x'ulû) = var(x'u). The contrast between genetic values cannot be predicted, and there is a disconnectedness, according to Lalo6 definitions (1993). A value of 1 (which is never reached) would indicate that the correlation between predicted and exact values was equal to 1, or that no more information could be obtained from the data. PEV IC and CD measure the discrepancy between the real situation and a reference situation. The values of the index of connectedness and of the criteria of precision are located between 0 and 1. The theoretical interpretation of these values is that the nearer a value is to 1, the better the situation would be. An IC of a contrast equal to 1 demonstrates that there is no influence of the fixed effects on the prediction of this contrast; a CD is the squared correlation between the predictor and the real value; these values are interpretable. However, a value of a PEV alone cannot be interpreted in itself. It must be compared with values of the same contrast in other situations, or with other contrasts. For instance, in case (ii) of the theoretical animal model example where the PEV between individual units was 2, this must be compared to the value of the same PEV in case (i) (PEV = 4), or the covariance between units must be considered. AN OPTIMIZATION PROBLEM Consider a model including the fixed effect 'herd' and a random effect 'sire'. The number of observations N is the same per herd (here, N = 60). There are two natural service intraherd sires (t observations per sire) and a reference sire (m observations per herd and sire) used in each herd, as shown in table V; N = 2 t+m.
The sires are not related and heritability equals 0.2. The problem is how to choose m and t in order to obtain the most precise genetic values of the ten natural service sires. In that context, where animals are unrelated, PEV and CD are equivalent. If normed contrasts x'u (such as x'x = 1) are considered, without loss of generality, the following results: An increase of CD then corresponds to a decrease of PEV, and the use of both methods leads to the same results. For this reason, we used IC and CD. IC, 0 1 , <P2, p l and p z were computed for the set of the ten natural service sires, and IC(x), O(x) and CD(x) were computed for a contrast between genetic levels of two herds, and with respect to different values of t and m. These results are given in table VI.
Criteria of connectedness IC, 0 1 and IC(x) increase with m, starting from strictly positive values and reaching their maximum value near 1 when m = 58 (table VI). <jJ 2 and !(x) also increase with m, but start from 0 when m = 0, exhibiting a disconnectedness, to a maximum value near 1 when m = 58. All these criteria favor the less incomplete design, which is also the design where the natural service sires have only one progeny. Whatever the criteria used, studying only the structure of the design was insufficient to judge the precision of an evaluation or to optimize a design.

Criteria of precision
Criteria of precision range from 0 when m = 0, exhibiting a disconnectedness, to several maxima (m = 20 for p 2 , m = 16 for P1 and m = 30 for CD(x)). It was not surprising that the maxima were different depending to the criteria because p 2 is more sensitive to a poor connectedness than p l , and reached its maximum value for a more connected design than p i . The contrast of genetic levels between herds was the less connected one, and it was most precise for a greater value of m. The values of the criteria then decreased; the enhancement of connectedness no longer compensated for the loss of information provided by the data. Unlike the indices of connectedness, the use of criteria of precision led to optima that were compromises between information from the data and the structure of the design.
Consider the contrast between genetic levels of two different herds, CD(x) _ 0.180 in two cases: -for t = 25, cjJ(x) = 0.317 (IC(x) = 0.527), with a poor level of connectedness, about two-thirds of the information is lost, due to the design structure: -for t = 5, !(x) = 0.863 (IC(x) = 0.986), the restrictive factor here is the amount of information that can be obtained from the data.
This conclusion is obvious without using these criteria on simple designs, but the interpretation of the indices needs to be as clear as possible when dealing with more complicated ones. LINKS BETWEEN IC, PEV, CD AND EXPECTED GENETIC PROGRESS Maximization of IC and genetic progress Hanocq et al (1966) showed in a simulation study that a high level of connectedness only slightly increases the genetic trend. In the extreme, if the factor 'year' is included in the model and the corresponding design is completely connected (IC = 1), all the sires must be used the same way in all the years (equation !17!).
