GW-SEM: A Statistical Package to Conduct Genome-Wide Structural Equation Modeling

Abstract

Improving the accuracy of phenotyping through the use of advanced psychometric tools will increase the power to find significant associations with genetic variants and expand the range of possible hypotheses that can be tested on a genome-wide scale. Multivariate methods, such as structural equation modeling (SEM), are valuable in the phenotypic analysis of psychiatric and substance use phenotypes, but these methods have not been integrated into standard genome-wide association analyses because fitting a SEM at each single nucleotide polymorphism (SNP) along the genome was hitherto considered to be too computationally demanding. By developing a method that can efficiently fit SEMs, it is possible to expand the set of models that can be tested. This is particularly necessary in psychiatric and behavioral genetics, where the statistical methods are often handicapped by phenotypes with large components of stochastic variance. Due to the enormous amount of data that genome-wide scans produce, the statistical methods used to analyze the data are relatively elementary and do not directly correspond with the rich theoretical development, and lack the potential to test more complex hypotheses about the measurement of, and interaction between, comorbid traits. In this paper, we present a method to test the association of a SNP with multiple phenotypes or a latent construct on a genome-wide basis using a diagonally weighted least squares (DWLS) estimator for four common SEMs: a one-factor model, a one-factor residuals model, a two-factor model, and a latent growth model. We demonstrate that the DWLS parameters and p-values strongly correspond with the more traditional full information maximum likelihood parameters and p-values. We also present the timing of simulations and power analyses and a comparison with and existing multivariate GWAS software package.

Appendix 1: syntax and application

A supplementary goal of GW-SEM is to create a user-friendly set of commands that researchers who may not be dedicated data analysts can use effectively. Therefore to demystify the process, in this section we explain the use of each of the principal functions.

The first step in the analysis is to calculate the SNP-invariant covariances. These calculations are conducted using the facCov() function:

facCov( dataset, VarNames, covariates)

where dataset is a dataframe in R, VarNames is a list of the variable names of the items, and covariates is a list of covariates. The function returns covariances, weights, and standard errors of all of the variances, covariances, means and thresholds for all of the items and covariates. Because this function runs quickly (even for a relatively large number of items), and is necessary for all subsequent functions, it is called directly by the other functions. Users can use this function to ensure that their data is properly organized, and to ensure that there are no peculiarities with any of the variables they plan on including in their analyses.

The second step in the analysis is to estimate the SNP-item and SNP-covariate covariances. These calculations are conducted using the snpCovs() function:

snpCovs(FacModelData, vars, covariates, SNPdata, output, zeroOne, runs, inc, start)

where FacModelData is the path to the text file with the item and covariate data, vars is a list of items, covariates is a list of covariates, SNPdata is the path to the text file with the SNP values, output is the prefix for the output files, zeroOne is a logical value indicating whether the first and second thresholds should be fixed at 0 and 1, freeing up parameters to estimate the mean and the variance following the liability-threshold Model (Mehta et al. 2004), runs is the number of batches of SNPs to be analyzed, inc is the number of SNPs included in each batch, and start, is the column in the SNP file of the first SNP to be sampled. The output from this function is saved in three separate files as specified by the output argument: the covariances, the weights and the standard errors.

The final step of the model fits the SEM using the gwasDWLS function:

gwasDWLS(itemData, snpCov, snpWei, VarNames, covariates, runs, output, inc)

where itemData is the path to the text file with the item and covariates, snpCov is the path to the text file with covariances between the SNPs and the item and covariates (calculated in the previous step), snpWei is the path to the text file with the weights, VarNames is a list of items, covariates is a list of covariates, runs is the number of batches of SNPs to be analyzed, output is the file name for the output file, and inc is the number of SNPs included in each batch. Due to identification restrictions, users must supply at least three items (indicators) for the latent factor. There is no minimum or maximum for the number of covariates that can be included in the analysis. Note that with these two lines of R code, it is possible to conduct the one-factor GWAS.

The next SEM is the residuals model. The syntax to fit the residuals model is:

snpCovs(FacModelData, vars, covariates, SNPdata, output, zeroOne, runs, inc, start)

resDWLS(itemData, snpCov, snpWei, VarNames, covariates, resids, factor, runs, output, inc)

As can be seen above, for the residuals model, the only two arguments that differ from the one-factor model are resids which is a list of the items to be regressed on the SNPs, and factor which is a logical value asking whether the latent factor is to be regressed on the SNPs. The other arguments operate in exactly the same way as with gwasDWLS. Further, the snpCovs function is equivalent for both the gwasDWLS and the resDWLS, making it possible to easily conduct additional analyses with minimal additional steps. Again, at least three items are required in order to provide an identified factor model.

The third model in the package is the two-factor SEM. The syntax to run the two-factor GWAS is:

snpCovs(FacModelData, vars, covariates, SNPdata, output, runs, inc, start)

twofacDWLS(itemData, snpCov, snpWei, f1Names, f2Names, covariates, runs, output, inc)

Again the snpCovs argument is identical to the previous GWAS models, and the only change in arguments from the gwasDWLS to the twofacDWLS is the addition of f1Names and f2Names, which are lists of the variable names that load on Factor 1 and Factor 2, respectively. These lists are not exclusive for generality but at least three items must be specified for each factor, with at least one item for each factor excluded from the alternative factor.

The last model included in the software is the LGM, depicted in Fig 1d. The syntax for the LGM GWAS is:

snpCovs(FacModelData, vars, covariates, SNPdata, output, zeroOne, runs, inc, start)

growDWLS(itemData, snpCov, snpWei, VarNames, covariates, quadratic, orthogonal, runs, output, inc)

The snpCovs function is again equivalent to the function described above, except that for categorical data, the zeroOne argument should be specified as TRUE, to facilitate the estimation of the LGM. As the LGM is particularly focused on mean and variance changes, this is an important feature of the covariance model. The only change in the growDWLS() function from the gwasDWLS() function is the inclusion of the quadratic and orthogonal arguments. The quadratic argument is a logical value asking whether to include a latent quadratic growth parameter. The orthogonal argument is a logical value asking whether to use the standard growth loadings or orthogonal contrasts. The other arguments are exactly the same as the gwasDWLS function.