2.13.1. Overview of Mixed Linear Models¶
Mixed linear models incorporate both “fixed effects” and “random effects” (that is, “mixed effects”). The independent variables in a linear regression may be thought of as fixed effects. To solve for the random effects in a mixed model, something should be known about the variances and covariances of these random effects.
The Mixed Model Equation¶
Suppose there are measurements of a phenotype which is influenced by fixed effects and instances of one random effect. The mixed linear model may be written as
where is an vector of observed phenotypes, is an matrix of fixed effects, and is a vector representing the coefficients of the fixed effects. is the unknown random effect, and is an matrix relating the instances of the random effect to the phenotypes. We assume
is the variance-covariance matrix for the random effect , and
so that
Examples of “fixed effects” may include the mean, one or more genotypic markers, and other additional covariates that may be analyzed.
Examples of a “random effect” are:
Polygenic effects from each of subgroupings, where the measurements have been grouped into subgroupings such as inbred strains. is then an incidence matrix relating subgroupings/strains to measurements, and should be a matrix showing the pairwise genetic relationship among the strains.
Polygenic effects from each of the samples, where there is just one measurement per sample. is then just the identity matrix , and should be a pairwise genetic relationship or kinship matrix among the samples.
Note
At this time, none of the SVS Mixed Linear Model Analysis tools supports organizing measurements into subgroupings such as inbred strains.
The parameters and are called the “variance components”, and are assumed to be unknown. To solve the mixed model equation, the variance components must first be estimated. Once this is done, a generalized least squares (GLS) procedure may be used to estimate .
More than One Random Effect¶
It is also possible for a mixed linear model to have more than one random effect. For instance, a mixed model with two random effects and would be defined by
where
and
and being the variance/covariance matrices for and respectively, so that
In this example, there are three variance components: , , and .
Finding the Variance Components¶
The SVS Mixed Linear Model Analysis tools use either one of two approaches to estimate the variance components:
EMMA. To estimate the variance components and for a mixed model with only one random effect, you may select to use the EMMA (Efficient Mixed Model Association) approach ([Kang2008]). This approach reduces the estimating problem to a maximization search in just one dimension, and will thus be reliable and fast for use cases with 10,000 or fewer samples.
(For sample sizes over 10,000 or 15,000, this method starts becoming limited by the time it takes to solve the eigenvalue/eigenvector problem for these matrices. While some of the Mixed Linear Model Analysis tools (specifically GBLUP and K-Fold cross validation using GBLUP) have a large-data capability which saves space in memory, you must still choose, when using this capability, the tradeoff between faster approximation of the eigenvalues and eigenvectors vs. more exact computation of these values.)
AI. Average Information (AI) ([Yang2011], [LeeSH2012]) is the other approach you may select. Multiple variance components may be estimated using this approach, and it is not limited for large sample sizes by the need to solve an eigenvalue/eigenvector problem.
However, neither the MLMM feature nor large-data capability is currently implemented in SVS for the AI approach.
Either the full likelihood (Maximum Likelihood ML) or the restricted likelihood (Restricted Maximum Likelihood REML) may be maximized. The restricted likelihood is defined as the full likelihood with the fixed effects integrated out. As stated in [Kang2008], “The restricted likelihood avoids a downward bias of maximum-likelihood estimates of variance components by taking into account the loss in degrees of freedom associated with fixed effects.”
Note
The SVS Mixed Linear Model Analysis tools always maximize the restricted likelihood (REML) rather than the full likelihood (ML) except when the Bayes Information Criterion (for the MLMM feature) is computed.
Finding the Variance Components Using EMMA¶
Suppose , , and , which is a function of . Under the null hypothesis, the full log-likelihood function can be formulated as
and the restricted log-likelihood function can be formulated as
The full-likelihood function is maximized when is , and the optimal variance component is for full likelihood and for restricted likelihood, where is a function of as well.
Using spectral decomposition, it is possible to find and such that
and
where , is , and is an eigenvector matrix corresponding to the nonzero eigenvalues. is an matrix corresponding to the zero eigenvalues. and are independent of .
Let . Then, finding the maximum-likelihood (ML) estimate is equivalent to optimizing
with respect to , and finding the restricted maximum-likelihood (REML) estimate is equivalent to optimizing
with respect to . (See the Appendix of [Kang2008] for the mathematical details, except that , not “” as that Appendix states.) These functions are continuous for if and only if all the eigenvalues are nonnegative (and, for , the eigenvalues are nonnegative). Otherwise, if the kinship matrix is not positive semidefinite, the likelihood will be ill-defined for a certain range of .
The derivatives of these functions, which may be used to find the local maxima for the functions themselves, are
and
The ML or REML may be searched for by subdividing the values of into 100 intervals, evenly in log space from to , and applying a method such as the Newton-Raphson algorithm or the secant method on or to all the intervals where the sign of the derivative function changes, then taking the optimal among all the stationary points and endpoints.
Note
The secant method is used by the SVS Mixed Linear Model Analysis tools.
Notice that evaluating or does not require a large number of matrix multiplications or inverses at each iteration as other methods typically do – instead, the EMMA technique computes spectral decomposition only once. Thus, using the grid search indicated above, the likelihood may be optimized globally with high confidence using much less computation.
Finding the Variance Components Using the Average Information (AI) Technique¶
Problem Restatement and Summary¶
Suppose we have random effects in our mixed model. We may then write our mixed model as
where is a vector of random genetic effects. In this model, the phenotypic variance is partitioned into the variance explained by each of the genetic factors and the residual variance. We have
where is the variance of the -th genetic factor with its corresponding variance/covarince matrix .
( can be a kinship matrix or a GBLUP Genomic Relationship Matrix (GRM). See The Genomic Relationship Matrix for more information about the GRM matrix.)
The variance components (for all ) and are unknown and need to be determined. To determine these variance components, SVS uses the Average Information (AI) technique to implement the restricted maximum likelihood (REML) method. This process may be summarized as follows:
SVS first initializes each component to (the somewhat arbitrary value of) divided by the number of total components .
Next, SVS uses one iteration of the expectation maximization (EM) algorithm (as adapted for the AI REML method).
