Candidate Gene Analysis for Quantitative Traits Using the Transmission Disequilibrium Test: The Example of the Melanocortin 4-Receptor in Pigs

Corresponding author: Jules Hernández-Sánchez, Roslin, Midlothian, EH25 9PS, Scotland, United Kingdom., jules.hernandez{at}bbsrc.ac.uk (E-mail)

Communicating editor: M. A. F. NOOR

	ABSTRACT

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Population-wide associations between loci due to linkage disequilibrium can be used to map quantitative trait loci (QTL) with high resolution. However, spurious associations between markers and QTL can also arise as a consequence of population stratification. Statistical methods that cannot differentiate between loci associations due to linkage disequilibria from those caused in other ways can render false-positive results. The transmission-disequilibrium test (TDT) is a robust test for detecting QTL. The TDT exploits within-family associations that are not affected by population stratification. However, some TDTs are formulated in a rigid form, with reduced potential applications. In this study we generalize TDT using mixed linear models to allow greater statistical flexibility. Allelic effects are estimated with two independent parameters: one exploiting the robust within-family information and the other the potentially biased between-family information. A significant difference between these two parameters can be used as evidence for spurious association. This methodology was then used to test the effects of the fourth melanocortin receptor (MC4R) on production traits in the pig. The new analyses supported the previously reported results; i.e., the studied polymorphism is either causal or in very strong linkage disequilibrium with the causal mutation, and provided no evidence for spurious association.

POPULATION-wide associations between loci due to linkage disequilibrium can be used in high-resolution mapping of quantitative trait loci (QTL). However, spurious associations between markers and QTL can also arise as a consequence of population stratification, for example due to admixture of two different populations. Associations between a genotype and a trait have been frequently tested, after corrections, with simple one-way ANOVA models, e.g., testing mean genotype differences directly. However, these types of analyses are prone to false-positive results due to confounding effects of population stratification/admixture (e.g., DENG et al. 2001 ). SPIELMAN et al. 1993 developed an allele-trait association test called the transmission disequilibrium test (TDT), which is robust to these confounding effects. Different TDTs have since been developed for dichotomous traits (SCHAID 1996 ; HORVATH and LAIRD 1998 ; LUNETTA et al. 2000 ; MARTIN et al. 2000 ; ZHAO et al. 2000 ) and for quantitative traits (ALLISON 1997 ; RABINOWITZ 1997 ; SZYDA et al. 1998 ). However, most of these TDTs have been formulated in a rather rigid form, hence reducing the scope for further statistical modeling. In addition, previously described TDTs make use of within-family information only.

This study presents a generalized version of TDT to obtain robust estimates of genetic effects within the statistically more flexible mixed-linear model context. This approach allows maximum-likelihood estimates of genetic effects to be obtained via residual maximum likelihood (REML). Additive allele substitution effects were estimated with two independent regression coefficients (b), the within-families coefficient b_TD (transmission disequilibrium) and the between-families coefficient b_PD (population disequilibrium). Moreover, the rejection of the null hypothesis b_TD = b_PD provides evidence for stratification/admixture and hence can be used to guard against false-positive results.

The analysis of the fourth melanocortin receptor (MC4R) locus on pig chromosome 1 constitutes a practical demonstration of this method. The interaction between melanocortins and their receptors (MC3R and MC4R) at the hypothalamus is one of the main neuro-endocrinological pathways controlling energy balance (WARDLAW 2001 ). In humans, different allelic variants of both MC4R and MC3R have been associated with obesity (VAISSE et al. 1998 ; YEO et al. 1998 ; HINNEY et al. 1999 ; LI et al. 2000 ). In pigs, the seventh transmembrane region of the MC4R locus contains a mutation at codon 298 that causes a change of aspartic acid for asparagine, i.e., Asp298Asn (KIM et al. 1999 ). This region is highly conserved across all four types of melanocortin receptors in humans (GANTZ et al. 1993 ), and it is also very conserved between pigs and humans (KIM et al. 2000 ). The Asp298Asn mutation has been associated with fatter and faster-growing pigs, having significant effects on back fat, days to 110 kg, test daily gain, and daily food intake in a study involving four different commercial pig lines from nucleus breeding farms (KIM et al. 1999 , KIM et al. 2000 ). However, the original analyses were performed with methods that are potentially biased in the presence of population stratification. Here we analyze an augmented data set using the new methodology to confirm and extend the original findings.

