Effect of Population Stratification on SNP-by-Environment Interaction

Abstract

Proportions of false positive rates in genome-wide association analysis (GWAS) are affected by population stratification, and if it is not correctly adjusted, the statistical analysis can produce the large false-negative finding. Therefore various approaches have been proposed to adjust such problems in genome-wide association studies. However in spite of its importance, a few studies have been conducted in genome-wide SNP-by-environment interaction studies. In this report, we illustrate in which scenarios can lead to the false-positive rates in association mapping and approach to maintain the overall type-1 error rate.

Genome-wide SNP-by-environment interaction analysis is affected by multiple factors such as heteroscedasticity of phenotypes by environment and population stratification (Park et al., 2018; Engelman et al., 2009). Population stratification can introduce bias into the statistical analyses of genetic data (Pritchard & Rosenberg, 1999; Devlin & Roeder, 1999, Won et al., 2009). In this report, for case/control studies and for population-based studies with quantitative phenotypes, we examine the effect of the population stratification on SNP-by-environment interaction analyses, using analytical and empirical results. Specifically, we are interested in scenarios in which the environmental effects on the disease phenotype of interest vary between populations, as such situations are frequently encountered in respiratory and cardiovascular diseases (Vollmer et al., 2000; Dransfield et al., 2006; Chaturvedi, 2003).

For our initial analytical considerations, we assume a study of unrelated individuals in which a quantitative phenotype without ascertainment is recorded. Using the simulation studies, we will later generalize our theoretical findings to case/control designs. For now, we further assume that our study consists of two different subpopulations, A and B. Subpopulation is denoted by the parameter $P_i,$ and is coded by 1 for subpopulation A and 0 for subpopulation B. The relative proportion of subpopulation A to the sample size is denoted by $\pi .$ The parameters Y_i, E_i, and G_i were used to indicate the quantitative phenotype of interest, the environment exposure variable and the genotype of subject i, respectively. We assume that the phenotype can be modelled by

$$E\left(Y_i\right)=\{\begin{array}{c}{\beta }_{00}+{\beta }_{10}{E}_i+{\beta }_{20}{G}_i+{\beta }_{30}{G}_i{E}_i,\text{if}{P}_i=0\\ {\beta }_{01}+{\beta }_{11}{E}_i+{\beta }_{21}{G}_i+{\beta }_{31}{G}_i{E}_i,\text{if}{P}_i=1\end{array},\dots$$ (1).

Here the parameters, ${\beta }_{20},{\beta }_{21},{\beta }_{30},$ and ${\beta }_{31},$ allow for different genetic main effects and different SNP-by-environment interactions in the distinct subpopulations.

Dudbridge and Fletcher have shown that the spuriously significant results for ${\beta }_{3j}$ (j = 0, 1) can be observed, even under the absence of SNP-by-environment interaction, if the environmental exposure $E_i$ is not independent of the genotype $G_i$ (Dudbridge, 2014). The environmental exposure variable $E_i$ is therefore assumed to be independent with $G_i$. Furthermore, we assume here the absence of a genetic main effect and a SNP-by-environment effect, e.g. ${\beta }_{2j}={\beta }_{3j}=0.$ Under these assumptions, the model of equation (1) simplifies to the null model

$$E\left(Y_i\right)=\{\begin{array}{c}{\beta }_{00}+{\beta }_{10}{E}_i,\text{if}{P}_i=0\\ {\beta }_{01}+{\beta }_{11}{E}_i,\text{if}{P}_i=1\end{array}\dots$$ (2).

We denote the minor allele frequencies of $G_i$ be ${\theta }_1$ and ${\theta }_0$ for subpopulations A and B, respectively. We model population stratification by assuming different population means and allele frequencies in each subpopulation, i.e. ${\beta }_{00}\ne {\beta }_{01}$ and ${\theta }_0\ne {\theta }_1.$ As we want to assess the effects of different environmental exposure effects in our null model (2), we set ${\beta }_{10}\ne {\beta }_{11}.$ As the subpopulation indicator variable $P_i$ is generally unobserved in applications, it has to be estimated, e.g. PC scores based on genetic relationship matrix (Price et al., 2006; Prokopenko et al., 2016; Schlauch et al., 2017).