Finally, repeated iterations of the Average Information (AI) algorithm itself are used until convergence is achieved. The criterion for convergence is that the log-likelihood changes by less than the threshold value of .
Definitions and Algorithm Details¶
Define as a vector of variance components
the matrix as
and as the log-likelihood function of the model, ignoring the constant:
We now define a matrix called which is the average of the observed and expected information matrices:
The vector of first derivatives of the log likelihood function with respect to each variance component is
The -th variance component is initialized to
This is updated by one iteration of the EM algorithm as
The EM algorithm is used as an initial step to determine the direction of the iteration updates because it is robust to poor starting values.
The remaining iterations use the Average Information algorithm itself, which consists of updating the vector of variance components as
to get from iteration to iteration .
Convergence is determined by whether , where is the log likelihood of the -th iteration.
During the iteration process, any component whose estimate is negative (that is, the estimate “has escaped from the parameter space”) is set to .
See [Yang2011] for further information.
Estimating the Variance of Heritability for One Random Effect¶
The formula for the estimate of the variance of heritability is derived as follows using a Taylor series expansion:
Now, using the fact that the pseudo-heritability (or narrow-sense heritability) is:
We can obtain the formula for the estimate of the variance of heritability:
Note that
So:
The formulas and methods for calculating the variance components can be found in Finding the Variance Components. is the (estimated) sampling variance/covariance of the (estimates of the) variance components, and may be computed as
where is the Average Information Matrix for
Solving the Mixed Model Equation Using EMMA¶
The generalized least squares (GLS) solution to
may now be obtained. Note that the variance of is
If we can find a matrix such that
we can substitute , , and to get
(The Cholesky decomposition of is one way to obtain such a matrix .) This equation can be solved for through ordinary least squares (OLS), because we have
The value of the residual sum of squares (RSS) from solving the transformed equation is the Mahalanobis RSS for the original equation .
Taking advantage of the eigendecomposition of performed in the EMMA algorithm, the computation of a valid can be simplified to
Using the Mixed Model for Association Studies¶
The Exact Model¶
Association studies are typically carried out by testing the hypothesis for each of loci, one at a time, on the basis of the model
where is the minor allele count of marker for individual , is a (fixed) effect size of marker , and are other fixed effects such as the intercept and any fixed covariates. The error term is
If we assume the individuals are unrelated and there is no dependence across the genotypes, the values will be independently and identically distributed (i.i.d.), and thus simple linear regressions will make appropriate inferences for the values of .
However, the variance of the first term of actually comes closer to being proportional to a matrix of the relatedness or kinship between samples. Thus, if we write
we see that the equation for reduces to the mixed-model equation
Note that strictly speaking, to use this equation, we should base not only the kinship, but also the variance components, upon all markers except for marker .
The EMMAX Approximations and Technique¶
Even using the EMMA technique, finding the kinship matrix for, variance components for and solving for for all would be a daunting task. However, we may make two approximations:
Let
approximate , and let
approximate . Then, we have
To solve this, we need to compute the kinship matrix just once, using all markers. That kinship matrix may then be used to solve this equation for every marker .
Find the variance components once – specifically, for the system of equations
using the kinship matrix which is computed just once for all markers . Then, use these variance components (for the variance of the which is ) and (for the variance of the which is ) to apply the GLS method for solving
for for every marker .
This technique, which is called EMMAX (EMMA eXpedited) and was published in [Kang2010], allows mixed models to be used for genome-wide association testing within a very reasonable amount of computing time.
Scaling the Kinship Matrix¶
The actual EMMAX technique, however, before using the kinship matrix , scales it (and thus effectively scales ) by an amount that will make the expectation of the estimated population variance of the (scaled) to be , just as the expectation of the estimated population variance for the is .
This is done by defining a scaling factor as
and multiplying by it to get
Here, and is a length vector of ones. is called a “Gower’s centering matrix”–it has the property that when you apply it to a vector to get , it will subtract the mean of the components of from each component of :
The reasoning for using this scaling factor is as follows:
Suppose we have a vector of elements . Estimate the population variance of these elements over all the samples . (This estimate is sometimes called the “sample variance”.) The (unbiased) estimate would be
where is the average of the components of .
However, another way to write this is
since
and
Two other ways to write this are
since is a scalar, and
since for any two matrices and .
We note that
therefore, the estimated population variance may be written as
Looking at as a random variable of dimensions and as a scalar random variable, let us write the expectation of the estimated population variance as:
But defines a relationship matrix among the elements of possible . (Note that the possible instances of themselves might not be “centered” – that is, the components of the may or may not have zero averages.) We now write the expected estimated population variance as
We wish to “scale to one” – that is, set to one by scaling appropriately. We do that by defining , where
and noting that
Note
The features that scale the kinship matrix before using it are:
Single-locus mixed model GWAS (EMMAX) (Performing Analysis)
Multi-locus mixed model GWAS (MLMM) (Performing Analysis)
Single-locus mixed model GWAS (EMMAX) with Interactions (Mixed Linear Model Analysis with Interactions)
The other SVS mixed-model features, including the GBLUP-related features, do not scale the kinship matrix before using it.
Note
When the EMMA algorithm for GBLUP is used, GBLUP will output what the computed pseudo-heritability would have been had the matrix been scaled.
Further Optimization When Covariates Are Present¶
The following technique is mentioned in passing in [Segura2012] and is used both in [Vilhjalmsson2012] and in the mixed-model tools of SVS.
If we have a mixed linear model with fixed-effect covariates , one particular “more interesting” fixed-effect covariate and a random-effect covariate for which the scaled relationship matrix is
and we have this model for many and we don’t need to find the covariate coefficients for any of these models, and we have a matrix such that (see Solving the Mixed Model Equation Using EMMA), we can perform the following optimization:
Solve the ordinary-least-squares (OLS) “null hypothesis” or reduced-model problem
to find as an estimate for .
Designate the (Mahalanobis) RSS obtained from solving this equation as
Perform the QR algorithm on to get
where and are the “thin”, “reduced”, or “economic” versions of and .