	METHODS

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Data:
Performance traits were recorded on four different commercial PIC pig lines in the same farm over a 5-year period (1993–1998). The traits were lifetime daily gain (LDG), test daily gain (TDG), daily food intake (DFI), and back fat depth (BF) at the 10th rib. All pigs were performance tested for growth over a fixed period of 12 weeks, during which they were fed ad lib. and weighed at the beginning (on-test) and at the end (off-test) of that period. TDG was calculated as off-test weight minus on-test weight divided by the number of days on test. LDG was calculated as off-test weight minus one (assumed the average birth weight) divided by the age of the pig (in days) at off-test. BF was measured ultrasonically in real time at off-test, and it was normalized with the natural log transformation. DFI was electronically recorded for some pigs over the testing period. The sample sizes by line and sex are given in Table 1. In this data set, there were 726 extra records of BF, 574 of TDG, and 44 of DFI, with respect to the data set analyzed by KIM et al. 2000 . The Asp298Asn substitution mutation is located within a TaqI restriction enzyme recognition site (KIM et al. 1999 ), which was used to generate a codominant restriction fragment length polymorphism (RFLP) to distinguish all three genotypic classes (KIM et al. 2000 ).

Statistical models:
The effects of the Asp298Asn mutation on pig production traits were estimated with the models used in KIM et al. 2000 , which are not robust to population stratification/admixture, and with new robust models. The latter models were a combination of the former models and a TDT (RABINOWITZ 1997 ). The polygenic variance was estimated, fitting sire as a random factor. This model is computationally faster than fitting each animal as a random factor, and the two models were very similar in terms of variance components estimation (results not shown).

ANOVA: The original models included sex, batch, line, and genotype as fixed factors and sire as random effect. BF records were analyzed both in the original scale and in the log-transformed scale. The skewness and kurtosis of the untransformed distribution of BF were 0.88 (±0.049) and 1.96 (±0.099), respectively. After transformation the distribution of BF became more normal (skewness = -0.03 ± 0.049, kurtosis = 0.32 ± 0.099). All two-way interactions between fixed factors were also included in the analyses. Nonsignificant factors were dropped out of the models using a backward elimination procedure. The coefficients for genotypes can be found in column A in Table 2. The analyses were performed with the REML procedure in GENSTAT (PAYNE et al. 2001 ). This method of estimating allele effects is not robust to stratification/admixture (HERNANDEZ-SANCHEZ et al. 2002B ). We refer to this as the ANOVA method.

Batches as random: There were 54 batch means to estimate when they were fitted as fixed effects in the models. To avoid this unnecessary loss of degrees of freedom, batches were fitted as a random term where direct comparisons were being made with the TDT, and their effect was accounted for with cubic splines. This procedure was feasible because all trait means followed a yearly cycle when plotted against batches. The correction uses up only 2 d.f.: one in fitting a linear regression across all batches and a second in estimating the residual variance around the previous line. The software used to run models with batch as random splines was ASREML (GILMOUR et al. 2001 ). This approach was also implemented in the TDT analyses (see below).

TDT: A robust analysis of the Asp298Asn mutation was performed with the same models but substituting genotype for two independent fixed covariates. One of these two covariates was based on a TDT (RABINOWITZ 1997 ). Given a biallelic marker, each individual's genotype received a coefficient equal to H(T - ¹/₂) + H(T - ¹/₂); where H = 1 if the individual's sire was heterozygous, or 0 otherwise; T = 1 if the sire had transmitted allele 1 to the individual, or 0 otherwise; and likewise, H and T for the individual's dam. These coefficients can be found in column TD in Table 2. The slope of this covariate, b_TD, is a robust estimate of additive substitution effects of alleles at the locus. Allelic effects were also estimated via a second regression coefficient sensitive to the effects of population structure, b_PD (L. L. G. JANSS, personal communication). The appropriate coefficients to estimate b_PD were obtained by subtracting column A from column TD in Table 2. We refer to this as the TDT method.

