GW-SEM 2.0: Efficient, Flexible, and Accessible Multivariate GWAS

Abstract

Most genome-wide association study (GWAS) analyses test the association between single-nucleotide polymorphisms (SNPs) and a single trait or outcome. While valuable second-step analyses of these associations (e.g., calculating genetic correlations between traits) are common, single-step multivariate analyses of GWAS data are rarely performed. This is unfortunate because multivariate analyses can reveal information which is irrevocably obscured in multi-step analysis. One simple example is the distinction between variance common to a set of measures, and variance specific to each. Neither GWAS of sum- or factor-scores, nor GWAS of the individual measures will deliver a clean picture of loci associated with each measure’s specific variance. While multivariate GWAS opens up a broad new landscape of feasible and informative analyses, its adoption has been slow, likely due to the heavy computational demands and difficulties specifying models it requires. Here we describe GW-SEM 2.0, which is designed to simplify model specification and overcome the inherent computational challenges associated with multivariate GWAS. In addition, GW-SEM 2.0 allows users to accurately model ordinal items, which are common in behavioral and psychological research, within a GWAS context. This new release enhances computational efficiency, allows users to select the fit function that is appropriate for their analyses, expands compatibility with standard genomic data formats, and outputs results for seamless reading into other standard post-GWAS processing software. To demonstrate GW-SEM’s utility, we conducted (1) a series of GWAS using three substance use frequency items from data in the UK Biobank, (2) a timing study for several predefined GWAS functions, and (3) a Type I Error rate study. Our multivariate GWAS analyses emphasize the utility of GW-SEM for identifying novel patterns of associations that vary considerably between genomic loci for specific substances, highlighting the importance of differentiating between substance-specific use behaviors and polysubstance use. The timing studies demonstrate that the analyses take a reasonable amount of time and show the cost of including additional items. The Type I Error rate study demonstrates that hypothesis tests for genetic associations with latent variable models follow the hypothesized uniform distribution. Taken together, we suggest that GW-SEM may provide substantially deeper insights into the underlying genomic architecture for multivariate behavioral and psychological systems than is currently possible with standard GWAS methods. The current release of GW-SEM 2.0 is available on CRAN (stable release) and GitHub (beta release), and tutorials are available on our github wiki (https://jpritikin.github.io/gwsem/).

Introduction

Genome-wide association studies (GWAS) of raw genomic data have greatly improved our understanding of the genetic etiology of complex traits, and are spurring progress towards effective prevention and treatment strategies in several medical disciplines (Cardon and Harris 2016; Nelson et al. 2015; Barrett et al. 2015). While univariate GWAS of psychiatric phenotypes report numerous independent associations with genomic variants (International Schizophrenia Consortium et al. 2009; Lee et al. 2018; Wray et al. 2018; Xue et al. 2018; Ripke et al. 2013; Yengo et al. 2018; Nagel et al. 2018), these results have yet to yield actionable interventions in psychiatry. This lack of progress towards interventions may be partly due to the inherent multivariate nature of psychiatric and psychological outcomes, the interplay between the symptoms that comprise a disorder, the environmental risk factors that cause them, and comorbidity among disorders. The aim of the present article is to demonstrate how GW-SEM 2.0 (Genome-wide Structural Equation Modeling) can address these issues by alleviating many of the bottlenecks in multivariate GWAS analyses.

Identifying associations with complex traits, such as psychiatric, psychological and behavioral phenotypes, is complicated by the fact that many traits cannot be directly measured and instead must be inferred from the presence of multiple, observable symptoms or indicators that are theorized to be the result of the latent pathology. This conceptualization makes the application of structural equation models (SEMs) especially appropriate for the analysis of complex traits. SEM is a powerful framework in which complex, multivariate, and interacting components can be jointly analyzed. Simple linear models, commonly available in GWAS software, break down once constructs can no longer be unambiguously assigned as either independent (X) or dependent (Y) variables. The flexible SEM framework allows correlational relationships between variables, variables to simultaneously be both independent and dependent, and variables may be either directly observed or latent constructs. SEM subsumes a wide variety of common statistical models including multiple linear or ordinal regression, factor analysis, path analysis, and mediation. The breadth and flexibility of SEM makes it particularly well-suited for GWAS of complex traits as it can easily incorporate environmental and etiological factors, and test hypotheses concerning symptoms, risk factors, and comorbid disorders.

Genome-wide data are large, with microarray datasets typically consisting of approximately 10 million directly assayed or imputed SNPs. To enable practical GWAS analysis, popular methods are mostly limited to very simple statistical methods, such as linear and logistic regression (Chang et al. 2015; Marchini et al. 2007). These methods cannot model the nuances of complex traits. Although several multivariate GWAS approaches exist, they too face limitations. Some multivariate analytical strategies use raw genomic data, but they tend to be modest extensions of univariate methods. For example, multivariate GWAS software that uses linear mixed modeling to analyze the optimal linear combination of a set of items is analogous to a sum-score or latent factor model (Zhou and Stephens 2012, 2014). While item-level information may be used, disorder-specific or symptom-specific parts of the variance are not accessible. Other multivariate methods , such as GenomicSEM, use summary statistics from prior GWAS analyses, thereby dramatically reducing the analytical burden (Grotzinger et al. 2019; van der Sluis et al. 2013). These approaches, however, are limited to exploring genetic correlations between traits, with relatively few opportunities for understanding environmental or phenotypic causal pathways. Thus there is a clear lack of software that can optimally integrate risk factors and comorbid disorders into GWAS analyses using raw genomic data.