Define
giving us
Transform the original equation by pre-multiplying it by to get
But
because the columns of are “orthogonal” and of “unit length” and so and . Thus, we have
may be re-written as
the last step being because
and, since ,
and
Thus,
Define as
so that
If we pre-multiply the original problem simply by , we get
which may be solved as an ordinary-least-squares (OLS) problem (Solving the Mixed Model Equation Using EMMA), making proportional to .
Meanwhile,
and thus is proportional to . This means that
may be treated as an ordinary-least-squares (OLS) problem. This may now be solved for all .
Note
For optimization, SVS pre-computes the matrix product and uses this product as one matrix to help perform all of the regressions involving .
Note
The matrix is the “annihilator matrix” for the null hypothesis problem
Designate the Mahalanobis Residual Sum of Squares (Mahalanobis RSS) for marker as
which is optimized to
This is the Residual Sum of Squares (RSS) value for the regression as transformed by pre-multiplying it by .
Note
If we have a (completely fixed-effect) linear model with covariates and one particular “more interesting” covariate ,
and we have this model for many and we don’t need to find the covariate coefficients for any of these models, we can perform the same kind of optimization.
Solve the ordinary-least-squares (OLS) “null hypothesis” or reduced-model problem
to find as an estimate for .
Perform the QR algorithm on to get
where and are the “thin”, “reduced”, or “economic” versions of and .
Define
Transform the original equation by pre-multiplying it by to get
But
because the columns of are “orthogonal” and of “unit length” and so and . Thus, we have
may be re-written as
because and .
Thus,
This is equivalent to the ordinary-least-squares (OLS) problem
where the variance of is proportional to . This is because
which we assume to be proportional to .
Note that the matrix is the “annihilator matrix” for the null hypothesis problem
The Mixed Model Equation for Gene-by-Environment Interactions¶
Note
This algorithm is distinct from the GBLUP-related algorithm discussed later in Incorporating Gene by Environment Interactions into GBLUP.
Suppose there are measurements of a phenotype which is influenced by fixed effects and instances of one random effect. Also suppose there are one or more (say, ) gene-by-environment effects, where the interaction is potentially with one “more interesting” predictor variable.
We model this with a full model and a reduced model. The full model is:
and the reduced model is:
where is an -vector of observed phenotypes, is an matrix of fixed covariates, is an matrix of fixed terms that are being tested for gene-environment interactions, is an -vector containing the current “more interesting” covariate or predictor variable that may be doing the interacting, and is an matrix containing the interaction terms created by multiplying the columns of element-by-element with (the “Predictor”). All of the above terms correspond to the terms with which they are shown and to whether they apply to the full model or to the reduced model. The and terms are random effect and error terms, respectively.
We approximate this using a technique like that used in the EMMAX method without interactions (The EMMAX Approximations and Technique). Specifically, we find the variance components once, using the parts of the above equations that are independent of altogether. This works as follows–we write
where “vc” stands for (finding the) “variance components”.
We assume that
is the variance-covariance matrix for the random effect , and that
so that
The EMMA technique (Finding the Variance Components Using EMMA and Solving the Mixed Model Equation Using EMMA) is then used to determine the variance components and , as well as a matrix (and its inverse) such that
It is now possible to go through every marker (“more interesting covariate”) and compute (as an EMMAX-type approximation) the following full and reduced models:
for the full model, where the term is assumed to be an error term proportional to the identity matrix, and
for the reduced model, where also the term is assumed to be an error term proportional to the identity matrix.
Optimization when Gene-Environment Interaction Terms Are Included¶
While SVS explicitly computes the full model mentioned above and outputs all of its values, it performs only an optimization of the reduced model computation, which is sufficient to determine the RSS of the reduced-model equation above (which is the approximate Mahalanobis RSS for the original reduced-model equation), and thus to determine the full-vs-reduced p-value for the complete model.
This reduced-model optimization is to solve
where , and is derived from performing the QR algorithm, as
Note
This optimization technique is detailed above in Further Optimization When Covariates Are Present. (Note that in that section, it is the full model that is being optimized, but here, we are optimizing the reduced model.) Please substitute for and for in that section.
Note
For the similar (completely fixed-effect) linear-model problem with full model
and reduced model
we may optimize computing the reduced model using the technique shown in the (completely fixed-effect) linear-model note to Further Optimization When Covariates Are Present, where that optimization is for a full model, and should be substituted for the in that note.
The reduced-model Mahalanobis RSS that we will use for marker thus becomes, in its optimized form,
Meanwhile, for the full-model Mahalanobis Residual Sum of Squares (Mahalanobis RSS) for marker , we use:
The Multi-Locus Mixed Model (MLMM)¶
For complex traits controlled by several large-effect loci, a single-locus test may not be appropriate, especially in the presence of population structure.
Therefore, [Segura2012] has proposed a simple stepwise mixed-model regression with forward inclusion and backward elimination of genotypic markers as fixed effect covariates. This method, called the Multi-Locus Mixed Model (MLMM), proceeds as follows:
Begin with an initial model that includes, as its fixed effects, only the intercept and any additional covariates you may have specified.
Using this model, perform an EMMAX scan through all markers (that you have not specified as additional covariates).
From the markers scanned above, select the most significant marker and add it to the model as a new fixed-effect covariate (“cofactor”), creating a new model.
Repeat (2) and (3) (forward inclusion) until a pre-specified maximum number of forward steps is reached or until forward inclusion must be stopped for some other reason (see note below).
For each selected marker in the current model, temporarily remove it from the fixed effects and perform an EMMAX scan over only that marker.
Eliminate, from the current model, the marker that came out as least significant using the above test. A new smaller model is created.
Repeat (5) and (6) (backward elimination) until only one selected marker is left.
The variance components are re-estimated between each forward and backward step, while the same kinship matrix is used throughout the calculations.
Note
The following are reasons why forward inclusion might be stopped early:
When the pseudo-heritability estimate has become close to zero.
When all of the variance is explained by the currently-selected fixed-effect covariate markers (cofactors) and any covariates you may have specified.
When the latest significant marker that has been selected is collinear with the previously-selected fixed-effect covariate markers (cofactors) and any covariates you may have specified.
When the number of active samples is no longer enough to continue analysis.
When the number of active (non-selected) markers is no longer enough to analyze the resulting models.
Model Criteria¶
The result of this stepwise regression is a series of models. Several model criteria have been explored by the authors of [Segura2012] to determine how appropriate any of the models is:
Bayes Information Criteria (BIC). This is calculated as , where is the full-model log-likelihood, is the number of model parameters (one for the intercept, one for , one for each marker covariate used in the particular MLMM model, and finally one for each additional covariate used in all of the models), and is the sample size/number of individuals.
Given any two estimated models, the model with the lower value of BIC is the preferred choice.
However, the authors of [Segura2012] believe this model is “too tolerant in the context of GWAS”.
Extended Bayes Information Criteria (Extended BIC). This is the BIC penalized by the model space dimension. Its formula is
where is the initial number of model parameters (one for the intercept, one for , and one for each additional covariate used in all of the models), and is the total number of models which can be formed using marker covariates under the assumption that these will only be selected from the best markers.
Modified Bayes Information Criteria (Modified BIC). This adds a different penalty based not only by the model space dimension, but also by how many overall markers there are to test. Its formula is
where is the total number of markers being tested in the current step.
Bonferroni Criterion. Only defined for models derived from forward selection, this selects the model with the most covariate marker loci for which the best p-value obtained from the preceding EMMAX scan was below the Bonferroni threshold. The (Bonferroni) threshold used is , where is the total number of markers being tested in the current step.
Multiple Bonferroni Criterion. This selects the model with the most covariate marker loci all of which have individual p-values below the Bonferroni threshold. Here, “individual p-value” is as explained in the note of Outputs from the Multi-Locus Mixed Linear Model (MLMM) Method. The (Bonferroni) threshold used is , where is the total number of markers being tested in the current step.
Multiple Posterior Probability of Association. This selects the model with the most covariate marker loci all of which have posterior probabilities of association above a PPA threshold. The threshold used is 0.5 . Posterior probabilities of association are based on Bayesian priors of for every marker (and for every step), where is the total number of markers being tested in the current step, and are computed as follows:
Find the Bayes factor for marker as
where and are the values of the Mahalanobis RSS for the base model and for testing with marker , respectively.
Determine the posterior odds and posterior probability as
and
Gender Correction for EMMAX and MLMM¶
Gender correction for EMMAX and MLMM consists simply of considering X Chromosome genotypes for males as either zero (major allele present) or one (minor allele present). No other gender correction is performed.
Note
If a genotypic spreadsheet is used, it is still expected that male X chromosomes are coded in this spreadsheet as “two major alles” (major allele present) vs. “two minor alleles” (minor allele present).
Genomic Best Linear Unbiased Predictors (GBLUP)¶
Problem Statement¶
Suppose we have the mixed model equation
over samples, with fixed effects specified by that include the intercept and any additional covariates you may have specified, and we assume that . Also suppose that the random effects are additive genetic merits or genomic breeding values associated with these samples, and that these may be formulated from autosomal markers as
where is an matrix, the exact construction of which will be discussed later in Overall Normalization and Normalization by Individual Marker, and is a vector for which is the allele substitution effect (ASE) for marker . For now, suffice it to say that matrix is derived from the individual genotypes, with dependent on the genotype for the -th sample at the -th locus. (For inclusion of non-autosomal markers, see Correcting the GRM for Gender Using Overall Normalization and Correcting the GRM for Gender Using Normalization by Individual Marker.)
Further assume that , which makes , and that , where is an (unknown) constant which is the component of variance associated with the ASE. (For inclusion of non-autosomal markers in the calculation of ASE values, see Correcting the ASE for the Gender Chromosome.)
This gives us
This gives us
and
Our object is to estimate both the genomic breeding value for every sample and the ASE for every marker.
We now discuss obtaining the matrix and from that, a normalization factor or normalization factors and a kinship matrix.
Overall Normalization¶
There are two ways to normalize the entries of matrix . The first method is Overall Normalization.
For Overall Normalization, we use the mixed model equation exactly as described above, namely
with set to , , or , depending upon whether the genotype for the -th sample at the -th locus is homozygous for the minor allele, heterozygous, or homozygous for the major allele, respectively. Here, and are the major and minor allele frequencies for marker , respectively.
Additionally, we write the sum of would-be variances over all the markers if each had been at Hardy-Weinberg equilibrium as
We now define the variance matrix for Overall Normalization as
which gives us
where we let .
Normalization by Individual Marker¶
The second normalization method is Normalization by Individual Marker. (This method is used by the software package GCTA [Yang2011].)
For Normalization by Individual Marker, we instead use
and
as our mixed-model equations, where
and is two, one, or zero, depending upon whether the genotype for the -th sample at the -th locus is homozygous for the minor allele, heterozygous, or homozygous for the major allele, respectively. Here also, and are the major and minor allele frequencies for marker , respectively.
This time, we define our variance matrix as
which gives us
where we let .
We note that if we define
we see that
where is the matrix defined above in Overall Normalization.
The Genomic Relationship Matrix¶
For either normalization method, we refer to the matrix ( or ) as the GBLUP Genomic Relationship Matrix (GRM). This matrix may be used as a kinship matrix for solving the GBLUP mixed-model equation, and or may be thought of as the variance component for .
Note
Because this method uses a kinship matrix based on genotypes rather than on actual ancestry, the results are referred to as “Genomic Best Linear Unbiased Predictors” rather than just “Best Linear Unbiased Predictors”.
Unlike some other SVS mixed-model analysis tools, SVS GBLUP does not scale its kinship matrix using a Gower’s centering matrix. Instead, the normalizing method used, whether overall or individual-marker, is considered sufficient to scale the GBLUP Genomic Relationship Matrix.
Because of how it is constructed, the GBLUP Genomic Relationship Matrix is itself a “centered matrix” and is (thus) singular. However, it is still a positive semidefinite matrix and will work well as a kinship matrix.
The GBLUP Genomic Relationship Matrix may either be constructed as a separate step by itself, separate from performing any analyses, or it may be computed “dynamically” or “on-the-fly” as the first part of a GBLUP analysis. Note that
If the Genomic Relationship Matrix is pre-computed separately from performing any analyses, genotypes from all samples will be used.
