Misspecification in Mixed-Model-Based Association Analysis

Abstract

Additive genetic variance in natural populations is commonly estimated using mixed models, in which the covariance of the genetic effects is modeled by a genetic similarity matrix derived from a dense set of markers. An important but usually implicit assumption is that the presence of any nonadditive genetic effect increases only the residual variance and does not affect estimates of additive genetic variance. Here we show that this is true only for panels of unrelated individuals. In the case that there is genetic relatedness, the combination of population structure and epistatic interactions can lead to inflated estimates of additive genetic variance.

MIXED models with random genetic effects have become an important tool for studying the genetic architecture of complex traits. The covariance of the genetic effects is assumed to be proportional to a genetic similarity matrix (GSM) based on a dense set of markers, which is equivalent to assuming additive effects for each standardized marker score. Under several additional assumptions, such as constant linkage disequilibrium, this gives unbiased estimates of additive genetic variance and narrow-sense heritability (Yang et al. 2010; Speed et al. 2012; Lee and Chow 2014; Speed and Balding 2015). The sampling variance of such heritability estimators has been studied in Visscher and Goddard (2014) and Kruijer et al. (2015). These results are, however, derived under the assumption that the model is correct, i.e., contains the true distribution of the data. Here we consider situations where this is not the case and argue that potential sources of bias may be identified by computing the parameter value $\tilde{\theta }$ that minimizes the Kullback–Leibler (KL) divergence $\mathrm{KL}\left(Q,P_{\theta }\right)={\displaystyle \int \log \left(Q/P_{\theta }\right)dQ}$ with respect to the true distribution Q. For n-dimensional Gaussian distributions $P=N\left(0,{\Sigma }_1\right)$ and $Q=N\left(0,{\Sigma }_0\right)$, the KL divergence equals

$$\text{KL}\left(\text{Q}|\text{P}\right)=\frac{1}{2}\left(tr\left({\Sigma }_1^{-1}{\Sigma }_0\right)+\mathit{log}\left(\left|{\Sigma }_1\right|/\left|{\Sigma }_0\right|\right)-n\right)$$

It is a well-known fact from statistics that in the case of misspecification, i.e., when Q is not contained in the model $\left\{P_{\theta }:\theta \mathrm{\in \Theta }\right\}$, the maximum-likelihood (ML) estimator converges to $\tilde{\theta }$ (Huber 1967; White 1982). Here we investigate misspecification in a mixed-model context, the covariance of the data being misspecified due to infinitesimal interactions or other nonadditive effects. We consider three different scenarios (A–C) and in each of them three different values of additive and nonadditive genetic variance. The total phenotypic variance is assumed to be known and equal to 1.

In scenario A, the phenotype $Y={\left(Y_1,\mathrm{\dots },Y_n\right)}^{\prime }$ of n individuals is modeled using the multivariate normal distribution

$$P_{{\sigma }_{\mathrm{A}}^2,{\sigma }_{\mathrm{E}}^2}=N\left(0,{\sigma }_{\mathrm{A}}^{2}K+{\sigma }_{\mathrm{E}}^2{I}_n\right),$$ (1)

where K is a marker-based GSM, $I_n$ is the identity matrix, ${\sigma }_{\mathrm{A}}^2\mathrm{\in }\left[\mathrm{0,1}\right]$ is the additive genetic variance, and ${\sigma }_{\mathrm{E}}^2=1\mathrm{-}{\sigma }_{\mathrm{A}}^2$ is the residual variance. We assume, however, that Q, the actual distribution of Y, is the zero mean normal distribution with covariance $0.4K+0.2\left(K\mathrm{\cdot }K\right)+0.4I_n$, $K\mathrm{\cdot }K$ being the Hadamard (entry-wise) product. The matrix $\left(K\mathrm{\cdot }K\right)$ is the covariance due to small epistatic interactions between all standardized marker scores (Supporting Information, File S1; see also Jiang and Reif 2015). Hence, the narrow- and broad-sense heritabilities are equal to, respectively, 0.4 and 0.6. In addition to this genetic architecture, we also consider the case where the covariance matrix of Y is $0.2K+0.1\left(K\mathrm{\cdot }K\right)+0.7I_n$ (i.e., $h^2=0.2$ and $H^2=0.3$) and $0.6K+0.3\left(K\mathrm{\cdot }K\right)+0.1I_n$ (i.e., $h^2=0.6$ and $H^2=0.9$).

