# MTG2

1. MTG is a computer program to provide Genomic Residual Maximum Likelihood (GREML) estimates for genetic and environmental variance and covariance across multiple traits. The program implements a multivariate linear mixed model and can fit complex covariance structures that can be derived from genomic information, i.e. multivariate version of GCTA GREML. The program also provides best liner unbiased prediction (BLUP) of additive genetic effects; either breeding values or predictions of genetic risk. MTG uses the direct average information algorithm (Lee and van der Werf; *Genet Sel Evol* 2006; 38:25-43). For more details of GREML and GBLUP, please see

Maier, R., et al. (2015) Joint analysis of psychiatric disorders increases accuracy of risk prediction for schizophrenia, bipolar disorder and major depression disorder. *The American Journal of Human Genetics* 96,283-294

2. We combined the direct AI algorithm with an eigen-decomposition of the genomic relationship matrix, as first proposed by Thompson and Shaw (*Biometrics* 1990; 46:399-413). We can apply the procedure to analysis of real data with univariate, multivariate and random regression linear mixed models with a single genetic covariance structure, and demonstrate that the computation efficiency can increase by > 1,000 fold compared with standard REML software based on Mixed Model Equations.The details of the procedure and application are in

Lee, SH and van der Werf, JHJ (2016) MTG2: An efficient algorithm for multivariate linear mixed model analysis based on genomic information. *Bioinformatics* 32, 1420-1422

3. We theoretically derived the relationship between the genomic prediction accuracy and population parameters, e.g. effective population size (*N*_{e}). We used a stochastic coalescence simulation and ral data analyses to verify the theory. This study shows that the area under the receiver operating characteristic curve (AUC) increased exponentially with decreasing *N*_{e}, e.g. from 0.6 with *N*_{e} =10,000 to 0.9 with *N*_{e} =100. It also shows that the top percentile of the estimated genetic profile scores had 23 times higher proportion of cases than the general population (with *N*_{e }= 100), which increased from 2 times higher proportion of cases (with *N*_{e} = 10000). (also see section 7, 8, 9 and 10 in the manual)

Lee, S.H. et al. (2017) Using information of relatives in genomic prediction to apply effective stratified medicine. Scientific Reports 7: 42091.

4. We present a theoretical framework for genomic prediction accuracy when the reference data consists of information sources with varying degrees of relationship to the target individuals. A reference set can contain both close and distant relatives as well as ‘unrelated’ individuals from the wider population. The various sources of information were modeled as different populations with different effective population sizes (*N*_{e}). With a similar amount of data available for each source, we show that close relatives can have a substantially larger effect on genomic prediction accuracy than lesser related individuals. When using multiple reference populations that have different degrees of relationship or/and have the imperfect genetic correlation (< 1) between reference populations, MTG2 can calculate a weighted prediction accuracy (see section 9.1 in the manual).

Lee et. al. (2017) Estimation of genomic prediction accuracy from reference populations with varying degrees of relationship. PLoS ONE 12(12): e0189775. http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0189775

5. We have developed multivariate reaction norm model (MRNM) to tackle genotype–environment (G–E) correlation and interaction problems. It is well known that G–E correlation causes spurious G–E interaction signals although there is few statistical tools to correct this bias. MRNM implemented in mtg2 (section 1.4) can unbiasedly estimate G–E interaction in the presence of G–E correlation and even have higher power to detect the interaction, compared to existing methods. It is also notable that MRNM is efficient to detect significant heterogeneity in the estimated residual variances across different environmental or covariate levels. For more detail, please see the following paper.

Ni et al. (2019) Genotype–covariate correlation and interaction disentangled by a whole-genome multivariate reaction norm model. Nature Communications 10: 2239.

6. CORE GREML (see chapter 15 in the manual and example 12) can estiamte correaltion between two random effects in the phenotypic analysis where the covariance structure between the random effects are not pre-defined, e.g. genome-transcriptome corerlation in the phenotypic analysis of a complex trait.

Zhou, Im and Lee (2020)** **CORE GREML: Estimating covariance between random effects in linear mixed models for genomic analyses of complex traits. Nature Communications 11: 4208.

7. GxEsum (GxEsum script, README and example) is to estimate genome-wide GxE based on GWAS summary statistics, which can be applied to a large sample size.

Shin and Lee (2020) GxEsum: a novel approach to estimate the phenotypic variance explained by genome-wide GxE interaction based on GWAS summary statistics for biobank-scale data. Genome Biology 22: 183.

8. Integrative analysis of genomic and exposomic data (IGE)

IGE is a whole-genome approach to the estimation of heritability and g x e interactions, which models variances explained by additive effects of exposomic variables, by exposome x exposome interactions, and by exposome x covariate (such as demographics) interactions; and covariance between genetic effects and exposomic effects (Table 3). Further, bivariate or multivariate IGE (i.e., simultaneously including two or more traits) can be feasibly performed using mtg2 version 2.18. Please see section 17 in the manual and exampleIGE below, which can be also found in the IGE GitHub.

The algorithms, theory, coalescence simulation functions are implemented in MTG2 software that can be downloaded from the link below. There are manual and examples.