Power for mapping quantitative trait loci in crosses between outbred lines in pigs

Power for mapping quantitative trait loci using crosses between segregat- ing populations was studied in pigs. Crossing generates gametic disequilibrium and increases heterozygosity. The condition for a heterozygosity among F individuals to be greater than in either line at crossing is that allele frequency should be lower than 1/2 in one line and higher than 1/2 in the other line. Maximum expected power and expected risk were used to compare hierarchical backcross (each boar mated to several sows; contrast within boars), hierarchical intercross (each boar mated to several sows; contrast within both boars and sows) and traditional intercross (each boar mated to one sow; contrast within both boars and sows). The use of hierarchical designs (back- cross and intercross) increased power for traits with low or intermediate heritabilities. For small QTL effects and low heritabilities the hierarchical backcross design gave the highest expected power but also the highest risk. There is not a general design which allocates resources in an optimum fashion across situations (heritabilities, QTL ef- fect, heterozygosity). A compromise between designs with high power and low risk is suggested. A hierarchical backcross design of 400 piglets and heterozygosity 0.68 requires between four (maximum expected power) and eight boars (minimum risk) to detect a QTL of 0.5 phenotypic standard deviations. Selection of extreme individuals in parental lines increased power up to 21 %. Commercial crosses are proposed as an alternative to experiments for QTL mapping. @ Inra/Elsevier, Paris gene mapping / quantitative trait locus / statistical power / hierarchical design / pig

1/2 in one line and higher than 1/2 in the other line. Maximum expected power and expected risk were used to compare hierarchical backcross (each boar mated to several sows; contrast within boars), hierarchical intercross (each boar mated to several sows; contrast within both boars and sows) and traditional intercross (each boar mated to one sow; contrast within both boars and sows). The use of hierarchical designs (backcross and intercross) increased power for traits with low or intermediate heritabilities.
For small QTL effects and low heritabilities the hierarchical backcross design gave the highest expected power but also the highest risk. There is not a general design which allocates resources in an optimum fashion across situations (heritabilities, QTL effect, heterozygosity). A compromise between designs with high power and low risk is suggested. A hierarchical backcross design of 400 piglets and heterozygosity 0.68 requires between four (maximum expected power) and eight boars (minimum risk) to detect a QTL of 0.5 phenotypic standard deviations. Selection of extreme individuals in parental lines increased power up to 21 %. Commercial crosses are proposed as an alternative to experiments for QTL mapping. @ Inra/Elsevier, Paris gene mapping / quantitative trait locus / statistical power / hierarchical design / pig Résumé -Puissance de détection des loci à effets quantitatifs dans les croisements entre lignées non consanguines chez le porc. On a étudié chez le porc la puissance de détection des loci à effets quantitatifs à partir de croisements entre populations en ségrégation. Le niveau d'hétérozygotie chez les individus F I est supérieur à celui de

INTRODUCTION
Crosses between inbred lines and analyses within outbred lines have been proposed for quantitative trait loci (QTL) detection. In the first approach, inbred lines are assumed to be fully homozygous for alternative alleles at both marker and QTL. Consequently, F l individuals are fully heterozygous and gametic disequilibrium is maximum. High heterozygosity allows high segregation and, therefore, high power. With maximum gametic disequilibrium, a specific allele at the marker is associated with a specific allele at the QTL in all families. Consequently, increased power is achieved because a lower number of contrasts is needed using across family analyses based on a larger number of observations per contrast.
In the second approach, analyses within outbred lines, both marker and QTL are segregating. Power in crosses between inbred lines is much higher than in analyses within outbred lines. A combination of both approaches has assumed that the lines are segregating at the marker but fixed for alternative alleles at the QTL [1, 2!. Another alternative, not explored yet, is to consider parental lines with different allele frequencies at the QTL because of different selection history.
Conventional types of crosses between inbred lines are intercross and backcross. Power of an intercross versus a backcross design is increased because the number of segregating rneioses in an intercross is twice as many as in a backcross. Experiments conducted in pigs have restrictions in family structure to a maximum of approximately ten piglets per litter. A small number of full-sibs may result in low power !9!. A hierarchical structure, such as a few boars mated to several sows each, allows larger subgroups of progeny inheriting alternative alleles from each boar, which may increase power. It has been proposed to use selective genotyping to increase power for QTL detection [10,11]. Selective genotyping is efficient when the cost of growing progeny is less than the cost of genotyping. Another use of selection, not considered yet, would be to use extreme individuals for the quantitative trait as parents to produce the F l . This approach increases the frequency of heterozygotes in the F l and, therefore, statistical power.
The objectives of this paper are: 1) to investigate the degree of disequilibrium generated when crossing two populations segregating at two loci; 2) to investigate under which conditions the heterozygosity (and consequently power) in the cross of two lines segregating at a QTL is higher than within either line; 3) to investigate power using hierarchical designs; 4) to investigate optimum allocation of resources for a given experiment size; and 5) to investigate the effect of selection of extreme individuals in the parental lines on heterozygosity and power.