Such a design surely cannot lead to any genetic progress, since animals born in different years would be bred from exactly the same sires in the same proportions.
Behavior of PEV and CD on a hypothetical animal model where animals are equally related As noted before, CD and IC are equivalent when dealing with contrasts involving unrelated animals, but they account for relationships differently. It would be interesting to see what the differences are when one method is compared to the other, particularly with respect to the genetic progress. Indeed, Kennedy and Trus (1993) wrote &dquo;... minimization of PEV does not necessarily maximize rate of genetic improvement because it may come at a cost of reduced selection intensity associated with selection among related as opposed to unrelated individuals&dquo;. We will use a hypothetical and unrealistic model to study the behavior of both indices according to the relationships between animals.
For a 'mean + animal' model, where the animals are equicorrelated with a relationship coefficient r, and the number n of animals is large, we have (cf, Here, PEV(x) and p l vary in exactly the same way according to r. To optimize the design with PEV (minimization of PEV(x)) or with CD (maximization of p i ) leads to a maximal r or a null r, respectively. The expression of the expected genetic progress is (cf, Appendix 3): where i P T = ip(l-r )°. 5 can be viewed as the reduced selection intensity associated with selection among related animals (Kennedy and Trus, 1993), and p i is the global criterion of precision. This expression is similar to the expression of the expected genetic progress in the case of a classical selection index and made on a large population of unrelated animals: where ip is the selection intensity and CD the coefficient of determination of the animal selection index. p i plays the same role in [21] as CD in (22]. The increase of r induces a decrease of R, initially because of the decrease in the selection intensity, as noted by Kennedy and Trus (1993), and secondly because of a decrease in the precision p i . At the same time, the PEV decreases. In this situation, PEV and genetic progress are in conflict.

CONCLUSION
Methods PEV and CD answer different questions. If the predicted value of a contrast is null, PEV allows the appreciation of the likelihood of this result. The probability that x'ul x' û =ü will be near 0 increases as PEV(x) decreases, because xlulx,!!=0 -N(O, PEV(x)). The CD permits the determination of whether the predicted value will be different from 0. In general terms, the probability that x'u will be different from 0 increases with CD(x), because x'f -N(O, CD(x)x'Axo, a 2 ).
PEV is more related to the likelihood of the hypothesis 'all the animals are equal', and CD could be linked to the power of the test 'are the animals different?'.
This distinction is very important, since the main aim of genetic evaluation is to discriminate between animals on the basis of their predicted genetic values, in order to select the best ones.
While both methods are equivalent when animals are unrelated, they can, however, be in conflict in other situations. Genetic relationships decrease the PEV, and also decrease the selection intensity and the genetic variability. PEV is minimized when var(x'ul x'û ) is a minimum, and CD is minimized when var(xulx ' û)jvar(x'u) is a minimum. PEV then favors contrasts between related animals, where var(x'ul x' û) is small, as CD accounts for the decrease of var(x'u). CD combines both aspects, genetic variability and PEV, and is therefore more related to genetic progress, as shown in the theoretical example in the previous section.
The problem of (dis)connectedness is formulated differently according to a priori knowledge about differences between the evaluated populations or genetic levels of management units. First, if the differences are known or supposed to be, can they be exhibited in the evaluation? This question can be answered by CD. Second, a priori, there are no differences. Disconnectedness is then only a source of a decrease of precision, and its study has no inherent interest. Its study may permit the choice of a strategy for precision increase, either by connectedness increase or by an increase of the amount of information provided by data. IC is not very appropriate to this kind of study, mainly because it does not always exhibit disconnectedness and because it decreases with the amount of information obtained from data. Large values of this index could be due either to a good connectedness or to poor information. Another index, devoted to the design structure and independent of the information obtained from the data, was suggested to minimize this drawback. To look only at the data structure is not sufficient. An orthogonal design could not lead to any genetic progress. A genetic evaluation must be precise and discriminatory. CD, which combines data structure and amount of information and also accounts for both PEV and genetic variability, is a good method to select for judging the precision of a genetic evaluation or to optimize corresponding designs.