In this paper we describe GW-SEM 2.0, an efficient multivariate GWAS software package for conducting a variety of complex statistical models that can provide nuanced insights into the genetic architecture of complex traits. GW-SEM 2.0 is freely available through CRAN (https://cran.r-project.org/package=gwsem). GW-SEM 2.0 builds on the prior release (GW-SEM 1.0; Verhulst et al. 2017), which was largely a proof-of-concept for the integration of SEM with genome-wide data. Sadly, the initial release was too computationally inefficient to be of widespread practical utility. GW-SEM 2.0 dramatically increases computational efficiency, expands modeling flexibility, and simplifies use by supporting common GWAS file formats. Here we describe these improvements and demonstrate their application with a GWAS of substance use frequency.

Enhancing computational efficiency

SEM analyses typically proceed by: (1) specifying a model, (2) providing data to test the model, and (3) optimizing parameter estimates to find the best fit of the model to the data. Fitting SEMs on a genome-wide scale is complicated by two major challenges: handling large genomic data files, and fitting the the same model to millions of different datasets (one for each genomic variant). Many GWAS studies analyze over 10 million SNPs; CPU time per variant must be kept to a minimum. To facilitate rapidly analyzing GWAS data within a SEM context, GW-SEM 2.0 splits the analytical process into a frontend, where the user specifies the model in R (R Core Team 2014), which passes the model to a high-performance C++ computational backend that users do not interact with directly. This makes the GW-SEM 2.0 interface similar to a traditional SEM analytical experience with the primary difference being that the model is fit millions of times.

The large size of genomic data files makes it cumbersome to manipulate, search, extract, and combine the data. To enhance efficiency, GW-SEM 2.0 takes the user’s model, generates a computational plan in OpenMx (Neale et al. 2016) that reads genomic data directly from common compressed formats, analyzes it, and writes the results to an output file. The cumulative effect of these optimizations make GW-SEM 2.0 approximately 100 times faster than the GW-SEM prototype (Verhulst et al. 2017).

Enhancing flexibility

With GW-SEM 2.0, users are empowered to specify models that test hypotheses of interest. In addition, broad trade-offs are available between statistical precision and computational efficiency. GW-SEM 2.0 achieves general utility through a combination of predefined models and features that allow infinitely customized modeling.

Predefined models

To help overcome the R language barrier (which can be a serious impediment for novice users), GW-SEM 1.0 provided a set of commonly-used predefined models for direct use with minimal effort. These models included One-Factor, One-Factor Residuals, and Two-Factor GWAS models (see Verhulst et al. 2017 for more details). In addition to these features, version 2.0 now includes a “standard” univariate GWAS function that allows users to fit simple regression models, functions for simultaneous genome-wide modeling of multiple dependent and multiple independent variables, and functions for gene-environment interaction analyses on a genome wide basis (see Verhulst et al., Under Review for more details). A set of detailed tutorials for using the predefined GWAS models can be found on GitHub (https://jpritikin.github.io/gwsem/); these include models that, to our knowledge, cannot be explored in other GWAS software.

Custom user-specified models

Beyond the predefined models, GW-SEM enables user to specify custom models that can take full advantage of the flexibility of OpenMx (Neale et al. 2016), a general purpose package for structural equation and other statistical modeling. Users can specify any SEM in OpenMx, or draw the path diagram in Onyx (von Oertzen et al. 2015), and then run the GWAS using GW-SEM. The primary requirement for specifying custom models is that the data must have a “snp” column to contain the data for the current genomic variant. The user can then incorporate the SNP onto one or more elements of the model in regression or covariance paths. It is essential that users provide a SEM that is identified. i.e., there is only one set of parameter estimates that minimize the fit function. A test of identification can be performed on the phenotypic level prior to conducting a GWAS in GW-SEM. As GW-SEM relies on the OpenMx architecture, OpenMx functions such as mxRun and mxCheckIdentification can be used to check model specification and numerically test model identification. As with any model-fitting analysis, using GWAS with a misspecified or statistically under-identified model will likely generate uninterpretable results. A detailed tutorial for constructing custom GWAS can be found on our GitHub wiki, https://jpritikin.github.io/gwsem/UserSpecifiedGWASModels.html.

Ordinal data

Many complex traits are assessed using ordinal measures, but GW-SEM is the only GWAS software package that can handle ordinal data appropriately (i.e. using a liability-threshold framework). Treating ordinal data as if they are continuous can severely attenuate associations among the phenotypes, and between the phenotypes and the genomic variants (Verhulst and Neale 2021). The problem is exacerbated when the thresholds are unevenly spaced along the underlying continuum of liability. This situation frequently occurs when symptoms or items are aggregated (e.g., sumor factor-scores) but few participants endorse any of the items. Reverse-J shaped distributions, with many individuals in the lowest category, are often generated in this way. GW-SEM allows users to specify whether a particular dependent variable is continuous, binary, or ordinal, and automatically generates an appropriate model for the types of data being analyzed. In the multivariate case, users can specify models with a mixture of binary, ordinal, and continuous variables. The liability-threshold model is used for binary or ordinal variables, linear regression is used for continuous variables.