Underlying genetic model and assumptions
The purpose of the experimental crossing is the mapping of QTL by the use of genetic markers such as microsatellites. It was assumed that recombination events did not occur between marker and QTL and that all offspring were informative for the marker. These simplifying assumptions were made to reduce the number of parameters to be shown in the results. The impact of these assumptions in practical scenarios is addressed in the discussion.
An underlying mixed inheritance additive model was assumed with an additive biallelic locus segregating at frequencies of the favourable allele p B and pc for lines B and C, respectively. The QTL was assumed to be in Hardy-Weinberg equilibrium and in gametic phase equilibrium with the polygene in each parental line. The variance attributable to the QTL in units of the residual phenotypic variance in line i was: vb(i) = 2p i (1 -Pi)a2 /(u! + a E ), where p i is the frequency of the favourable allele in line i, cr is the QTL effect, and (a A 2 + 0 ,2 E is the residual phenotypic variance with 0 ,2 A and OF2 E being the additive variance attributable to the polygene component and the environmental variance, respectively. The heritability comprising QTL and polygene variation in parental line i was: where h! is the residual heritability attributable to the polygene component, and has value a A 2 /(U 2 A + 0,2 E ). Parameters h! and V 2 Q are usually not known and only estimates of h 2 (i) are available for some traits. Different heritabilities in the two parental populations were not assumed because of the magnitude of the sampling variance of the estimates of heritability and because parental populations may have been raised under different environmental conditions. For simplicity, power computation in this paper was carried out for a given QTL effect and constant residual heritabilities among F l individuals. Therefore, results are valid for a variety of situations with respect to the parental populations segregating at a QTL, not only at a different frequency, but also with a different heritability (comprising QTL and polygene). The highest possible heritability (QTL and polygene) in the parental populations (denoted as h 2 (max)) leading to a given heterozygosity at the QTL among F l individuals was computed as an indication of the values of heritability in practical situations. The impact of this assumption on power is addressed in the Discussion.
Three levels of heterozygosity in the F l were considered: 1, 0.68 and 0.32. A heterozygosity of 1 occurs when the QTL is fixed for alternative alleles. A heterozygosity of 0.68 may occur for a range of situations in parental populations (e.g. PB = 0.8; pc = 0.2 and p B = 0.68; pc = 0). A heterozygosity of 0.32 represents the situation where the two lines segregate at the same frequency (e.g. p B = 0.8; pc = 0.8) or when one allele is absent from one population and segregating at a frequency lower than 0.5 in the other population (e.g. p B = 0.32; p c = 0.0). The advantages of crossing experiments aimed at QTL mapping are: 1) linkage disequilibrium between QTL and marker alleles in the F l ; and 2) increased heterozygosity among F l individuals.

Gametic disequilibrium in the cross of two outbred lines
If the two populations are fixed for alternative alleles at both marker and QTL, then gametic disequilibrium is maximum in the F l . Therefore, power is increased with respect to within-family analysis. If the two parental populations are segregating at the same frequency, then the resulting F l population is in linkage equilibrium and no benefits are expected from crossing. If the genes are segregating at different frequencies in the parental populations then some degree of disequilibrium will be generated. For a more general description of this problem, gametic disequilibrium will be computed at two loci, M and N, that can be either two markers or one marker and one QTL. Gametic disequilibrium between markers can be estimated if estimates of allele frequencies in the parental lines at crossing are available. Let f 6M and fb,! be vectors of allele frequencies at locus M (with k alleles) and at locus N (with g alleles) in parental population B. Similarly, f eM and f!,V are the corresponding vectors of allele frequencies at loci M and N in parental line C. Assuming that the two loci are in linkage equilibrium within the line then the matrix with gametic frequencies at each pair of alleles for line B is: The matrix of gametic frequencies in line C is F eMjV = f 'M f!!,. The matrix with gametic frequencies for each pair of alleles at loci M and N in the F l of the cross between lines B and C is given by: F beM N = 1/2(F bMN + F EMN )- The allele frequencies for loci M and N in the F l cross are fbcm ! F bcMN 1 1 and f6!N = 1' F beMN , respectively. In these equations 1 represents the unit vector. The gametic disequilibrium matrix is then obtained simply by D  Therefore, S2 MN measures the general degree of association between alleles at loci M and N. The value of 52,,,1,! ranges between 0 and 1. For example, Q MN is 1 when lines at crossing are fixed for alternative alleles and disequilibrium is maximum and Q MN is 0 when the lines at crossing are segregating at the same frequency for each allele and disequilibrium is null.

