Comparison of mixedmodel approaches for association mapping in rapeseed, potato, sugar beet, maize, and Arabidopsis
 Benjamin Stich^{1} and
 Albrecht E Melchinger^{1}Email author
DOI: 10.1186/147121641094
© Stich and Melchinger; licensee BioMed Central Ltd. 2009
Received: 03 June 2008
Accepted: 27 February 2009
Published: 27 February 2009
Abstract
Background
In recent years, several attempts have been made in plant genetics to detect QTL by using association mapping methods. The objectives of this study were to (i) evaluate various methods for association mapping in five plant species and (ii) for three traits in each of the plant species compare the T_{ opt }, the restricted maximum likelihood (REML) estimate of the conditional probability that two genotypes carry at the same locus alleles that are identical in state but not identical by descent. In order to compare the association mapping methods based on scenarios with realistic estimates of population structure and familial relatedness, we analyzed phenotypic and genotypic data of rapeseed, potato, sugar beet, maize, and Arabidopsis. For the same reason, QTL effects were simulated on top of the observed phenotypic values when examining the adjusted power for QTL detection.
Results
The correlation between the T_{ opt }values identified using REML deviance profiles and profiles of the mean of squared difference between observed and expected P values was 0.83.
Conclusion
The mixedmodel association mapping approaches using a kinship matrix, which was based on T_{ opt }, were more appropriate for association mapping than the recently proposed QK method with respect to the adherence to the nominal α level and the adjusted power for QTL detection. Furthermore, we showed that T_{ opt }differs considerably among the five plant species but only marginally among different traits.
Background
Artificially induced variations, such as mutations, have been successfully used for gene identification in genetic and physiological studies [1]. Development of DNA markers, however, has made it possible to study the naturally occuring allelic variation underlying complex traits [2, 3]. In many plant species, the approaches for detecting quantitative trait loci (QTL) relied so far on segregating populations derived from crosses between inbred lines. These QTL detection procedures, commonly referred to as linkage mapping, have major limitations, that include high costs [4] and a poor resolution in detecting QTL. Moreover, with biparental crosses of inbred lines only two alleles at any given locus can be studied simultaneously [5]. Association mapping methods, which have been successfully applied in human genetics to detect genes coding for human diseases [6], promise to overcome these limitations [7]. Therefore, in recent years several attempts have been made in a plant genetics context to detect QTL by using such methods [7–10].
Linkage disequilibrium (LD) in linkage mapping populations is caused by genetic linkage [9]. In contrast, LD in association mapping populations can also be the consequence of population structure, relatedness, genetic drift, and selection [5, 11]. Therefore, the success of association mapping efforts depends on the ability to separate LD due to linkage from LD due to other causes. To correct for LD caused by population structure, linear models accounting for subpopulation effects [8] or a logistic regression ratio test [12, 13] were proposed. Owing to the large germplasm sets required for dissecting complex traits [14], the probability of including partially related individuals increases. This applies in particular when genotypes selected from plant breeding populations are used for association mapping [7, 9, 13]. However, the abovementioned approaches fail to adhere to the nominal α level if the germplasm set analyzed comprises related individuals [13].
The recently proposed QK mixedmodel for association mapping promises to correct for LD caused by population structure and familial relatedness [15]. The authors demonstrated the suitability of their new method for association mapping in two allogamous species, humans and maize. The suitability of the QK method, however, has to be evaluated in plant species with different reproduction systems covering a wide range of population structure and familial relatedness.
In contrast to coancestry coefficients calculated from pedigree records, markerbased kinship estimates may account for the effects of deviations from expected parental contributions to progeny due to selection or genetic drift [16]. Therefore, markerbased kinship estimates might be more appropriate for association mapping approaches than coancestry coefficients calculated from pedigree records [15, 17]. A difficulty with calculation of markerbased kinship estimates is the definition of unrelated individuals [18]. The markerbased kinship matrix might be determined based on the definition that random pairs of genotypes are unrelated [15] or that pairs of genotypes are unrelated if they have no allele in common [17]. However, both definitions seem to be arbitrary. Recently, it was proposed to estimate by restricted maximum likelihood (REML) T_{ opt }, the conditional probability that marker alleles are alike in state, given that they are not identical by descent [19], using genotypic and phenotypic data [20]. However, no study compared this estimation of unrelated individuals among plant species with different reproduction systems as well as for various phenotypic traits.
The objectives of our study were to evaluate various methods for association mapping with respect to their adherence to the nominal α level and the adjusted power for QTL detection based on (i) empirical data sets and (ii) computer simulations in five plant species with different reproduction systems. Second, we compared T_{ opt }for three traits in each of the plant species.
Methods
With computer simulations it is hardly possible to simulate data sets showing a population structure and familial relatedness comparable to that of empirical data sets. Nevertheless, to compare association mapping methods with respect to their adherence to the nominal a level based on scenarios with realistic estimates of population structure and familial relatedness, we analyzed phenotypic and genotypic data of rapeseed, potato, sugar beet, maize, and Arabidopsis. For the same reason, QTL effects were simulated on top of the observed phenotypic values when examining the adjusted power for QTL detection.
Plant materials, phenotypic data, and molecular markers
In each of the five plant species, with the exception of Arabidopsis thaliana, we selected three traits with different genetic complexity (presumably low, medium, and high). Detailed descriptions of the examined data sets are available as Additional file 1.
Rapeseed (Brassica napus L.)
We studied a total of n = 136 rapeseed inbreds, proprietary to Norddeutsche Pflanzenzucht HansGeorg Lembke KG (Holtsee, Germany). The entries were evaluated for thousand kernel weight (TKW; g), oil content (OC; %), and oil yield (OY; t/ha). All entries were fingerprinted with m = 59 genomewide distributed simple sequence repeat markers by SaatenUnion Resistenzlabor GmbH (Hovedissen, Germany) following standard protocols.
Potato (Solanum tuberosum L.)
Our study was based on the phenotypic and genotypic data evaluated earlier [21]. Briefly, the n = 184 tetraploid potato clones from the breeding programs of BöhmNordkartoffel Agrarproduktion OHG (Lüneburg, Germany) and SakaRagis Pflanzenzucht GbR (Windeby, Germany) were evaluated for Globodera pallida St. resistance (GPR) [22]. Our statistical analyses were based on the square root of the number of visible nematode cysts. Furthermore, the area under the disease progress curve [23] was used as measure for P. infestans resistance (PIR). In addition, plant maturity (PM) was evaluated in uninfected plants, using a 1 to 9 scale (1 = very early, 9 = very late). All entries were fingerprinted with m = 31 genomewide distributed simple sequence repeat markers [21] by the potato genome analysis group of the Max Planck Institute for Plant Breeding Research (Cologne, Germany). For 21 markers the allele dosage was scored based on relative band intensities.
Sugar beet (Beta vulgaris L.)
We analyzed a total of n = 178 sugar beet inbreds of the pollen parent heterotic pool, proprietary to KWS SAAT AG (Einbeck, Germany). The testcross progenies of these entries with an inbred of the seed parent heterotic pool were evaluated in a series of plant breeding trials. Data were recorded for amino nitrogen (AN) [24], beet yield (BY), and corrected sugar yield (CSY) [25] in % of the mean performance of checks. All entries were fingerprinted with 59 simple sequence repeat markers and 41 single nucleotide polymorphism markers (m = 100), both randomly distributed across the sugar beet genome. The fingerprinting was done by KWS SAAT AG following standard protocols.
Maize (Zea mays L.)
Our study was based on the phenotypic and genotypic data analyzed earlier [15]. In short, the n = 277 maize inbreds representing worldwide genetic diversity were evaluated for ear height (EH; cm), ear diameter (ED; cm), and days to pollen shed (DPS). For all inbreds, genotypic data of m = 553 genomewide distributed single nucleotide polymorphism markers was available.
Arabidopsis thaliana L.
Our study was based on the n = 95 Arabidopsis thaliana L. inbreds for which phenotypic information was available [17]. These inbreds represent worldwide genetic diversity of Arabidopsis. We examined the normalized gene expression of FLOWERING LOCUS C (FLC) and FRIGIDA (FRI) as well as the number of days from germination to first opening of flowers under long day conditions with vernalisation treatment (LDV). For these inbreds, resequencing data of m = 876 genomewide distributed short fragments was available [26]. To reduce the computational load, we used only the central single nucleotide polymorphism marker of each fragment.
The anonymised data sets of rapeseed, potato, and sugar beet are available upon request from the authors.
Statistical analyses
The empirical type I error rate of associationmapping approaches based on adjusted entry means (twostep approaches) is only slightly higher than that of approaches in which the phenotypic data analysis and the association analysis were performed in one step (onestep approaches) [20]. Therefore, in a first step we analyzed the phenotypic data and calculated adjusted entry means (rapeseed, potato) or entry means (sugar beet, maize, and Arabidopsis) M_{ i }for each individual under consideration (Additional file 2). These estimates were then used in a second step for the association analyses.
Association analyses
Methods used for association mapping and the corresponding statistical models.
Method  Statistical model  Population structure matrix D  Kinship matrix K 

