Likelihood ratio test for genetic association study with case–control data under Probit model

ABSTRACT

Probit and Logit models are the most popular for binary disease statusing in genetic association studies. They are equally used and nearly exchangeable in the analysis of prospectively collected data. However, no strong inferences were made based on Probit models for the retrospectively collected case–control data, especially in the presence of random effects. This paper systematically investigates the performance of Probit mixed-effects models for case–control data. We find that the retrospective likelihood has a closed-form, which motivates the development of likelihood ratio tests for genetic association. Specifically, we developed four likelihood ratio tests based on whether the disease prevalence is completely unavailable, partly available, or completely available. We show that their limiting distribution without a genetic effect is an equal mixture of two chi-square distributions with degrees of freedom 1 and 2, respectively. Our simulations indicate that they can have a remarkable power gain against the popular Logit-model-based score tests, and the disease prevalence information can enhance the power of the likelihood ratio tests. After analyzing a Kenya malaria data, we found out that the proposed test produces a significant result on the association of the gene ABO with malaria, whereas the commonly used competitors fail.

1. Introduction

In epidemiology, clinical trials, and genetic association studies, retrospective (or case–control) and prospective studies are two popular approaches to explore whether certain factors are associated with a disease. In prospective studies, researchers first recruit a fixed number of individuals for a certain disease before recording all individuals' information after a period, including both the covariates and disease status. Let X denote a vector of clinical covariates, and the disease status for generic individuals is D = 0 for a non-disease and 1 for a disease. Prospective data are independent and identically distributed (iid) from the joint distribution of $(D,X)$. A retrospective study begins by identifying many non-diseases and diseases and then draws samples separately from these two groups and records their covariates. The retrospective data with the same disease status D are iid from the conditional distribution of X given D, while those with different disease statuses are no longer identically distributed. Prospective and retrospective data have different joint distributions; therefore, they should be dealt with using different inference methods. An advantage of retrospective over prospective studies is that they can save money, time, or/and effort, especially for rare diseases such as cancer with a prevalence of 0.01%. The price is that the disease prevalence or mortality rate cannot be consistently estimated from retrospective data, which is generally not the case for prospective data.

The Logit and Probit models are the most popular to fit binary regression based on prospectively collected data. These are equally used and nearly interchangeable, though Logit distributions have slightly flatter tails. The Logit model is desirable if one prefers log odds interpretations. In contrast, the Probit model is more natural if one believes that the binary outcome depends deterministically on a latent Gaussian variable. Specifically, we let $Z=X^{\mathrm{\top }}\beta +ϵ$, where $ϵ\sim N(0,{\sigma }^2)$. The Probit model supposes that the binary outcome variable D = 1 when Z>0 and is 0 otherwise. The latent variable Z is called the liability and the resulting Probit model is the well-known genetic ‘liability-threshold model’ introduced by Pearson and Lee [11]. The disease of interest is assumed to be present $(D=1)$ in individuals whose liabilities are above 0. The celebrated selection biased sampling model by Heckman [1] is based on the liability model due to its preferred interpretation of the potential wage.

When interest is examining within-cluster associations in binary data using Logit or Probit models with mixed effects, there is a theoretical grounding that prefers Probit models. Pearson [10] showed that if multivariate normal data were generated and discretized, the correlations between the underlying variables and their joint distribution are still statistically identifiable. In contrast, Logit mixed-effects models have an identifiable variance in the random effects, but their correlations or joint distribution are not. This is because the joint distribution of the random effects is a mixture of normal and logistic random variables and it loses the property of being fully specified by its mean and covariance matrix.

To the best of our knowledge, only the Logit model is used in existing literature for case and control studies. The theoretical foundation for the justification of Logit models based on case–control data was established in a seminal paper by Prentice and Pyke [12]. They showed that one may use the prospective logistic likelihood to make inferences on the odds ratio parameter even if the data are collected retrospectively. The intercept in the logistic regression model plays a multiplicative role; thus, it can be treated together with disease prevalence as one independent parameter, which facilitates statistical analysis. This is not the case for Probit models where the intercept and disease prevalences have to be treated as two independent parameters. However, case–control data indicate that the disease prevalence is not estimable in general, and the ‘exchangeability’ between the Probit and Logit models (those with mixed effects) does not exist. This motivates investigating the performance of Probit models when analyzing case–control data. We focus on the problem of testing whether gene expression information, such as multiple single nucleotide polymorphisms (SNPs) within a gene or region, has a significant effect on the disease status after adjusting for environmental variables.

