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- Genetics
- v.178(3); 2008 Mar
- PMC2278052

# Comparison of Mixed-Model Approaches for Association Mapping

^{*}Jens Möhring,

^{†}Hans-Peter Piepho,

^{†}Martin Heckenberger,

^{*}Edward S. Buckler,

^{‡}

^{§}

^{**}and Albrecht E. Melchinger

^{*,}

^{1}

^{*}Institute for Plant Breeding, Seed Science, and Population Genetics and

^{†}Institute for Crop Production and Grassland Research, University of Hohenheim, 70593 Stuttgart, Germany,

^{‡}Institute for Genomic Diversity and

^{§}Department of Plant Breeding and Genetics, Cornell University, Ithaca, New York 14853 and

^{**}United States Department of Agriculture–Agricultural Research Service, Ithaca, New York 14853

^{1}

*Corresponding author:*Institute for Plant Breeding, Seed Science, and Population Genetics, University of Hohenheim, Fruwirthstrasse 21, 70599 Stuttgart, Germany.E-mail: ed.miehnehoh-inu@regnihclem

## Abstract

Association-mapping methods promise to overcome the limitations of linkage-mapping methods. The main objectives of this study were to (i) evaluate various methods for association mapping in the autogamous species wheat using an empirical data set, (ii) determine a marker-based kinship matrix using a restricted maximum-likelihood (REML) estimate of the probability of two alleles at the same locus being identical in state but not identical by descent, and (iii) compare the results of association-mapping approaches based on adjusted entry means (two-step approaches) with the results of approaches in which the phenotypic data analysis and the association analysis were performed in one step (one-step approaches). On the basis of the phenotypic and genotypic data of 303 soft winter wheat (*Triticum aestivum* L.) inbreds, various association-mapping methods were evaluated. Spearman's rank correlation between *P*-values calculated on the basis of one- and two-stage association-mapping methods ranged from 0.63 to 0.93. The mixed-model association-mapping approaches using a kinship matrix estimated by REML are more appropriate for association mapping than the recently proposed *QK* method with respect to (i) the adherence to the nominal α-level and (ii) the adjusted power for detection of quantitative trait loci. Furthermore, we showed that our data set could be analyzed by using two-step approaches of the proposed association-mapping method without substantially increasing the empirical type I error rate in comparison to the corresponding one-step approaches.

ESTIMATION of the positions and effects of quantitative trait loci (QTL) is of central importance for marker-assisted selection. In plant genetics, this has so far been accomplished by applying classical linkage-mapping methods. Besides high costs (Parisseaux and Bernardo 2004), their major limitations are a poor resolution in detecting QTL and the fact that with biparental crosses of inbred lines only two alleles at any given locus can be studied simultaneously (Flint-Garcia *et al.* 2003). Association-mapping methods, which have been successfully applied in human genetics to detect genes coding for human diseases (*e.g.*, Ozaki *et al.* 2002), promise to overcome these limitations (Kraakman *et al.* 2004). Therefore, in plant genetics several attempts were made for detecting QTL by using such methods (*e.g.*, Kraakman *et al.* 2004; Olsen *et al.* 2004).

Application of association-mapping approaches in plants is complicated by the population structure present in most germplasm sets (Flint-Garcia *et al.* 2003). To overcome this problem, linear models with fixed effects for subpopulations (*e.g.*, Breseghello and Sorrells 2006) or a logistic regression-ratio test (Pritchard *et al.* 2000b; Thornsberry *et al.* 2001) can be employed. Owing to the large germplasm sets required for dissecting complex traits, the probability increases that partially related individuals are included. This applies in particular when genotypes selected from plant-breeding populations are used for association mapping (*e.g.*, Thornsberry *et al.* 2001; Kraakman *et al.* 2004). The above-mentioned approaches fail to adhere to the nominal α-level, however, if the germplasm set under consideration comprises related individuals (*cf.* Thornsberry *et al.* 2001).

Recently, Yu *et al.* (2006) proposed the *QK* mixed-model association-mapping approach that promises to correct for linkage disequilibrium (LD) caused by population structure and familial relatedness. The authors demonstrated the suitability of their new method for association mapping in humans and maize. Besides natural populations of *Arabidopsis thaliana* (*cf.* Zhao *et al.* 2007), the suitability of the *QK* method has to be evaluated in breeding germplasm of autogamous species, because their population structure is presumably high and levels of familial relatedness are diverse (*cf.* Garris *et al.* 2005).