Heterozygosity in the cross of two outbred lines
Consider two outbred lines B and C segregating at a biallelic QTL with alleles Q and q. Assuming random mating, the frequency of heterozygous individuals at the QTL among the F l individuals is: It is shown in Appendix 1 that the necessary condition for increased heterozygosity with respect to either line at crossing is that one line is segregating at a frequency of the favourable allele greater than 1/2 and the other line at an allele frequency lower than 1/2.

QTL mapping designs for the cross between two outbred lines
Power computation was carried out assuming that allele frequencies at the marker were not very different in the two parental populations, and consequently, gametic disequilibrium was small and ignored. However, the increased heterozygosity in the F 1 boars can be used to study segregation, within families, of a QTL associated to a marker by recording performance and inheritance of alternative marker alleles in the next generation (backcross or intercross depending on the mating of F l boars to one of the parental lines or F l sows, respectively). Therefore, the approach that can be followed is the same as in QTL mapping within outbred lines but with an increased heterozygosity among F l individuals. Three alternative designs are considered in this section: hierarchical backcross design, traditional intercross design and hierarchical intercross design. Contrasts in hierarchical designs can use large subgroups of progeny of a mixture of half-and full-sibs within boar.

Hierarchical backcross design
In a hierarchical backcross design, b boars from the F l are mated to s sows each to produce p piglets per litter. Sows can be from any parental line. A statistical model to analyse the data could be: where y2!kc is the lth observation of phenotype on a piglet with marker allele k inherited from boar i when mated to sow j, bo i is the fixed effect of boar i, so2! is the fixed effect of sow j mated to boar i, m ijk is the fixed effect of marker allele k inherited from boar i, and e ijkl is the residual random error. Model (2) is a three level hierarchical design in which marker alleles are nested within sows, which in turn are nested within boar. For simplicity, sows and boars were assumed fixed. It is not strictly correct because the model does not account for relationships between animals.
Power for model (2) was computed following the x 2 approach of Geldermann [5] and Weller et al. [14] comparing the square of the difference between the two progeny subgroups inheriting alternative alleles from their boar (SDP) for each boar family to the expected squared difference under the null hypothesis. Therefore, the alternative hypothesis is a QTL linked to the marker and the null hypothesis is the absence of a QTL linked to the marker. For a more detailed description of the method and discussion of the assumptions, see Weller et al. !14!. Briefly, the assumptions are complete linkage between a fully informative marker and a QTL, and the analyses are carried out within families. For simplicity, it is assumed that there is no common environmental variance and litter size is fixed at 10.
The statistical tests to reject or to accept the null hypothesis are based on the distribution of the SDP. The summed SDP values divided by the squared standard error of the contrast (SE 2 ) follow a central k 2 distribution with b degrees of freedom under the null hypothesis for a large sample size or when phenotypic variance is known. For the alternative hypothesis E(SDP/SE 2 ) N x 2 (nc, b), where nc is the non-centrality parameter of a noncentral x2 distribution with b degrees of freedom. The value of the noncentrality parameter is nc = b he BC 0!/SE!, where 0 = a + 6(l -2 p B ) and 0 = c! + 6(1 -2p c ), for backcrosses with lines B and C, respectively (Appendix 2), a is half the difference between the two alternative homozygotes, 6 is the dominance deviation, and SE! = 4 [(1 + (1/4)shrl/2h!)/(sp)] [equation (A3) in Appendix 3!.
Following Weller et al. [14], the power to detect a segregating QTL was computed as 1 -! = 1 ! p[x 2 (nc, b) < T], where 1 3 is the probability of committing a type 2 error, [p(x 2 (nc, b) < T] is the probability of a x 2 value under the alternative hypothesis (non-central X 2 with parameter nc and b degrees of freedom) less than T, with T being the value of the central x 2 (b) for a given significance level of committing a type 1 error.