Somewhat counter-intuitively, but consistent with an additive interpretation of genetic effects, SNPs are treated as a continuous linear contrast and not as ordinal variables. For an additive genetic model, scoring genotypes on a linear scale corresponds with a simple mean difference between genotypes. Individuals with the same genotype may express different phenotypes, but this residual variation is usually assumed equal within genotypes at that locus.

Gene-environment interaction (GxE)

Modeling GxE is very popular in twin, family, and adoption studies. The general idea is that the effects of the environment depend on an individual’s genotype, or that genetic effects are sensitive to environmental stimuli. To our knowledge, testing for GxE at a locus-by-locus level has not been popular in GWAS, perhaps due to the lack of GWAS software that can control for different measurement precision at different points on the liability continuum. The current release of GW-SEM includes a suite of functions for GxE analyses. Users may specify GxE models for continuous, binary, ordinal, and latent variables, synthesize the results, and process them with other common post-GWAS software. We have written a companion paper that outlines the methodological details for conducting GxE GWAS analyses in GW-SEM 2.0 (Verhulst et al., Under Review), as the discussion requires several nuanced considerations that extend beyond the scope of the current paper.

Choice of optimizer

Optimizer choice is often overlooked when specifying and fitting statistical models, leading many researchers to rely on software defaults. To obtain parameter estimates within a reasonable time bound, it is important to choose an estimator suitable for the data and model used (Rubin 1976). In the current version of GW-SEM, we provide two popular estimators: Full Information Maximum Likelihood (FIML) and Weighted Least Squares [WLS; Jöreskog and Moustaki (2001)]. The relative strengths and weaknesses of FIML and WLS rest primarily on computational speed and the patterns of missing data in the sample (Pritikin et al. 2018; Bradley 1973).

Missing phenotypic data can result in a variety of statistical problems ranging from decreases in statistical power to biased parameter estimates, depending on the specific pattern of missingness (Pritikin et al. 2018; Bradley 1973). Accurately modeling the process that generates the observed pattern of missing data can dramatically improve many aspects of the statistical estimation procedure by increasing power and decreasing parameter bias. Specifically, if observations are missing completely at random (MCAR), parameter estimates will be less precise in proportion to the amount of missingness, but they will remain unbiased (Bradley 1973). Because traditional linear modeling techniques, such as regression, focus on explaining variation in a single dependent variable, many available techniques utilize list-wise deletion, where individuals with any observed variable missing is excluded from the analysis. In regression models, excluding individuals with any missing data can bias the parameter estimates if the independent variables with missing data are predictive of the dependent variable (Jones 1996). In multivariate settings with numerous variables, the probability that an individual’s response is missing for at least one measured variable increases to the point where list-wise deletion becomes extremely restrictive. GW-SEM 2.0 can utilize all available data, even if individuals are only assessed on a single item. If the data are MCAR, the resulting parameter estimates are unbiased if either the WLS or FIML estimator is used.

However, missing data can be the result of planned restrictions at the time of data collection. In psychological assessments, screening questions allow interviewers (or software that automates skip-outs) to decide whether specific follow-up questions are relevant or can be reasonably posed. This pattern of missing data, where missing values are a function of one or more measured variables in the data, is referred to as missing at random (MAR). To illustrate, if a research question focuses on measuring quantity of marijuana use in a given month, it is necessary to know whether an individual has ever engaged in any marijuana use in their lifetime. In other words, measuring the quantity of marijuana use depends on having tried it in the first place. Since WLS relies on univariate and bivariate information only, information from screening questions can only influence adjacent parts of the model. Consequently, data with MAR can result in biased parameter estimates (Jöreskog 1990; Lee et al. 1992; Muthén 1984). By contrast, FIML is asymptotically unbiased when data are MAR (Enders and Bandalos 2001), and is therefore preferred to WLS in such cases, assuming that the increase in computational time can be tolerated (Hagenaars 1988; Asparouhov and Muthén 2010).

Enhancing accessibility

Common genetic data file formats

GW-SEM 1.0 required the genomic data to be provided as an R dataframe, with individuals on rows and genomic variants on the columns (i.e., a flat file). Although this format could be constructed with PLINK’s *.raw formatting, it is rarely used in the genomic sciences. Instead, GW-SEM 2.0 can directly read from many of the common genomic data files. Specifically, we support binary genomic files including: .BED (PLINK; Purcell et al. 2007), .PGEN (PLINK2; Chang et al. 2015) and .BGEN (the default format for UKB data; Band and Marchini 2018). Other popular data formats, such as .VCF ( Li et al. 2009), can easily be transformed into these binary file formats. Furthermore, because of how variants are incorporated into the statistical model, GW-SEM supports both hard-called and dosage-formatting for genotypes.

Integration with common post-GWAS software

To enhance utility, GW-SEM output has been reformatted so that it can be read directly by standard post-GWAS software. This feature facilitates, for example, the generation of publication-quality figures. Running genetic associations in GW-SEM is only one part of the work-flow, and high-quality tools already exist for interrogating and presenting GWAS results. Specifically, we constructed application programming interfaces (APIs) to present the results using: qqman (Turner 2014) to generate Manhattan and Quantile-Quantile plots, LDSC (Bulik-Sullivan et al. 2015a, b) to estimate SNP-heritability, LDhub (Zheng et al. 2017) to estimate genetic correlations, and locus-zoom (Pruim et al. 2010) to provide detailed regional association plots. Constructing these APIs makes it virtually effortless to integrate summary results from GW-SEM into existing data analysis pipelines.

Illustrations: using GW-SEM 2.0 to analyze substance use frequency

To highlight some of the advantages and added features in GW-SEM 2.0, we conducted a set of latent variable GWASs using three substance use frequency items. Our goal for these illustrations was to show that GW-SEM 2.0 can (1) treat ordinal variables appropriately and that doing so has a substantial impact on the GWAS results, and (2) test meaningful theoretical models that uncover patterns of association that cannot be seen with current GWAS packages.

Software

All of the software used in the demonstrations can be downloaded and installed across most operating systems. Stable releases of the software can be obtained directly through R using the install.packages function. As we continue to add new features, beta releases can be downloaded from GitHub, https://github.com/jpritikin/gwsem.

Data

The phenotypic and genomic data come from the UK Biobank (UKB; Allen et al. 2014; Sudlow et al. 2015; application number 40967). For our demonstration, we only included participants of self-reported Northern European ancestry who were unrelated to other participants in the sample (N = 378,130). Participant’s age, sex, and the first 10 ancestry principle components were included as covariates. All the genomic data for the demonstration were formatted using PLINK 2.0 (.PGEN/.PSAM/.PVAR files; Chang et al. 2015), and directly read by GW-SEM. We analyzed data from the imputed SNPs with a minor allele frequency larger than .01. We did not filter by imputation information quality. As nearly 10 million association tests were conducted, there was a small proportion of model failures, which we excluded after the analyses (i.e. implausibly large coefficients, or parameters that did not move from the starting values, etc.).

Substance use frequency items.

We analyzed three substance use frequency items from the UKB that assessed how often individuals used tobacco, marijuana, and alcohol. All of the items were measured using ordinal self-report response options. The specific wording for each item are presented in Table 1. As is typical with psychopathology indicators, there is a skew to the responses with the plurality at the non-user side of the distribution for each item.

Table 1.

Question wording for the Tobacco, Cannabis, and alcohol frequency items and response options

Item wording	Response option
	0	1	2	3	4	5
In the past, how often have you smoked tobacco?	I have never smoked	Just tried once or twice	Smoked occasionally	Smoked on most or all days
Have you taken cannabis, even if it was a long time ago?	No	Yes, 1–2 times	Yes, 3–10 times	Yes, 11–100 times	Yes, more than 100 times
About how often do you drink alcohol?	Never	Special occasions only	One to three times a month	Once or twice a week	Three or four times a week	Daily or almost daily

Statistical models

We used three of GW-SEM’s pre-defined GWAS functions: a standard GWAS, a three-item latent variable GWAS, and a three-item residuals GWAS (Fig. 1). First, to demonstrate the importance of analyzing ordinal variables assuming a liability threshold model, we conducted two single item GWAS models for alcohol use frequency: once treating the variable as continuous and once treating the variable as ordinal.

Fig. 1.

A general path diagram of two latent variables model fit in the current analyses. The One-Factor model includes the solid red line from the SNP to the latent Substance Use Frequency variable, but excludes the dotted blue paths from SNP to the substance use phenotypes. The residuals model is the converse, including the dashed blue paths, but not the solid red one. We did not fit a model with both red and blue paths; this model is not identified unless one of the four paths is fixed to a constant value such as zero. Both models include the linear effects of Age, Sex, and 10 PCs on the observed nicotine, cannabis and alcohol items, but these covariates are omitted from the Figure to simplify our presentation. The factor loadings for the ordinal analysis are presented in a bold red font, and the factor loadings for the continuous analysis are presented in an italic purple font (Color figure online)

Second, we fitted the One-Factor model to the three substance use frequency items, modeling them as either ordinal or continuous indicators of the latent variable. Each SNP was regressed on the latent factor (the red solid line, Fig. 1). Third, we fitted the One-Factor Residuals model, a modification of the One-Factor model that regressed the individual substance use frequency items onto the SNP (the blue dashed lines, Fig. 1) and omitted the SNP to latent factor path. All statistical analyses used the weighted least squares (WLS) fit function. Standard post-GWAS software was used to visualize our results: Manhattan plots were created using qqman (Turner 2014), and genomic regions of interest were plotted with association results using LocusZoom (Pruim et al. 2010). Follow-up LDSR analyses were conducted to examine any potential difference in the $h_{SNP}^2$ across the latent variable and item residuals for the continuous and ordinal specifications of the variables. We also estimated SNP-based genetic correlations (r_g–SNP) between the latent variable and item residuals with variables from the GWAS & Sequencing Consortium of Alcohol and Nicotine use (GSCAN) project: smoking initiation and drinks per week (Bulik-Sullivan et al. 2015a, b).

Next, to showcase the speeds of the key models we timed a single item GWAS; a one factor GWAS with 3 and 5 items; and a two factor GWAS with 6 items (3 items per factor) and 10 items (5 items per factor) with continuous, binary, and ordinal dependent variables. Specifically, we simulated a phenotype, or set of phenotypes, with no associations with the genotypes and used the single item, One-Factor, and Two-Factor Models to match the sample sizes from the UKB analyses. We then regressed the phenotypes on the first 10,000 on chromosome 22 from the UK Biobank imputed genomic data. For realism, we regressed the simulated items on age, sex, and the first 10 principle components from the UK Biobank sample. We ran complimentary analyses for simulated continuous, binary, and three-category ordinal phenotypes. For the One-Factor model, we conducted analyses with 3 items and with 5 items. For the Two-Factor Model, we conducted analyses with 6 items (2 factors by 3 items) and 10 items (2 factors by 5 items). This results in 15 sets of analyses for the timing study. The analyses were run on the Texas A&M Ada cluster, and timing may vary with other computing clusters.

Finally, we assessed the Type I Error rates of the predefined GWAS functions. Similar to the timing studies, we simulated a phenotype, or set of phenotypes, with no associations with the genotypes and used the continuous single item, One-Factor, and Two-Factor Models to match the sample sizes from the UKB analyses, running the association analyses for SNPs on chromosome 22 with a minor allele frequency greater than .05.

Illustration 1: ordinal vs. continuous item modeling

We conducted GWAS analyses of the alcohol use frequency item in GW-SEM, treating the item as either ordinal or continuous. An advantage of GW-SEM is its ability to model ordinal data using a probit liability-threshold model. To contrast the effects of treating an ordinal item as if it were continuous, versus assuming it follows a liability-threshold model, we conducted two single item GWASs. The alcohol frequency item was directly regressed on the SNPs using standard linear regression methods in the continuous case, and ordered probit regression in the ordinal case.

Both the z-statistics and the p-values were highly positively correlated across the two methods. This resemblance is expected, given that the same data were used to conduct both analyses. The main difference is that the effect sizes and the standard errors were slightly larger when the data were treated as ordinal compared with when the data were treated as continuous. Specifically, when the variance of the alcohol use frequency item is equated across analyses, the coefficients from the ordinal analysis were on average 5.3% larger. We would not expect major differences between the effect sizes for continuous or ordinal data methods because (Verhulst and Neale 2021): (a) alcohol is a commonly consumed substance (and therefore not as skewed toward nonusers), and (b) there are six frequency categories (not two or three categories). However, treating the data as ordinal is appropriate; the narrower standard error of the continuous method may inflate the false positive rate.

Although efforts at locating positive signals (or ‘hits’) may have succeeded with either approach, the different effect sizes found by the two methods could have important downstream consequences. Differences in the magnitude of the effect sizes would likely influence polygenic scores. Moreover, the polygenic scores derived from the two approaches may associate to a different degree with other variables. For indicators with fewer categories and more skewed distribution, the assumption of normality is more severely violated. Such violations can profoundly attenuate estimates of the correlations between ordinal variables if they are treated as continuous measures (Verhulst and Neale 2021).

Illustration 2: latent variables with ordinal vs continuous items

We conducted a series of multivariate GWAS analyses using the One-Factor model and the residuals models with the tobacco, cannabis, and alcohol frequency items as indicators of the latent variable. To illustrate the differences between treating the items as ordinal and continuous for a multivariate application, we fit the One-Factor and residuals substance use frequency models. First we treated the items as ordinal indicators, using polychoric correlations (Olsson 1979) to summarize covariances between the items. Second, the items were treated as continuous, summarizing them with Pearson product-moment correlations. While the polychoric approach takes slightly more computer time, it is consistent with the item’s data generating process. As we demonstrate below, the polychoric method yields: (i) parameter estimates that are more easily interpreted; (ii) more significant genetic associations; and (iii) larger genetic correlations with existing GWAS studies. The advantages of using the liability-threshold framework for ordinal data carry over to the multivariate case.

Before looking at the genetic associations, it is instructive to examine the standardized factor loadings for the ordinal and continuous analyses, which we present in Fig. 1. As in any factor analysis, the pattern of factor loadings defines the phenotype and profoundly affects the interpretation of any subsequent genetic associations. In the ordinal data analyses, the factor loadings (bold red font) are larger than those for the continuous analysis (italicized purple font), especially for cannabis use frequency. This implies that the ordinal latent factor is accounting for more variation in the observed indicators. Furthermore, the latent factor is primarily defined by the tobacco and cannabis use items, and more faintly by the alcohol use item.

The genetic associations for the ordinal and continuous analyses reveal several genome-wide significant signals coming from broadly consistent genomic regions, although the p values are more significant in the ordinal analysis. The Manhattan plots of the − log₁₀ p values for the ordinal and continuous one factor models are presented in Figure 2. The − log₁₀ p value peaks from the ordinal model (Fig. 2a) are notably higher than those of the continuous model (Fig. 2b).

Fig. 2.

Manhattan Plots for the association tests from the latent variable and the residuals models of substance use frequency with ordinal items for an analysis of 378,130 subjects. The x axis presents the genomic position (Chromosomes 1–22) and the y axis presents statistical significance as −log₁₀(p – value) . The threshold for statistical significance accounting for multiple testing is shown by the red horizontal line ( p = 5 × 10⁸ ), while the blue horizontal line shows the suggestive level of statistical significance ( p = 1 × 10⁵ ) (Color figure online)

To delve deeper into the differences between latent factor models that treat the items as ordinal vs continuous, the lead SNPs for the associations with the latent factor and item residuals are presented in Table 2. Notably, the associations for the ordinal models are substantially more statistically significant than the same SNPs in the continuous models. Specifically, in some cases, the ordinal estimate is genome-wide significant (p < 5 × 10⁻⁸) while the continuous estimate is only suggestively significant (1 × 10⁻⁵ > p and p > 5 × 10⁻⁸).

Table 2.

Genome Wide significant associations for the latent factor or the specific item residuals in the ordinal and continuous factor models

Chr	Gene	Position	SNP ID	Ordinal		Continuous
				Beta	P	Beta	P
Latent factor
11	NCAM1	112912387	rs3802848	−0.052	1.78 × 10−18	−0.046	1.04 × 10−15
10	CYP17A1	104670832	rs28606370	−0.040	5.47 × 10−11	−0.036	1.76 × 10−9
3	CADM2	85899045	rs796814530	−0.044	1.66 × 10−10	−0.035	1.73 × 10−7
16	EIF3CL	28529461	rs542594336	0.033	2.13 × 10−8	0.025	2.20 × 10−5
Tobacco residual
11	NCAM1	112910881	rs113332425	0.060	4.52 × 10−19	0.040	9.70 × 10−15
10	CYP17A1	104941112	rs35525740	0.042	6.89 × 10−10	0.027	1.86 × 10−7
3	LOC101928135	34800554	rs2080428	0.054	1.22 × 10−8	0.029	9.19 × 10−5
Cannabis residual
3	CADM2	85508499	rs1368740	−0.064	9.06 × 10−12	−0.027	9.75 × 10−11
6	HLA Region	31566168	rs3130631	0.074	8.61 × 10−10	0.026	3.04 × 10−6
22	TNRC6B	40692635	rs761508279	−0.063	4.84 × 10−9	−0.025	2.93 × 10−7
16	EIF3CL	28529461	rs542594336	0.047	2.42 × 10−8	0.019	7.54 × 10−7
10	No Gene	83215811	rs72821472	−0.062	2.59 × 10−8	−0.024	1.02 × 10−6
10	CYP17A1	104652815	rs34643607	−0.047	2.34 × 10−7	−0.022	1.83 × 10−7
Alcohol residual
4	ADH1B	100248642	rs145452708	−0.170	1.57 × 10−18	−0.051	3.36 × 10−2
4	KLB	39418301	rs772218928	−0.036	6.54 × 10−15	−0.036	1.14 × 10−9
2	GCKR	27739880	2:27739880:CT:C	0.031	1.05 × 10−11	0.031	1.32 × 10−7
3	FOXP1	71580180	rs746573926	0.025	3.82 × 10−8	0.026	7.61 × 10−6
16	EIF3CL	28529461	rs542594336	−0.025	3.77 × 10−8	−0.025	1.29 × 10−5

The One Factor and Residuals models were run separately for the ordinal and continuous analyses. The associations that are presented in the table represent the lead SNP in the genomic region from the relevant analysis

The abbreviations on column headings are as follows: Chr chromosome, Gene most likely/proximate gene for the association, Position genomic location, SNP ID the rsID number or vcf identifier for the genomic variant, Beta the unstandardized regression coefficient, P the p value for the association test

The results of the One-Factor and Residuals models allow us to distinguish common from specific contributions to trait variation. Here we explore the complexity of the genomic associations for the cell adhesion molecule 2 gene (CADM2). CADM2 has been previously associated with risky behaviors such as drug use propensity (Bidwell et al. 2015a, b). The latent factors in the One-Factor and Residuals models are consistent with an overall drug use propensity, and likely index other risk behaviors. The pattern of associations with the latent factor (Fig. 3a) and the item residuals (Fig. 3b–d) paint a complex picture of the relationship between this region and substance use frequency. The genetic associations with the latent variable and the tobacco use frequency residual are stronger around the anti-sense locus at the 3’ region of the CADM2 gene, whereas the strongest associations for cannabis, and to a lesser extent alcohol, are closer to the 5’ region of the gene. Thus, while the SNP associations appear in the same broad region (CADM2), the pattern of observed associations reveals a complex set of associations that does not fit into a simple unidimensional interpretation.

Fig. 3.

LocusZoom Plot of CADM2 (Cell Adhesion Molecule 2) on Chromosome 3 for the association tests from the latent factor and the residuals of the three substance use frequency with ordinal items for an analysis of 378,130 subjects. The x axis presents the genomic position along Chromosome 3 along with the location of CADM2 and other proximate genes. The y axis presents statistical significance as −log₁₀ (p – value). To enhance comparisons between the four associations, the LocusZoom plots are centered on the SNP rs77552899

To further investigate the genetic architecture of substance use behaviors, we used LD Score Regression to estimate the $h_{SNP}^2$ of the latent factors and the item residuals as well as the genetic correlations with the GWAS summary statistics for Smoking Initiation (SI) and Drinks Per Week (DPW) from the GSCAN study (GWAS & Sequencing Consortium of Alcohol and Nicotine use; Liu et al. 2019). The $h_{SNP}^2$ results are presented in Table 3, and the genetic correlation results are presented in Table 4.

Table 3.

SNP-based heritability estimates for the One-Factor and Residuals Models (Standard Errors in parentheses)

	SNP heritability ($h_{SNP}^2$)
	Ordinal items	Continuous items
Latent factor	0.045 (0.002)	0.062 (0.001)
Tobacco residual	0.042 (0.002)	0.058 (0.001)
Cannabis residual	0.029 (0.002)	0.065 (0.002)
Alcohol residual	0.032 (0.002)	0.047 (0.001)

Table 4.

SNP-based genetic correlations with the items smoking initiation and drinks per week items from GSCAN (Standard errors in parentheses)

	Ordinal items		Continuous items
	Smoking initiation	Drinks per week	Smoking initiation	Drinks per week
Latent factor	0.609 (0.027)	0.469 (0.038)	0.477 (0.019)	0.310 (0.026)
Tobacco residual	0.699 (0.025)	0.322 (0.040)	0.520 (0.019)	0.231 (0.026)
Cannabis residual	0.321 (0.037)	0.391 (0.047)	0.203 (0.020)	0.232 (0.028)
Alcohol residual	0.103 (0.032)	0.704 (0.043)	0.059 (0.018)	0.431 (0.026)

Table 3 shows that the h²_SNP estimates for the continuous method are consistently and significantly larger than those for the ordinal method. We had expected slightly larger h²_SNP estimates for the ordinal models, which was not found, possibly due to sampling variation. In both approaches, heritability estimates are similar: small but statistically distinguishable from zero. When estimating h²_SNP for binary variables, the GWAS summary statistics are transformed to a liability scale within the LDSC algorithm (Bulik-Sullivan et al. 2015a, b). This procedure does not take place when ordinal variables have three or more categories, which may affect estimates of h²_SNP from LDSC. Also notable is that all eight LD Score regression analyses yielded intercept values significantly lower than 1. This situation implies that the Type I error rate is not inflated for the latent factor GWAS associations, but may be slightly below nominal alpha levels due to lack of statistical power. LD Score intercepts that are significantly lower than one may result from excessive genomic control (Devlin and Roeder 1999), or from other statistical corrections that do not affect the effect sizes of the associations but instead correct the test statistics and the p-values.

Table 4 shows the LD Score genetic correlations between the One-Factor and Residuals models and two external validators, the smoking initiation and drinks per week from GSCAN. As expected, both the latent factor and the tobacco residual have high genetic correlations with GSCAN smoking initiation. Similarly, the alcohol residual correlates highly with the drinks-per-week item. The latent variables’ genetic correlations with these two validators are lower than those of the residuals. Given that the latent factor is strongly associated with the tobacco frequency item, it is expected that the latent factor would be more associated with the smoking initiation variable than the drinks per week results. Further, the tobacco and alcohol residuals are more strongly associated with the GSCAN summary statistics for the same substance. Notably, the ordinal GWAS results correlate substantially more with the GSCAN results than do the continuous ones.

Timing study for the GW-SEM GWAS functions

GW-SEM 2.0 is substantially faster than its predecessor. Table 5 shows the mean times per SNP for 10,000 SNPs for each of the 15 GWAS models. Unsurprisingly, models with more items take longer to fit, and continuous models are faster than binary and ordinal models. While approximately 10 s per SNP seems slow, GWAS analyses are embarrassingly parallel, and can be sped up by running chromosomes, or sections thereof, separately. Additional speed-ups can be obtained by restricting analyses to common variants (minor allele frequencies greater than .05). Note also that these analyses regress each item on 12 covariates as would be done in standard GWAS analyses, which increases computation time.

Table 5.

Results from the timing study for the GWAS functions (seconds per SNP)

	Continuous	Binary	Ordinal
Single item GWAS	0.32	0.75	0.81
One-factor GWAS (3 items)	0.46	1.32	1.29
One-factor GWAS (5 items)	1.02	4.01	3.32
Two-factor GWAS (6 items)	1.47	5.05	5.08
Two-factor GWAS (10 items)	4.48	10.44	10.59

Type I error rate study for the GW-SEM

The results from GW-SEM appear to identify more genome-wide significant loci than other methods, such as sum-scores. This finding could occur under two scenarios. On the one hand, we would expect more significant genetic associations if latent variable models have more statistical power to detect the genomic signals. On the other hand, it is possible that the Type I error rate for latent variable GWAS analyses are inflated such that we are simply more likely to observe smaller p values. To disentangle these two possibilities we undertook a simulation study to estimate the likelihood of observing a significant genetic association by chance alone, i.e. we did not simulate associations between the genotypes and the latent variable(s). The results of the Type I Error rate study are presented in Fig. 4. As can be seen in the figure, the p values from the one factor model and each factor from the two factor model follow a uniform distribution, as would be expected under the null hypothesis. Accordingly, there does not appear to be any inflation of the Type I Error rate.

Fig. 4.

Histogram of the p values for the genetic associations assuming the null hypothesis is true for the One- and Two-Factor models

Discussion

The aim of this article was to present GW-SEM 2.0: an easy-to-use, flexible, and rapid GWAS software package for modeling genomic variation in and covariation between multiple phenotypes. Ease of use is provided by functions to construct popular types of models, and by interfaces that facilitate data acquisition and subsequent graphical or tabular post-processing of results. It is flexible because users can go beyond the set of pre-programmed models and apply GWAS within any appropriate structural equation model for their data. The use of weighted least squares as a fit function for ordinal data eases the computational burden of FIML in two important ways. First, the covariance matrix of the outcome measures and their weight matrix are only calculated once per SNP. Second, FIML for ordinal data rapidly becomes computationally intractable as the number of variables, m, increases, due to the need to numerically integrate the m-dimensional multivariate normal distribution. GW-SEM provides elegant solutions to all these issues.

From a scientific methodology perspective, GWAS is essentially hypothesis-free. Any of the 3.1 billion SNPs in the human genome could, in principle, be associated with an outcome of interest. The inductive approach, assaying all variants and controlling Type I error via multiple testing correction, was partly spurred by the poor replicability of candidate gene studies (Duncan and Keller 2011). Structural equation modeling, by contrast, is strongly hypothesis driven. GW-SEM is therefore a hybrid that merges exploratory association methods with a hypothesis-driven framework for the assessment of multiple phenotypes and their putative risk factors. Specifically, the same hypothesis-driven phenotypic model is fit to every SNP without a priori expectations regarding the association for any given genomic variant. The driving philosophy here is that a single-step analysis often has advantages over two-step (or more) methods. A multi-step approach, such as estimating common and specific factor scores and applying univariate GWAS to the scores typically loses information, with potential increases in false positive and false negative association signals.

A further advantage of GW-SEM is that it uses the liability-threshold model for ordinal data. To our knowledge, GW-SEM 2.0 is the only GWAS software package that can treat ordinal data with three or more categories in this way. The alternatives seem to require that users either (i) treat ordinal data as continuous, likely underestimating correlations between measures (Verhulst and Neale 2021); or (ii) recode the data into a binary variable, which likely discards useful information and reduces statistical power. Furthermore, aggregating ordinal items to a sum- or factor-scores can generate reverse-J shaped distributions that violate the normality assumptions of linear models. Staying within the SEM framework and directly analyzing the variables or items that are measured avoids these pitfalls.

GW-SEM makes it possible to obtain and interpret GWAS findings within multivariate phenotypic models for complex behavioral and psychological traits. Such nuanced modeling does not seem possible with other software. The SEMs that can be fit in GW-SEM have been extensively applied in prior epidemiological, psychological, and psychiatric studies but have not been used in a GWAS context. This integration makes genetic results directly applicable to existing theories, dramatically reducing the gap between genomics and complex trait assessment. GW-SEM provides pre-defined functions for statistical models that are used widely for complex traits. Users can expand the set of pre-defined models and tailor their model to best match the phenotypes being studied.

Applying GWAS within a structural equation model requires more computer time than a simple univariate GWAS. However, the increase in computational time can easily be managed because each SNP could, in principle, be analyzed on a different compute node. Cloud computing provides an inexpensive approach to massively parallel analyses (Pritikin et al. 2019). Naturally, computer time increases with the number of observed items or variables. While timing may appear slow, it is important to keep in mind that a latent variable analysis with three items effectively runs three GWASs simultaneously.

As GWAS designs begin to accommodate more complex statistical models, the number of potential models that can be specified increases dramatically. GW-SEM users should have a fully conceptualized SEM prior to using GW-SEM for analysis. It would be inefficient to use GW-SEM to conduct exploratory phenotypic analyses that incorporate genomic variants. Instead, researchers should have a general understanding of the phenotypic model and the potential hypotheses that would be substantively interesting for their specific phenotype. This previously estimated “measurement model” can then be modified to explore the role of SNPs in generating variation and covariation.

Applications to drug use data

The complex pattern of results we present highlights the intrinsic difficulties of interpreting the association between genomic variants and substance use phenotypes. Several important findings have emerged. In the illustrations, we highlight the importance of modeling ordinal data appropriately. When ordinal data are treated as continuous, as is the case for most GWAS software packages, the observed associations are biased towards zero, making the associations less statistically significant. This has clear downstream implications for subsequent analyses. In the second illustration, we show that by working within a SEM framework, it is possible to delve deeper into the multivariate relationships that drive genetic associations to provide novel insights beyond what other GWAS software can offer. By treating complex behaviors as an interconnected multivariate structure, we can enhance our understanding of both direct genetic associations with specific behaviors as well as features of a pleiotropic system.

Conclusion

The proliferation of genomic data provides unparalleled opportunities to test novel hypotheses about the genetic etiology of psychiatric and substance abuse disorders, the comorbidity within and between behavioral and physical disorders, and genetic risk factors for mental and physical health. While standard univariate GWAS have successfully detected genetic associations with a range of psychological traits, significant gaps remain in our understanding. More work is needed to weave these findings into more complete description of the genetic architecture of behavioral and psychological risk factors for adverse health outcomes. Many of the remaining questions concern the complex interplay between symptoms, phenotypes, and environmental contexts. These can only be answered through multivariate analysis.

We designed GW-SEM to begin addressing these issues. The current release of GW-SEM 2.0, which is available on both the CRAN and GitHub websites, is a massive improvement over the previous software release. We have also built a series of tutorials that are available on the GW-SEM GitHub page (https://jpritikin.github.io/gwsem/) that cover each of the pre-defined GWAS functions, as well as a tutorial for constructing custom user-defined GWAS models. As GWAS data are disseminated to a broader set of analysts, especially those with psychometric training, the need for analytical designs that can accommodate complex statistical models will grow exponentially.

Supplementary Material

The online version contains supplementary material available at https://doi.org/10.1007/s10519-021-10043-1.