In contrast to coancestry coefficients calculated from pedigree records, marker-based kinship estimates may account for the effects of deviations from expected parental contributions to progeny due to selection or genetic drift (Bernardo *et al.* 1996). Therefore, marker-based kinship estimates underlying the studies of Yu *et al.* (2006) and Zhao *et al.* (2007) might be more appropriate for association-mapping approaches than coancestry coefficients calculated from pedigree records. A difficulty with calculation of marker-based kinship estimates arises regarding the definition of unrelated individuals (Bernardo 1993). The marker-based kinship matrix underlying the study of Yu *et al.* (2006) was determined on the basis of the definition that random pairs of inbreds are unrelated, whereas Zhao *et al.* (2007) defined pairs of inbreds that do not share any allele as unrelated. However, both definitions are rather arbitrary. Therefore, we propose to estimate the conditional probability that marker alleles are alike in state, given that they are not identical by descent (Lynch 1988), by restricted maximum likelihood (REML).

As a first step of all earlier association-mapping studies in a plant genetics context, phenotypic data were analyzed and entry means or adjusted entry means were calculated for each individual of the population under consideration. These estimates were then used in a second step for the actual association analysis. Such two-stage procedures generally account neither for heteroscedasticity (heterogeneity in experimental errors) nor for possible covariances among the adjusted entry means (Cullis *et al.* 1998). These problems can be overcome by applying a one-stage association-mapping approach in which the phenotypic data analysis and the association analysis are performed in one step.

The objectives of our research were to (i) evaluate various methods for association mapping in the autogamous species wheat using an empirical data set, (ii) determine a marker-based kinship matrix based on a REML estimate of the probability that two inbreds carry alleles at the same locus that are identical in state but not identical by descent, and (iii) compare the results of one- and two-stage approaches for various association-mapping methods.

## MATERIALS AND METHODS

#### Plant materials, field experiments, and molecular marker analyses:

A total of 303 soft winter wheat (*Triticum aestivum* L.) inbreds developed by Lochow-Petkus (Bergen-Wohlde, Germany) were used for this study. For 194 entries, pedigree information up to the great-grandparents was available, whereas for the other 109 entries no pedigree records were available. In 2005, all 303 entries were evaluated for grain yield in a series of five breeding trials at four to six locations, with the number of entries per trial ranging from 36 to 110. The experimental design for each trial was a lattice design with two to four replications per location. Two of the 303 entries were evaluated as common entries in each lattice.

All 303 entries as well as five wheat cultivars, which are unrelated by pedigree to the 303 entries, were fingerprinted by Lochow-Petkus following standard protocols with 36 simple sequence repeat markers and one single-nucleotide polymorphism marker. The 37 marker loci were randomly distributed across 19 of the 21 wheat chromosomes. Map positions of all markers were determined on the basis of the linkage map of Lochow-Petkus (unpublished data).

#### Statistical analyses:

##### Phenotypic data analyses:

The phenotypic data were analyzed on the basis of the statistical model

where *y _{ijkno}* was the phenotypic observation for the

*i*th entry at the

*j*th location in the

*o*th incomplete block of the

*n*th replicate of the

*k*th trial, μ was an intercept term,

*g*was the genetic effect of the

_{i}*i*th entry,

*l*was the effect of the

_{j}*j*th location,

*t*was the effect of the

_{kj}*k*th trial at the

*j*th location,

*r*was the effect of the

_{njk}*n*th replicate of the

*k*th trial at the

*j*th location,

*b*was the effect of the

_{onjk}*o*th incomplete block of the

*n*th replication of the

*k*th trial at the

*j*th location, and

*e*was the residual. Error variances were assumed to be heterogeneous among locations. For estimation of variance components, all effects were considered as random.

_{ijkno}For estimating entry means, we regarded *g _{i}* as fixed and all other effects as random (Patterson 1997). Over all trials, an adjusted entry mean

*M*was calculated for each of the 303 entries as

_{i}where and denote the generalized least-squares estimates of μ and *g _{i}*, respectively.

##### Two-stage association analyses:

On the basis of 10 different statistical models (summarized in Table 1), adjusted entry means *M _{i}* of the 303 entries were used to calculate a

*P*-value for the association of each of the 37 marker loci with the phenotypic trait.

**Mixed-model methods used for association mapping and the corresponding statistical models for the two-stage association approaches analyzed in this study**

The first model was an ANOVA model of the form

where *M _{ip}* was the adjusted entry mean of the

*i*th entry carrying allele

*p*,

*a*the effect of allele

_{p}*p*, and

*e*the residual.