Traditional intercross design
In the intercross design, inbred lines B and C are crossed to produce the F l boars and sows that are intercrossed among themselves. In the traditional intercross design b boars are mated to one sow each. A linear model allowing testing for the marker-(aTL effects is: where Y ij k is the hth observation on piglet k with marker genotype j inherited in family i, f z is the fixed effect of family i, m ij is the fixed effect of marker-QTL genotype j inherited in family i ( j = 1 to 3; homozygous for either allele and heterozygous) and e ijk is the residual error. Model (3) would be the one to use in the analysis of real data. However, to take advantage of the X 2 approach of Geldermann [5] and Weller et al. [14] for power calculation, the following linear model can be used: where !z!!l is the lth phenotypic observation of piglet inheriting marker allele k from the sow and allele j from the boar, mb ij is the fixed effect of marker allele j inherited from the boar, m 8 i k is the fixed effect of marker allele k inherited from the sow, and ei!xl is the residual error.
Model (4) is linearly equivalent to model (3) under the assumption of a fully additive underlying genetic model and no sexual imprinting. The distribution under the alternative hypothesis of E(SDP/SE 2 ) is approximately x z (nc, b) and X 2 (nc, s) for boars and sows, respectively. Assuming that the hypothesis being tested is the same either within boars or within sows, the distribution of the sum of '¿,(SDP /SE 2 ) for both boars and sows follows a x z (2nc, b+s) under the alternative hypothesis. The sum of variables having non-central x z distributions jointly independent also follows a non-central X2 distribution with degrees of freedom equal to the sum of the degrees of freedom of the former non-central X2 2 and non-centrality parameter equal to the sum of the non-centrality parameters of the former variables having non-central x z . The non-centrality parameter of the x z has a value 2nc = (b + s) he BC (p /SE 2 , where 0 = a + 6(l -PB -p C ), Similarly to a hierarchical backcross design, power to detect a segregating QTL was computed as 1 -(3 = 1 -p[x2(2nc, 2b) < T], where /3 is the probability of committing a type 2 error, !p(2(2nc, 2b) < T] is the probability ofax 2 value under the alternative hypothesis (non-central X2 with parameter 2nc and 2b degrees of freedom) less than T, with T being the value of the central X2 (2b) for a given significance level of committing a type 1 error.

Hierarchical intercross design
In a hierarchical intercross design, b boars from the F l are mated to s sows each from the F l to produce p piglets per litter. Power calculation can be carried out using the linear model: where Y ij klm is the mth observation of phenotype on a piglet with marker allele k inherited from boar i mated to sow j, bo i is the fixed effect of boar i, so ij is the fixed effect of sow j mated to boar i, mb ik is the fixed effect of marker allele k inherited from boar i, ms ijl is the fixed effect of marker allele I inherited from sow ij and e ijklm is the residual error. Power calculation using model (5)  Assuming independence, the distribution of the sum of E(SDP/SE 2 ) for both boars and sows follows a x 2 (nc b +nc.&dquo; b(1+s)) under the alternative hypothesis. Under the null hypothesis, the distribution of the sum of E(SDP/SE 2 ) for both boars and sows follows a central X 2 (b( + s)). Power to detect a segregating QTL was computed as 1 -(3 = 1 ! p!x2(ncb + nc s , b(1 + s)) < T], where /3 is the probability of committing a type 2 error, (p(x 2 (nc h + nc,, b(1 + s)) < T] is the probability of a x 2 value under the alternative hypothesis (non-central x 2 with parameter nci, + nc s and b(1 + s) degrees of freedom) less than T, with T being the value of the central x 2 (b(1 + s)) for a given significance level of committing a type 1 error.

Design of experiments using crosses between outbred lines
Most experimental costs are in raising, genotyping and recording of a given number of slaughter pigs in the F 2 or backcross. Therefore, the parameters to be chosen by the researcher are the number of boars, the number of sows and the number of piglets per sow. In practice, it is convenient for the handling of the experiment to fix the number of piglets per sow. It is also economical since the use of few piglets per sow would increase the cost in raising a large number of sows. It will be assumed from now on that the number of piglets per litter in the experiment is fixed at ten and the question is how to make optimal use of different numbers of boars and sows in the F l for a given experiment size (total number of slaughter pigs).
Parameters of interest are the expected power (EP) and its standard deviation (SD): where pr(i) is the probability according to the binomial distribution of having i heterozygous boars [assuming a frequency of heterozygous boars given by equation (1)], and P i is the power with i boars computed according to the previous sections. Expected power is utilized for each design to account for random sampling of the boars. SD can be used as a measurement of the variation in power. Comparison of alternative crossing designs can be done by comparing expected power and its standard deviation at a fixed experiment size. The approach used in this paper for optimum allocation of resources is based on the repeated computation of expected power for all possible combinations of the number of boars and sows at a given experiment size assuming constant litter size of ten piglets. The power of the mating structure having the highest expected power will be called maximum expected power (MEP). The functions CINV and PROBCHI of SAS [12] were used to compute central and non-central x probabilities, respectively. Another parameter of interest in planning experiments is expected risk (ER) which we define as the probability of having power lower than 0.5 due to sampling among F l individuals in the segregating population: 2.6. Selection in parental lines to increase power for QTL mapping Selection of high ranking individuals in one parental line and of low ranking individuals in another parental line can be used to increase statistical power for the experiment when the lines are not fixed for alternative alleles. Assume a biallelic QTL segregating in the two parental lines. Consider a hierarchical backcross design in which high and low ranking individuals selected in the parental lines are randomly mated to produce F l , which are again randomly mated to one of the unselected parental lines. Two effects would occur in this scheme: 1) an increase in the frequency of heterozygous individuals in the F r which would increase the power to detect QTL; and 2) an increase in genetic variance of the backcross offspring due to linkage disequilibrium [3], which would decrease power to detect QTL.
2.6.1. Frequency of heterozygotes among offspring of high and low ranking parents Computation of the frequency of heterozygous F i individuals was carried out by using integrals simultaneously of the three normal distributions corresponding to the three genotypes at the QTL. The selected proportion (p, 5 ) for the high line is related to the proportion selected among individuals with genotypes QQ, Qq and qq (p QQ , poq and pqq) by the equation: with f(xIG i ) being the normal density given genotype (Gi = QQ, Qq or qq). Values p QQ , PQ q, and P qq can be found for a unique truncation point, t, analytically. In the examples considered in this paper Simpson's rule [13] was used for increasing values of the abscissa until a value of t was found satisfying the above equation for a given p s . Computation of expected genotype frequencies among high ranking individuals after selection in the high line was carried out by The same procedure was followed to obtain expected genotype frequencies among low ranking individuals in the low line (pQ Q , L , PQ q L' p 9 9 , L ) using the same proportion selected, p s , as for the high line. Computing the expected frequency of heterozygous F l offspring resulting from the cross of selected parents from each line was carried out with equation (3) of Gomez-Raya and Gibson [6]. It assumes random mating and utilizes the frequencies among extreme individuals from each of the two parental lines.