Generating parental genotype data:
The TDT method requires parental genotype data, which in the MC4R data set were mostly missing. Missing parental genotypes were generated using Gibbs sampling (e.g., WANG et al. 1994 ; SORENSEN and GIANOLA 2002 ). Gibbs sampling was equivalent to integrating over all genotype probabilities of parents with missing genotypes. Missing parental genotypes were sampled conditional on genotypes of progeny and other relatives. The sampling algorithm has been reported previously in the literature (GUO and THOMPSON 1992 ; JANSS et al. 1995 ). Convergence was reached after 10³ realizations after a burn-in period of 100 realizations. The autocorrelation in Gibbs sampling was minimized by sampling 1 realization every 50 consecutive ones; hence a total of 50 x 10⁴ realizations were generated. The actual integration over missing genotype probabilities was carried out by analyzing the MC4R data set after each sampled realization and averaging results (i.e., P values) across all realizations.

Simulation:
A simulation study was carried out to investigate the properties of b_PD and b_TD. Population stratification was generated by sampling from two separated populations with different allele frequencies and analyzing the data jointly. Each population was characterized by: (1) 15 unrelated full-sib families and four offspring per family (60 progeny in total); (2) random mating; (3) a QTL and a linked neutral marker, both biallelic and with allele frequency fixed to 0.9 in one population, and frequencies, at both loci, of 0.9, 0.7, 0.5, 0.3, and 0.1 in the other population; (4) recombination rate between loci of c = 0 or ¹/₂; (5) standardized linkage disequilibrium in parents (D' = D/D_max; LEWONTIN 1988 ) of either 0 or 1; (6) residual variance of 1 and polygenic variance 0; and (7) QTL explaining 5% (and in some cases 10%) of the total phenotypic variance. There were no interpopulation matings and all analyses were performed at the marker locus.

	RESULTS

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Properties of b_PD and b_TD in analyses of simulated data:
The power of estimating genetic effects through b_PD and b_TD in the simulated data is shown in Fig 1, where F-ratios are plotted against the level of stratification S (allele frequency difference between populations) across four different scenarios. Each dot in Fig 1 is the average of 100 replicates. The significance threshold is based on the tabulated nominal 5% threshold and is shown as a straight line. Fig 1A shows the results after analyzing a marker totally unlinked (c = ¹/₂) and with no association in the population (D' = 0) to a QTL. In this situation, any significant effect is a type I error or due to bias. The F_TD (i.e., the F-ratio testing whether a significant amount of the total variation is explained by b_TD) is approximately 1 across all S values, which indicates that the marker did not have a significant effect on the trait. On the contrary, F_PD (i.e., the corresponding F-ratio test for b_PD) appeared positively correlated to the level of S. The effect was significant when S ≥ 0.6. In this case, spurious disequilibrium increases as stratification increases, and b_PD cannot distinguish between this sort of disequilibrium and disequilibrium due to linkage.

View larger version (0K): In this window In a new window Download PPT slide   Figure A1. (A) Expected mean squares (EMS) for bTD, estimated with a simple linear regression (see text), when frequencies of allele A in two different populations are P1 and P2. (B) EMS for bPD.

Fig 1B shows the results after analyzing a marker totally unlinked (c = 0.5) but in complete disequlibrium (D' = 1) in the parents. Here, F_PD is always >F_TD, and this difference increases with S. Moreover, the value of F_PD was >1 (2) even without stratification (S = 0). This can be explained by considering that, on average, the level of disequilibrium among offspring was one-half, because D' is expected to be halved every generation assuming no linkage and random mating. This feature suggests that b_PD could detect an effect given sufficient linkage disequilibrium between a marker and a QTL, even if these two loci are totally unlinked, whereas b_TD needs the joint occurrence of linkage and linkage disquilibrium to estimate an effect.

Fig 1C shows how F_PD rapidly increases for S > 0.4, while F_TD remains constant and equal to one regardless of S. The marker was totally linked to the QTL locus in this set of simulations (c = 0). Despite having simulated no linkage disequilibrium (D' = 0) within each population, mixing two populations with different allele frequencies will cause haplotype frequencies to depart from the expected equilibrium frequencies. In this simple scenario, the fact that F_TD = 1 across all values of S even when c = 0 demonstrates the robustness of the test based on b_TD.

Finally, Fig 1D shows the effect of effectively analyzing the QTL itself (D' = 1 and c = 0). The power of estimating b_PD increases monotonically with S, a fact consistently observed in all previous graphs. However, the power of estimating b_TD reaches its maximum at intermediate levels of S. This is so because intermediate allele frequencies [e.g., at 0.5, represented by S = 0.9 (population 1) - 0.5 (population 2) = 0.4 in Fig 1D] are associated with a higher proportion of heterozygous parents, providing better information to estimate b_TD.