_{ip}The statistical model underlying our mixed-model association-mapping approaches (Table 1) was

where *v _{u}* was the effect of the

*u*th column of the population structure matrix

**D**and was the residual genetic effect of the

*i*th entry. The matrix

**D**, which comprised

*z*linear independent columns, differed among the various association-mapping methods (Table 1) and, thus, this matrix is described in the sections detailing the individual methods. The variance of the random effects and

**e**= {

*e*

_{1,1},…,

*e*

_{303,12}} was assumed to be and , where

**K**was a 303 × 303 matrix of kinship coefficients that define the degree of genetic covariance between all pairs of entries. was the genetic variance and was the residual variance, both estimated by REML. For a direct comparison of our results to those of Yu

*et al.*(2006),

**R**

_{1}was a 303 × 303 matrix in which the off-diagonal elements were 0 and the diagonal elements were reciprocals of the number of phenotypic observations underlying each adjusted entry mean. In a second association-mapping approach, instead of matrix

**R**

_{1}we used matrix

**R**

_{2}, in which the diagonal elements were calculated as the square of the standard errors of the adjusted entry means

**M**(Piepho and Möhring 2007).

For the *QK* mixed-model method (Yu *et al.* 2006), the population structure matrix **Q** was calculated by the software STRUCTURE (Pritchard *et al.* 2000a), which gives for each individual under consideration the probability of membership in each of the *z* + 1 subpopulations. In our investigations, the set of 303 entries was analyzed by setting *z* from 0 to 13 in each of five repetitions. For each run of STRUCTURE, the burn-in time as well as the iteration number for the Markov chain Monte Carlo algorithm was set to 100,000, following the suggestion of Whitt and Buckler (2003).

Plant populations often comprise related and/or admixed entries (Camus-Kulandaivelu *et al.* 2007). Therefore, we used the *ad hoc* criterion described by Evanno *et al.* (2005) to estimate the number of subpopulations, as it promises to reliably detect the true number of subpopulations also in complex genetic situations. The *z* + 1 columns of the **Q** matrix add up to one and, thus, only the first *z* columns were used as a **D** matrix in the *QK* method to achieve linear independence. Furthermore, in accordance with Yu *et al.* (2006) the kinship matrix **K** was calculated on the basis of the 37 marker loci using the software package SPAGeDi (Hardy and Vekemans 2002), where negative kinship values between inbreds are set to 0.

The *PK* method was based on the same kinship matrix **K** as used for the *QK* method. Following Zhao *et al.* (2007), however, the first eight principal components of an allele-frequency matrix, which explain altogether 36.8% of the variance, were used as a **D** matrix of the *PK* method (Table 1).

The *K* and *G* methods were based on mixed models that do not include any *v _{u}* effects (Table 1). The

*K*method uses the same kinship matrix

**K**as used for the

*QK*method. For the

*G*method, we estimated the

**K**matrix for all 303 inbreds on the basis of the available pedigree records, according to the rules described by Falconer and Mackay (1996), and using PROC INBREED in SAS (Sas Institute 2004). The coancestry coefficient between inbreds with unknown relationship was set to 0 (Bernardo 1993).

Bernardo (1993) proposed calculating the kinship coefficient *K _{ij}* between inbreds

*i*and

*j*(

*i.e.*, the probability that inbreds

*i*and

*j*carry alleles at the same locus that are identical by descent) on the basis of marker data according to

where *S _{ij}* is the proportion of marker loci with shared variants between inbreds

*i*and

*j*and

*T*is the average probability that a variant from one parent of inbred

_{ij}*i*and a variant from one parent of inbred

*j*are alike in state, given that they are not identical by descent. Thus,

*T*is a function of the proportion of variants common to unrelated inbreds and is specific for each pair of inbreds (Lynch 1988). In practice, the value of

_{ij}*T*is unknown.

_{ij}Our *K*_{unrel} method uses a matrix **K _{unrel}** based on one

*T*value for all pairs of inbreds obtained as the average

*S*between each of the five wheat cultivars and the 303 entries, as proposed by Lynch (1988) and Melchinger

_{ij}*et al.*(1991).

The *QK _{T}*,

*PK*, and

_{T}*K*methods were based on a matrix

_{T}**K**

_{T}that was calculated according to

We examined *T* = 0, 0.025, …, 0.975 to obtain a REML estimate of *T*. Negative kinship values between inbreds were set to 0.