Increased variance among the offspring of F l individuals
The effect of selection of high and low ranking individuals on genetic variance and on the estimation of heritability is discussed by Bulmer [4] and Gomez-Raya et al. !7!. Briefly, if selection of high and low ranking individuals is carried out in the same population then genetic variance increases in the selected group of parents QAS = (1 + !s!)o'!; where QA is the genetic variance for the polygenic component in the unselected population, k, 5 = [z 4>(z)]/ps, z is the absolute value of standard normal deviates at cutoff points, 4>( z) is the ordinate and r is accuracy of evaluation. The above equation can also be used when crossing high ranking individuals from one parental line with low ranking individuals from the other parental line as long as the parental populations have the same genetic variance and the same polygenic heritability and allowance is made for the different selection intensities for each genotype at the QTL and for each line. For simplicity, an approximation to account for different selection pressures across genotypes was made by weighting the values of z and 4 >(z) according to the proportion selected from each genotype and line in the computation of k 8 by where zi&dquo; and 4>(z;&dquo;) are the absolute values of standard normal deviate and ordinate at truncation point for genotype i and line m (i had values 1, 2 and 3 for genotypes QQ, Qq and qq, respectively).
An exact computation would be feasible by using all possible groups resulting in the F l offspring and by including the variation of the means corresponding to the different groups. This variation is small, particularly for QTL of small effect, and the approach used accounts for the increased variance among offspring from selected parents. Following Bulmer (3!, the genetic variance in the F l generation is 0' A (Fl) 2 (1 + (1/2)ksr2)u!. The disequilibrium is halved in the next generation since parents in the F l are chosen randomly. The genotypic variance among backcross offspring from randomly mating F l individuals to one of the parental lines (unselected) is: The heritability among backcross offspring is also increased by selection of high and low ranking individuals in the parental lines: Power calculation was carried out as described in the corresponding section but using the increased frequency of heterozygotes and heritability as computed above. The estimates of the gene effect are biased if selection of parents is used.
Correction for selection bias in the estimates of the gene effect can be achieved after accounting for the increased variance among backcross offspring by where a * is the estimate of the gene effect from the experiment using selection, and a is the estimate of the gene effect after correction for selection bias. Both & * and a are in phenotypic standard deviations units. Table I shows the gametic disequilibrium parameter for alternative number of alleles segregating in the parental populations. The disequilibrium is high when alleles at each of the two loci are fixed in one population but they are rare in the other population. If the alleles at the two loci are segregating at a similar frequency in the two populations then disequilibrium is small. In this situation, the analysis could be performed within families allowing for different alleles at the marker to be associated with the same allele at the QTL in different families.