In addition to the aforementioned D and X, we let Y denote a vector of genetic variables. The goal is to test whether the genetic effect of Y on the disease status D exists after adjusting for the covariates X. Suppose that conditioning on the covariate information $(X,Y)$, the disease status D is modeled with the following liability model $D=I(Z>0)$, where the liability is $Z=\alpha +X^{\mathrm{\top }}\beta +Y^{\mathrm{\top }}{\gamma }^{\ast }+\epsilon$. Here, α, β, and ${\gamma }^{\ast }$ are unknown parameters, and ε is an error term. Different distribution assumptions on the error ε lead to various models for the disease status D. For example, Logit and Probit models are implied when ε follows logistic and normal distributions, respectively. The linear function $Y^{\mathrm{\top }}{\gamma }^{\ast }$ of Y in the model may not be flexible enough to capture more realistic scenarios where different genetic markers convey non-uniform risk levels (magnitude and/or direction). Lee et al. [4] and Sun et al. [13] proposed utilizing a random effects model

$$Z=\alpha +X^{\mathrm{\top }}\beta +Y^{\mathrm{\top }}{\gamma }^{\ast }+{\theta }^{1/2}{Y}^{\mathrm{\top }}v+\epsilon ,$$

where ${\gamma }^{\ast }$ and $v=(v_1,\dots ,v_q{)}^{\mathrm{\top }}$ denote the fixed and random effects, respectively, and θ is a scalar. For simplicity, we assume that ${\gamma }^{\ast }=\gamma \mathbb{1}$ where γ is a scalar and $\mathbb{1}$ is a vector of all ones and that the random effects of the $v_i$'s are iid as N(0, 1) and independent of ε. Testing the non-existence of the genetic effect is equivalent to testing $H_0:\gamma =0\mathrm{\&}\theta =0$.

The random effect v is not observable in general. Given data on $(D,X,Y)$, inference is usually made through the conditional distribution $\mathrm{pr}(Z|X,Y)$. If ε is assumed to follow a logistic distribution, $\mathrm{pr}(Z|X,Y)$ does not have a closed-form but involves a computationally intensive multivariate integral, making the subsequent likelihood-based inference formidable or intractable. Therefore, score tests, which avoid complicated integrals, become increasingly popular, and several score tests have been developed in genetic association studies. Popular examples include adaptive burden tests (e.g. [5]), sequence kernel association tests (SKAT; Wu et al. [15], SKAT-O ; Lee et al. [4]), and mixed-effects score tests (MiST; Sun et al. [13]) to name a few. Lee et al. [3] provided a more thorough review. As all these tests are independent of disease prevalence, such information is not helpful to improve the power. With retrospective case–control data, Liu et al. [6] developed score tests based on the retrospective likelihood under an independent-random-effect assumption and showed that the testing power of their tests can be improved by incorporating disease prevalence information in genetic association studies.

Although score tests are convenient and have good power for local alternatives, they may lose significant power compared with likelihood ratio tests when the alternative is far from the null. Fortunately, when $\epsilon \sim N(0,{\sigma }^2)$, the conditional distribution $\mathrm{pr}(Z|X,Y)$ has a closed-form of

$$Z|(X,Y)\sim N(\alpha +X^{\mathrm{\top }}\beta +\gamma Y^{\mathrm{\top }}\mathbb{1},{\sigma }^2+\theta Y^{\mathrm{\top }}Y),$$ (1)

where the variance ${\sigma }^2$ is set to 1 to ensure identifiability of the model parameters. This is a remarkable advantage of Probit models (normal errors) against Logit models (logistic errors). The closed-form of $\mathrm{pr}(Z|X,Y)$ not only makes it feasible but also motivates the derivation of likelihood ratio tests to investigate their performance in genetic association studies.