For all these genetic architectures, $\left(K\mathrm{\cdot }K\right)$ does not equal the identity matrix $I_n$, and Q is therefore not contained in model (1). Hence, the ML estimator will not converge to Q, but rather to the point (${\tilde{\mathit{\sigma }}}_{\mathrm{{\rm A}}}^{\mathrm{2}},{\tilde{\mathit{\sigma }}}_{\mathrm{{\rm E}}}^{\mathrm{2}}$) minimizing the KL divergence, $\mathrm{KL}\left(Q,P_{{\sigma }_{\mathrm{A}}^2,{\sigma }_{\mathrm{E}}^2}\right)$. For genetic similarity matrices derived from published data in maize, rice, and Arabidopsis, ${\tilde{\sigma }}_{\mathrm{A}}^2$ ranges between 0.47 and 0.53, given a true value of 0.4 (Table 1). Similar bias occurs when ${\sigma }_{\mathrm{A}}^2=0.2$ and ${\sigma }_{\mathrm{A}}^2=\mathrm{0.6.}$ Hence, the presence of epistatic interactions leads to inflated estimates of additive genetic variance. For a panel of simulated unrelated individuals, ${\tilde{\sigma }}_{\mathrm{A}}^2$ equals the true value of ${\sigma }_{\mathrm{A}}^2$, which is due to the much smaller off-diagonal elements of K, making $K\mathrm{\cdot }K$ almost indistinguishable from $I_n$.

Table 1 .

Values of the additive genetic variance $\left({\tilde{\sigma }}_{\mathrm{A}}^2\right)$ minimizing the Kullback–Leibler divergence KL(Q, P) with respect to the true distribution (Q) of scenarios A–C, with P contained in models (1)–(3)

Population/source	Species	Size (n)	Scenario A	Scenario B	Scenario C
${\sigma }_{\mathrm{A}}^2=0.2,H^2=0.3$
Swedish regmap	Arabidopsis thaliana	298	0.26 (0.111)	0.27 (0.052)	0.26 (0.076)
Hapmap	A. thaliana	350	0.23 (0.164)	0.29 (0.048)	0.23 (0.116)
Van Heerwaarden et al. (2012)	Zea mays	400	0.25 (0.096)	0.27 (0.045)	0.25 (0.066)
Zhao et al. (2011)	Oryza sativa	413	0.26 (0.075)	0.25 (0.044)	0.26 (0.060)
Unrelated individuals	Simulated	3000	0.20 (0.067)
${\sigma }_{\mathrm{A}}^2=0.4,H^2=0.6$
Swedish regmap	A. thaliana	298	0.53 (0.101)	0.58 (0.034)	0.53 (0.075)
Hapmap	A. thaliana	350	0.47 (0.176)	0.60 (0.030)	0.48 (0.136)
Van Heerwaarden et al. (2012)	Z. mays	400	0.50 (0.092)	0.58 (0.029)	0.50 (0.069)
Zhao et al. (2011)	O. sativa	413	0.51 (0.098)	0.52 (0.032)	0.50 (0.102)
Unrelated individuals	Simulated	3000	0.40 (0.066)
${\sigma }_{\mathrm{A}}^2=0.6,H^2=0.9$
Swedish regmap	A. thaliana	298	0.78 (0.086)	0.89 (0.011)	0.77 (0.083)
Hapmap	A. thaliana	350	0.71 (0.160)	0.90 (0.008)	0.71 (0.149)
Van Heerwaarden et al. (2012)	Z. mays	400	0.75 (0.079)	0.88 (0.019)	0.66 (0.163)
Zhao et al. (2011)	O. sativa	413	0.73 (0.156)	0.77 (0.162)	0.57 (0.494)
Unrelated individuals	Simulated	3000	0.60 (0.062)

Minimization was performed by evaluating KL divergence on the grid 0, 0.01, …, 1 for all variance components, under the constraint they sum to one. Standard errors (in parentheses) were calculated as the square root of the asymptotic variance (White 1982, theorem 3.2). Five populations were considered: the Arabidopsis Hapmap and Swedish regmap (Horton et al. 2012; Kruijer et al. 2015), the rice population from Zhao et al. (2011), the maize population of van Heerwaarden et al. (2012), and a simulated population (File S1). In scenarios B and C there are r = 2 replicates of each genotype.

In scenario B, a plant trait is phenotyped on r genetically identical replicates. Following Kruijer et al. (2015), the observations $Y={\left(Y_{11},\mathrm{\dots },Y_{nr}\right)}^{\prime }$ are modeled by the normal distribution

$$P_{{\sigma }_{\mathrm{A}}^2,{\sigma }_{\mathrm{E}}^2}=N\left(0,{\sigma }_{\mathrm{A}}^{2}ZK{Z}^{\prime }+{\sigma }_{\mathrm{E}}^2{I}_{nr}\right),$$ (2)

Z being an incidence matrix assigning plants to genotypes. The true distribution Q is multivariate normal with covariance $0.4ZK{Z}^{\prime }+0.2Z{Z}^{\prime }+0.4I_{nr};$ i.e., there are nonadditive (not necessarily epistatic) effects with independent $N\left(\mathrm{0,0.2}\right)$ distributions. Such effects could be due to, for example, genotype–environment interaction. As in scenario A, we also consider a genetic architecture with $h^2=0.2$ and $H^2=0.3$ (i.e., covariance $0.2ZK{Z}^{\prime }+0.1Z{Z}^{\prime }+0.7I_{nr}$) and a genetic architecture with $h^2=0.6$ and $H^2=\mathrm{0.9.}$ In contrast to model (1) (where $Z=I_n$ and $r=1$), $Z{Z}^{\prime }$ is different from $I_{nr}$, and Q is not contained in model (2). Again, the value ${\tilde{\sigma }}_{\mathrm{A}}^2$ minimizing KL divergence is substantially larger than the true value (Table 1), and additive genetic variance will tend to be overestimated. Intuitively, this is because the block structure $Z{Z}^{\prime }$ is better captured by $ZK{Z}^{\prime }$ than by the diagonal residual.