##### One-stage association analyses:

Phenotypic data analyses and association analyses were performed in one step, on the basis of the model

where, except for *a _{p}* and

*v*, all effects were regarded as random and error variances were assumed to be heterogeneous among locations. Var(

_{u}**g***) and

**D**were modeled by the same nine methods as in the two-stage analysis (Table 1).

##### Power simulations:

Because of the high computational effort of the one-stage association analyses, our power simulations were conducted only for the two-stage association approaches. For each of the examined methods (ANOVA, *QK*, *PK*, *K*, *G*, *K*_{unrel}, *QK*_{0.70}, *PK*_{0.70}, *K*_{0.70}, and *K*_{0.35}) the empirical type I error rate α* was calculated on the basis of the *P*-values observed for the 37 marker loci in a scenario without simulated QTL (α = 0.05). In our study, we examined the power to detect a QTL of interest, which (i) explained a fraction of the phenotypic variance and (ii) was in complete LD with one marker locus, as follows. The QTL effect *G _{r}*, calculated as

*r*= 0.1 times the standard deviation of the vector of adjusted entry means

**M**of the 303 wheat inbreds, was assigned in consecutive simulation runs to each of the detected 202 marker alleles whereas all other alleles were assigned the genotypic effect 0. In each simulation run, the genotypic value of each entry

*i*was calculated by summing up the QTL effects of the alleles and the adjusted entry mean

*M*. The above-mentioned two-stage association-mapping methods were run on the inbreds' genotypic values to determine whether the QTL can be detected. To adjust the association-mapping methods for their different empirical type I error rates α*, we calculated the adjusted power as the proportion of QTL detected at (Yu

_{i}*et al.*2006). In addition to

*r*= 0.1, we examined

*r*= 0.2, 0.3,…, 2.

The percentage (π) of the total phenotypic variation explained by a QTL effect *G _{r}* was calculated as

where *s* was the sample size and *q* the allele frequency of the QTL (Yu *et al.* 2006).

##### Measures for comparison of association-mapping methods:

Under the assumption that the random markers *m* = 1, 2,…, 37 in our study are unlinked to functional polymorphisms controlling yield, it is expected that the *P*-values observed for an association-mapping approach are uniformly distributed (*cf.* Yu *et al.* 2006). Therefore, for the *P*-values observed for all marker loci and association-mapping methods, expected *P*-values were calculated as *r*(*x _{m}*)/37, where

*r*(

*x*) is the rank of the

_{m}*P*-value

*x*observed for the

_{m}*m*th marker locus. Association-mapping methods that adhere to the nominal α-level show a uniform distribution of

*P*-values,

*i.e.*, a diagonal line in the plot of observed

*vs.*expected

*P*-values. The mean of the squared difference (MSD) between observed and expected

*P*-values of all marker loci was therefore calculated as a measure for the deviation of the observed

*P*-values from the uniform distribution. High MSD values indicate a strong deviation of the observed

*P*-values from the uniform distribution, which suggests that the empirical type I error rate of these approaches is considerably higher than the nominal α-level.

Computer simulations were performed to examine which difference in MSD values between two association-mapping methods could be expected purely by chance. The simulation accounted for correlation among *P*-values of two methods as follows. Pairs of *P*-values were drawn from a bivariate beta distribution (Magnussen 2004) with parameters α = β = 1 and correlation equal to the observed correlation *C*_{obs} for a pair of methods. Thus, the marginal distribution of *P*-values for a method was uniform, and the correlation among methods equaled *C*_{obs}. In each simulation run, the difference of the MSD value for both methods was calculated. This procedure was repeated 100,000 times and the 95% quantile of the MSD difference was determined. We investigated the following four pairs of two-stage association approaches: (i) *QK*/ANOVA, (ii) *QK*/*K*, (iii) *QK*/*G*, and (iv) *QK*/*QK*_{0.70}.

For methods *QK _{T}*,

*PK*, and

_{T}*K*, we profiled the deviance for

_{T}*T*. Spearman's rank correlation was calculated between the observed

*P*-values of one- and two-stage association-mapping approaches.

All mixed-model calculations were performed with ASReml release 2.0 (Gilmour *et al.* 2006).

## RESULTS

For grain yield, the adjusted entry means *M _{i}* of the 303 elite inbreds varied between 7.52 and 9.60

*t*ha

^{−1}, with an average of 8.66

*t*ha

^{−1}. The genotypic variance was 0.085

*t*

^{2}ha

^{−2}and the genotype × environment variance was 0.090

*t*

^{2}ha

^{−2}. For the different locations, the error variance ranged from 0.081 to 0.200

*t*

^{2}ha

^{−2}.

The total number of marker alleles detected for the 37 loci was 202, with the number of alleles per marker locus ranging from 2 to 12. The average number of alleles per locus was 5.5. In principal coordinate analysis based on Rogers' distance estimates of the 303 entries as well as five wheat cultivars, the first two principal coordinates explained 14.6 and 10.7% of the molecular variance (Figure 1). With respect to these two principal coordinates, no clear grouping of inbreds could be detected. The model-based approach of STRUCTURE revealed eight subpopulations.