In this paper, we model the disease status from a Probit mixed-effects model and develop retrospective likelihood ratio tests when the overall genetic effect does not exist. In general, disease prevalence is not estimable from case–control data but may be publicly available. We construct four likelihood ratio tests based on the accessibility of auxiliary information on disease prevalence. Under the null hypothesis that there is no genetic effect, these tests asymptotically follow a chi-squared mixture distribution with known proportions. If the Probit model is constructed through a liability-threshold for the within-cluster associations, we find that the proposed tests are much easier to calculate numerically than their counterparts under Logit models. Our extensive simulation studies indicate that the proposed likelihood ratio tests have significant power gain against existing score tests, especially when the alternative is far from the null. In addition, the proposed likelihood ratio tests do not seem sensitive to the prevalence. Even when the prevalence is replaced with the proportion of cases in the sample, they have acceptable type I errors and minimal power loss. One more advantage of the proposed tests against the existing score tests is that they do not require variance estimation; an inaccurate variance estimate may downplay the performance of score tests.

The rest of this paper is organized as follows. Section 2 introduces the proposed retrospective likelihood tests and establishes their limiting distribution. Section 3 contains extensive simulation results to evaluate the finite-sample performance of the proposed tests and compares them to existing score tests. In Section 4, a Kenya malaria data set is analyzed for illustration. We end in Section 5 with a discussion. For clarity, all proofs are relegated to the supplementary material.

2. Semiparametric likelihood

Suppose we have n independent observations of case–control data from Model (1). Without loss of generality, we assume that the first $n_1$ observations are the cases $\{(x_i,y_i,D_i=1):i=1,\dots ,n_1\}$, and the remaining $n_0=n-n_1$ observations are controls, i.e. $\{(x_j,y_j,D_j=0):j=n_1+1,\dots ,n\}$. Let $\xi =(\theta ,\gamma ,\alpha ,{\beta }^{\mathrm{\top }}{)}^{\mathrm{\top }}$ and $A(\xi )=A(x,y,\xi )=(\alpha +x^{\mathrm{\top }}\beta +y^{\mathrm{\top }}\mathbb{1}\cdot \gamma )/\sqrt{1+y^{\mathrm{\top }}y\cdot \theta }.$ Conditionally,

$$\mathrm{pr}(D=1|x,y)=\mathrm{\Phi }(A(\xi ))$$

is still a Probit model, where $\mathrm{\Phi }(x)$ represents the standard Gaussian cumulative probability distribution function.

Let $F(x,y)$ be the joint density function of X and Y and $\eta =\mathrm{pr}(D=1)=\int \mathrm{\Phi }(A(\xi ))\mathrm{d}F(x,y)$ be the prevalence of the disease of interest. The retrospective likelihood based on the case–control data is

$$\begin{array}{rl}\tilde{L}(\xi ,\eta ,F)& =\prod _{i=1}^n[\{\mathrm{pr}(x_i,y_i|D=1){\}}^{D_i}\{\mathrm{pr}(x_i,y_i|D=0){\}}^{1-D_i}]\\ & =\prod _{i=1}^n[{\left\{\frac{\mathrm{\Phi }(A_i(\xi ))}{\eta }\right\}}^{D_i}{\left\{\frac{1-\mathrm{\Phi }(A_i(\xi ))}{1-\eta }\right\}}^{1-D_i}\mathrm{d}F(x_i,y_i)],\end{array}$$

where $A_i(\xi )=A(x_i,y_i,\xi )$. The corresponding log-likelihood is

$$\begin{array}{rl}\tilde{\ell }(\xi ,\eta ,F)& =\sum _{i=1}^n[D_i\log \{\mathrm{\Phi }(A_i(\xi ))\}+(1-D_i)\log \{1-\mathrm{\Phi }(A_i(\xi ))\}]\\ & +\sum _{i=1}^n\log \{dF(x_i,y_i)\}-n_1\log \eta -n_0\log (1-\eta ).\end{array}$$