Analyses of MC4R data:
ANOVA method: Results were obtained from analyses of the data from the four lines separately and from an overall analysis of the combined data from all lines. Striking differences between genotypes were detected for BF (P < 0.001), TDG (P < 0.001), and LDG (P < 0.001) in the overall (i.e., all lines together) analysis (Table 3). Batches were treated as fixed effects in these analyses to allow us to compare results directly with the findings of KIM et al. 2000 . Genotypic differences within all pig lines were confirmed for BF (ranging from P < 0.001 in lines A and C to P < 0.02 in line B) and also within some lines for TDG (ranging from P < 0.001 in line C to P > 0.8 in line B) and LDG (ranging from P < 0.01 in line C to P > 0.5 in line D). No significant effect of the Asp298Asn substitution mutation on DFI was found in this analysis, in spite of the significant difference of 0.17 kg between both homozygotes (P < 0.01) reported by KIM et al. 2000 . In this study, the estimated difference in DFI between the two homozygous genotypes, although not significant, was in the same direction as that previously reported at 0.1 kg [standard error of the difference (SED) = 0.063] in the overall analysis.

TDT method: Only results from the overall analyses of the combined data set are shown. The estimation of b_TD and b_PD via REML was done on the basis of 10³ replications, where each replicate used a different population of parental genotypes generated with Gibbs sampling. The average P values across these analyses are shown in Table 4. The ASREML software does not perform hypothesis testing for fixed effects in the model although it provides t-values and residual degrees of freedom. The P values associated with genotypic contrasts were obtained from the t-distribution. Two independent null hypotheses were of interest: H¹₀, b_PD = 0, and H²₀, b_TD = 0; both were rejected for all traits except DFI [H¹₀, (BF) P < 0.0001, (TDG) P < 0.01, (LDG) P < 0.01, and (DFI) P > 0.5; H²₀, (BF) P < 0.0001, (TDG) P < 0.001, (LDG) P < 0.01, and (DFI) P > 0.2]. For the sake of comparison, Table 4 also incorporates results from the ANOVA method (reported as regression coefficients rather than genotype means as in Table 3). Both the ANOVA and the TDT methods yielded similar results, although the former was a slightly more powerful analysis (the standard errors were generally smaller). Results in Table 3 and Table 4 are of similar magnitude.

	DISCUSSION

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The study demonstrated that the mean additive effect of allele substitution calculated with a one-way ANOVA model can be decomposed into the within- and between-family effects. These two effects can be estimated via a flexible REML analysis (PATTERSON and THOMPSON 1971 ) as the regression coefficients b_TD and b_PD, respectively, using a mixed-linear model that can also incorporate other fixed and random effects.

Parental genotypes are needed to estimate b_TD and b_PD. This information was not available for the MC4R data analyzed and was generated via Gibbs sampling (1000 realizations). The new methodology was tested via simulation and real data analysis of the effect of the MC4R gene on pig production traits.

The simulation results can be summarized in three main points. First, b_TD extracts information from the within- and b_PD from the between-family genetic variances, ²_WF and ²_BF, respectively. Second, b_TD is robust and b_PD is biased in the presence of population admixture/stratification. Third, there is generally more power to detect a significant b_PD 0 than a significant b_TD 0.

Population stratification/admixture increases ²_BF and not ²_WF. As a consequence, estimates of b_PD may be biased in the presence of stratification/admixture, whereas estimates of b_TD are not as the simulation results demonstrate. Tests such as the TDT are robust because they exploit only ²_WF. However, the one-way ANOVA model uses both ²_WF and ²_BF, from which a pooled estimate between b_PD and b_TD is obtained, and because of the latter component of variance, this pooled estimate of allelic effects may be biased if there is population admixture/stratification. In spite of this potential bias producing false-positive results (e.g., Fig 1A and Fig B), robust methods such as the TDT have seldom been used in animal breeding (some exceptions are BINK et al. 2000 and HERNANDEZ-SANCHEZ et al. 2002A ). This may be because TDT is viewed as a not very powerful method (e.g., Fig 1D, with severe limitations in some circumstances, e.g., low frequency of heterozygous parents).