RESULTS
Maximum expected power and its standard deviation for three different crossing designs in a variety of situations (QTL effect, residual heritability, experiment size) for heterozygosity 1, 0.68 and 0.32, are given in tables II, 111 and IV, respectively. The heritability including QTL and polygenic variation (h 2 (max)) of the parental population having the highest possible value is given in these tables. It can be observed that residual heritability has a small effect on power for the range of heritabilities in the parental populations. Therefore, power figures can be considered as a good approximation when accurate estimates of heritability in the parental population are not available.
On the other hand, a reduction in the frequency of heterozygotes among F r individuals diminished the power for any situation considered. For example, for residual heritability 0.2 and QTL effect 0.5 in an experiment with 200 piglets in a hierarchical backcross design, the maximum expected power with a significance level of 0.05 fell from 0.85 to 0.31 for heterozygosity 1 and 0.32, respectively. Power using hierarchical designs (backcross and intercross) increases in all cases with the exception of traits with high heritability when compared to traditional intercross designs (tables 77-7 V). Maximum expected power using hierarchical backcross designs is larger than using hierarchical intercross designs when the QTL effect is small. The opposite occurs for large QTL effects.
The use of a larger experiment size increases power in all cases. For example, the average maximum power across QTL size and residual heritability (table II) for heterozygosity 1 in hierarchical backcross designs is 0.60, 0.79 and 0.91 for experiment sizes 200, 400 and 800, respectively. Similarly, for hierarchical intercross designs, average maximum power is 0.58, 0.71 and 0.81 for experiment sizes of 200, 400 and 800, respectively. Variation in power due to the sampling in the F l should also be considered. Hierarchical backcross designs show a much larger variation than either intercross designs (tables III and II!. An example of how power, standard deviation of power and expected risk change for alternative mating designs is given in table V. In this example, heterozygosity is 0.68, residual heritability of the trait is 0.2 and QTL effect is 0.5 phenotypic standard deviations with experiment size of 400 piglets. The resulting expected power utilizing a hierarchical backcross design is maximum when using two boars and 20 sows (0.79). The corresponding standard deviation of power and expected risk are 0.27 and 10 %, respectively. A compromise between expected power and risk can be taken. For example, the use of five boars has a low expected risk of 4 % with an expected power of 0.74. It is more obvious with a hierarchical intercross design where expected power using between one and ten boars is very similar (0.81 to 0.83).
However, the expected risk is only low when using four or more boars.
Hierarchical and traditional intercross designs gave identical results at heritability 0.5 (tables 11-I!. This is because traditional intercross is an extreme case of hierarchical intercross (one sow per boar) which is the best allocation of resources at high heritabilities. Table VI shows the number of boars, expected power and risk for designs having maximum expected power and minimum risk in hierarchical intercross and backcross designs. The residual heritabilities are 0.10, 0.20 and 0.50 and the QTL size ranges from 0.3 to 0.6 phenotypic standard deviations in an experiment of 400 piglets with F z heterozygosity of 0.68. It can be observed that the numbers of boars for maximum expected power and minimum risk are different in many instances but the values are not very far apart. It should also be noticed that the optimal allocation of resources in the experiment varies with the QTL effect. A small number of boars is required for small QTL effects which in turn would also have high risk.
Heterozygosity and power using phenotypic selection of extreme individuals in a hierarchical backcross design with 400 piglets, residual heritability 0.20 and varying percentage selected are depicted in figures 1 and 2, respectively. Parental lines were segregating at frequencies of 0.8 (selected to increase) and 0.2 (selected to decrease). A percentage selected of 100 is given for comparison with the unselected case. Percentage selected of 0.5 % was the lowest attempted. Heterozygosity increases with increasing selection pressure in each line and with the size of the QTL. Following the same pattern, power increases with decreasing percentage selected being up to 21 % more ( figure 2). The increase is small if the power in the unselected situation is close to 0 or 1.