To evaluate the effect of population stratification on standard genetic association analysis, we consider three different regression models. In Model 1, we test only for a genetic main effect with adjustment of the population stratification for it. Model 2 assesses the presence of a genetic main effect and a SNP-by-environment interaction with adjustment for the population stratification for the former but not for the latter. In contrast to standard analysis approaches, in Model 3, we include environment-by-population substructure interaction variables, and test for a genetic main effect and SNP-by-environment interaction with adjustment of population stratification for both. Analytically we derived the expected biases for coefficients of SNP-by-environment interaction under the null model (2). The potential biases for those models are summarized in Table 1 (see Appendix for proof). In Table 1, we assumed that there are two subpopulations (P_i = 0, 1), $E_i\perp P_i$ and $\pi =\mathrm{0.5.}$ We also assumed that there is no SNP-by-environment interaction, and the effect of environment differ by subpopulations. If $E_i\perp P_i$ is not satisfied, the bias may become larger. Result for $\pi \ne 0.5$ can be found from Appendix. Table 1 indicates that SNP-by-environment effect can be inflated by population stratification if ${\beta }_{11}\ne {\beta }_{12},$ and thus $P_i{E}_i$ should be incorporated as covariates for the regression model. In particular, the bias is proportional to ${\alpha }_4={\beta }_{11}-{\beta }_{10},$ and it is also a function of $p_0$ and $p_1.$ It increases if differences between ${\beta }_{11}$ and ${\beta }_{10}$ and/or ${\theta }_0$ and ${\theta }_1$ becomes larger. Furthermore by Lobach’s result the absence of both SNP-by-environment effect and environment-by-population substructure interaction can bias the main SNP effect for case-control studies under the population stratification as well (Lobach, 2018).

Table 1. Biases of SNP and SNP-by-environment interaction effects for quantitative phenotypes.

We assumed that $\text{E}\left(E_i\right)=0,$ $E_i\perp G_i$ and $E_i\perp P_i,$ and $\pi =\mathrm{0.5.}$ We also assumed that there is no SNP-by-environment interaction, and the effect of environment differ by subpopulations.

	Fitted regression model	Biases of the coefficient of $G_i$	Biases of the coefficient of $G_i{E}_i$
M1:	$E\left(Y_i\right)={\alpha }_0^1+{\alpha }_1^1{P}_i+{\alpha }_2^1{E}_i+{\alpha }_3^1{G}_i$	0
M2:	$E\left(Y_i\right)={\alpha }_0^2+{\alpha }_1^2{P}_i+{\alpha }_2^2{E}_i+{\alpha }_3^2{G}_i+{\alpha }_4^2{G}_i{E}_i$	0	$\frac{{\alpha }_4\left({\theta }_1-{\theta }_0\right)}{4\left({\theta }_1+{\theta }_0-2{\theta }_1{\theta }_0\right)}$
M3:	$E\left(Y_i\right)={\alpha }_0^3+{\alpha }_1^3{P}_i+{\alpha }_2^3{E}_i+{\alpha }_3^3{G}_i+{\alpha }_4^3{P}_i{E}_i+{\alpha }_5^3{G}_i{E}_i$	0	0

The effect of population substructure on association analysis was evaluated with simulation studies. Assuming Hardy-Weinberg equilibrium, genotypes were generated by drawing from B(2, minor allele frequencies). We assumed the admixture of 2 subpopulations, and each subject was assigned to the one of the 2 subpopulations with 50% probability. The allele frequencies for each SNP in the two subpopulations were generated by the Balding-Nichols model (Balding & Nichols, 1995). If we denote the allele frequency in an ancestral population by p, p was generated from U(0.1,0.9), and the marker allele frequencies for the two subpopulations were independently sampled from beta (p(1−F_ST)/F_ST, (1−p)(1−F_ST)/F_ST) for the whole markers in each replicate of the simulated GWAS. A survey reported F_ST estimates with a median of 0.008 among Europeans, and the corresponding values are 0.027 and 0.12 among Africans and Asians respectively (Cavalli-Sforza et al., 1993). In our simulations, the value for Wright’s F_ST was assumed to be 0.005, 0.01, or 0.02. 10,000 SNPs with no effect on phenotypes were generated and they were used to calculate the genetic relationship matrix which was used to estimate ten PC scores (Price et al., 2006).

We considered dichotomous and quantitative phenotypes. Quantitative phenotypes were generated by summing environmental effects, additive polygenic effect, subpopulation-specific effects, and random errors as follows:

$$Y_i={\beta }_{0j}+A_i+{\beta }_{1j}·f\left(E_i\right)+{\epsilon }_{i}if{P}_i=j,j=0,\mathrm{1...}$$ (3).