Scenario C is a combination of scenarios A and B. To avoid the misspecification occurring in scenario B, the model

$$P_{{\sigma }_{\mathrm{A}}^2,{\sigma }_{\mathrm{G}}^2,{\sigma }_{\mathrm{E}}^2}=N\left(0,{\sigma }_{\mathrm{A}}^{2}ZK{Z}^{\prime }+{\sigma }_{\mathrm{G}}^{2}Z{Z}^{\prime }+{\sigma }_{\mathrm{E}}^2{I}_N\right)$$ (3)

is considered, extending (2) with independent nonadditive effects. This model has been used in the analysis of field trials (Oakey et al. 2006, 2007), as well as genomic prediction (Gianola and van Kaam 2008; Howard et al. 2014; Jarquin et al. 2014). If in fact the nonadditive effects have covariance $K\mathrm{\cdot }K$ (as in scenario A) and ${\sigma }_{\mathrm{A}}^2=0.4$, the data have covariance $0.4ZK{Z}^{\prime }+0.2Z\left(K\mathrm{\cdot }K\right)Z^{\prime }+0.4I_{nr}$. As in scenarios A and B, the ${\tilde{\sigma }}_{\mathrm{A}}^2$ minimizing KL divergence is larger than the true value (Table 1), except for the rice population of Zhao et al. (2011) with $H^2=\mathrm{0.9.}$ In the latter case, ${\tilde{\sigma }}_{\mathrm{E}}^2$ was on average 0.14, while its bias was at most 0.01 for all other populations and heritability levels.

In addition to the minimization of KL divergence we analyzed ML estimates for simulated traits, in which case the phenotypic variance is unknown (File S2). For most populations and heritability levels, the bias of additive genetic variance estimates $\left({\tilde{\sigma }}_{\mathrm{A}}^2\right)$ is similar to what was found by minimizing KL divergence in models (1)–(3). Differences are largest for the population of Zhao et al. (2011), where the total phenotypic variance is consistently overestimated.

The bias we identified here by statistical arguments and simulations has important implications, in particular for immortal populations, for which genetically identical replicates are available (e.g., Arabidopsis thaliana, agronomic crops, bacteria, and fungi). Typically there is strong population structure and often only several hundreds of different genotypes are phenotyped. One can analyze such data at the individual level [model (2)] or at the level of genotypic means [model (1), with ${\sigma }_{\mathrm{E}}^2$’s divided by the number of replicates]. Kruijer et al. (2015) showed that in the latter type of analysis, standard errors of heritability estimates can be huge, and recommended model (2) for both heritability estimation and genomic prediction. Here we have shown that in the presence of nonadditive effects, this model is likely to overestimate additive genetic variance. If, however, the nonadditive effects are due to epistatic interactions, analysis at the genotypic means level [model (1)] will, apart from the large sampling variance, also give inflated estimates of additive genetic variance. This is a rather realistic scenario, since epistasis may be an important part of the genetic architecture (Mackay 2014), and several other types of nonadditive effects can be ruled out or minimized for immortal populations: e.g., genotype–environment interactions are unlikely in homogeneous controlled environments with adequate randomization, and dominance effects are absent when using inbred lines. Inflated heritability estimates may also affect the performance of G-BLUP, although the loss in accuracy is considerably smaller than in the case where heritability is underestimated (Kruijer et al. 2015).

Interestingly, the inflation of additive genetic variance is not due to any nonlinearity or absence of main effects (as in, e.g., Culverhouse et al. 2002; Song et al. 2010; Zuk et al. 2012), but rather to the population structure present in the epistatic GSM, which to some extent resembles the structure of the GSM for the additive effects. At the same time, it is this structure that makes the epistatic GSM distinguishable from the diagonal error. This suggests that epistatic interactions are easier to model in structured populations; i.e., sampling variance of epistatic variance components may not be as large as in unstructured human populations (Yang et al. 2011). Expressions for the asymptotic variance in a model with both additive and epistatic effects (File S3) indicate that this is indeed the case. More generally, the inflation of heritability estimates due to misspecification illustrates the difficulty of modeling and estimating genetic effects. As recently pointed out by Speed and Balding (2015) this is already challenging for the additive genetic effects, in the sense that depending on the genetic architecture, different GSMs may be appropriate. Indeed, the potential bias resulting from an inappropriate GSM could be assessed by evaluating KL divergence with respect to the true model, as is the case for alternatives for the epistatic GSM considered here.