The fact that b_TD and b_PD exploit different sources of information can be intuitively appreciated by inspecting the coefficients in Table 2. First, the coefficients required in the estimation of b_TD are weights given to all individuals with records (e.g., offspring) according to their genotypes and to the genotypes of their parents (i.e., family type). Hence, b_TD is the slope of the regression of phenotypes onto explanatory variables that combine both offspring's genotypes and family type. Second, the coefficients required in the estimation of b_PD can alternatively be obtained as , where G_ij = ¹/₂, 0, or -¹/₂ if the genotype of the jth parent in the ith family is 11, 12, or 22, respectively. Therefore b_PD is the slope of the regression of phenotypes onto family type.

More explicitly, let us model y_ij, the phenotype of the jth individual having the ith QTL genotype, as , where µ is the population mean, g_i is the effect of the ith QTL genotype on the jth offspring, and a_ij and e_ij are the polygenic and residual random terms drawn from two independent normal distributions with zero means and variances ²_A and ²_e, respectively. Let there be random mating, no population stratification, and only additive genetic effects at the QTL locus. Under these circumstances, the total additive genetic variance splits equally between and within families; therefore and , where , and ²_Q is the variance due to the QTL. If a statistical model explains all ²_Q, then the additive genetic effects of the QTL will be fully accounted for. However, if the model estimates additive effects only through either b_TD or b_PD, then ²_Q will be only partially explained. For example, if b_TD is the only estimator of QTL effect, then and as only within-family variation can be used to estimate b_TD. If, on the other hand, b_PD is the only estimator of QTL effect, then and , as only between-family variation can be used to estimate b_PD. These changes in the within- and between-family variances due to estimation of b_TD or b_PD were validated via computer simulations (data not shown).

The situation is more complex when there is stratification within a population. The Appendix shows how the proportion of the total phenotypic variation explained, fitting a linear regression, differed when either b_TD or b_PD was estimated in the presence of population stratification.

Simulation results showed that testing the null hypothesis b_PD = 0 rather than b_TD = 0 tends to produce higher F-ratios when it is not true (Fig 1). Nevertheless, the analysis of MC4R data showed the opposite effect (i.e., t-tests for b_TD were always higher than those for b_PD in Table 4). This result is possible when there is no admixture/stratification, because b_TD is expected to be equivalent to b_PD. Moreover, in simulations, the only between-family component was one-half of the total QTL variance, whereas in reality other factors (e.g., litter and sow effects) may also increase ²_BF. The analysis of real data showed that b_TD was very similar to b_PD across all traits and, furthermore, that both estimates were also similar to allelic effects obtained with ANOVA (Table 4). This suggests that there was no significant stratification/admixture in the population and that there were equivalent amounts of genetic information between and within families.

Where estimates of b_TD and b_PD differ, a t-test could be used to test the null hypothesis b_TD = b_PD. If no evidence to reject the null hypothesis were found, i.e., no evidence for admixture/stratification in the population, then a more powerful one-way ANOVA could be safely implemented within the mixed-linear model to estimate allele effects. Otherwise, robust approaches such as the one proposed here, i.e., estimating b_TD or other TDTs, should be considered as the only reliable methods of analysis.

A complication of this approach applied to many real data sets, including the MC4R data used as an example, is generating missing parental genotypes. A Gibbs sampling technique to generate 10³ replicates was readily implemented (and we know they were accurate enough because 10 times more replicates did not affect the outcome) in this simple scenario: i.e., a single biallelic marker, a maximum of three generations, and few missing data. Other robust tests that do not require parental genotypes, e.g., sib-TDTs (e.g., SCHAID and ROWLAND 1999 ; SPIELMAN and EWENS 1998 ), were found to be less powerful than the method outlined here in additional analyses of simulated data [e.g., F_sib = 5.5 vs. F_TD-PD = 7.7 (based on 1000 replicates)]. Analyzing more complex or unbalanced data sets will probably demand more realizations of the Gibbs sampler and it will present an additional problem: testing fixed effects in REML, because the asymptotic properties of the t-test or the Wald test could not be guaranteed (see KENWARD and ROGER 1997 ; WELHAM and THOMPSON 1997 ; ELSTON 1998 ). If few fixed effects are fitted in the model, then maximum likelihood rather than REML can be used to test allele effects via likelihood-ratio tests.

There is strong evidence from both this study and KIM et al. 2000 that the substitution-mutation Asp298Asn in the MC4R gene affects production traits in the pig, e.g., BF and growth (TDG, LDG). These effects were reestimated with extra records with respect to the original work. Significant effects were found for BF, TDG, and LDG across all lines (P < 0.001), although not for all pig lines (ranging from P < 0.001 to P > 0.8).