DISCUSSION
Most previous research in QTL mapping has assumed that lines at crossing are fixed for alternative alleles at the QTL [1,2,8]. The availability of divergent inbred lines for traits of economical interest is a limiting factor in these studies.
Domesticated breeds of pigs have been undergoing artificial selection to increase growth rate and litter size, for example. Only for research purposes, selection criterion has been low growth rate or reduced litter size in experimental populations. Therefore, it seems more reasonable to assume that the lines at crossing differ in allele frequency at the QTL rather than that they are fixed for alternative alleles.
The benefits of crossing lines are to increase gametic disequilibrium between marker and QTL alleles and to increase heterozygosity among F l individuals. Gametic disequilibrium generated at crossing is high when the allele frequency at the two loci (either two markers or one marker and one QTL) is very different in the two parental lines. Gametic disequilibrium between two linked markers could be estimated in chromosomal fragments by the use of the disequilibrium parameter proposed in this paper. It would require that markers are not tightly linked so parental lines could be assumed to be in linkage equilibrium. If the gametic disequilibrium is low (e.g. Q < 0.2) then the analysis could be carried out within families as considered in this paper with only a small loss in power. If the gametic disequilibrium is high (e.g. S2 > 0.8) then a maximum likelihood approach could be developed allowing the same marker allele to be associated with the same QTL allele only in some families. Consequently, power figures as given in this study represent a lower bound of the achievable power in those cases. More work is needed to assess the gain in power in situations where disequilibrium parameter has intermediate values.
Power for QTL detection in experiments involving crosses between two outbred lines is higher than using within-line experiments when the frequency of the favourable allele is higher than 1/2 in one parental line and lower than 1/2 in another parental line. The increased power can be attributed to the higher frequency of heterozygous F l individuals than in either parental breed. The larger the difference in allele frequency between the two parental populations, the larger is the increase in power.
It was assumed in the computation of power that residual heritability was constant for varying QTL effects and for a given heterozygosity in the F l . This represents a variety of situations in which parental populations can be segregating for a QTL at a different frequency and having a different heritability. The approach taken in this paper was to compute also the heritability (comprising QTL and polygene variation) of the parental line with the highest possible value for a given heterozygosity in the F l . At low residual heritability, the contribution of large QTL to the heritability is high. For example, for residual heritability 0.1 and QTL effect of 0.6 phenotypic standard deviations, the heritability is 0.22 when the allele frequency in the parental population is 0.68 (table 111). In spite of the large contribution of the QTL variation to the heritability, changes in power for different heritability values are small. Discrepancies in power comparing residual heritability 0.1 and 0.2 for the same QTL size range between 0.00 and 0.05 (table 111). Therefore, power figures, as given in this paper, are well approximated when precise estimates of heritabilities from parental population are not available. The approximation is particularly good when the QTL has a small effect.
On the other hand, it was assumed when computing power that recombination between QTL and marker did not occur. In most instances, recombination would occur. If the recombination fraction between the marker and the QTL is c then the number of offspring should be increased in proportion to 1/( 1-2c? to obtain the same power as with complete linkage !14!. Power could be increased by using interval mapping instead of the use of a single marker !10!. Most of the relevant information in interval mapping comes from the non-recombinant individuals. The number of offspring required to obtain power as given in this paper should be increased in a proportion 1/(1-c'), where c' is the recombination fraction between the two markers. That is, around 25 % more offspring are needed to achieve power as given in this paper for interval mapping between two markers with a recombination fraction of 0.20. A larger experiment size will also be needed to achieve the same power figures as given in this paper when the marker is not fully informative. The use of interval mapping can mitigate the lack of informative offspring by utilizing information corresponding to nearby informative flanking markers.
The models used in this paper ignore the possibility of a common full-sib family component such as may occur for some traits of economic interest in pigs. Power for mapping QTL affecting those traits would be reduced !9!. The approach used in this study to compute power could be modified to account for this problem by incorporating a term corresponding to the common family component in the standard error of the contrasts.
It has also been assumed that the heritabilities in parental populations as well as within the offspring of F l remain unchanged. The frequency of heterozygotes at other loci affecting the trait could increase genetic variance among F l individuals and, therefore, heritability. One generation of random mating is enough to restore Hardy-Weinberg equilibrium as would occur after intercrossing F l individuals. However, changes in the genetic variance among offspring resulting from the hierarchical backcross design may occur. If there are other QTL affecting the trait showing dominance or epistasis, then changes in the phenotypic variance and heritability could also occur. Therefore, departures from the power as computed in this paper would be expected for traits that show heterosis such as reproductive traits.
For a given power, a backcross requires about twice as many progeny as an intercross design because the number of segregating meioses is doubled in the intercross [10]. Experimental design for mapping QTL in pigs has the limitation that the number of piglets per litter has a maximum around ten. This limits the size of the subgroups of progeny where the contrasts are carried out. Hierarchical backcross and intercross designs can, however, be used in pigs and allow contrasts within boars with larger subgroups of progeny. Power of intercross versus backcross depends on both heritability and QTL effect when using hierarchical designs. At high heritability levels, the hierarchical intercross design is more powerful than the hierarchical backcross design. The opposite occurs at low heritability values.