**...**

For the examined levels of *T*, the MSD between observed and expected *P*-values for the *QK _{T}* and

*PK*methods ranged from 0.002 to 0.035 (Figure 2). By comparison, the MSD was higher for the

_{T}*K*method and varied for the various levels of

_{T}*T*between 0.010 and 0.090. The deviances for the three methods

*QK*,

_{T}*PK*, and

_{T}*K*ranged from ~ −270 to ~ −350, with smallest values observed for

_{T}*T*= 0.775.

*P*-values as well as deviance for different two-stage association-mapping methods depending on threshold

*T*.

The MSD between observed and expected *P*-values of the *QK* and *PK* methods was 0.010 (Table 2), which was 10 times lower than that of the ANOVA approach (0.100). For the *K*, *G*, and *K*_{unrel} methods, the MSDs were 0.016, 0.077, and 0.013, respectively. The computer simulations that are based on the correlated beta distribution for the four pairs of association methods *QK*/ANOVA, *QK*/*K*, *QK*/*G*, and *QK*/*QK*_{0.70} resulted in a 95% quantile of MSD differences of 0.009, 0.006, 0.008, and 0.004, respectively. The trend observed for the MSD of the mixed-model approaches based on the **R**_{2} matrix was the same as that found for the approaches based on the **R**_{1} matrix.

**Mean of the squared differences (MSD) between observed and expected**

*P*-values for various mixed-model association- mapping methods as well as Spearman's rank correlation coefficient ρ between the*P*-values of one- and two-stage association-mapping**...**

The adjusted power to detect QTL of all association-mapping approaches increased with increasing size of the genetic effect assigned to an allele (Figure 4). For small as well as large genetic effects the slope of the power curve was flat, whereas for genetic effects of medium size the slope was steep. For all examined sizes of genetic effects, the adjusted power of the *QK*_{0.70} and *PK*_{0.70} methods was higher than that of the *QK* and *PK* methods. In comparison with the other association-mapping methods, the ANOVA method and the *G* method showed the lowest adjusted power to detect QTL for all examined sizes of genetic effects.

*G*. The percentage of phenotypic variation explained by a QTL was calculated for an allele frequency of 0.2.

_{r}**...**

The MSD between observed and expected *P*-values for the one-stage association-mapping methods ranged from 0.002 (*QK*_{0.70}) to 0.091 (*PK*). The trend observed for these approaches was similar to that found for the two-stage approaches (Table 2). Spearman's rank correlation between the *P*-values of one- and two-stage association analyses ranged from 0.63 to 0.87 for the nine mixed-model methods based on the **R**_{1} matrix. Likewise, the correlation ranged from 0.64 to 0.93 for association-mapping approaches based on the **R**_{2} matrix.

## DISCUSSION

#### Phenotypic data analyses in association-mapping approaches:

Previous association-mapping approaches in a plant genetics context were mostly based on entry means (*e.g.*, Aranzana *et al.* 2005; Yu *et al.* 2006). The more complex is the phenotypic trait under consideration, however, the more elaborate are the field designs as well as phenotypic data analyses that are required. Therefore, we used an efficient method for calculating adjusted entry means *M _{i}* (Smith

*et al.*2001). In this analysis, error variances were assumed to be heterogeneous among locations. The statistical model is easily extended to other settings,

*e.g.*, heterogeneous block and replication variances or modeling of spatial heterogeneity at the plot level (nearest-neighbor analyses; Moreau

*et al.*1999).

#### Comparison of various association-mapping approaches:

Investigations on the adjusted power to detect QTL as well as on the type I error rate of association-mapping approaches based on empirical data require that the marker loci are unlinked to polymorphisms controlling the trait under consideration. In this study this assumption seems to be reasonable for two reasons. First, findings of Breseghello and Sorrells (2006) suggest that LD in winter wheat inbreds decays within 5 cM, which is considerably shorter than the average marker distance in our study. Second, the 37 marker loci were randomly selected from the wheat genome. Consequently, our study was based on the assumption that no polymorphisms affecting yield were present in a region of 370 cM, which corresponds to only 10% of the wheat genome (Quarrie *et al.* 2005). Similar to other studies comparing association-mapping approaches based on empirical data (*e.g.*, Yu *et al.* 2006; Zhao *et al.* 2007), however, we cannot rule out the possibility that some markers might be linked to functional polymorphisms of the trait under consideration.

Similar to other studies (*e.g.*, Yu *et al.* 2006; Zhao *et al.* 2007), we used the same markers for estimation of population structure as well as familial relatedness as were used for calculating the MSD between observed and expected *P*-values. Theoretical considerations suggest that by this procedure that the MSDs between observed and expected *P*-values are underestimated for markers that were not included in the estimation of population structure and familial relatedness. However, this issue did not influence our conclusions regarding the eligibility of various methods for association mapping, because they were compared on the basis of the same set of markers.