We adopt Owen's [8,9] empirical likelihood to handle the non-parametric $F(x,y)$. In the principle of empirical likelihood, $F(x,y)$ is modeled using a discrete distribution, which assigns a probability $p_i$ to $(x_i,y_i)$. Reasonable constraints on $p_i$'s include $p_i\ge 0$, $\sum _{i=1}^n{p}_i=1$, and $\sum _{i=1}^n{p}_i\{\mathrm{\Phi }(A_i(\xi ))-\eta \}=0$, where the last constraint holds because of the definition of η. Given ξ and η, the log-likelihood $\tilde{\ell }(\xi ,\eta ,F)$ has its maximum at

$${\hat{p}}_i=\frac{1}{n[1+\lambda \{\mathrm{\Phi }(A_i(\xi ))-\eta \}]},$$

where $\lambda =\lambda (\xi ,\eta )$ is the solution to $\sum _{i=1}^n\{\mathrm{\Phi }(A_i(\xi ))-\eta \}/[1+\lambda \{\mathrm{\Phi }(A_i(\xi ))-\eta \}]=0$. This leads to the profile semi-parametric log-likelihood (up to a constant that does not depend on ξ or η).

$$\begin{array}{rl}\ell (\xi ,\eta )& =\sum _{i=1}^n[D_i\log \{\mathrm{\Phi }(A_i(\xi ))\}+(1-D_i)\log \{1-\mathrm{\Phi }(A_i(\xi ))\}]\\ & -\sum _{i=1}^n\log [1+\lambda \{\mathrm{\Phi }(A_i(\xi ))-\eta \}]-n_1\log \eta -n_0\log (1-\eta ).\end{array}$$ (2)

As case–control data do not contain any prevalence information, making inferences based directly on the likelihood in Equation (2) is misleading. To overcome this, we consider two cases:

1. the true value or estimate of η is available, and

2. a known distribution of η is available.

Ideally, the prevalence η is known and we proceed by inserting it into Equation (2). This is often the case for many common diseases such as hypertension. For example, Wang et al. [14] showed that the hypertension prevalence for Chinese male adults, 2012–2015, was 24.5%. Instead, if only a prevalence proportion $\hat{\eta }$ from an independent cohort study is available, the plug-in method is still applicable. If no information about η is available, we propose taking the case–control data as if it were prospective and replacing η by $n_1/n$.

2.1. Case I

When the disease prevalence η is known, $\ell (\xi ,\eta )$ in Equation (2) can be written as

$$\begin{array}{rl}\ell (\xi )& =\sum _{i=1}^n[D_i\log \{\mathrm{\Phi }(A_i(\xi ))\}+(1-D_i)\log \{1-\mathrm{\Phi }(A_i(\xi ))\}]\\ & -\sum _{i=1}^n\log [1+\lambda \{\mathrm{\Phi }(A_i(\xi ))-\eta \}].\end{array}$$

We denote the maximum likelihood estimator as $\hat{\xi }=\arg \underset{\xi }{max}\ell (\xi ),$ and define the empirical log-likelihood ratio test of $(\theta ,\gamma )$ as

$$R(\theta ,\gamma )=2\{\underset{\theta \ge 0,\gamma ,\alpha ,\beta }{sup}\ell (\xi )-\underset{\alpha ,\beta }{sup}\ell (\xi )\}.$$

Theorem 2.1 establishes the limiting distributions of $\hat{\xi }$ and the likelihood ratio $R(\theta ,\gamma )$. For ease of exposition, let ${\xi }_0$ be the true ξ, and $A_0=A(X,Y,{\xi }_0)$. We use ${\mathrm{\nabla }}_{\xi }$ to denote the differentiation operator with respect to ξ, and we use $\varphi (x)$ to denote the probability density function of the standard normal distribution. We define

$$W=V_{11}-V_{12}{V}_{22}^{-1}{V}_{21},$$ (3)