ANOVA  ${M}_{i}=\mu +{\alpha}^{\prime}{X}_{i}+{e}_{i}$     
K  ${M}_{i}=\mu +{\alpha}^{\prime}{X}_{i}+{g}_{i}^{\ast}+{e}_{i}$    SPAGeDi 
Q_{1}K  ${M}_{i}=\mu +{\alpha}^{\prime}{X}_{i}+{\displaystyle {\sum}_{u=1}^{z}{D}_{iu}{v}_{u}+}{g}_{i}^{\ast}+{e}_{i}$  STRUCTURE; ΔK criterion  SPAGeDi 
Q_{2}K  ${M}_{i}=\mu +{\alpha}^{\prime}{X}_{i}+{\displaystyle {\sum}_{u=1}^{z}{D}_{iu}{v}_{u}+}{g}_{i}^{\ast}+{e}_{i}$  STRUCTURE; Log likelihood  SPAGeDi 
PK  ${M}_{i}=\mu +{\alpha}^{\prime}{X}_{i}+{\displaystyle {\sum}_{u=1}^{z}{D}_{iu}{v}_{u}+}{g}_{i}^{\ast}+{e}_{i}$  Principal components; explaining simultaneously 25% of the variance  SPAGeDi 
K_{ T }  ${M}_{i}=\mu +{\alpha}^{\prime}{X}_{i}+{g}_{i}^{\ast}+{e}_{i}$    ${K}_{Tij}=\frac{{S}_{ij}1}{1T}+1$; T = 0,0.025, ..., 0.975 
Q_{1}K_{ T }  ${M}_{i}=\mu +{\alpha}^{\prime}{X}_{i}+{\displaystyle {\sum}_{u=1}^{z}{D}_{iu}{v}_{u}+}{g}_{i}^{\ast}+{e}_{i}$  STRUCTURE; ΔK criterion  T = 0,0.025, ..., 0.975 
Q_{2}K_{ T }  ${M}_{i}=\mu +{\alpha}^{\prime}{X}_{i}+{\displaystyle {\sum}_{u=1}^{z}{D}_{iu}{v}_{u}+}{g}_{i}^{\ast}+{e}_{i}$  STRUCTURE; Log likelihood  T = 0,0.025, ..., 0.975 
PK_{ T }  ${M}_{i}=\mu +{\alpha}^{\prime}{X}_{i}+{\displaystyle {\sum}_{u=1}^{z}{D}_{iu}{v}_{u}+}{g}_{i}^{\ast}+{e}_{i}$  Principal components; explaining simultaneously 25% of the variance  T = 0,0.025, ..., 0.975 
Description of the examined data sets.
Parameter  Rapeseed  Potato  Sugar beet  Maize  Arabidopsis 

n  136  184  178  277  95 
Entry type  Inbred line  Noninbred clone  Inbred line  Inbred line  Inbred line 
Phenotypic data  
Trait 1  Thousand kernel weight  Resistance to G. pallida  Amino nitrogen  Ear height  Norm. gene expression of FLC 
Abbrev.  TKW  GPR  AN  EH  FLC 
Unit  g  $\sqrt{\text{No}\text{.ofnematodecysts}}$  %  cm  % 
h ^{2}  0.78  0.98  0.89     
Range M_{ i }  3.0–4.6  0.4–19.5  71.1–226.2  8–136  0.021–6.270 
Trait 2  Oil content  Resistance to P. infestans  Beet yield  Ear diamter  Norm. gene expression of FRI 
Abbrev.  OC  PIR  BY  ED  FRI 
Unit  %  Area under disease progress curve  %  mm  % 
h ^{2}  0.81  0.77  0.90     
Range M_{ i }  46.1–51.7  6.4–165.1  84.8–113.6  23.7–46.4  0.211–4.386 
Trait 3  Oil yield  Plant maturity  Corrected sugar yield  Days to pollen shed  Flowering time 
Abbrev.  OY  PM  CSY  DPS  LDV 
Unit  t/ha  Rating scale 1 to 9  %  No. of days  No. of days 
h ^{2}  0.50  0.94  0.81     
Range M_{ i }  2.2–3.0  3.4–9.5  87.8–108.7  54.5–82.5  18.7–55.7 
Genotypic data  
Type of markers  SSRs  SSRs  SSRs & SNPs  SNPs  SNPs 
m  59  31  100  553  876 
Avg. allele freq.  0.37  0.18  0.30  0.50  0.50 
where α were the effects of allele substitution of the marker under study, x_{ i }a column vector with the number of copies of the corresponding alleles, and e_{ i }the residual.
where v_{ u }was the effect of the u th column of the population structure matrix D and ${g}_{i}^{\ast}$ was the residual genetic effect of the i th entry. The matrix D, which comprised z linear independent columns, differed among the examined mixedmodel association mapping methods (Table 1), which is why it is described in the paragraphs on the individual methods. The variances of the random effects g* = $\{{g}_{1}^{\ast},\mathrm{...},{g}_{n}^{\ast}\}$ and e = {e_{1}, ..., e_{ n }} were assumed to be Var(g*) = $2K{\sigma}_{{g}^{\ast}}^{2}$ and Var(e) = $R{\sigma}_{r}^{2}$, where K was a n × n matrix of kinship coefficients that define the degree of genetic covariance between all pairs of entries. ${\sigma}_{{g}^{\ast}}^{2}$ was the residual genetic variance and ${\sigma}_{r}^{2}$ the residual variance, both estimated by REML. R was an n × n matrix in which the offdiagonal elements were 0 and the diagonal elements were reciprocals of the number of phenotypic observations underlying each entry mean or adjusted entry mean [15].
The K method was based on the above described mixedmodel with the difference that it did not include any v_{ u }effects (Table 1). The kinship matrix K was calculated based on all marker data using the software package SPAGeDi [28], where negative kinship values between entries were set to 0.
The Q_{1}K and Q_{2}K methods were based on the above described mixedmodel. For these two methods, the population structure matrices Q_{1} and Q_{2}, which were calculated using software STRUCTURE [29] and described in the following paragraphs, were used as D matrix. In our investigations, the set of n entries was analyzed by setting z from 0 to 14 in each of five repetitions. For each run of STRUCTURE, the burnin time as well as the iteration number for the Markov Chain Monte Carlo algorithm were set to 100 000 [30].
For the Q_{1} matrix, the number of subpopulations was estimated based on the adhoc criterion ΔK [31]. In contrast, for the Q_{2} matrix, we used the run with the highest log likelihood to and the lowest number of subpopulations [32]. The z + 1 columns of both, the Q_{1} and Q_{2} matrix, add up to one and, thus, only the first z columns were used as D matrix of the Q_{1}K and Q_{2}K method, respectively, to achieve linear independence. The Q_{1}K and Q_{2}K methods were based on the same kinship matrix K as used for the K method.
We used the first p principal components of an allele frequency matrix as D matrix of the PK method (Table 1) [17]. p was chosen in such a way that the explained variance of the first p principal components was about 25%. The PK method was based on the same kinship matrix K as used for the K method.
where S_{ ij }is the proportion of marker loci with shared variants between inbreds i and j [20]. We examined T = 0, 0.025, ..., 0.975 to obtain a REML estimate of T, which is the conditional probability that marker alleles are alike in state, given that they are not identical by descent.
Measures for comparison of association mapping methods
The mean squared difference (MSD) between observed and expected P values of all marker loci was calculated as measure for the adherence to the nominal α level [20]. High MSD values indicate that the empirical type I error rate of these approaches is considerably higher than the nominal α level. Computer simulations were performed based on a bivariate betadistribution [33] to examine which difference in MSD values between two association mapping methods could be expected purely by chance [20]. For each trait of each plant species, we investigated five pairs of association mapping approaches (i) Q_{1}K/ANOVA, (ii) Q_{1}K/K, (iii) Q_{1}K/Q_{2}K, (iv) Q_{1}K/PK, and (v) Q_{1}K/${Q}_{1}{\text{K}}_{{T}_{opt}}$.
For each of the five plant species, the Pearson correlation coefficient between the observed P values of all association mapping methods was calculated for the trait with medium genetic complexity.
Power simulations
The power to detect a biallelic QTL of interest, which explained a fraction of the phenotypic variance and was in complete LD with one marker locus, was examined as described in detail previously [20]. Briefly, the QTL effect G_{ r }, calculated as r = 0.1 multiplied by the standard deviation of the vector of adjusted entry means m = (M_{1}, M_{ i }, ..., M_{ n }) of the n entries, was assigned in consecutive simulation runs to each of the detected marker alleles whereas all other alleles were assigned the genotypic effect 0. In each simulation run, the phenotypic value of each entry i was calculated by summing up the QTL effect of the alleles and the adjusted entry mean M_{ i }. All association mapping methods were run on the phenotypic values of the entries to determine whether the QTL can be detected. To adjust the association mapping methods for their different empirical type I error rates, we calculated the adjusted power as the proportion of QTL detected, based on the nominal α for which the empirical type I error rate α* was 0.05. In addition to r = 0.1, we examined r = 0.4, 0.7, ..., 1.9. The percentage (π) of the total phenotypic variation explained by a QTL effect G_{ r }was calculated [15].
All mixedmodel calculations were performed with ASReml release 2.0 [34].
Results
For each trait examined in the current study, considerable variation was observed for the entry means or adjusted entry means M_{ i }(Table 2). The total number of marker alleles detected for rapeseed, potato, sugar beet, maize, and Arabidopsis was 331, 158, 176, 1106, and 1752, respectively. The average allele frequency ranged from 0.18 for potato to 0.50 for maize and Arabidopsis.
The modelbased approach of STRUCTURE revealed z + 1 = two, two, two, five, and six subpopulations for rapeseed, potato, sugar beet, maize, and Arabidopsis, respectively, when using the adhoc criterion ΔK. In contrast, based on SBC, the number of subpopulations revealed by STRUCTURE was 11, 15, 10, 15, and 5. For rapeseed, potato, sugar beet, maize, and Arabidopsis, the minimum number of principal components p explaining simultaneously 25% of the variance was 4, 5, 4, 13, and 8, respectively.
Mean of squared differences (MSD) between observed and expected P values for various association mapping methods in five plant species.
Method  Rapeseed  

TKW  OC  OY  
ANOVA  0.0624  0.0326  0.0523 
K  0.0098  0.0053  0.0016 
Q_{1}K  0.0021  0.0047  0.0061 
Q_{2}K  0.0013  0.0010  0.0192 
PK  0.0008  0.0007  0.0026 
Potato  
GPR  PIR  PM  
ANOVA  0.1928  0.0947  0.1534 
K  0.0499  0.0162  0.0604 
Q_{1}K  0.0122  0.0179  0.0389 
Q_{2}K  0.0181  0.0017  0.0063 
PK  0.0189  0.0183  0.0422 
Sugar beet  
AN  BY  CSY  
ANOVA  0.1526  0.1625  0.1533 
K  0.0136  0.0191  0.0173 
Q_{1}K  0.0060  0.0239  0.0051 
Q_{2}K  0.0118  0.0167  0.0081 
PK  0.0090  0.0137  0.0065 
Maize  
EH  ED  DPS  
ANOVA  0.0333  0.0147  0.0909 
K  0.0003  0.0002  0.0006 
Q_{1}K  0.0002  0.0003  0.0002 
Q_{2}K  0.0002  0.0002  0.0003 
PK  0.0003  0.0005  0.0002 
Arabidopsis  
FLC  FRI  LDV  
ANOVA  0.0040  0.0004  0.0070 
K  0.0006  0.0022  0.0013 
Q_{1}K  0.0026  0.0033  0.0017 
Q_{2}K  0.0019  0.0022  0.0013 
PK  0.0021  0.0034  0.0018 
T values for which the lowest deviance or the lowest mean of squared differences between observed and expected P values were found for various association mapping methods in five plant species.
Mixedmodel method  deviance  MSD  deviance  MSD  deviance  MSD 

Rapeseed  
TKW  OC  OY  
K_{ T }  0.725  0.350 (0.0068)  0.800  0.475 (0.0006)  0.750  0.400 (0.0011) 
Q_{1}K_{ T }  0.775  0.700 (0.0039)  0.800  0.425 (0.0007)  0.775  0.700 (0.0011) 
Q_{2}K_{ T }  0.700  0.825 (0.0009)  0.800  0.850 (0.0007)  0.900  0.900 (0.0128) 
PK_{ T }  0.725  0.700 (0.0006)  0.800  0.725 (0.0004)  0.750  0.750 (0.0019) 
Potato  
GPR  PIR  PM  
K_{ T }  0.525  0.500 (0.0065)  0.625  0.550 (0.0034)  0.475  0.475 (0.0082) 
Q_{1}K_{ T }  0.600  0.600 (0.0054)  0.625  0.550 (0.0033)  0.475  0.525 (0.0086) 
Q_{2}K_{ T }  0.600  0.600 (0.0091)  0.625  0.500 (0.0010)  0.625  0.525 (0.0031) 
PK_{ T }  0.575  0.550 (0.0121)  0.625  0.550 (0.0048)  0.475  0.475 (0.0153) 
Sugar beet  
AN  BY  CSY  
K_{ T }  0.575  0.325 (0.0022)  0.475  0.300 (0.0022)  0.475  0.350 (0.0012) 
Q_{1}K_{ T }  0.575  0.325 (0.0023)  0.450  0.300 (0.0059)  0.475  0.375 (0.0006) 
Q_{2}K_{ T }  0.475  0.325 (0.0029)  0.475  0.275 (0.0021)  0.575  0.350 (0.0009) 
PK_{ T }  0.475  0.300 (0.0019)  0.350  0.300 (0.0034)  0.475  0.350 (0.0009) 
Maize  
EH  ED  DPS  
K_{ T }  0.575  0.450 (0.0002)  0.575  0.575 (0.0001)  0.600  0.575 (0.0002) 
Q_{1}K_{ T }  0.575  0.500 (0.0001)  0.725  0.575 (0.0003)  0.600  0.575 (0.0001) 
Q_{2}K_{ T }  0.875  0.475 (0.0003)  0.725  0.525 (0.0001)  0.600  0.525 (0.0001) 
PK_{ T }  0.875  0.475 (0.0002)  0.725  0.525 (0.0001)  0.600  0.600 (0.0001) 
Arabidopsis  
FLC  FRI  LDV  
K_{ T }  0.875  0.875 (0.0004)  0.825  0.975 (0.0007)  0.875  0.900 (0.0034) 
Q_{1}K_{ T }  0.875  0.975 (0.0023)  0.875  0.800 (0.0028)  0.875  0.900 (0.0020) 
Q_{2}K_{ T }  0.875  0.950 (0.0006)  0.875  0.800 (0.0018)  0.875  0.900 (0.0017) 
PK_{ T }  0.875  0.775 (0.0015)  0.925  0.925 (0.0025)  0.900  0.900 (0.0011) 
The 95% quantile of differences in MSD calculated for the five pairs of association methods Q_{1}K/ANOVA, Q_{1}K/K, Q_{1}K/Q_{2}K, Q_{1}K/PK, and Q_{1}K/${Q}_{1}{\text{K}}_{{T}_{opt}}$ was highest for potato and ranged from 0.0041 to 0.0114 (Additional file 5). For Arabidopsis, the 95% quantile of differences in MSD was lowest and varied from 0.0001 to 0.0004.
The Pearson correlation coefficient between the observed P values of all examined association mapping methods ranged from 0.05 to 0.99 (Additional file 6)
Discussion
Assumptions underlying the comparison of association mapping approaches using empirical data sets
Simulation of data sets mimicing the population structure and familial relatedness of empirical data sets is hardly possible. However, only with such data sets a reliable assessment of the performance of different association mapping approaches is possible. Therefore, our study was based on empirical data sets.
Investigations on the type I error rate and on the adjusted power to detect QTL of association mapping approaches using empirical data sets require that the examined marker loci are unlinked to polymorphisms controlling the expression of the trait under consideration. In the present study, this assumption seems to be reasonable as for the five plant species examined the available marker density was considerably lower than that required for genomewide association mapping. Similarly to other studies comparing association mapping approaches based on empirical data [15, 17], however, we cannot rule out the possibility that some markers might be linked to functional polymorphisms of the traits under consideration.
In accordance with previous studies [15, 17], 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 MSD values calculated in this way might underestimate the MSD values for markers which are not included in the estimation of population structure and familial relatedness such as markers in candidate genes. However, our computer simulations on the Arabidopsis dataset, in which the half of the available markers were used for estimation of population structure and familial relatedness and the remaining markers for calculation of the MSD values, suggested that this underestimation is negligible (data not shown). This result indicates that association mapping methods, for which we observed MSD values close to zero, will also adhere to the nominal α level in empirical association mapping experiments.
Our power simulations assumed a QTL allele which is in complete LD with one marker allele. This assumption allows the comparison of results from various plant species irrespective of the available number of markers. However, it maximizes the power for QTL detection. In most empirical studies no markers are available which 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 influencing the detection of the QTL of interest, which was neglected in our power simulations, are additional QTL that are linked to the QTL of interest. Incomplete LD between marker and QTL as well as additional linked QTL are expected to alter the power of QTL detection in all association mapping methods to a similar 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.
Comparison of various association mapping approaches
ANOVA approach
A frequently used method for association mapping in a plant genetics context is the ANOVA approach [10]. This approach was therefore used in our study as reference method. Under the assumption that the random marker loci in our study are unlinked to the polymorphisms controlling the expression of the traits under consideration, association mapping methods that adhere to the nominal α level show a uniform distribution of P values, i.e., a MSD value close to zero. With the exception of the normalized gene expression data of the FRI gene in Arabidopsis, we observed a nonuniform distribution of P values in the ANOVA approach of all traits (Table 3). This finding is in accordance with the results of previous studies [15, 17, 20] and indicates that the ANOVA approach is inappropriate for association mapping in the examined plant species, because the resulting proportion of spurious markerphenotype associations is considerably higher than the nominal type I error rate.
QK approach
The recently proposed QK mixedmodel association mapping method promises to correct for multiple levels of relatedness [15]. The MSD between observed and expected P values found for the Q_{1}K and Q_{2}K methods of all examined traits was considerably lower than that observed for the ANOVA approach (Table 3). Furthermore, this difference in MSD values was considerably larger than the 95% quantile observed based on the computer simulations (Additional file 5). These findings suggest the advantage of the Q_{1}K and Q_{2}K methods over the ANOVA method for association mapping not only in maize and Arabidopsis for which similar results were previously reported [15, 17] but also in rapeseed, potato, and sugar beet.
For estimation of the number of subpopulations using STRUCTURE [29], ΔK, an ad hoc criterion related to the second order rate of change in the log likelihood of data, was proposed [31]. In other studies, the number of subpopulations z+1 was chosen in such a way that a further increase in z did not considerably improve the log likelihood of data [35]. We used these two criteria to estimate the number of subpopulations for the Q_{1} and Q_{2} matrices.
For some traits, we observed a smaller MSD value for the Q_{1}K than for the Q_{2}K method, whereas the opposite was true for the other traits (Table 3). Furthermore, with few exceptions, these differences were smaller than the corresponding 95% quantiles observed in our computer simulations on the correlated betadistribution (Additional file 5). These findings demonstrate that the association mapping models based on the two population structure matrices, Q_{1} and Q_{2}, are equally appropriate for association mapping with respect to (i) adherence to the nominal α level as well as (ii) the adjusted power for QTL detection.
Despite promising results for the Q_{1}K and Q_{2}K association mapping approaches, these methods have several drawbacks, as previously discussed [20]. Therefore, we examined two association mapping methods which were not based on the population structure matrix from STRUCTURE. For the PK mixedmodel association mapping approach, the Q_{1} or Q_{2} matrix from STRUCTURE was replaced by a matrix comprising p principal components (Table 1). In contrast, the K method was based on a mixedmodel which does not include any v_{ u }effects.
PK approach
The MSD between observed and expected P values, which was found for this method, was similar to those observed for the Q_{1}K and Q_{2}K methods (Table 3). Furthermore, all three methods yielded a similar adjusted power of QTL detection across the examined plant species (Fig. 2). These findings were in accordance with those of previous studies [17, 20], suggesting that the PK approach is a promising alternative to the Q_{1}K and Q_{2}K methods.
K approach
For the K approach, we observed for most examined traits a higher MSD value than for the mixedmodel methods Q_{1}K, Q_{2}K, and PK. The opposite result was observed with respect to the adjusted power of QTL detection (Fig. 2).
These results indicated that the K approach was less appropriate for association mapping than the approaches based on the integration of fixed effects in the statistical model. This conclusion may be explained by the fact that the software package SPAGeDi [28] used 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 results in a kinship matrix for which a large number of pairwise kinship estimates are negative. It was proposed to replace these negative values by 0, because such pairs of individuals are less related than random pairs of individuals [15]. This approach, however, ignores information on the structure of unrelated individuals, which was captured in the kinship matrix, and consequently necessitates the inclusion of fixed effects in the mixedmodel. Therefore, we examined mixedmodel association mapping approaches which are based on K matrices calculated for different thresholds T [20].
Approaches based on K matrices calculated for different values of T
The values of T_{ opt }calculated for the current data sets using the REML approach, which might also be used to infer the probability of identity by descent for genotypes with no pedigree information available, were not always identical with those identified based on the MSD profiles (Table 4). Across all plant species, traits, and association mapping methods, however, the correlation between the T_{ opt }value identified based on both approaches was 0.83 (Additional file 4). This result suggested that for association mapping approaches the T_{ opt }value might be identified using the REML approach because it is associated with a lower computational load. The REMLbased deviance, used to estimate T_{ opt }, however, can only be compared among models which are based on the same set of fixed effects. Therefore, we used the MSD between observed and expected P values for comparison of the Q_{1}K_{ T }, Q_{2}K_{ T }, PK_{ T }, and K_{ T }method and furthermore used the T_{ opt }values identified based on this criterion.
The MSD values observed for the association mapping approaches based on the T_{ opt }value, were considerably lower than that of the corresponding association mapping approaches based on the K matrix from SPAGeDi, for all examined plant species and traits (Table 4). Furthermore, the adjusted power observed for the former approaches was for most examined traits higher than that observed for the latter approaches. These findings suggest that methods based on a kinship matrix calculated for the T_{ opt }value are more appropriate for association mapping than the corresponding association mapping approaches which are based on the K matrix from SPAGeDi. Nevertheless, the MSD values observed for the association mapping methods, which include fixed effects such as the ${Q}_{1}{\text{K}}_{{T}_{opt}}$, ${Q}_{2}{\text{K}}_{{T}_{opt}}$, or ${\text{PK}}_{{T}_{opt}}$, were lower than that of the ${\text{K}}_{{T}_{opt}}$. Therefore, in our study the ${Q}_{1}{\text{K}}_{{T}_{opt}}$, ${Q}_{2}{\text{K}}_{{T}_{opt}}$ or ${\text{PK}}_{{T}_{opt}}$ are the most appropriate methods for association mapping.
Comparison of the properties of association mapping approaches among plant species and traits
MSD values
The MSD values observed for potato and sugar beet across all association mapping methods were considerably higher than those for maize and Arabidopsis, whereas those for rapeseed were of medium size (Table 3). This may be due to the low number of random molecular markers available in our study for potato, sugar beet, and rapeseed. Thereby, not very precise estimation of population structure is possible which in turn increases the MSD values.
To examine this issue in more detail, random markers were selected in replicated simulation runs from maize and Arabidopsis linkage maps in such a way that the total number of alleles of the selected markers corresponds to those observed for the other three species. All association mapping methods were then run with these markers. Our results (data not shown) suggested that the low number of random molecular markers for potato, sugar beet, and rape seed only partially explains the observed differences in MSD values.
Another factor that explains the observed difference in MSD values among the plant species is the difference in the extent of population structure and relatedness present in the examined genetic materials. This difference in population structure and relatedness may partly be due to the fact that the entries of the examined plant species differ in their origin. While the Arabidopsis entries were selected from natural populations, the entries of the other four plant species were chosen from plant breeding programs. Because entries selected from plant breeding programs have a complex ancestry, the extent of population structure and relatedness in such germplasm sets is expected to be higher than in germplasm sets consisting of entries selected from natural populations.
In addition, the difference in the extent of population structure and relatedness between rapeseed, potato, sugar beet, and maize can be explained by the different sampling strategies underlying the examined genetic materials. The entries of the maize data set represent worldwide genetic diversity, whereas the genetic materials of rapeseed, potato, and sugar beet were sampled from commercial plant breeding programs. Theoretical considerations suggest that this increases the probability of including partially related entries.
Furthermore, the difference in the extent of population structure and relatedness between rapeseed, potato, sugar beet, and maize may partly be due to the different reproduction systems and types of varieties bred in a particular crop. For entries from hybrid breeding programs [11] such as sugarbeet and maize, distinct subpoulations are expected. In contrast, when line or clonal varieties are bred, as in the case of rapeseed and potato, no distinct subpopulations are expected to develop as population structure is disregarded when choosing the parents of a cross. Nevertheless, this procedure is expected to generate diverse levels of familial relatedness [36].
Adjusted power for QTL detection
Across all examined statistical methods for association mapping, considerable differences in the adjusted power for QTL detection were observed for the five examined plant species (Fig. 2). The adjusted power is influenced by (i) the size of the QTL effect G_{ r }, (ii) the extent of LD between marker allele and QTL allele, (iii) the number of entries n, (iv) the QTL allele frequency, and (v) the heritability of the trait under consideration. Our power simulations assumed the same QTL effects for all plant species and a QTL allele which is in complete LD with one marker allele. These two factors cannot contribute to the observed difference in adjusted power for QTL detection among the examined plant species.
High adjusted power for the maize data set with its high number of entries and a low adjusted power for the Arabidopsis data set with a low number of entries indicated that differences in the number of entries n have a large influence on the observed differences in adjusted power among the examined plant species. This explanation is supported by results of previous studies [37]. In contrast, the small difference in adjusted power for QTL detection between sugar beet and potato data sets, which comprised a similar number of entries but differed in their average allele frequency, suggested that variation in this factor caused only small differences in the adjusted power.
In our study, heritability estimates were only available for two plant species and, thus, no inferences can be made about the contribution of this factor to differences in the adjusted power for QTL detection. However, results from previous studies suggested that increasing heritability has the potential to considerably increase the power for QTL detection [14].
T_{ opt }
The optimum T values identified in our study differed considerably among the various plant species (Table 4). This finding may be due to the difference in the extent of population structure and familial relatedness among the examined plant species as described above. The influence of population structure and familial relatedness on the optimum T value can be explained by the fact that lower values for T reduce the number of negative pairwise kinship estimates in the kinship matrix K_{ T }. Thereby the use of information concerning the structure of unrelated individuals, which was comprised in the kinship matrix K_{ T }, is improved and decreases the MSD values.
In comparison with the large differences among the optimum T values identified for different plant species, differences in the optimum T values for different traits of the same species were only small (Table 4). This finding might be explained by the fact that differences in the optimum T values identified for different traits of the same plant species can only be due to differences in the extent of population structure and relatedness for the traits under consideration generated by natural or artificial selection. Therefore, one optimum T value might be calculated across all traits of one species to improve the precision of this value. However, this requires further research on the standard error of the optimum T values.
Conclusion
Our study suggests that the QK method [15] is not only appropriate for association mapping in humans, maize, and Arabidopsis but also in rapeseed, potato, and sugar beet. Furthermore, our results indicate that the estimation of the number of subpopulations based on the two criteria, ΔK and SBC, results in different numbers of subpopulations. Nevertheless, the association mapping models which are based on these two population structure matrices are equally appropriate with respect to adherence to the nominal α level as well as the adjusted power for QTL detection. Furthermore, we recommend replacing the K matrix of the Q_{1}K, Q_{2}K, and PK approach by a K_{ T }matrix, which is based on a REML estimate of the conditional probability that two inbreds carry alleles at the same locus which are identical in state but not identical by descent and, thus, increase the adherence to the nominal α level. Finally, we showed that the T_{ opt }value estimated in this way differs considerably among the five plant species but only a little for the different traits within species.
Abbreviations
 AN:

amino nitrogen
 BY:

beet yield
 CSY:

corrected sugar yield
 ED:

ear diameter
 EH:

ear height
 FLC:

FLOWERING LOCUS C
 FRI:

FRIGIDA
 GPR:

Globodera pallida resistance
 LD:

linkage disequilibrium
 LDV:

long day conditions with vernalisation treatment
 MSD:

mean of squared difference
 OC:

oil content
 OY:

oil yield
 PIR:

Phytophthora infestans resistance
 PM:

plant maturity
 QTL:

quantitative trait locus
 REML:

restricted maximum likelihood
 SBC:

Schwarz Bayesian criterion
 TKW:

thousand kernel weight.
Declarations
Acknowledgements
This research was conducted within the Breeding and Informatics (BRAIN) project of the Genome Analysis of the Plant Biological System (GABI) initiative http://www.gabi.de. We thank the breeding companies Norddeutsche Pflanzenzucht HansGeorg Lembke KG (Holtsee, Germany), BöhmNordkartoffel Agrarproduktion OHG (Lüneburg, Germany), SakaRagis Pflanzenzucht GbR (Windeby, Germany), and KWS SAAT AG (Einbeck, Germany) as well as the potato genome analysis group of the Max Planck Institute for Plant Breeding Research (Cologne, Germany) for providing phenotypic and genotypic data. The authors appreciate the editorial work of Prof. Dr. B.S. Dhillon and Dr. J. Muminović, whose suggestions considerably improved the style of the manuscript. The authors thank three anonymous reviewers for their valuable suggestions.
Authors’ Affiliations
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