DFI was the only trait not significant in this study (P > 0.2) that was significant in the study of KIM et al. 2000 , even after using their statistical models. The lack of effect on DFI was unexpected given previously reported results (P < 0.01) and other reported associations of MCR with appetite and feeding behavior in macaques (KOEGLER et al. 2001 ), mice (BUTLER et al. 2000 ), and layer chicks (TACHIBANA et al. 2001 ). One possible explanation for not detecting an effect on DFI is the lack of power due to a small data set (n = 275). Although there were less data (n = 231) in the study of KIM et al. 2000 than in this study, the difference is small and additional statistical noise may have been introduced through factors such as data structure and/or chance. An independent study that used a larger data set (n = 619) found a significant (P < 0.05) additive gene effect of 0.075 kg/day on DFI (G. PLASTOW, personal communication).

Further experiments are needed to ascertain whether the Asp298Asn substitution mutation at the MC4R locus is causative. For example, one could test the effect of this and other polymorphisms on the same exon/gene in different segregating populations. Final proof will require molecular experiments, e.g., testing whether different molecular pathways are activated in independent cell cultures having different Asp298Asn variants and/or creating mouse models for testing this polymorphism on identical genetic backgrounds.

This study demonstrates that TDT can be implemented within the REML framework. As a guideline, to maximize the power in an association analysis, one should test associations with ANOVA (within a REML or maximum-likelihood framework) only after having checked that b_TD and b_PD are not significantly different from each other, i.e., making sure no false-positive results are being caused by population stratification.

	ACKNOWLEDGMENTS

We thank Dr. R. Pong-Wong for his advice on Gibbs sampling; Dr. O. Southwood, D. Waddington, and anonymous referees for helpful comments that improved earlier versions of this manuscript; and especially Dr. L. L. G. Janss for sharing his ideas with us. We acknowledge Kwan Suk Kim and Max Rothschild (Iowa State University) for providing the MC4R data set and the Biotechnology and Biological Sciences Research Council and PIC International Group for funding this project.

Manuscript received August 15, 2002; Accepted for publication January 23, 2003.

	APPENDIX

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IMPACT OF STRATIFICATION ON b_TD AND b_PD
Population stratification affects both the estimation of the effect via b_PD and the power of that estimation (SE of b_PD). We assume the simplest scenario where a population is divided into two subpopulations of equal size, where mating is at random within each subpopulation, and there is no mating across subpopulations.

The expected mean square of a linear model that regresses phenotypes onto a single explanatory variable X is , where B is the expected regression parameter and ²_e is the residual variance (SOKAL and ROHLF 1995 ). When n phenotypes are simulated with gene effect (B) and , then . Let us assume that p_i is the frequency of allele i at the trait locus in subpopulation 1 and q_i is the equivalent frequency in subpopulation 2. Furthermore, let us assume that p₁₁ = (p₁)² is the frequency of genotype 11 in subpopulation 1 and q₁₁ is the frequency of the equivalent genotype in subpopulation 2. It can be shown that E[PD] = (p₁ - p₂ + q₁ - q₂)/2 and that . The same process is followed to develop the expected mean squares when using TD; thus , and . These two predictions were very similar to simulation results (not shown). We can see the effect of stratification on E[MSR]_TD and E[MSR]_PD for n = 100 in Fig 1A and Fig B. Fig 1A takes a hill-type shape (seen from above) where the highest point is at the center and gradually decays in all directions away from the center. Fig 1B takes a valley-type shape (seen from above) where the lowest points are on the diagonal (in fact, Fig 1A and Fig 1B are identical on the diagonal) and quickly rising away from the diagonal. If there is no stratification, e.g., on the diagonal passing through points p₁ = q₁, then E[MSR]_TD = E[MSR]_PD, and thus both regression lines are identical. However, when there is stratification (e.g., p₁ q₁), E[MSR]_PD increases and E[MSR]_TD decreases. The effect on E[MSR]_TD is not due to stratification itself but rather to a reduction of the information content due to a decrease in the frequency of heterozygous genotypes. Although we have shown the effect of analyzing the trait locus itself, this effect is transferred to a marker as a function of linkage disequilibrium between both loci.

	LITERATURE CITED

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