where

$$\begin{array}{rl}{V}_{11}& =\rho (1-\rho )\mathbb{E}\left[\frac{{\varphi }^2(A_0)\cdot ({\mathrm{\nabla }}_{\xi }{A}_0{)}^{\otimes 2}}{\mathrm{\Phi }(A_{i0})\{1-\mathrm{\Phi }(A_0)\}\{(\rho -\eta )\mathrm{\Phi }(A_0)+(1-\rho )\eta \}}\right],\\ V_{12}& =V_{21}^{\mathrm{\top }}=\eta (1-\eta )\mathbb{E}\left\{\frac{\varphi (A_0)\cdot ({\mathrm{\nabla }}_{\xi }{A}_0)}{(\rho -\eta )\mathrm{\Phi }(A_0)+(1-\rho )\eta }\right\},\\ V_{22}& =-\eta (1-\eta )\mathbb{E}\left[\frac{\{\mathrm{\Phi }(A_0)-\eta {\}}^2}{(\rho -\eta )\mathrm{\Phi }(A_0)+(1-\rho )\eta }\right],\end{array}$$

and $U^{\otimes 2}=U{U}^{\mathrm{\top }}$ for a vector or matrix U.

Theorem 2.1

Suppose there exists a constant $\rho \in (0,1)$ such that $n_1/n=\rho +o(1)$ as n goes to infinity. Assume that $\mathrm{pr}(D=1|x,y)=\mathrm{\Phi }(A(\xi ))\in (0,1)$ and the disease prevalence $\eta \in (0,1)$ is known. Let ${\xi }_0$ denote the truth value of ξ. If W as defined in Equation (3) is nonsingular, then as n goes to infinity, (a) $n^{1/2}(\hat{\xi }-{\xi }_0)\stackrel{d}{\to }N(0,W^{-1})$, where ${\theta }_0>0$ is the true value of θ and $\stackrel{d}{\to }$ stands for convergence in distribution, and (b) $R(0,0)\stackrel{d}{\to }0.5{\chi }_1^2+0.5{\chi }_2^2$ under the null hypothesis $H_0:\gamma =0\mathrm{\&}\theta =0$.

If only a prevalence proportion $\hat{\eta }$ instead of the true prevalence η is available from an external cohort study, the large-sample properties of the likelihood ratio become more complicated in general. Nevertheless, the results from Theorem 2.1 still hold if n is negligible compared with the size of the cohort data. If no information about the prevalence is available, we propose replacing η with $n_1/n$, which is the proportion of cases in the data. There are no current theoretical results for the case when $n_1/n$ does not converge to η. Our simulation results indicate that with this choice of η, the resulting likelihood ratio test generally has a controlled type I error and desirable power, although it may have potential power loss. Result (b) from Theorem 2.1 indicates that an approximate critical value c of the likelihood ratio test $R(0,0)$ for $H_0:\gamma =0\mathrm{\&}\theta =0$ at the significance level $\alpha \in (0,0.5)$ is the solution to

$$\{P({\chi }_1^2\le c)+P({\chi }_2^2\le c)\}/2=1-\alpha ,$$ (4)

where ${\chi }_k^2$ denotes a random variable following the ${\chi }_k^2$ distribution.

2.2. Case II

In the case of unknown prevalence, we consider an alternative way to incorporate the prevalence proportion $\hat{\eta }$ based on an external cohort data of size m. The available information indicates that there are $m\hat{\eta }$ diseases and $m(1-\hat{\eta })$ non-diseases in the cohort. That is, the cohort has a likelihood contribution of

$${\eta }^{m\hat{\eta }}(1-\eta {)}^{m(1-\hat{\eta })}.$$

Taking this into account, we define an augmented log-likelihood as

$$\begin{array}{rl}{\ell }_A(\xi ,\eta )& =\sum _{i=1}^n\{D_i\log \{\mathrm{\Phi }(A_i(\xi ))\}+(1-D_i)\log \{1-\mathrm{\Phi }(A_i(\xi ))\}\\ & -\log [1+\lambda \{\mathrm{\Phi }(A_i(\xi ))-\eta \}]\}-n_1\log \eta -n_0\log (1-\eta )\\ & +(m\hat{\eta })\log (\eta )+m(1-\hat{\eta })\log (1-\eta ).\end{array}$$