Our power simulations assumed a QTL that is in complete LD with one marker locus (Yu *et al.* 2006). This assumption maximizes the power for QTL detection. In most empirical studies, however, no markers are available that are in complete LD with the QTL. Therefore, for such studies, a lower power for QTL detection is expected depending on the extent of LD between marker and QTL. A further factor hampering the detection of the QTL of interest, which was neglected in our power simulations, is additional QTL that are linked to the QTL of interest. The incomplete LD between marker and QTL as well as additional linked QTL, however, is expected to reduce the power for QTL detection of all association-mapping methods to the same extent. Therefore, no influence on our conclusions regarding the ranking of various methods for association mapping is expected with respect to the assumptions made in our power simulations.

##### ANOVA approach:

A frequently used method for association mapping in a plant genetics context is the ANOVA approach (*e.g.*, Kraakman *et al.* 2004; Olsen *et al.* 2004), which was used in the current study as a reference method. Under the assumption that the low number of random marker loci in our study is unlinked to the polymorphisms controlling grain yield, association-mapping methods that adhere to the nominal α-level show a uniform distribution of *P*-values. By contrast, we observed a nonuniform distribution of *P*-values with the ANOVA approach (Figure 3). This finding indicates that this method is inappropriate for association mapping in our germplasm set, because it results in a proportion of spurious marker–phenotype associations that is considerably higher than the nominal type I error rate.

In addition to the nonuniform distribution of *P*-values with the ANOVA approach, STRUCTURE revealed eight subpopulations. Consequently, absence of distinct subpopulations in the principal coordinate analysis does not necessarily imply that population structure can be neglected in the association-mapping approach. This might be explained by the fact that the current study was based on germplasm from a line-breeding program of an autogamous species. In contrast to germplasm from hybrid-breeding programs (*cf.* Stich *et al.* 2005), no distinct subpopulations are expected for such germplasm as population structure is disregarded when choosing the parents of a cross. Nevertheless, line breeding generates high levels of population structure and diverse levels of familial relatedness (*cf.* Garris *et al.* 2005).

##### QK approach:

Recently, Yu *et al.* (2006) proposed a new association-mapping approach called the *QK* method. The MSD between observed and expected *P*-values that was found for this method was ~10 times lower than that observed for the ANOVA approach (Table 2), and this difference was considerably larger than the 95% quantile observed in our computer simulations on the correlated beta distribution. This underlines the advantage of the *QK* method over the ANOVA method for association mapping not only in allogamous species such as humans and maize, as suggested by the results of Yu *et al.* (2006), but also in the autogamous species wheat. Similar findings were reported by Zhao *et al.* (2007) for *A. thaliana*.

An association test frequently used in a plant genetics context is the logistic regression-ratio test (Pritchard *et al.* 2000b; Thornsberry *et al.* 2001). The null hypothesis of this test states that the molecular marker under consideration is associated with population structure, whereas under the alternative it is associated both with population structure and with the phenotypic variation. The logistic regression-ratio test and the EIGENSTRAT method (Price *et al.* 2006), recently proposed in a human genetics context, as well as linear models with fixed effects for subpopulations, however, correct only for LD caused by population stratification. The *QK* method, which allows the modeling of population structure and also of familial relatedness, proved to be superior to this class of association-mapping methods with respect to the adherence to the nominal α-level as well as to the adjusted power for QTL detection (*e.g.*, Yu *et al.* 2006; Zhao *et al.* 2007). Therefore, the logistic regression-ratio test and the EIGENSTRAT method, as well as linear models with fixed effects for subpopulations, were not examined in our study.

In our study, the difference between observed and expected *P*-values for the *QK* method was slightly higher than that in the study of Yu *et al.* (2006). This might be explained by (i) less precise kinship estimates resulting from the lower marker density underlying our study and (ii) high levels of population structure and diverse levels of familial relatedness expected in germplasm of an autogamous species (*cf.* Garris *et al.* 2005) selected from plant-breeding programs. These issues did not influence our conclusions regarding the ranking of various methods for association mapping, because they were compared on the basis of the same data set.

Despite promising results for the *QK* association-mapping approach, this method has several drawbacks. Estimation of the **Q** matrix using STRUCTURE is computationally demanding (Balding 2006; Price *et al.* 2006). Even more problematic is that STRUCTURE was designed for unrelated individuals that belong to populations in Hardy–Weinberg equilibrium (Pritchard *et al.* 2000a). For germplasm sets of most species, however, these assumptions are not met and, thus, results of STRUCTURE demand careful interpretation (*cf.* Camus-Kulandaivelu *et al.* 2007). Because of these issues, we examined the *PK* mixed-model association-mapping approach in which the **Q** matrix from STRUCTURE was replaced by a matrix comprising the first eight principal components from the allele-frequency matrix.

##### PK approach:

The MSD between observed and expected *P*-values, which was found for this method, was similar to that observed for the *QK* approach (Figure 3). Furthermore, both methods yielded a similar adjusted power of QTL detection (Figure 4). In accordance with Zhao *et al.* (2007), these findings suggested that the *PK* method is a promising alternative to the *QK* method.

The *QK* method as well as the *PK* method is based on the integration of the fixed effects in the association-mapping model. This leads to a loss of degrees of freedom, which is mainly a problem if the number of entries is low. Furthermore, such approaches hamper the detection of loci contributing to phenotypic differences among subpopulations, because the differences between subpopulations are disregarded in the estimation of the genotypic effects of the loci under consideration. Because of these issues, we examined mixed-model association-mapping approaches that are not based on the assignment of individuals to subpopulations.

##### G approach:

In plant-breeding populations, extensive information about pedigree relationships is available. In our study, pedigree records were used to calculate the **K** matrix for the *G* mixed-model approach. Despite the fact that pedigree information was lacking for about one-third of all inbreds, the MSD between observed and expected *P*-values was slightly lower for this method than that for the ANOVA (Figure 3). The difference in MSD between these two methods was slightly higher when comparing them on the basis of a data set comprising only entries with available pedigree records (data not shown). These observations suggested that for our data set the *G* method is more appropriate for association mapping than the ANOVA approach.

Nevertheless, the MSD between observed and expected *P*-values for the *G* method was considerably higher than those of the *QK* and *PK* methods irrespective of whether the complete data set (Table 2) or a data set comprising only entries with available pedigree records (data not shown) was used. The opposite was true for the adjusted power of QTL detection (Figure 4). These observations suggest that in our study the *G* method was less appropriate for association mapping than the *QK* and *PK* methods. This might be explained by (i) incomplete or wrong pedigree records and (ii) differences between actual coancestry and coancestry computed from pedigree records due to selection and genetic drift (Bernardo 1993; Schut *et al.* 1997; Tams *et al.* 2004).

##### K approach:

For the mixed-model association-mapping approach *K*, we observed a lower value for the MSD between observed and expected *P*-values than that calculated for the *G* method irrespective of whether the complete data set (Table 2) or a data set comprising only entries with available pedigree records (data not shown) was used. This observation indicated that kinship coefficients estimated from molecular marker data are more appropriate than coancestry coefficients calculated from pedigree records. Nevertheless, for the *K* method the MSD was higher than that observed for the *QK* method as well as *PK* method, and our results on the correlated beta distribution suggested that this difference is considerably larger than what is expected at random. This result might be explained by the fact that the software package SPAGeDi (Hardy and Vekemans 2002), proposed in the study of Yu *et al.* (2006) for calculation of the kinship coefficients, assumes that random pairs of individuals of the germplasm set under consideration are unrelated and assigns them a kinship coefficient of 0.

This definition of unrelated individuals seems to be arbitrary. Furthermore, it results in a kinship matrix for which a large number of pairwise kinship estimates are negative. Yu *et al.* (2006) replaced these negative values by 0, arguing that such pairs of individuals are less related than random pairs of individuals. This approach ignores information on the structure of unrelated individuals, which was composed in the kinship matrix, and consequently necessitates the inclusion of the **Q** matrix from STRUCTURE in the mixed model. This suggests examining mixed-model association-mapping approaches that are based on **K** matrices calculated for different thresholds *T*.

#### Approaches based on K matrices calculated for different values of *T*:

For the *QK _{T}*,

*PK*, and

_{T}*K*methods, the optimum value of

_{T}*T*, which was calculated for the current data set using a REML approach, was 0.775 (Figure 2). The value of

*T*estimated in this way was in good accordance with the optimum

*T*identified using the MSD profiles. This observation suggested that for association-mapping approaches the optimum

*T*value might be identified using a REML approach.

Because the REML-based deviance, used to estimate *T*, can be compared only among models that are based on the same set of fixed effects, we used the MSD between observed and expected *P*-values for comparison of the *QK _{T}*,

*PK*, and

_{T}*K*method. The MSD profiles of

_{T}*QK*and

_{T}*PK*had their global minimum at

_{T}*T*= 0.70, while that of the

*K*method was found for

_{T}*T*= 0.35 (Figure 2). This observation might be explained by the fact that for an association-mapping model, which is not based on the assignment of individuals to subpopulations, lower values for

*T*reduce the number of negative pairwise kinship estimates. Thereby, the use of information concerning the structure of unrelated individuals, which was composed in the kinship matrix

**K**

_{T}, is improved.

The MSD observed for the *K*_{0.35} method was slightly lower than that of the *QK* as well as the *PK* method (Table 2). The opposite was true for the adjusted power of QTL detection (Figure 4). These findings suggested that the *K*_{0.35} method, which is based on the optimum **K**_{T} matrix, performed slightly better than the *QK* and *PK* methods. Furthermore, the *K*_{0.35} method avoids the previously described shortcomings of association-mapping methods that are based on the assignment of individuals to subpopulations. By contrast, the MSD of methods *QK*_{0.70} and *PK*_{0.70} is considerably lower than that of the *K*_{0.35} method, whereas higher values for the adjusted power of QTL detection were observed for the former. Therefore, the *QK*_{0.70} and *PK*_{0.70} methods were the most appropriate methods for association mapping in the examined data set.

##### K_{unrel} approach:

Lynch (1988) and Melchinger *et al.* (1991) proposed to estimate *T* as the average proportion of marker loci with shared variants between two sets of genotypes: (i) the entries and (ii) genotypes that are unrelated by pedigree to the entries. The *T* value calculated in the current study on the basis of five wheat cultivars, which are unrelated by pedigree to the 303 entries, was 0.30. This value is in good accordance with the *T* value of 0.35 estimated on the basis of the MSD profile for the *K _{T}* method, suggesting that this approach might be used in studies on genetic diversity where no phenotypic data are available. The MSD for

*K*

_{0.35}, however, was lower than that of

*K*

_{unrel}. Furthermore, the optimum

*T*for the

*QK*and

_{T}*PK*methods was considerably higher than that estimated on the basis of genotypes unrelated by pedigree. These observations indicated that in association-mapping studies and especially in studies requiring fixed subpopulation effects, estimation of

_{T}*T*based on MSD or likelihood profiles are more promising than estimation based on genotypes unrelated by pedigree alone.

#### Comparison of one- and two-stage association-mapping approaches:

In all types of genetic mapping experiments, the one-step approach, in which the phenotypic and genotypic data analysis is performed in one step, is fully efficient (Cullis *et al.* 1998). Consequently, *P*-values calculated for the marker loci under consideration on the basis of such a statistical model are the reference values (Piepho and Pillen 2004). To our knowledge, however, only two-stage association-mapping approaches were applied in all earlier association-mapping studies with plants, *i.e.*, entry means or adjusted entry means were calculated in the first step and then used for association mapping in the second step. Therefore, we compared one- and two-stage association-mapping approaches.

The lowest MSD values among the one-stage association-mapping approaches were observed for the *QK*_{0.70}, *PK*_{0.70}, *K*_{0.70}, and *K*_{0.35} methods, which were also the most appropriate methods for two-step association mapping (Table 2). For these methods, the MSD of the one-stage approaches was lower than that for the corresponding two-stage association approaches, indicating that in our data set the former were more appropriate for association mapping than the latter, although the differences were rather small. Furthermore, for the association-mapping methods based on **K**_{T} matrices, high correlation coefficients between *P*-values calculated for all marker loci on the basis of two-stage association-mapping approaches and the corresponding one-stage association approaches were found. These observations suggest that our data set could be analyzed by two-step association-mapping methods, using **K**_{T} without increasing the empirical type I error rate too much in comparison to the corresponding one-step approaches.

#### Conclusions:

The results of our study indicate that the ANOVA approach is inappropriate for association mapping in the examined germplasm set. Furthermore, our observations suggest that the *QK* method is appropriate for association mapping not only in allogamous species such as humans and maize (Yu *et al.* 2006), but also in the autogamous species wheat, when the examined data set is similar in size compared to that of our study. Nevertheless, we recommend replacing the **K** matrix of the *QK* and *PK* approaches by a **K**_{T} matrix, which is based on a REML estimate of the probability that two inbreds carry alleles at the same locus that are identical in state but not identical by descent and, thus, increases (i) the adherence to the nominal α-level as well as (ii) the adjusted power of QTL detection. Finally, we showed that our data set might be analyzed using the newly proposed two-step association-mapping method without increasing the empirical type I error rate too much in comparison to the corresponding one-step approaches.

## Acknowledgments

We thank Lochow-Petkus for providing phenotypic and genotypic data. The authors appreciate the editorial work of J. Muminović, whose suggestions considerably improved the style of the manuscript. The authors thank the associate editor and two anonymous reviewers for their valuable suggestions. This research was conducted within the Breeding and Informatics project of the Genome Analysis of the Plant Biological System (GABI) initiative (http://www.gabi.de). E.S.B. was supported by the National Science Foundation (DBI-0321467).

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