Here $A_i,$ $E_i$ and ${\epsilon }_i$ indicate additive polygenetic effects, and observed environment effects, and unobserved environment effect respectively. $A_i,$ and ${\epsilon }_i$ were generated from $\text{N}\left(0,{\sigma }_A^2\right)$, and $\text{N}\left(0,{\sigma }^2=1\right),$ respectively. For E_i, random values were generated from $\text{N}\left(0,{\sigma }_E^2=1\right).$ Effect of E_i can be non-linear and thus it was transformed with the function $f\left(x\right).$ We considered $f\left(x\right)=x$ or $x^2.$ To generate the population stratification, we assumed that ${\beta }_{00}={\beta }_{01}+0.5$ and ${\beta }_{10}={\beta }_{11}+\mathrm{0.2.}$ ${\beta }_{01}$ was assumed to be 0, and ${\beta }_{11}$ was obtained by setting the relative proportions of variances explained by additive polygenic effects $(h_A^2),$ and observed environmental factors ($h_E^2$). We let

$$h_A^2=\frac{{\sigma }_A^2}{{\sigma }_A^2+0.5\left({\beta }_{11}^2+{\beta }_{10}^2\right){\sigma }_E^2+{\sigma }^2},h_E^2=\frac{0.5\left({\beta }_{11}^2+{\beta }_{10}^2\right){\sigma }_E^2}{{\sigma }_A^2+0.5\left({\beta }_{11}^2+{\beta }_{10}^2\right){\sigma }_E^2+{\sigma }^2}\mathrm{...}$$ (4).

$\left(h_A^2,h_E^2\right)$ were set to be (0.3,0.1), (0.3,0.3), or (0.3,0.5). ${\beta }_{11}$′s obtained from the equation (4) for those becomes 0.295, 0.760, and 1.478 respectively. For each replicate, we generated 1,000 subjects, and considered three different models in Table 1. In particular, $P_i$ is generally unknown, and 10 PC scores estimated from the genetic relationship matrix were considered as follows (Price et al., 2006):

$$\text{M1}:\text{E}\left(Y_i\right)={\alpha }_0^1+{\sum }_{l=1}^{10}{\alpha }_{1l}^{1}P{C}_{il}+{\alpha }_2^1{E}_i+{\alpha }_3^1{G}_i.$$

$$\text{M2}:E\left(Y_i\right)={\alpha }_0^2+{\sum }_{l=1}^{10}{\alpha }_{1l}^{2}P{C}_{il}+{\alpha }_2^2{E}_i+{\alpha }_3^2{G}_i+{\alpha }_4^2{G}_i{E}_i$$

$$\text{M3}:E\left(Y_i\right)={\alpha }_0^3+{\sum }_{l=1}^{10}{\alpha }_{1l}^{3}P{C}_{il}+{\alpha }_2^3{E}_i+{\alpha }_3^3{G}_i+{\sum }_{l=1}^{10}{\alpha }_{4l}^{3}P{C}_{il}{E}_i+{\alpha }_5^3{G}_i{E}_i$$

Each model was fitted with the multiple linear regressions, and empirical type-1 error rates at the 0.05 significance level and variance inflation factors were estimated with 5,000 replicates.

For dichotomous phenotypes, we first generated the unobserved latent variables, and they were transformed to the dichotomous phenotypes. Unobserved latent variables were assumed to be continuous and were generated with the equation (3). We assumed that ${\beta }_{00}={\beta }_{01}+1$ and ${\beta }_{10}={\beta }_{11}+1$, but except this, the other parameters for unobserved latent variables were set to be the same as for quantitative phenotypes. Subjects whose unobserved latent variables are larger than the certain threshold were assumed to be affected, and otherwise to be unaffected. Disease prevalence was assumed to be 0.2, and threshold was accordingly chosen. To mimic the sampling bias in case-control studies, we generated genotypes and dichotomous phenotypes for 10,000 subjects, and then 500 affected and 500 unaffected subjects were randomly selected. For dichotomous phenotypes, we used logistic regression and fitted the same mean models as for quantitative phenotypes (M1, M2 and M3). All simulations were done with the statistical software, R version 3.5.1, and Figures were generated with Rex version 3.0 (R Core Team, 2014; RexSoft, 2018).

Table 2, 3, and Figure 1 are results from the quantitative phenotypes. Table 2 and 3 shows the inflation of statistical inference, and Figure 1 shows the bias of the estimates. Table 2 shows the empirical type-1 error rate estimates at the 0.05 significance level with 5,000 replicates. Our simulation results suggest that the statistical tests for ${\alpha }_3^1$, ${\alpha }_3^2$ and ${\alpha }_3^3$ in M1, M2 and M3 correctly controls the type-1 error rates, respectively, which indicates the statistical inference for the main effect is valid as long as PC scores are included as covariates. Heterogeneous effects of environment among subpopulations do not affect the validity of the main effect of SNPs. However, the statistical test for ${\alpha }_4^2$ in M2 is inflated, while ${\alpha }_4^2$ in M3 preserves the nominal significance level. If $f\left(E_i\right)=E_i^2,$ both tests become inflated and the amount of inflation becomes much larger. We can therefore conclude that the statistically valid SNP-by-environment interaction analysis requires the inclusion of correctly specified PC-by-environment interaction relationship into the regression model as covariates. It should be noted that the false negative rates affected by population substructure is adjusted by the proposed method as well. Table 3 and Figure 1 show the variance inflation factors (VIFs) and the bias of coefficients for the main SNP effect and SNP-by-environment effect. For Figure 1, we assumed that ${\theta }_0$ and ${\theta }_1$ were assumed to be 0.1 and 0.3, respectively. We selected the values for ${\theta }_0$ and ${\theta }_1,$ as they seem to be realistic choices based on our experience in applications and similar parameter choices in other studies (Panarella & Burkett, 2019). Different values for both can be utilized but the similar patterns are expected as long as $\frac{\left({\theta }_1-{\theta }_0\right)}{{\theta }_1+{\theta }_0-2{\theta }_1{\theta }_0}$ is same. Both results show that the variances of main effects for ${\alpha }_4^2$ in M2 are inflated, and its coefficients are biased, while other estimates remain unbiased. Interestingly, Table 3 shows that the VIFs are positively related with $F_{ST}$ and $h_E^2,$ but the amount of biases seem constant, which correspond to our derivation in Table 1. Table 3 and Figure 1 also show results when effect of environment on phenotypes is nonlinear. Both VIFs and biases of ${\alpha }_5^3$ and ${\alpha }_4^2$ in M3 and M2 are inflated. Furthermore, the biases of ${\alpha }_5^3$ and ${\alpha }_4^2$ are not constant, and positively related with $F_{ST}$ and $h_E^2$. Thus, PC-by-non-linear-environment interaction term need to be included as covariates if any non-linear relationship between environment and phenotype is expected.

Table 2. Empirical type-1 error rates at the 0.05 significance level for continuous phenotypes.

Empirical type-1 error rates at the 0.05 significance level were estimated with 5,000 replicates. ${\alpha }_3^1$ indicates the coefficient of the main SNP effect in Model 1. ${\alpha }_3^2$ and ${\alpha }_4^2$ indicate the coefficients of the main SNP effect and SNP-by-environment interaction effect in Model 2. ${\alpha }_3^3$ and ${\alpha }_5^3$ indicate the coefficients of the main SNP effect and SNP-by-environment interaction effect in Model 3. Any inflated results were shown in a bold type.

$f\left(E_i\right)$	$h_E^2$	$F_{ST}$	${\alpha }_3^1$	${\alpha }_3^2$	${\alpha }_4^2$	${\alpha }_3^3$	${\alpha }_5^3$
$E_i$	0.1	0.005	0.0524	0.0494	0.0568	0.0512	0.0486
		0.01	0.047	0.0516	0.0634	0.0516	0.05
		0.02	0.0566	0.0514	0.0714	0.0506	0.0484
	0.3	0.005	0.0476	0.0452	0.0634	0.0466	0.0542
		0.01	0.0482	0.0436	0.0486	0.0434	0.0424
		0.02	0.0464	0.0436	0.0666	0.0436	0.0498
	0.5	0.005	0.0554	0.0506	0.0512	0.051	0.046
		0.01	0.0504	0.047	0.0602	0.0466	0.052
		0.02	0.0456	0.0532	0.0656	0.053	0.0528
$E_i^2$	0.1	0.005	0.0572	0.055	0.1436	0.0542	0.1392
		0.01	0.0516	0.0512	0.1422	0.0518	0.1416
		0.02	0.049	0.0492	0.1744	0.0474	0.1694
	0.3	0.05	0.047	0.0496	0.2556	0.0492	0.254
		0.01	0.05	0.0522	0.2478	0.0518	0.2506
		0.02	0.0508	0.0528	0.2872	0.053	0.2864
	0.5	0.05	0.053	0.0546	0.3004	0.0546	0.3084
		0.01	0.0518	0.0532	0.3106	0.0546	0.3156
		0.02	0.0474	0.0496	0.2822	0.0484	0.2718

Table 3. VIFs for continuous phenotypes.

VIFs were estimated with 5,000 replicates. ${\alpha }_3^1$ indicates the coefficient of the main SNP effect in Model 1. ${\alpha }_3^2$ and ${\alpha }_4^2$ indicate the coefficients of the main SNP effect and SNP-by-environment interaction effect in Model 2. ${\alpha }_3^3$ and ${\alpha }_5^3$ indicate the coefficients of the main SNP effect and SNP-by-environment interaction effect in Model 3. Any inflated results were shown in a bold type.

$f\left(E_i\right)$	$h_E^2$	$F_{ST}$	${\alpha }_3^1$	${\alpha }_3^2$	${\alpha }_4^2$	${\alpha }_3^3$	${\alpha }_5^3$
$E_i$	0.1	0.005	1.027386	1.012797	1.019713	1.015115	0.9939585
		0.01	1.003105	1.008678	1.0730363	0.998719	1.0097014
		0.02	1.004055	1.049463	1.1521499	1.05569	1.0210178
	0.3	0.005	0.991464	1.001888	1.0939963	0.995082	1.0860828
		0.01	1.006365	0.957771	0.9864476	0.953706	0.8886005
		0.02	0.980962	0.980874	1.2150878	0.998315	0.9941031
	0.5	0.005	0.981666	1.051838	1.0637439	1.061499	1.0519922
		0.01	1.019816	0.956978	1.0972295	0.939544	1.0082697
		0.02	0.975011	0.998073	1.1210838	1.016236	1.0227772
$E_i^2$	0.1	0.005	1.022543	1.009262	1.774122	1.022223	1.770374
		0.01	0.975239	0.994675	1.80265	1.001162	1.782905
		0.02	1.048757	1.049851	2.149673	1.056037	2.132268
	0.3	0.05	0.997586	1.009116	2.88706	1.006533	2.901799
		0.01	0.978008	0.972002	2.919534	0.974188	2.958553
		0.02	0.998619	0.989772	3.179439	0.989597	3.268996
	0.5	0.05	0.990459	0.983499	3.588286	0.989325	3.650458
		0.01	1.002896	1.030339	3.796575	1.037167	3.751741
		0.02	1.034665	1.051726	3.289446	1.059519	3.259833

Figure 1. Bias of beta coefficients for continuous phenotypes.

θ₁ and θ₀ were fixed as 0.3 and 0.1 respectively, and boxplots of estimated main SNP effect and SNP-by-environment interaction effect were provided with results from 5,000 replicates. G_in_M1 indicates the coefficient of the main SNP effect in Model 1. G_in_M2 and GE_in_M2 indicate the coefficients of the main SNP effect and SNP-by-environment interaction effect in Model 2. G_in_M3 and GE_in_M3 indicate the coefficients of the main SNP effect and SNP-by-environment interaction effect in Model 3. Dashed horizontal line indicates the expected bias derived by results from the Table 1.

Results from dichotomous phenotypes are shown in Table 4, 5 and Figure 2. Table 4 shows that statistical inferences for ${\alpha }_3^2$ and ${\alpha }_4^2$ are inflated if $f\left(E_i\right)=E_i.$ For quantitative phenotypes, the statistical inference for ${\alpha }_3^2$ was not inflated and it is attributable to the sampling bias as was shown by Lobach (Lobach, 2018). If $f\left(E_i\right)=E_i^2,$ inflation for statistical inference is observed for ${\alpha }_3^2,$ ${\alpha }_4^2,$ ${\alpha }_3^3$ and ${\alpha }_5^3,$ and the amount of inflation becomes much larger than those for $f\left(E_i\right)=E_i.$ Table 5 shows VIFs and this result corresponds to those for Table 4. Last Figure 2 shows biases of estimated regression coefficients. Results for $f\left(E_i\right)=E_i$ shows that estimates for ${\alpha }_3^2,$ and ${\alpha }_4^2$ are biased. Interestingly estimated coefficients have no biases if $f\left(E_i\right)=E_i^2.$ Heterogeneity of non-linear environmental effects may not affect the biases of estimated regression coefficients for dichotomous phenotypes, but further investigation is necessary.

Table 4. Empirical type-1 error rates at the 0.05 significance level for dichotomous phenotypes.

Empirical type-1 error rates at the 0.05 significance level were estimated with 5,000 replicates. ${\alpha }_3^1$ indicates the coefficient of the main SNP effect in Model 1. ${\alpha }_3^2$ and ${\alpha }_4^2$ indicate the coefficients of the main SNP effect and SNP-by-environment interaction effect in Model 2. ${\alpha }_3^3$ and ${\alpha }_5^3$ indicate the coefficients of the main SNP effect and SNP-by-environment interaction effect in Model 3. Any inflated results were shown in a bold type.

$f\left(E_i\right)$	$h_E^2$	$F_{ST}$	${\alpha }_3^1$	${\alpha }_3^2$	${\alpha }_4^2$	${\alpha }_3^3$	${\alpha }_5^3$
$E_i$	0.1	0.005	0.0519	0.0583	0.0977	0.05	0.0534
		0.01	0.0461	0.0552	0.0695	0.0471	0.0484
		0.02	0.0481	0.057	0.1287	0.0471	0.0489
	0.3	0.005	0.0492	0.0468	0.0782	0.0412	0.0454
		0.01	0.0497	0.0498	0.0602	0.0449	0.0477
		0.02	0.0554	0.0603	0.0956	0.0482	0.0473
	0.5	0.005	0.0493	0.0432	0.0611	0.0393	0.0457
		0.01	0.0446	0.0409	0.0595	0.0387	0.0496
		0.02	0.0542	0.0484	0.0721	0.0406	0.0418
$E_i^2$	0.1	0.005	0.0493	0.0491	0.0549	0.0504	0.0553
		0.01	0.0495	0.0506	0.053	0.0501	0.0543
		0.02	0.0489	0.0502	0.0509	0.0505	0.0522
	0.3	0.05	0.0483	0.0497	0.0575	0.0499	0.0574
		0.01	0.0481	0.0473	0.0578	0.0479	0.0596
		0.02	0.0526	0.054	0.0603	0.0551	0.0617
	0.5	0.05	0.0518	0.0527	0.0574	0.053	0.0584
		0.01	0.0439	0.0457	0.0557	0.0452	0.0553
		0.02	0.0529	0.0551	0.0553	0.0568	0.0558

Table 5. VIFs for dichotomous phenotypes.

VIFs were estimated with 5,000 replicates. ${\alpha }_3^1$ indicates the coefficient of the main SNP effect in Model 1. ${\alpha }_3^2,$ and ${\alpha }_4^2$ indicate the coefficients of the main SNP effect and SNP-by-environment interaction effect in Model 2. ${\alpha }_3^3$ and ${\alpha }_5^3$ indicate the coefficients of the main SNP effect and SNP-by-environment interaction effect in Model 3. Any inflated results were shown in a bold type.

$f\left(E_i\right)$	$h_E^2$	$F_{ST}$	${\alpha }_3^1$	${\alpha }_3^2$	${\alpha }_4^2$	${\alpha }_3^3$	${\alpha }_5^3$
$E_i$	0.1	0.005	0.990454	1.082145	1.421195	1.002839	1.019046
		0.01	0.981201	1.047026	1.133424	0.988776	0.994216
		0.02	0.993684	1.083341	1.760198	0.973748	0.993886
	0.3	0.005	0.965876	0.993789	1.200121	0.963688	0.977955
		0.01	0.997249	0.989646	1.05349	0.970484	0.966718
		0.02	1.069564	1.096656	1.420798	1.015194	0.966029
	0.5	0.005	1.031723	0.958122	1.092199	0.941847	0.939609
		0.01	0.954557	0.908889	1.026855	0.937842	0.955819
		0.02	1.039274	0.966197	1.139419	0.926031	0.964834
$E_i^2$	0.1	0.005	0.987105	0.998609	1.024468	0.99927	1.024458
		0.01	1.009694	1.009511	1.03768	1.013913	1.048363
		0.02	1.025191	1.028215	1.027291	1.020519	1.026361
	0.3	0.05	0.978876	1.00101	1.125861	1.000982	1.147405
		0.01	1.008625	1.008496	1.048979	1.00748	1.059942
		0.02	1.074007	1.089734	1.098727	1.080247	1.122698
	0.5	0.05	1.073888	1.067541	1.071861	1.061183	1.078626
		0.01	0.966745	0.971529	1.020514	0.97017	1.028622
		0.02	1.077112	1.09292	1.048648	1.087723	1.031459

Figure 2. Bias of beta coefficients for dichotomous phenotypes.

θ₁ and θ₀ were fixed as 0.3 and 0.1 respectively, and boxplots of estimated main SNP effect and SNP-by-environment interaction effect were provided with results from 5,000 replicates. G_in_M1 indicates the coefficient of the main SNP effect in Model 1. G_in_M2 and GE_in_M2 indicate the coefficients of the main SNP effect and SNP-by-environment interaction effect in Model 2.

We conducted GWAS of forced expiratory volume in 1 s (FEV1) and chronic obstructive pulmonary disease (COPD) status in the African American (AA) sample of COPDGene cohort data. All participants had at least 10 pack-years (PY) of smoking, and their ages were between 45 and 80 years. Details of the study protocol and data forms are available at www.copdgene.org (Regan et al., 2010) and SNP data underwent standard quality control measures (Cho et al., 2010). After QC of SNPs and subjects, 3,300 AAs genotyped for 626,477 SNPs remained. We included age, sex, body mass index, height, PY and sex×age as covariates. Top 10 principal component (PC) scores were estimated from the genetic relationship matrix (Song et al., 2018). Population stratification was adjusted with 5 different sets of covariates as follows:

• S1: no PC adjustment

• S2: PC1, …., PC10

• S3: PC1, …., PC10, PC1×PY

• S4: PC1, …., PC10, PC1×PY, PC2×PY

• S5: PC1, …., PC10, PC1×PY, …, PC10×PY

Wald tests for SNP and SNP×PY terms were conducted for FEV1 and COPD status, and VIFs for each test are provided in Table 6. Results show that VIFs for SNP are close to 1 as long as 10 PCs are included as covariate. However, when looking at the VIFs for SNP×PY, incorporation of 10 PCs are not sufficient, and PC1×PY should be additionally included as a covariate. For both FEV1 and COPD status, the best results were found in model S4. Our analysis findings illustrate the importance of adjustment with PC1×PY in SNP-by-environment interaction analyses as covariates.

Table 6. Analyses of COPDgene data.

Age, sex, BMI, height, pack-year (PY) and sex×age were included as covariates. Wald tests for SNP and SNP×PY were conducted for FEV1 and COPD status, and VIFs were provided.

	FEV1		COPD status
PC-related covariates	SNP	SNP×PY	SNP	SNP×PY
No PC adjustment	1.144121	1.096484	1.057125	1.06318
PC1+…+PC10	1.016645	1.125281	1.034342	1.076399
(PC1+…+PC10)+PC1×PY	0.972402	1.057972	1.007371	1.036103
(PC1+…+PC10)+PC1×PY+PC2×PY	0.969252	1.053921	1.011908	1.042859
(PC1+…+PC10)+(PC1×PY+…+PC5×PY)	0.973651	1.059613	1.023406	1.060539
(PC1+…+PC10)+(PC1×PY+…+PC10×PY)	0.980094	1.070457	1.031335	1.075239

In summary, the SNP-by-environment interaction analysis is affected by population stratification and our results indicate its adjustment requires the careful investigation about the relationship between environment and phenotypes.

However in spite of the important finding about the population substructure in SNP-by-environment interaction analysis in this report, there are some limitation in our results. First, in our derivation, we assume that the observed value of environment is independent with genotypes and population substructure. The most of statistical methods for SNP-by-environment interaction analysis requires the former independence (Dudbridge & Fletcher, 2014). The significant interaction effect between genotypes and environmental exposure can be obtained, even under its absence, and has been often verified before SNP-by-environment interaction analysis. The first independence assumption is therefore practically reasonable. However, the second independence assumption may not be preserved in many scenarios. For instance, the amount of smoking often differs by the subpopulation. In such case, the estimator for bias derived in the appendix can also be biased, and some modifications are required. Second, we assume that population substructure can be appropriately be described by the PC scores, regardless of the amount of Wright’s F_ST. This assumption is supported by several studies (Price et al., 2016). However, if local population admixture exists in certain genomic location, PC scores cannot represent such population substructure (Lin et al., 2007). If there is a local population admixture, the family-based association analyses such as FBAT may be more appropriate analysis approaches (Won et al., 2009). Third, we showed that inflated results of SNP-by-environment interaction analysis can be caused by the effect of environmental exposures and minor allele frequencies that differ by subpopulation. As it is not straightforward to check whether the effect of environment and minor allele frequencies differ by subpopulation, we propose statistical significance testing for both interaction variables in the analysis. However, significance will depend on the sample size as well as the actual differences by subpopulation. Alternatively, we therefore suggest to check whether VIFs are close to 1 when the PC-by-environment interaction term is included in the analysis. It is important to note that if this interaction is included in the absence of the interaction, its inclusion may lead to reduced statistical power, but it does not introduce bias or inflated type-1 errors. These limitations will be investigated in our future research.

APPENDIX

For simplicity, we consider the quantitative phenotypes. We assume that there is no genetic effect and $Y_i$ is modeled by

$$Y_i={\beta }_{0j}+{\beta }_{1j}{E}_i+{\epsilon }_i,{\epsilon }_i~N\left(0,{\sigma }^2\right)\text{if}{P}_i=j,j=1,2\dots$$ (4′).

We assume that $E_i$ is centered, that is $\text{E}\left(E_i\right)=0,$ and $E_i\perp G_i$ and $E_i\perp P_i.$ We assume that there are two different subpopulations, A and B, and it is denoted by $P_i.$ $P_i$ is 1 for subpopulation A and 0 for subpopulation B. The proportion of A is denoted by $\pi .$ The model (4) is equivalent to

$$Y_i={\alpha }_0+{\alpha }_1{P}_i+{\alpha }_2{E}_i+{\alpha }_4{P}_i{E}_i+{\epsilon }_i,{\epsilon }_i~N\left(0,{\sigma }^2\right).$$

Here it should be noted that ${\alpha }_0={\beta }_{00}, {\alpha }_1={\beta }_{01}-{\beta }_{00},{\alpha }_2={\beta }_{10}$ and ${\alpha }_4={\beta }_{11}-{\beta }_{10}.$ We fit three different regression models as follows:

$$\text{M1}:E\left(Y_i\right)={\alpha }_0^1+{\alpha }_1^1{P}_i+{\alpha }_2^1{E}_i+{\alpha }_3^1{G}_i$$

$$\text{M2}:E\left(Y_i\right)={\alpha }_0^2+{\alpha }_1^2{P}_i+{\alpha }_2^2{E}_i+{\alpha }_3^2{G}_i+{\alpha }_4^2{G}_i{E}_i$$

$$\text{M3}:E\left(Y_i\right)={\alpha }_0^3+{\alpha }_1^3{P}_i+{\alpha }_2^3{E}_i+{\alpha }_3^3{G}_i+{\alpha }_4^3{P}_i{E}_i+{\alpha }_5^3{G}_i{E}_i$$

For M3, it can be simply derived that both ${\alpha }_3^1$ and ${\alpha }_5^1$ have no bias. For M2, ${\alpha }_4^2$ indicates the SNP-by-environment interaction effect and by Frisch–Waugh–Lovell theorem (Frisch & Waugh, 1933; Lovell, 1963), we can get

$$E({\widehat{\alpha }}_4^2)=0+\frac{\text{cov}\left({\alpha }_4{P}_i{E}_i+{\epsilon }_i,G_i{E}_i\right)}{\text{var}\left(E_i{G}_i\right)}=0+{\alpha }_4\frac{\text{cov}\left(P_i,G_i\right)}{\text{var}\left(G_i\right)}$$

1. $$\begin{gathered} \text{cov}\left(P_i,G_i\right)=\text{E}\left(\text{E}(P_i{G}_i|P_i)\right)-\text{E}\left(P_i\right)\text{E}\left(E\left(G_i\text{|}{P}_i\right)\right)=2{\theta }_1\pi -2\pi \left({\theta }_1\pi \\ +{\theta }_0\left(1-\pi \right)\right)=2\pi \left(1-\pi \right)\left({\theta }_1-{\theta }_0\right) \end{gathered}$$
2. $$\begin{gathered} \text{var}\left(X_i\right)=\text{var}\left(\text{E}\right(G_i\left|P_i\right))+\text{E}(\text{var}\left(G_i\right|P_i\left)\right)=4({\theta }_1-{\theta }_0{)}^2\pi (1-\pi )+ \\ 2{\theta }_1(1-{\theta }_1)\pi +2{\theta }_0(1-{\theta }_0\left)\right(1-\pi ) \end{gathered}$$

Therefore the bias becomes

$$E\left({\widehat{\alpha }}_4^2\right)=\frac{{\alpha }_4\left({\theta }_1-{\theta }_0\right)}{2\left[2{\left({\theta }_1-{\theta }_0\right)}^2+{\theta }_1\left(1-{\theta }_1\right)/\left(1-\pi \right)+{\theta }_0\left(1-{\theta }_0\right)/\pi \right]}.$$

If $\pi =1/2,$ it is equal to

$$\frac{{\alpha }_4\left({\theta }_1-{\theta }_0\right)}{4\left({\theta }_1+{\theta }_0-2{\theta }_1{\theta }_0\right)}.$$

Last for M3, ${\alpha }_3^3$ indicates the main effect of SNP, and by Frisch–Waugh–Lovell theorem (Frisch & Waugh, 1933; Lovell, 1963), we can get

$$E\left({\widehat{\alpha }}_3^3\right)=0+\frac{\text{cov}\left({\alpha }_3^3{P}_i{E}_i+{\epsilon }_i,G_i\right)}{\text{var}\left(G_i\right)}=0+{\alpha }_3^3\frac{\text{cov}\left(P_i{E}_i,G_i\right)}{\text{var}\left(G_i\right)}$$

1. $$\begin{gathered} \text{cov}\left(P_i{E}_i,G_i\right)=\text{E}\left(P_i{E}_i{G}_i\right)-\text{E}\left(P_i{E}_i\right)\text{E}\left(G_i\right)= \\ \text{E}\left(E_i\right)\text{E}\left(P_i{G}_i\right)-\text{E}\left(E_i\right)\text{E}\left(P_i\right)\text{E}\left(G_i\right)=0 \end{gathered}$$
2. $$\begin{gathered} \text{var}\left(G_i\right)=\text{var}\left(\text{E}\right(G_i\left|P_i\right))+\text{E}(\text{var}\left(G_i\right|P_i\left)\right)=4({\theta }_1-{\theta }_0{)}^2\pi (1-\pi )+ \\ 2{\theta }_1(1-{\theta }_1)\pi +2{\theta }_0(1-{\theta }_0\left)\right(1-\pi ) \end{gathered}$$

Therefore the bias becomes

$$E\left({\widehat{\alpha }}_3^3\right)=0.$$