An interesting result was that power for small QTL effects was higher in a hierarchical backcross than in a hierarchical intercross. The only difference between the two designs is that the hierarchical backcross ignores meioses from sows. In a hierarchical intercross, contrasts between alternative alleles inherited from sows are small with a large variation given the small QTL effect and the small progeny group (ten piglets). However, the total number of degrees of freedom may be high because of the large number of sows. On the contrary, in a hierarchical backcross each of the few boars has large subgroups of progeny which means that contrasts have low variation and hypothesis testing is carried out with a low number of degrees of freedom. As a consequence of the above, power may be reduced in the intercross.
There is not a general experimental design which can allocate resources in an optimum way if the experiment size is fixed. Power in experiments with a low number of boars is higher for small size of QTL and high heritability (table V7). The opposite occurs for QTL with large size and low heritability.
Risk measured as the probability of power lower than 0.5 due to sampling of F, boars is high with a low number of boars. If the frequency of heterozygous F l individuals is low then the risk increases and the power decreases (results not shown). A compromise between expected power and risk should be made.
A possibility for experiments searching for QTL affecting traits with known heritability is first to decide the potential QTL size and F l heterozygosity detectable for a given experiment size (tables 77-7 V). Second is to choose a design for that QTL size with maximum expected power but conditional on an expected risk of less than a value of, say, 0.05. For example, for traits with low heritability and using hierarchical backcross designs with an experiment size of 400 piglets, the detectable QTL sizes are 0.3, 0.3 and 0.5 for a heterozygosity of 1, 0.68 and 0.32, respectively (tables 11 IV). Between four (maximum expected power) and eight boars (minimum risk) could be chosen for a heterozygosity of 0.68 and a QTL effect of 0.5 phenotypic standard deviations (table VI). If the heterozygosity is lower then the QTL will be very difficult to detect even using a design with optimal allocation of resources.
Dam and sire lines currently utilized for commercial production of pigs (e.g. Large White, Landrace and Duroc, in Norway) can be considered as parental lines which are crossed to produce F l individuals with high frequency of heterozygotes. Genotyping both boars (Fr ) and piglets (backcross to Landrace) and recording of traits in the slaughter pigs can be used to map QTL by contrasting inheritance of alternative alleles with large subgroups of progeny. Utilizing both sire and dam lines would also be feasible in a similar fashion to a hierarchical intercross design. However, the cross between F l boars and sows is not currently used by the industry. The swine industry in other countries often uses four-way crosses. For example, line A is crossed to line B to produce F l (AB), line C is crossed to line D to produce F l (CD). The cross F l (AB) x F l (CD) would also show increased heterozygosity and power with respect to experiments within line if the allele frequency is higher than 1/2 among F i (AB) and lower than 1/2 among F l (CD) individuals and vice versa. The use of commercial pigs for QTL mapping has two advantages: 1) saving in the cost of running the experiment since the carcasses can be sold at current prices in the market; and 2) findings can immediately be used in markerassisted selection in the commercial stocks.
Selective genotyping involving growing a large population but genotyping only those individuals whose phenotypes deviate far from the mean has been proposed to increase power for QTL detection [10,11]. In this paper an alternative use of selection to increase power is proposed. Selection of extreme individuals in the parental lines segregating at a QTL can be used to increase heterozygosity in the F l and consequently, power of the experiment. It has been shown that selection is more useful with experimental designs having intermediate values of power without selection. The increase in power is up to 21 %. A higher power would be feasible if high selection intensities are practised, for example, by screening large populations for extreme individuals. The main restriction with either use of selection (extreme individuals in the parental lines or extreme progeny) is that in many instances, QTL mapping experiments are carried out for several traits, which makes impractical the use of selection.
In this section, the conditions for increased heterozygosity in the cross between two segregating populations at a QTL is studied. Consider two outbred lines B and C segregating at a biallelic QTL with a favourable allele frequency p B and pc for lines B and C, respectively. Under Hardy-Weinberg equilibrium, the frequency of heterozygous individuals at the QTL in breed B is: Similarly, the frequency of heterozygotes among F l individuals in the cross is Rearranging the above equation where d = PB -P c, the difference in allele frequency between the two breeds.
The amount d(2p B -1) is the excess of heterozygotes in the cross with respect to line B. The same reasoning can be used to obtain the frequency of heterozygotes in the cross as a function of d and p C : The following can be concluded by inspection of equations A1 and A2. i) If p c > 1/2 and PB > 1/2 then d (2 p B -1) is positive and d (1 &mdash; 2 p!) is negative.
For i or ii, the heterozygosity in the offspring of a cross is not higher than in the parental line (B or C) with the highest heterozygosity. iii) If p c > 1/2 and p B < 1/2 then d (2 p B -1) is positive and d (1 -2 pc) is positive. iv) Ifpc < 1/2 and p B > 1/2 then d (2 p B -1) is positive and d (1 -2 p!) is positive.
For iii or iv, the heterozygosity in the offspring of a cross is higher than the heterozygosity within either line, B or C. Therefore, crossing between outbred lines segregating at a QTL with a frequency larger than 1/2 in one line and lower than 1/2 in the other leads to increased heterozygosity in F l , which increases statistical power for QTL mapping with respect to analysis within line.

APPENDIX 2: Contrasts in backcross and intercross designs
Outbred lines B and C are segregating at a biallelic QTL (with alleles Q and q) with favourable allele (Q) at frequencies PB and p c for lines B and C, respectively. Offspring from a heterozygous (Qq) F l boar inheriting allele Q can be QQ (with genetic value a) and Qq (with genetic value 6). Offspring inheriting q can be qQ (with genetic value 6) and qq (with genetic value -a).
The genotype frequencies of each type of offspring depending on the mating of F l individuals with parental line B (backcross BC-B), parental line C (backcross BC-C), or with other F l individuals (intercross) are: