Overestimation of the genetic variance explained by markers in linkage disequilibrium with causative genes or QTL

The overestimation problem described here was originally discovered in a reanalysis of data from genetic studies in plants where REML estimates of exceeded REML estimates of broad-sense heritability (H²) and REML estimates of p and exceeded 1.0, the theoretical upper limit for these parameters (Table 1). We initially suspected that selection bias might be the culprit [68, 69, 82–84] but concluded that selection bias alone could not explain or . Although proof was lacking and the bias was non-obvious, we hypothesized that many estimates in the theoretical range (0.0 ≤ p ≤ 1.0) must also be upwardly biased. The proof was found through algebraic analyses of the ANOVA estimators of , , and for balanced and unbalanced data (S1, S2 and S4 Texts). Although variance components are commonly estimated using REML, as was done in the analyses shown throughout this paper, algebraic analyses of ANOVA expected mean squares (EMSs) identified the source of the bias and yielded explicit algebraic solutions for bias correcting ANOVA and REML estimates of p and .

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. REML estimates of marker-associated variance (), the fraction of the genetic variance explained by markers (), and marker heritability () from random marker effects analyses and coefficients of determination (R²) from Type II and Type III fixed marker effects analyses for large effect loci identified in cattle, sunflower, and strawberry studies.

https://doi.org/10.1371/journal.pgen.1009762.t001

The source of the bias was identified by expressing the estimator of p as a function of the ANOVA estimators of and for balanced data and algebraically simplifying the equations. The linear mixed models (LMMs) and ANOVA estimators of the variance components needed to show this are described here. We start with the analysis of a single marker locus in an experiment where entries (e.g., individuals, families, or strains) are replicated, can be estimated, and the data for entries and markers are balanced. Extensions for one to three marker loci with unbalanced data are shown in S1, S2 and S3 Texts. Two LMMs are needed for estimating , , p, and . Consider a study where n_G entries are phenotyped for a normally distributed quantitative trait using a balanced completely randomized study design with r_G replications/entry, n_M marker genotypes/locus, and r_M replications/marker genotype. The LMM needed for estimating (the between entry variance component) is: (1) where y_jk is the jk^th phenotypic observation, μ is the population mean, G_j is the random effect of the j^th entry, ϵ_jk is the random effect of the jk^th residual, , , j = 1, 2, …, n_G, and k = 1, 2, …, r_G. Suppose entries are genotyped for a single marker locus (M) in linkage disequilibrium with a gene or QTL affecting the quantitative phenotype (y_jk). The between entry source of variation from LMM (1) can be partitioned into marker (M) and entry nested in marker (G : M) sources of variation (this is the residual genetic variation among entries not explained by markers in the model). The LMM for estimating and is: (2) where y_ijk is the ijk^th phenotypic observation, M_i is the random effect of the i^th marker genotype at locus M, G : M_i(j) is the random effect of the j^th entry nested in the i^th marker genotype, ϵ_ijk is the random effect of the ijk^th residual, i = 1, 2, 3, , , and .

The ANOVA estimator of the between-entry variance component () from LMM (1) with balanced data is: (3) where MS_G = SS_G/df_G is the between entry mean square, SS_G is the between entry sum of squares, df_G = n_G − 1 is the between entry degrees of freedom, is the residual mean square, SS_ϵ is the residual sum of squares, df_ϵ = n_G(r_G − 1) − 1 is the residual degrees of freedom, is the residual variance component, and r_G is the number of replications per entry [74]. The between-entry variance component has a theoretical genetic interpretation when entries are progeny with genetic relationships known from pedigrees, e.g., monozygotic twins, full-sib families, or recombinant inbred lines [24, 25, 30]. ANOVA estimators of the marker locus M and entry nested in M variance components from LMM (2) with balanced data are: (4) and (5) respectively, where n_G:M is the number of entries nested in each marker genotype, , , MS_G:M is the entry nested in M mean square, and MS_M is the mean square for marker locus M. The residuals in LMMs (1) and (2) are identical when the data are balanced (). Hence, for a single marker locus with balanced data, the ANOVA estimator of p is: (6) and the ANOVA estimator of broad-sense marker heritability on an entry-mean basis is: (7) where is the phenotypic variance on an entry-mean basis [25, 76].

The overestimation of p and was not obvious from inspection of ANOVA estimators (6) and (7). The source of the bias was discovered by substituting SS_M + SS_G:M for SS_G in the ANOVA estimator of from (3) and simplifying: (8) where the fraction k_M is source of the bias, 0 < k_M < 1, r_M is the number of replications per marker genotype, n_G:M is the number of entries nested in marker loci, SS_M is the marker sum of squares, df_M is the marker degrees of freedom, r_M is the number of replicates of each marker genotype, SS_G:M is the entry nested in marker sum of squares, and df_G:M is the entry nested in marker degrees of freedom. The term k_M in (8) depends on degrees of freedom and n_G:M and is hereafter referred to as the k_M bias coefficient, where the subscript M indexes the intralocus and interlocus effects of marker loci.

Eq (8) shows that the sum of ANOVA estimates of and from LMM (1) are greater than the ANOVA estimate of from LMM (2): (9)

Although the SS for sources of variation in these LMMs are additive (SS_M + SS_G:M = SS_G), the mean squares are not (MS_M + MS_G:M ≠ MS_G). Because , the sum from LMM (2) overestimates by a factor of . The ANOVA estimators of p and from analyses of LMMs (1) and (2) are upwardly biased because is multiplied by the fraction k_M in their denominators, and not the numerators: (10) and (11)

Substituting for in the denominators of p and decreases but does not eliminate the bias because is multiplied by k_M in the denominator (S1 Fig). For a single marker with balanced data, we found that: (12) and (13) where 0 < k_M < 1. Hence, the bias is caused by the k_M multiplier in the expected values of the ANOVA estimators of p and . As shown later, simulation analyses confirmed that (9) and (13) accurately predict the upward bias caused by k_M. Moreover, we concluded that the bias could be corrected by multiplying ANOVA or REML estimates of by k_M in the numerators of p and estimates.

Genetic models with unbalanced genotypic data

We started with the special case of balanced data, which seldom arises in practice, but develop results here for the general case of unbalanced data. Following the same approach as that shown above for a single locus with balanced data, we found k_M coefficients for bias-correcting ANOVA and REML estimates of p and for analyses of one to three marker loci with unbalanced genotypic data (S1, S2 and S3 Texts). For a single marker locus with unbalanced genotypic data, we found: (14) where n_G is the number of entries, df_G are the degrees of freedom for entries, and is the number of entries nested in the h^th marker genotype (S1 Text). This simplifies to (12) for a single marker locus with balanced genotypic data.

The k_M coefficients become slightly more complicated as the number of marker loci increases but nevertheless follow a predictable algebraic pattern, e.g., for a two locus genetic model, see equations (S10)-(S12) in S2 Text. Similarly, for a three locus genetic model, see equations (S19)-(S25) in S3 Text. k_M is greater (k_M bias is proportionally smaller) for interaction than main effects, e.g., for two marker loci, k_M1 < k_M1×M2 < 1 and k_M2 < k_M1×M2 < 1, where k_M1 is the coefficient for M₁, k_M2 is the coefficient for M₂, and is the coefficient for the M₁ × M₂ epistatic interaction (S2 Text). k_M for the two-locus interaction (k_M1×M2) is larger than k_M for the individual marker loci (k_M1 and k_M2) because the denominator (df_G r_G) is constant, whereas the numerators increase and approach the denominator as the degrees of freedom for marker effects increase. Therefore, the upward bias is proportionally smaller for the M1 × M2 variance component than the M₁ or M₂ variance components for a two locus genetic model. Similarly, for a three locus genetic model, the upward bias is proportionally smaller for the M₁ × M₂ × M₃ interaction variance component than the two-way interaction variance components (M₁ × M₂, M₁ × M₃, and M₂ × M₃). These results naturally extend to genetic models with more than three loci. Algebraic results are only shown for three marker loci because we found that the the k_M bias problem can be directly solved using average semivariance estimation methods when analyzing more complex genetic models (see below). Although certainly not limited to three marker loci, the methods described herein are primarily designed to study the effects of one to a few genes with large effects, e.g., BRCA2 [52], BTA19 [43], and the examples shown in Tables 1 and 2, and not to replace GWAS or QTL mapping.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Type I, II, and III sums of squares for fixed effect analyses of markers associated with QTL identified in GWAS and QTL mapping experiments in cattle and sunflower.

https://doi.org/10.1371/journal.pgen.1009762.t002

Study designs without replications or repeated measures of individuals or families

LMMs (1) and (2) arise in study designs where entries (individuals, families, or strains) are replicated, e.g., in studies with domesticated plants, biological replicates of half-sib or full-sib families, doubled haploid or recombinant inbred lines, or testcross hybrids are commonly phenotyped [24, 25, 31, 76, 87, 88] (see the sunflower example in Table 1). These same LMMs apply to study designs for monozygotic twins in humans and other mammals and clonally replicated individuals in asexually propagated plants, e.g., cassava (Manihot esculenta), strawberry (Fragaria × ananassa), and apple (Malus × domestica) (see the strawberry examples in Table 1). The extension of the proposed k_M bias correction solutions to LMMs with repeated measures is straightforward and should have applications in studies where large effect loci are important determinants of the genetic variation underlying quantitative traits in both replicable and unreplicable organisms or populations [88–94].

When entries are unreplicated, the random error or residual source of variation in LMM (2) disappears ( becomes the residual) and , , and p cannot be estimated; however, the marker heritability can be estimated using the phenotypic variance among unreplicated individuals (). As before, this variance component ratio is upwardly biased by the factor k_M (see the cattle example in Table 1). Without the insights gained from the algebra shown in equations (10), (S3), (S9), and (S18), and S1, S2 and S3 Texts, the bias would not be obvious unless one or more estimates of marker heritability exceeded 1, which only happens when the loci under study have very large effects. That was exactly how we originally discovered the bias problem in the first place (Table 1). The bias is systematic and ubiquitous but not immediately obvious when estimates fall within the expected range (). The same bias correction solutions we proposed for study designs with replications of entries can be applied in study designs where entries are unreplicated. When unreplicated entries are genotyped with a dense genome-wide of markers, be estimated using a genomic or pedigree relationship matrix [92, 95–97], which yields an estimate of p.

Average semivariance estimation directly solves the bias problem

The AMV methods proposed above for bias correcting ANOVA or REML estimates of p and are straightforward to apply in practice because they are the methods widely described in textbooks and implemented in popular statistical software packages, e.g., the R package ‘lme4’ and SAS package ‘GLIMMIX’ [78, 98]. Here we show that the bias problem can be directly solved by applying average semivariance (ASV) estimation methods [73]. As before, we start by showing results for a single marker locus with balanced genotypic data. AMV notation and estimators are reformulated in matrix notation here to build the foundation for describing ASV notation and estimators. The input for both are the adjusted entry-level means () from LMM (1) stored in an n_G-element vector. These are the best linear unbiased estimates (BLUEs) for entries [73, 99]. The LMM equivalent to (2) for the entry-level means analysis of the effect of a single marker locus (M) is: (15) where is the phenotypic mean for the ij^th entry, μ is the population mean, M_i is the random effect of the i^th marker genotype, , G : M_i(j) is the random effect of entries nested in M, , is the residual error, and . The residual variance-covariance matrix (R) is estimated in the first stage of a two-stage analysis [99–101]. The between-entry variance can be partitioned into and with individual variance-covariance matrices G_c defined by the genetic model, e.g., different main and interaction effects among marker loci.

The AMV estimator of the phenotypic (total) variance among observations for LMM (15) is: (16) where V is the variance-covariance matrix of the phenotypic observations, n_G is the number of entries, tr(V) is the trace of V, is the marginal variance explained by the c^th genetic factor in the model (e.g., M and G : M), Z_c are design matrices for the c genetic factors, is the AMV estimator of the residual variance, and R is the residual variance-covariance matrix. The AMV estimator of the genetic variance among entries (G) is: (17) where is a n_G identity matrix. From LMM (15), the AMV estimator of the variance associated with a single marker locus with balanced data is: (18) where , , is a n_M identity matrix, is an n_G:M-element unit vector, and u_M is a vector of random effects for M. The AMV estimator of the variance associated with the residual genetic variation among entries nested in M is: (19) where u_G:M is a vector of random entry nested in M effects and is a n_G identity matrix. Hence, the AMV estimators of and are identical to ANOVA estimators (4) and (5), respectively, with entry means as input for the former and original observations as input for the latter.

ASV, or the average variance of differences among observations, leads to a definition of the total variance that provides a natural way to account for the heterogeneity of variance and covariance among observations [73, 102]. ASV can be defined for any variance-covariance structure in a generalized LMM and allows for missing and unbalanced data [73]. The ASV estimator of total variance is half the average variance of pairwise differences among entries and can be partitioned into independent sources of variance, e.g., genetic and non-genetic or residual: (20) where is the idempotent matrix used for column-wise mean-centering, is an n_G × n_G identity matrix, and is an n_G × n_G unit matrix [73]. accounts for the variance and covariance of the phenotypic observations. From (20), is the variance explained by the c^th genetic factor (u_c), where c indexes genetic factors, the genetic factors are marker locus effects and entries nested in marker locus effects, and is the residual variance. The variance explained by the c^th genetic factor is , e.g., for a single marker locus M, . , , and the biases of these ASV estimators are defined in S4 Text.

The ASV estimator of the genetic variance among entries (G) is: (21) where is a n_G identity matrix. Hence, from Eqs (8), (17) and (21), AMV and ASV estimators of the between-entry variance component () are equivalent (). The ASV estimator of the variance associated with M is: (22) where k_M = df_M n_G:M/df_G is the bias correction coefficient, , df_G = n_G − 1, df_M = n_M − 1, and df_G:M = df_G − df_M. This definition of the k_M-bias coefficient is identical to the earlier definition with r_G factored out (see Eq 12). Eq (22) shows that the ASV estimator of is corrected by the fraction k_M, which correctly scales the estimate of to the genetic variance and yields unbiased estimates of p and . From Eqs (9) and (22), we found that by the factor k_M. The ASV estimator of the variance associated with G : M is: (23)

The ASV estimator of p for a single marker locus (M) is: (24)

Similarly, the ASV estimator of for a single marker locus is: (25) where is the phenotypic variance on an entry-mean basis [25]. From these results, we found that: (26) and showed that ASV estimators of p and are unbiased (automatically corrected for k_M).

Computer simulations confirmed that ASV-REML estimates of p and are unbiased

Computer simulations confirmed that AMV-REML estimates of p (6) and (7) are upwardly biased by the factor k_M and that ASV-REML estimates of these parameters form (24) and (25) are unbiased (Figs 1 and 2). The mean of AMV-REML estimates of p and from 21 different simulation study designs (S1 Table) were identical to those predicted by the k_M coefficients shown in S1, S2 and S3 Texts. Several insights arose from the simulation analyses. First, the bias caused by k_M increased as increased but was proportionally constant for different (Fig 1). These results show that the overestimation of p and is greatest for genes and gene-gene interactions with large effects (Fig 1). Their effects could be inflated by selection bias over and above k_M bias [67, 68, 82, 83, 103]; hence, we concluded that k_M-bias and selection bias could operate in combination to inflate estimates of the contribution of a locus to the heritable variation in a population (S1, S2 and S3 Texts). Moreover, because the bias increases as the effect of the locus increases, we concluded that the overestimation problem is worst for large-effect QTL (Fig 1). Second, k_M bias was greater for unbalanced than balanced data (Fig 1D and 1E). The effect of unbalanced data was more extreme for the F₂ simulation (Fig 1D) where the expected genotypic ratio was 1 AA: 2 Aa: 1 aa than for simulations where 10 or 33% of the observations were randomly missing for markers with roughly equal numbers of replicates/marker genotype (Fig 1E and 1F). Third, the F₂ and missing data simulations further showed that the precision of estimates of these parameters decreased as the genotypic data imbalance increased. Even though bias-corrected AMV and ASV estimates of these parameters are unbiased, the sampling variances among the simulated F₂ samples were larger than observed for the 10 and 33% missing data samples and yielded a small percentage of estimates slightly greater than 1.0 (Fig 1D). For the other simulation study designs (Fig 1), none of the ASV estimates exceeded 1.0. The sample variances of p and can be estimated using data resampling methods, e.g., bootstrapping [104], or the estimators we developed using the Delta method (S5 Text) [25, 105, 106]. Equations (S44) and (S45) in S5 Text show that ASV estimates are more precise than AMV estimates by a factor of . These predictions perfectly aligned with the empirical bootstrap estimates. Fourth, the relative biases were not affected by the number of replications of entries or the number of entries, although the precision of estimates increased as n_G and increased (Fig 2). Predictably, the number of entries (n_G) dramatically affected the precision of estimates of (Fig 2C and 2D). The relative biases were not affected by r_G or ; however, the sampling variances were strongly affected by and decreased as increased (Fig 2E and 2F and S2 Fig).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Accuracy of AMV and ASV estimators of marker heritability.

AMV and ASV estimates of are shown for 1,000 segregating populations simulated for different numbers of entries (n_G individuals, families, or strains), five replications/entry (r_G = 5), true marker heritability () ranging from 0 to 1, and one to three marker loci with three genotypes/marker locus (n_M1 = 3). AMV estimates of marker heritability (; red highlighted observations) and ASV estimates of marker heritability (; blue highlighted observations) are shown for: (A) one locus with balanced data for n_G = 540 entries (study design 1); (B) two marker loci with interaction (M1, M2, and M1×M2) and balanced data for n_G = 540 (study design 2); (C) three marker loci with interactions (M1, M2, M3, M1×M2, M1×M3, M2×M3, and M1×M2×M3) and balanced data for n_G = 540 (study design 3); (D) an population segregating 1:2:1 for one marker locus with r_G:M = 135 entries for both homozygotes and r_G:M = 270 heterozygous entries, and n_G = 540 (study design 4); (E) one locus with 10% randomly missing data among 540 entries (study design 5); and (F) one locus with 33% randomly missing data among 540 entries (study design 6). Study design details are shown in S1 Table.

https://doi.org/10.1371/journal.pgen.1009762.g001

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Effect of r_G, n_G, and on the relative bias of AMV and ASV estimators of .

(A and B) Phenotypic observations were simulated for 1,000 populations segregating for a single marker locus with three genotypes (n_M = 3), n_G = 900 progeny, and r_G = 1, 2, 5, 10, or 20 (study designs 7–11). The marker locus was assumed to be in complete linkage disequilibrium with a single QTL that explains 50% of the phenotypic variance (). (A) Distribution of the relative biases of AMV estimates of for different r_G. The relative bias was identical for different r_G. (B) Distribution of the relative biases of ASV estimates of for different r_G. The relative bias was identical for different r_G. (C and D) Phenotypic observations were simulated for 1,000 populations segregating for a single marker locus with three genotypes (n_M = 3), five replications/entry (r_G = 5), and n_G = 450, 900, 1,800, 3,600, or 7,200 entries/population (study designs 12–16). The marker locus was assumed to be in complete linkage disequilibrium with a single QTL that explains 50% of the phenotypic variance (). (C) Distribution of the relative biases of AMV estimates of for different n_G. The relative bias was identical across the variables tested. (D) Distribution of the relative biases of ASV estimates of for different n_G. The relative bias () was identical across the variables tested. (E and F) Phenotypic observations were simulated for 1,000 populations segregating for a single marker locus with three genotypes (n_M = 3), five replications/entry (r_G = 5), and n_G = 450 entries/population. The marker locus was assumed to be in complete linkage disequilibrium with a single QTL that explains 5–95% of the phenotypic variance ( to 0.95 (study designs 17–21). (E) Distribution of the relative biases of AMV estimates of for different . The relative bias was identical across the variables tested. (F) Distribution of the relative biases of ASV estimates of for different . The relative bias was identical across the variables tested.

https://doi.org/10.1371/journal.pgen.1009762.g002

GWAS example: A single marker locus with highly unbalanced genotypic data

The bias-correction methods described above are illustrated here for highly unbalanced genotypic data from a GWAS experiment. Variance components were estimated for two SNP markers (AX493 and AX396) in LD with a gene (FW1) conferring resistance to Fusarium wilt in a strawberry (Fragaria × ananassa) GWAS population (n_G = 564) genotyped with a genome-wide framework of SNP markers [86]. Both SNP markers had highly significant GWAS effects with −log₁₀(p) = 6.61 × 10⁻³¹ for AX493 and 2.95 × 10⁻²²² for AX396. Genotype frequencies were highly unbalanced for both markers with a scarcity of AA homozygotes (2.8%) for AX396 (16AA : 177Aa : 371aa) and a 1 : 2 : 1 ratio for AX493 (141AA : 282Aa : 141aa). For both loci, the minor allele frequency was >0.05. The k_M for these data (k_AX493 = 0.62 and k_AX396 = 0.47) were calculated as shown in S1 Text. The AMV-REML estimate of for AX396 exceeded 1.0, a telltale sign of k_M-bias (Table 1). AMV-REML estimates of and for both SNP markers were double or nearly double their bias-corrected ASV-REML estimates (Table 1). The bias-corrected estimate of marker heritability for AX396 was 0.62, versus 1.33 for the uncorrected estimate. Even with bias-correction, the sum of ASV-REML estimates of and for AX493 was slightly greater than the ASV-REML estimate of . This result was consistent with findings for highly unbalanced marker genotypic data in our simulation studies where a certain fraction of bias-corrected estimates exceeded the theoretical limit for heritability because of decreased precision (Fig 1). The k_M-bias problem would not necessarily have been detected in the analysis of AX396 because the p and estimates fell within the expected range, e.g., (Table 1). Although both SNP markers were closely associated with FW1, they accounted for dramatically different fractions of genetic variance because of historic recombination and because neither are causal DNA variants or in complete LD with causal DNA variants [17, 19, 86, 107].

QTL mapping example: Three marker loci with slightly unbalanced genotypic data

Statistics are shown here for an analysis of three marker loci (BR, PHY, and HYP) affecting seed oil content in a sunflower (Helianthus annuus) RIL population using LMM (27) [53]. The genotypic data were only slightly unbalanced and the three marker loci were identified by QTL mapping. The k_M needed for bias-correcting AMV-REML estimates of p and are shown in S3 Text (Table 1). The AMV-REML estimates of p and were nearly double the bias-corrected ASV-REML estimates, e.g., the AMV-REML estimate of for the three-locus genetic model (0.79) was nearly two-fold greater than the ASV-REML estimate (0.41) (Table 1). Similarly, the AMV-REML estimate of p for the BR locus (0.54) was slightly more than double the bias-corrected (ASV-REML) estimate (0.26). Hence, the uncorrected REML estimates of p and grossly inflated the predicted contributions of the three marker loci to genetic variation for seed oil content (Table 1).

GWAS example: Three marker loci with unbalanced genotypic data and unreplicated entries

The application of bias-correction is illustrated here for a genetic model with three marker loci, highly unbalanced genotypic data, and a single phenotypic observation per individual— and p could not be estimated for this example because individuals were unreplicated. Variance components were estimated for three SNP markers (rs10, rs45, and rs20) on chromosomes 2, 6, and 22, respectively, affecting white spotting (%) in a Holstein–Friesian cattle (Bos taurus) population (n_G = 2, 973) [85]. These SNP markers had the largest effects among those predicted to be in LD with genes affecting white spotting. The genotypic frequencies were 50AA : 586Aa : 2, 337aa for rs10, 78AA : 736Aa : 2, 159aa for rs45, and 237AA : 976Aa : 1, 760aa for rs20. The k_M for these data (k_rs10 = 0.35, k_rs45 = 0.41, and k₂₀ = 0.54) were calculated as shown in S3 Text. The uncorrected AMV-REML estimate of for the three-locus genetic model (0.76) was substantially greater than the bias-corrected ASV-REML estimate (0.37) (Table 1). Similar differences were observed for the three marker loci.

Candidate gene analysis: Fixed or random, BLUE or BLUP?

Our study was partly motivated by inconsistencies in the statistical approaches applied in candidate gene and other complex trait analyses when testing hypotheses and fitting genetic models for multiple large-effect loci. With the high densities of genome-wide markers commonly assayed in gene finding studies, investigators often identify markers tightly linked to candidate or known causal genes, as exemplified by diverse real world examples [17, 19, 33, 34, 37, 38, 40, 42, 43, 52, 54, 108]. The candidate marker loci are nearly always initially identified by genome-wide searches using sequential (marker-by-marker) approaches [56, 72, 75, 79, 109, 110]. Complicated and often misunderstood problems arise in the estimation and interpretation of statistics from sequential fixed effect analyses when the data are unbalanced [79, 111, 112]. Most importantly, there are multiple model fitting and analysis options (Type I, II, and III ANOVA) and the reduction in error sums of squares (SSE), test statistics, and parameter estimates differ among them, a problem that disappears when the data are balanced or when single large effect loci are discovered [79, 111–113]. Our review of the literature uncovered substantial variation and inconsistencies in the statistical approaches applied to the problem of fitting multilocus genetic models, testing multilocus genetic hypotheses, and calculating best linear unbiased estimates (BLUEs) from a fixed effects analysis of marker loci.

The problems that arise in fixed effect analyses of unbalanced data profoundly affect parameter estimates and statistical inferences but have not been universally recognized or addressed in complex trait analyses [79, 112]. We reanalyzed the cattle and sunflower examples with markers as fixed effects (Table 2) to show this, illustrate the challenges and nuances of fixed effects analyses of unbalanced data, and facilitate comparisons between random and fixed effects analyses of marker loci [56, 56, 75, 79, 109–112]. Following the discovery of statistically significant marker-trait associations from a marker-by-marker genome-wide scan, the natural progression would be to analyze multilocus genetic models where the effects of the discovered loci are simultaneously corrected for the effects of other discovered loci [79, 112], as shown in our multilocus analysis examples (Tables 1 and 2). This is straightforward when the genotypic data are balanced or nearly balanced (as in the sunflower example) but more complicated and convoluted when the genotypic data are unbalanced (as in the cattle example) [75, 79, 111, 112]. Although methods for fixed effect analyses of factorial treatment designs (multilocus genetic models) with unbalanced data are well known [56, 79, 109, 110, 112], there are several model fitting and parameter estimation variations that can lead to dramatically different parameter estimates and statistical inferences. This is perfectly illustrated by the cattle example where the coefficients of determination (analogous but not identical to ) from Type I, II, and III analyses were substantially different from each other and from estimates from the random effects analysis (Tables 1 and 2). The differences and ambiguities among the different fixed effects approaches disappear when the random effects approach is applied to the problem.

The analysis of markers as random effects in multilocus analyses of known or candidate genes with large effects with ASV, although historically uncommon, simultaneously yields unbiased estimates of the variance component ratios investigated in the present study (p and ) and best linear unbiased predictors (BLUPs) of the additive and dominance effects of the causative loci identified by marker associations, in addition to solving the often ambiguous problems that arise in fixed effects analyses of unbalanced data [32, 75, 77, 79, 112, 113]. As discussed in depth below and illustrated through a reanalysis of the cattle and sunflower examples (Table 2), the random effects approach we described (ASV with REML estimation of the variance components) yields accurate estimates of the underlying genetic parameters (variance component ratios and BLUPs of marker effects) from a single unambiguous generalized linear mixed model analysis, whereas wildly different parameter estimates can arise among the multitude of fixed effects analyses that investigators might elect to apply in practice when the underlying genotypic and phenotypic data are unbalanced (Tables 1 and 2).

As substantiated by our simulation analyses (Figs 1 and 2), ASV with REML estimation of the underlying variance components yields accurate estimates of p and for marker loci and interactions between marker loci, both individually and collectively, and BLUPs of the the additive and dominance effects of marker loci [76, 113–115]. When the genotypic data are unbalanced, the order with which marker and marker × marker effects enter the genetic model profoundly affects parameter estimates and statistical inferences in fixed effect analyses [56, 72, 74, 116]. To illustrate this, the main effects of marker loci A, B, and C were estimated for the six possible Type I ANOVA orders of the three loci (ABC, ACB, BAC, BCA, CAB, and CBA) (Table 2). Predictably, the reduction in the error sums of squares for a particular locus differed for each Type I order in the cattle example: the Type I SS ranged from 591.4 to 3,552.3 for rs10, 4,880.5 to 9,504.4 for rs20, and 4,259.6 to 8,384.8 for rs45. The R², or PVE, estimates for marker loci were radically different among the six Type I ANOVA and Type II and III analyses. The Type I SS were, in addition, significantly greater than the Type III SS for nearly every factor. Although Type III statistics are commonly estimated and reported in analyses of factorial treatment designs with unbalanced data, there are compelling arguments for estimating Type II statistics [109, 110]; nevertheless, as we have argued, the fixed effects approach is unnecessary.

Broadly speaking, the large effect loci segregating in a population are typically necessary but not sufficient for predicting genetic merit or disease risks but are often important enough to warrant deeper study and, in animal and plant breeding, direct selection via MAS or direct modelling in genome selection applications [21, 32, 57]. The BLUP (random marker effects) approach we applied was designed to align the study of loci with large and highly predictive effects with the BLUP approaches commonly applied to genomic prediction problems that are agnostic or indifferent to the effects of individual loci, the so-called “black box” of genomic prediction [6, 7, 20, 21, 88, 117–121]. The predictive markers associated with large effect marker loci can be integrated into the genome-wide framework of marker loci applied in genomic prediction or incorporated as fixed effects when estimating GEBVs or PRSs [21, 54, 57–61]. One of the greatest strengths of the random effects (BLUP) approach is that the genetic parameters can be estimated from a single REML analysis free of the challenges and uncertainty associated with the fixed effects model building process [79, 109, 110, 112]. Finally, if our conclusions are correct, the complex trait analysis literature is riddled with overestimates of the genotypic and phenotypic variances explained by specific genes or QTL.

Simulation studies

We used computer simulation to estimate the bias and assess the accuracy of uncorrected and bias-corrected REML estimates of p and for 21 study designs (S1 Table and S4 Text). Phenotypic observations (y_ijk) for LMMs (1) and (2) were simulated for n_M = 3 genotypes/marker locus and 21 combinations of study design variables (n_G, r_G, r_M, and H²) with balanced or unbalanced data (S1 Table). Simulations were performed to assess the accuracy of REML estimates of p and for 21 study designs with 1,000 replicates per study design (S1 Table). The phenotypic observations for each sample were obtained by generating random normal variables for entries, markers, and residuals using the R function rnorm() with known means and variances [122] as described by [123, 124]. The simulated random effects of entries, markers, and replications in LMMs (1) and (2) were summed to obtain n = n_G r_G phenotypic observations for each study design. Variance components for the random effects in LMMs (1) and (2) were estimated using the REML function implemented in and assess the accuracy of AMV and ASV estimators of p and . For study designs 1–6, the true marker heritability randomly varied from 0 to 1. Study designs 1–6 demonstrate how different numbers of marker loci (m) and unbalanced data affect estimates of p and (Fig 1; S1 Table). For study designs 5 and 6, we randomly deleted 10 and 33% of the phenotypic observations, respectively, to create unbalanced data. For study designs 7–21, the true variances of the independent variables were fixed for all samples, which allowed us to estimate the bias and relative bias associated with the different estimators (the biases are shown in S4 Text). Study designs 7–21 illustrate how r_G, n_G, and affected the biases and relative biases of p and (Fig 2; S1 Table). The variance components were estimated using REML in the lme4::lmer() v1.1–21 [78] package in R v4.0.2 [122]. We estimated the sample variances of AMV and ASV estimates of p for each study design (S1 Table). Finally, we developed estimators of the sampling variances of p and using the delta method [25, 106], as shown in S5 Text.

Estimation examples

To illustrate the application of bias-correction methods and the differences between AMV and bias-corrected AMV estimates of p and , we reanalyzed data from a GWAS study in cattle (Bos taurus), a QTL mapping study in oilseed sunflower (Helianthus annuus L.) [53], and a GWAS study of Fusarium wilt resistance in strawberry (Fragaria × ananassa Duchesne ex Rozier) [86]. For the sunflower study, two replications (r_G = 2) of n_G = 146 recombinant inbred lines (RILs) were phenotyped for seed oil concentration (g/kg) and genotyped for three marker loci (BR, PHY, and HYP) with two homozygous marker genotypes/locus [53]. For the cattle study, unreplicated entries (r_G = 1; n_G = 2, 973) were phenotyped for white spotting (%) and genotyped for three marker loci (rs10, rs45, rs20) with three marker genotypes per locus [85]. LMM (2) expanded to three marker loci with all possible interactions among marker loci is: (27) where BR_h is the h^th effect of the BR locus, PHY_i is the i^th effect of the PHY locus, HYP_j is the j^th effect of the HYP locus, G : (BR × PHY × HYP)_hij(k) is the k^th effect of entries nested in the hij^th BR × PHY × HYP interaction, and ϵ_hijkl is the hijkl^th residual effect. The data for RILs were balanced, whereas the data for marker genotypes were slightly unbalanced. Each of the eight BR × PHY × HYP homozygotes were observed in the RIL population; however, the number of entries nested in each marker genotype (n_G:M) varied from n_G:BR = 81 : 65, n_G:PHY = 60 : 86, and n_G:HYP = 70 : 76. Variance components for LMMs (1) and (27) were estimated using the REML method in lme4::lmer() [78]. The marker-associated genetic variances for individual marker loci and two- and three-way interactions among marker loci were bias-corrected using the formula described in S1, S2 and S3 Texts.

For the strawberry study, four replications (r_G = 4) of 565 entries (n_G = 565) from a genome-wide association study (GWAS) were phenotyped for resistance to Fusarium wilt and genotyped for single nucleotide polymorphism (SNP) markers in LD with FW1, a dominant gene conferring resistance to Fusarium oxysporum f.sp. fragariae, the causal pathogen [86]. The replications were asexually propagated clones of individuals; hence, the expected causal variance among individuals was equal to the total genetic variation in the population, analogous to monozygotic twins [25]. Genetic parameters were estimated for two SNP markers (AX493 and AX396) that were tightly linked to FW1 [86]. The genotypic data for both markers were highly unbalanced. Genotype numbers were 141 AA : 282 Aa : 141 aa for AX493 and 16 AA : 177 Aa : 371 aa for AX396, where A and a are alternate SNP alleles. The variance components were estimated for LMMs (1) and (2) using REML method implemented in the R package lme4::lmer() [78]. REML estimates of the marker-associated genetic variances for both marker loci were bias-corrected using the approach described in S1 Text.

For the cattle study, we used a model similar to (27) for the analysis. However, because entries are unreplicated in this experiment, we cannot include the entries nested in the three-way marker interaction (G : M) term because it has the same levels as the residual. The LMM for this case study is: (28) where rs10_h is the h^th effect of the peak SNP (rs109979909) on chromosome 2, rs45_i is the i^th effect of the peak SNP on chromosome 6 (rs451683615), rs20_j is the j^th effect of the peak SNP on chromosome 22 (rs209784468), and G : (rs10×rs45×rs20)_hij(k) is the hij(k)^th residual effect comprising residual genetic effects G : M and residual error. In this experiment, there were k entries and k observations, and because of this we cannot fit LMM (1) without incorporating pedigree or genomic relatedness. In this single case, we estimate from the log transformed data to use in the denominator of .

We used the SAS package PROC GLM [77] for Type I and III analyses of the sunflower and cattle data with marker loci as fixed effects. Type I analyses were done for the six possible orders of main effects (ABC, ACB, BAC, BCA, CAB, and CBA) and a single order for marker × marker interactions (A × B, A × C, B × C, and A × B × C), where A, B, and C are the three marker loci (factors). For the ABC order, the reduction SS for the main effects were R(A | μ), R(B |μ, A), and R(C | μ, A, B), where μ is the population mean. Similarly, for the ACB order, the reduction SS for the main effects were R(A | μ), R(C |μ, A), and R(B | μ, A, C), and so on for the other four orders (BAC, BCA, CAB, and CBA). C, A × B, A × C, B × C, A × B × C), the reduction SS for the main effect of B was R(B | B, C, A × B, A × C, B × C, A × B × C). For comparison, the Type III reduction SS for the main effects were R(A | μ, B, C, A × B, A × C, B × C, and A × B × C), R(B | μ, A, C, A × B, A × C, B × C, and A × B × C), and R(C | μ, A, B, A × B, A × C, B × C, and A × B × C).