If, on the other hand, the GRM is computed “dynamically” as a part of a GBLUP analysis, genotypes from any samples for which the phenotype or any covariates or gene-by-environment interaction variables involved in that analysis have a missing value will not used for computing the GRM.
The exception is that when predicting for missing values (Genomic Prediction) and the GRM is computed “dynamically”, genotype data from samples in the “prediction set” (those with a missing phenotype) that have missing covariate values is still used to compute the GRM. (But not data from samples with missing values for gene-by-environment interaction variables.)
Finding the Genomic Best Linear Unbiased Predictors Using EMMA¶
For either matrix ( or ), and using the EMMA technique (Finding the Variance Components Using EMMA), we can find , , and The second of Henderson’s mixed-model equations, as modified to accommodate singular , is
This may be rewritten as
This gives us
Noting the following equalities,
we may write
as a solution for the genomic BLUP.
In SVS, this is computationally streamlined by finding
then computing .
Finding the Allele Substitution Effect (ASE) Using EMMA and Overall Normalization¶
If we now define
where is the matrix for Overall Normalization, we find that
which makes a solution for the ASE.
In SVS, this is computationally streamlined by finding
where is the version of for Overall Normalization.
Finding the ASE Using EMMA and Individual Marker Normalization¶
If we now define
where is the matrix for Normalization by Individual Marker, we find that
This makes a best linear unbiased predictor (BLUP) for (as defined in Normalization by Individual Marker).
To obtain the ASE, we need to scale the elements of appropriately. Define the -th element of by dividing element of by . This calculation is equivalent to writing
We see, however, that
Thus, we see that is a solution for the ASE.
In SVS, we computationally streamline this by finding
where is the version of for Individual Marker Normalization.
Finding the Genomic Best Linear Unbiased Predictors and ASE Using AI¶
For any variance component , we find
where the variance components and the matrix are defined in Finding the Variance Components Using the Average Information (AI) Technique. From this, we can compute
for any variance component.
For any set of markers that contribute exclusively to a genomic relationship matrix (GRM) for just one variance component , where is completely generated from as or as , and or is also generated exclusively from set of markers, we may compute an allele substitution effect (ASE). This ASE value is either
if Overall Normalization is being used, or
if Normalize by Individual Marker is being used.
Note
The Gene-by-Environment Interaction feature Incorporating Gene by Environment Interactions into GBLUP (available only when using the Average Information algorithm) generates one or more GRM matrices that do not fit the proper criteria for generating ASE values. This is because these matrices are not generated directly from or for any set of markers.
Normalizing the ASE Itself Using EMMA¶
An additional output when using EMMA, for either GRM normalization choice, consists of a normalization of the allele substitution effects themselves. To do this, each ASE is divided by the SNP Standard Deviation, which is the square root of the component of variance associated with the ASE. is reconstructed by dividing the additive genetic variance by the sum of would-be variances over all the markers if each had been at Hardy-Weinberg equilibrium:
The normalized ASE is then:
Incorporating Gene by Environment Interactions into GBLUP¶
Note
This algorithm is distinct from the EMMAX-related algorithm discussed in The Mixed Model Equation for Gene-by-Environment Interactions.
Suppose there are one or more gene-by-environment interactions we wish to test for, where for each genotype-environment interaction effect , there is an environmental variable which specifies two or more categories, being categorical (each environment being a category) or binary (defining two environments). We may specify the mixed model as
with
where includes all environmental variables as separate fixed effects and is the vector of genotype-environment interaction effects corresponding to . We use for all pairs of individuals in the same environment as each other, as defined by , and for all pairs of individuals in separate environments.
The variance components , , and must be found using the Average Information (AI) REML algorithm.
Genetic Correlation Using GBLUP¶
Suppose we have two phenotypes, and , which are dependent on some genotypic data. We may write a standard bivariate linear mixed model for these as
and
where and are polygenic effects influencing the traits and , respectively.
We may alternatively think of this model as
If is the Genomic Relationship Matrix (GRM) for the genotypic data, we may write the variance/covariance matrix () for this model as
where is the covariance between and . may also be written as
from which we see we have five variance components.
SVS uses an adaptation of the AI REML algorithm to estimate these, with two variations:
Simply estimate these five variance components,
Or instead use
where is theoretically zero, but actually may exist as a residual covariance.
Designate the following names for the above variance/covariance matrices:
and, where used,
This gives us, for the two variations of this algorithm,
and
Alternative names for the variance components that are used by the outputs of SVS, along with the interpretations for these variance components, are
V(G1) () Variance of due to random effects
V(G2) () Variance of due to random effects
C(G12) () Covariance between and due to random effects
V(E1) () Error variance for
V(E2) () Error variance for and, if necessary,
C(E12) () (Residual) covariance of and due to error terms.
Please see [LeeSH2012] for further details about this algorithm.
Correcting GBLUP for Gender¶
When using the GBLUP features of SVS, there are several steps that may be taken to correct for gender:
Correct the gemomic relationship matrix (GRM) for gender by using male-specific and female-specific values for the genotypes of the gender chromosome, along with modifying the normalization (for either normalization method).
Correct the gender-chromosome allele-substitution effect (ASE) values to match the gender corrections done for the GRM. (To do this requires defining separate ASE values for males vs. females.)
When AI REML is being used, and thus multiple GRM matrices may be analyzed, one GRM matrix (presumably gender-corrected) may be used for the gender chromosome and a different GRM matrix may be used for the other chromosomes.
Correcting the GRM for Gender¶
For markers within the gender chromosome (usually X), there are two factors:
The number of copies of the alternate allele at any locus for males can only be zero or one.
There are three assumptions that can be made about how much effect a gender chromosome allele has in males vs. how much effect a gender chromosome allele has in females. These assumptions go by the name of Dosage Compensation, and are:
Full dosage compensation–Each allele in females has only half the effect that an allele has in males.
Equal X-linked genetic variance for males and females–This occurs when each allele in females has times as much effect as each allele in males.
No dosage compensation–Each allele has a similar effect on the trait.
For a further discussion of Dosage Compensation, please see A Further Note About Dosage Compensation below, as well as [Yang2011].
Correcting the GRM for Gender Using Overall Normalization¶
We first compute a combined value for the major and minor allele frequencies and for gender-chromosome marker and for all samples based on the total frequencies of the alleles over all occurrences of the gender chromosome, as
and
where and are the number of male samples and the number of female samples, respectively, and , , and are the major and minor allele frequencies for male samples only and female samples only, respectively.
To begin with, we use the following entries for matrix for the gender chromosome:
For females, we use half of what we would have used for autosomes, to compensate for the fact that males have only one gender chromosome. Specifically, we use , , or for , depending upon whether the genotype for the -th sample at the -th locus is homozygous for the minor allele, heterozygous, or homozygous for the major allele, respectively.
For males, we use or for , depending upon whether the genotype for the -th sample at the -th locus contains the minor (gender-chromosome) allele or the major (gender-chromosome) allele, respectively. The parameter is the dosage compensation factor. Its values are
for Full dosage compensation
for Equal X-linked genetic variance for males and females
for No dosage compensation.
Entries of for markers outside of the gender chromosome are computed the same as otherwise.
The remaining differences are that:
Contributions to the GRM from gender-chromosome genotypes of female subjects are multiplied by . This applies to contributions from both and , so that the contribution to the covariance between two female subjects (or to the variance of one female subject) is multiplied by two.
For contributions to the overall normalization factor from gender-chromosome markers, we use
where and are the fraction of the samples that are male and female, respectively.
Note
, which is approximated by , is the variance of the male-sample genotype data at marker , as coded into matrix , and , which is approximated by , is the would-be variance, at Hardy-Weinberg equilibrium, of the female-sample genotype data at marker , as coded into matrix and afterward multiplied by .
The remaining steps for computing a GRM and computing are the same as they would be otherwise.
To show computing the gender-corrected GRM as a matrix formula, we have to do some factoring out of our matrix. Without loss of generality, we may consider the matrix to be partitioned into
where and are the male and female entries for the matrix . We also define matrices , , and to be diagonal with entries of for autosomal-chromosome markers and of the following values for each gender-chromosome marker :
Then, we can write
to show that contributions to the variance from gender-chromosome genotypes of female subjects (from each of and ) are multiplied by .
The next two formulas, for either REML algorithm, remain the same. We still compute, for EMMA,
and
as before, and for AI and for any variance component , we compute
and
as before, where matrix , as defined in Finding the Variance Components Using the Average Information (AI) Technique, is based on the gender-corrected .
Correcting the GRM for Gender Using Normalization by Individual Marker¶
In this case, we start with using the following entries for matrix for the gender chromosome (where later on, matrix will be built using matrix ):
For females, we use , , or for , depending upon whether the genotype for the -th sample at the -th locus is homozygous for the minor allele, heterozygous, or homozygous for the major allele, respectively. This is the same as would be for a non-gender chromosome (that is, , where is two, one, or zero).
For males, we use the same values as the male values for Overall Normalization, namely, or for , depending upon whether the genotype for the -th sample at the -th locus contains the minor (gender-chromosome) allele or the major (gender-chromosome) allele, respectively. Just as for Overall Normalization, the values for the dosage compensation factor are
for Full dosage compensation
for Equal X-linked genetic variance for males and females
for No dosage compensation.
Entries of for markers outside of the gender chromosome are computed the same as otherwise.
We next compute the normalizations, which are different between genders. We must use two different matrices ( and ) to represent the normalizations in matrix form.
Now, the normalizations themselves (and matrices and ) may be computed as follows:
For females, the normalization for gender-chromosome marker (and the -th diagonal element for matrix ) is
For males, the normalization for gender-chromosome marker (and the -th diagonal element for matrix ) is
For non-gender chromosomes, the normalizations for both male and female samples are the same as they would have been without gender correction.
To show computing the gender-corrected GRM using Normalization by Individual Marker as a matrix formula, we note that without loss of generality, we may consider the matrix to be partitioned as
where and are the male and female entries for the gender chromosome and and are the male and female entries for the remaining (autosomal) chromosomes.
Also, we note that without loss of generality, we may partition matrices and as
and
We now may represent a gender-corrected matrix as
or
Using this , we compute
Now, for EMMA, we compute
and
as before, and for AI and for any variance component , we compute
and
as before, where matrix , as defined in Finding the Variance Components Using the Average Information (AI) Technique, is based on the gender-corrected .
Correcting the ASE for the Gender Chromosome¶
When there is a designated gender chromosome, the allele substitution effects (ASE) are computed separately for females and males, although the results will only be different between the genders for the gender chromosome markers.
To perform the separate ASE computations for either normalization method, we may consider, without loss of generality, as we did in Correcting the GRM for Gender Using Normalization by Individual Marker, the matrix to be partitioned into
where and are the male and female entries for the matrix , and and are the male and female entries for the gender chromosome and and are the male and female entries for the remaining chromosomes.
Also, for either normalization method, we may consider, without loss of generality, to be partitioned into
where and are the values of corresponding to male and female samples, respectively.
To find what the gender-corrected ASE values should be, we write out the formula for the random effects for each normalization method more explicitly, in terms of the values of .
Overall Normalization¶
For Overall Normalization, recalling that we have
we may write the random effects by sample as
Based on the definitions in Correcting the GRM for Gender Using Overall Normalization, and bearing in mind how we have partitioned , we write out the entries in matrix notation as
Noting that
and
this gives us
Thus,
Now define
and finally define the allele substitution effects (ASE) for Overall Normalization as
This gives us
But
and
We now can confirm that for Overall Normalization,
Note
The above definitions of and may be rewritten as
From this, we see that
In other words, the ASE values for Overall Normalization for gender-chromosome markers are in proportion to each other, but are not the same as each other. Two sets of ASE values must be used.
Normalization by Individual Marker¶
For Normalization by Individual Marker, we first represent the gender-corrected matrix as
as was done in Correcting the GRM for Gender Using Normalization by Individual Marker, using the same notations. We next write out some representations of :
Noting that all of the -type matrices are symmetric, we get the following:
Representing the GRM as
we now represent the sample-based random effects as
Now define
and finally define the allele substitutions effects (ASE) for Normalization by Individual Marker as
This gives us
But
and
So, we can now confirm that for Normalization by Individual Marker,
Note
The above definitions of and may be rewritten as
This is and multiplying the same vector
Since the normalizations for any marker in the gender chromosome, and the diagonal elements of and corresponding to marker , are and for females and males, respectively, we see that
Therefore,
In other words, it is also true that for Normalization by Individual Marker, the ASE values for gender-chromosome markers are in proportion to each other, but are not the same as each other. Two sets of ASE values must be used here, also.
A Further Note About Dosage Compensation¶
While [Yang2011] discusses this subject in detail, let’s answer one question–why are these factors used for dosage compensation? Let’s look at the individual covariance terms:
Covariance Terms¶
As implied above, the contribution from marker to the covariance between two female subjects and for Normalization by Individual Marker is the GRM variance component times the expression
where and are the number of minor alleles at marker for (female) subjects and , and is the (overall) minor allele frequency for marker .
Meanwhile, the contribution from marker to the covariance between two male subjects and for Normalization by Individual Marker is, instead, the GRM variance component times
where is the dosage compensation factor, and are the number of gender-chromosome minor alleles (0 or 1) at marker for (male) subjects and , and (as above) is the (overall) minor allele frequency for marker .
For Overall Normalization, the male terms used in are the same as those used for Normalization by Individual Marker. Also, the contributions to the Overall Normalization factor from the gender chromosome for males are analogous to the normalization terms for Normalization by Individual Marker, namely
Therefore, if you were to use Overall Normalization to compute a GRM for only one marker in the gender chromosome, and only for male subjects, your contribution from that marker to the covariance between two (male) subjects would be the GRM variance component times
which is equal, for this data, to the male/male covariance term for Normalization by Individual Marker.
To match this, for Overall Normalization, the female terms used in are twice as small as the corresponding autosomal terms. However, the contributions to the variance for the female gender chromosome are multiplied by for each female term (and thus by two if both terms are female). Thus, for an all-female gender-chromosome GRM computed on only one marker using Overall Normalization, the contribution from that marker to the covariance between two (female) subjects and would be the GRM variance component times
This is equal to
which is equal, for this data, to the female/female covariance term for Normalization by Individual Marker.
Comparison of Dosage Compensation Factors¶
Let’s (focus on Normalization by Individual Marker and) write the male/male covariance term for the three levels of dosage compensation, also giving these expressions a common denominator.
For Full Dosage Compensation, we have
For Equal X-Linked Variance, we have
And for No Dosage Compensation, we have
Comparing these with the covariance between two female subjects, we see that
For Full Dosage Compensation (“each allele in females has only half the effect that an allele has in males”), we can compare the female term with the male term.
For Equal X-Linked Variance, we find ourselves comparing the female term with the male term, using the common denominator above. You might think of this as a type of “Partial Dosage Compensation”.
Finally, for No Dosage Compensation (“each allele has a similar effect on the trait”), we can compare the female term with the male term.
This leads to the second question, “How is the Equal X-Linked Variance assumption ‘Equal’?” The answer is that in each of the expressions
and
the dosage is normalized by the would-be variance of the dosage for the whole marker if all the subjects were of the gender in question. This is not true for the male/male covariance term for either of the other two Dosage Compensation assumptions.
Separating Out Gender-Chromosome-Linked Effects¶
If you select AI as your REML algorithm, SVS allows you to separate out the gender-chromosome-linked effects from the non-gender-chromosome-linked effects by fitting the model
where is a vector of genetic effects attributable to the gender chromosome, with
and
where and are the genomic relationship matrices (GRM’s) for the gender chromosome and for all other chromosomes, respectively, so that
Note
You may wish to test for the most appropriate Dosage Compensation for the gender-chromosome GRM for this model by running the model three times, each time with a different Dosage Compensation for , then comparing the likelihoods of the model-fitting under each of the three assumptions.
Genomic Prediction¶
Sometimes, it is desired to predict the random effects (genomic merit/genomic breeding values) for samples for which there is genotypic data but no phenotype data, based on other samples for which phenotype data, genotypic data and, if applicable, covariate data does exist.
Additionally, it is sometimes desired to predict the phenotypes themselves for those samples with missing phenotypes, so long as the samples have not only genotypic data but also, if applicable, covariate data.
Call the samples for which there are phenotype values the “training set”, and the others the “validation set”. Assume all samples have genotypic data, imputed or otherwise, for all markers. Also assume all samples in the training set have valid covariate data, if there are covariates being used.
To predict the random effects and the missing phenotypes, we do the following:
Without loss of generality, imagine the samples of the training set all come first, before any samples of the validation set. Define , where the width and height of is , the width of the zero matrix is , and the height of the zero matrix is . Also partition , , and according to training vs. validation, as
Either compute the genomic relationship matrix or import a pre-computed genomic relationship matrix based on all samples (both training and validation sets).
Note
If correcting for gender has been selected, the modifications for computing and remain the same as noted above in Correcting the GRM for Gender Using Overall Normalization and Correcting the GRM for Gender Using Normalization by Individual Marker.
Use Restricted Maximum Likelihood (REML) on the mixed model for the training set, which may be expressed as
where and , to determine the variance components and the values of .
If EMMA is being used as the REML algorithm (Finding the Variance Components Using EMMA), it will also determine and the inverse of for the training set.
Noting that the form of the second of Henderson’s mixed-model equations, as modified to accommodate singular , is, for ,
we obtain
or
Noting the following equalities,
We may write
as a solution for the genomic BLUP.
If we now define, for Overall Normalization,
where is the matrix for Overall Normalization, we find that
which makes a solution for the ASE. The computational streamlining becomes computing as
computing as
and finally computing and the ASE as before.
Here, is calculated for all samples, namely both training samples and validation samples.
For Normalization by Individual Marker, if we now define
and
where is defined as in Normalization by Individual Marker, and is calculated for all samples, namely both training samples and validation samples, we find that
and that
which makes a solution for the ASE. The computational streamlining becomes computing as
computing as
and finally computing and the ASE as before.
If Average Information is being used as the REML algorithm (Finding the Variance Components Using the Average Information (AI) Technique), we may express the total random effect as the sum of the random effects of the individual components
Given this, we express the mixed model for the training set as
with
for either normalization method. Along with the variance components ( and ), the AI algorithm will find the matrix and the values of for the training set. From these are found
and
for each variance component , as well as
for each variance component, with the total random effect estimate being
For any variance component for which the matrix is completely generated from a set of markers as or as , where or is also generated exclusively from set of markers, we may compute an allele substitution effect (ASE) value for any marker within such a set. The vector of these values is
for Overall Normalization, or
for Normalization by Individual Marker.
Finally, noting that the full mixed-model problem is
or
we predict the validation phenotypes
from (the intercept and) any validation covariates and the predicted values . If there are missing validation covariates, the corresponding validation phenotypes are not predicted.
Note
As a “bonus”, we also use the same prediction algorithm to “re-predict” the training phenotypes as
Note
If correcting for gender has been selected, the modifications for partitioning and either or (both of which involve all samples, both training samples and validation samples) and computing the ASE ( or ) remain the same as noted above in Correcting the GRM for Gender Using Overall Normalization, Correcting the GRM for Gender Using Normalization by Individual Marker, and Correcting the ASE for the Gender Chromosome.
Bayes C and C-pi¶
Problem Statement¶
Suppose that we have the following mixed model equation to describe the relationship between the phenotypes of our samples, their genotypes, and fixed and random effects.
over samples, with fixed effects in specified in that include the intercept and any additional covariates you may have specified. Also, the random effects, are additive genetic merits or genomic breeding values associated with each sample.
We can define in terms of the genotypes of each sample over autosomal markers and the allele substitution effects.
where is an x matrix containing the genotypes of each sample. is 0, 1, or 2, depending upon whether the genotype for the -th sample at the -th locus is homozygous for the major allele, heterozygous, or homozygous for the minor allele, respectively. is a vector for which is the allele substitution effect (ASE) for marker .
Estimating the Model Parameters¶
The two Bayesian methods for fitting this mixed model implemented in SVS are Bayes C and Bayes C [Habier2011]. The only difference between the two methods is the assumptions about .
is the prior probability that a SNP has no effect on the phenotype, the value of is treated as unknown and is estimated in the Bayes C method and is treated as known with a value of 0.9 [Neves2012] in Bayes C.
The Bayesian approach uses prior probabilities, beliefs about the parameters before data is analyzed, and the conditional probabilities, a probability density for a parameter based on the given data, to construct full conditional posterior probabilities. These posterior probabilities are then sampled from to make estimates about the model parameters.
SVS uses a single-site Gibbs sampler.
Prior and Posterior Distributions¶
There are six parameters that are sampled each iteration of the Gibbs sampler.
The parameters are:
: the prior probability that a SNP has no effect.
: the allele substitution effect (ASE) at loci .
: the component of variance associated with the ASE.
: the component of variance associated with the error term.
: the component of the -th fixed effect.
: whether a marker is included in the current iteration.
The prior distributions for the parameters are as follows [Fernando2009a], [Sorensen2002]:
Parameter |
Prior |
---|---|
Where is the number of iterations for the Gibbs sampler, , , [Fernando2009b] and is defined as [Habier2011]:
where is the initial value of and is defined as [Habier2011]:
where is:
or the sum of would-be variances over all the markers if each had been at Hardy-Weinberg equilibrium. And is the additive-genetic variance explained by SNPs [Habier2011] and we can set it as 0.05 [Fernando2009b].
The prior of is constant and not proper, however the posterior is. The initial value for the first is the mean of the phenotype values and the rest are set to 0.
The posterior distributions for these parameters are [Fernando2009a], [Sorensen2002]:
Parameter |
Posterior |
---|---|
where is the number of markers included in this iteration (see Deciding when to include a marker) and is the scale factor for the posterior distribution and it is set at 1. And is [Habier2011]:
and is defined as [Habier2011]:
where is defined as above and is updated each iteration with the new value of .
Note
A prime in the subscript mean “all but this.” For example, when sampling we remove all the fixed effects and their coefficients except for the current one.
The posterior of can be better explained by [Fernando2009a]:
Deciding when to include a marker¶
Because of the difficulty in sampling directly from the distribution we use the log-likelihoods, find the probability that is 1, and sample from a uniform distribution, , to determine if a marker will be included in the current iteration, if we do not include a marker, then will be set to 0.
The log-likelihoods are defined as [Fernando2009b]:
We can then define the probability that is 1 as [Fernando2009b]:
Finding the ASE and Genomic Estimated Breeding Values¶
To find the our estimates for and we take the average over all iterations:
for all and .
The ASE values will then be the new values.
To find the genomic estimated breeding values (GEBV) we use the ASE values:
, , and are found in the same way as and , taking the average value over all iterations.
Gender Correction¶
To correct for gender, we take the following steps:
For markers within the X chromosome, we use the following entries for matrix :
For females, we encode them the same as non-X-chromosome markers.
For males, we use 0 or 1 for , depending upon whether the genotype for the -th sample at the -th locus contains the major (X-chromosome) allele or the minor (X-chromosome) allele, respectively.
The other entries of are left the same.
To compute , we continue to use as the expected-variance term for most markers. For X-chromosome markers, however, we use
where and are the fraction of the samples that are male and female, respectively.
We still compute
as before.
During the sampling phase of the Gibbs sampler, the and matrices are split into male and female sections and separate samples are taken.
The non-X-Chromosome ASE values are added (from the males and females) to get the final ASE values.
Normalizing the ASE¶
To normalize the ASE values we do:
Standardizing Phenotype Values¶
Phenotype values will be standardized to prevent from becoming too large and disrupting the results.
Values will be set to their z-score:
The result spreadsheet by marker will have the original phenotypes and the standardized phenotypes, if phenotype prediction is chosen, then there will be a column of predicted phenotypes and transformed back predicted phenotypes.
Genomic Prediction¶
To predict the phenotype of samples for which there is genotypic data but no phenotypic data (will be known as our validation set) we can run the Gibbs sampler with just the data from the samples with known phenotypic and genotypic data (the training set).
After running the Gibbs sampler with just the training set data we can estimate the GEBVs for all samples with:
where is estimated with just the training samples.
And we can predict the phenotypes for all samples (training and validation) with:
However, if there are missing covariate values the phenotypes cannot be predicted for the sample and it will be dropped from the result spreadsheet.