We denote the maximum likelihood estimators as $({\hat{\xi }}^A,{\hat{\eta }}^A)=\arg \underset{(\xi ,\eta )}{max}{\ell }_A(\xi ,\eta ),$ and the empirical log-likelihood ratio functions of $(\xi ,\eta )$ as

$$R^A(\theta ,\gamma )=2\{\underset{\theta \ge 0,\gamma ,\alpha ,\beta ,\eta }{sup}{\ell }_A(\xi ,\eta )-\underset{\alpha ,\beta ,\eta }{sup}{\ell }_A(\xi ,\eta )\}.$$

Before obtaining the asymptotics, we give some notations that are used in Theorem 2.2. Let ${\xi }_0$ and ${\eta }_0$ be the true ξ and η, respectively. We define

$$\begin{array}{rl}{V}_{11}^A& =\rho (1-\rho ){\mathbb{E}}_{(X,Y)}\left[\frac{{\varphi }^2(A_{i0})\cdot ({\mathrm{\nabla }}_{\xi }{A}_{i0}{)}^{\otimes 2}}{\mathrm{\Phi }(A_{i0})\{1-\mathrm{\Phi }(A_{i0})\}\{(\rho -{\eta }_0)\mathrm{\Phi }(A_{i0})+(1-\rho ){\eta }_0\}}\right],\\ V_{12}^A& =V_{21}^{A\mathrm{\top }}=\frac{(\rho -{\eta }_0{)}^2}{{\eta }_0(1-{\eta }_0)}{\mathbb{E}}_{(X,Y)}\left\{\frac{\varphi (A_{i0})\cdot ({\mathrm{\nabla }}_{\xi }{A}_{i0})}{(\rho -{\eta }_0)\mathrm{\Phi }(A_{i0})+(1-\rho ){\eta }_0}\right\},\\ V_{13}^A& =V_{31}^{A\mathrm{\top }}={\eta }_0(1-{\eta }_0){\mathbb{E}}_{(X,Y)}\left\{\frac{\varphi (A_{i0})\cdot ({\mathrm{\nabla }}_{\xi }{A}_{i0})}{(\rho -{\eta }_0)\mathrm{\Phi }(A_{i0})+(1-\rho ){\eta }_0}\right\},\\ V_{22}^A& =-\frac{(\rho -{\eta }_0{)}^2}{{\eta }_0(1-{\eta }_0)}{\mathbb{E}}_{(X,Y)}\left\{\frac{1}{(\rho -{\eta }_0)\mathrm{\Phi }(A_{i0})+(1-\rho ){\eta }_0}\right\}\\ & -\frac{{\eta }_0^2+\rho (1-2{\eta }_0)}{\{{\eta }_0(1-{\eta }_0){\}}^2}+\frac{{\rho }_{\ast }}{{\eta }_0(1-{\eta }_0)},\\ V_{23}^A& =V_{32}^A=-{\eta }_0(1-{\eta }_0){\mathbb{E}}_{(X,Y)}\left\{\frac{1}{(\rho -{\eta }_0)\mathrm{\Phi }(A_{i0})+(1-\rho ){\eta }_0}\right\},\\ V_{33}^A& =-{\eta }_0(1-{\eta }_0){\mathbb{E}}_{(X,Y)}\left[\frac{\{\mathrm{\Phi }(A_{i0})-{\eta }_0{\}}^2}{(\rho -{\eta }_0)\mathrm{\Phi }(A_{i0})+(1-\rho ){\eta }_0}\right],\end{array}$$

where ${\rho }_{\ast }=m/n+o(1)$. Let

$$W^A=\left(\begin{array}{cc}{V}_{11}^A-V_{13}^A(V_{33}^A{)}^{-1}{V}_{31}^A& V_{12}^A-V_{13}^A(V_{33}^A{)}^{-1}{V}_{32}^A\\ V_{21}^A-V_{23}^A(V_{33}^A{)}^{-1}{V}_{31}^A& V_{22}^A-V_{23}^A(V_{33}^A{)}^{-1}{V}_{32}^A\end{array}\right).$$ (5)

Theorem 2.2

Assume that $n_1/n=\rho +o(1)$ for $\rho \in (0,1)$ and $\mathrm{pr}(D=1|x,y)=\mathrm{\Phi }(A(\xi ))\in (0,1)$. Suppose that a prevalence proportion $\hat{\eta }$ based on an external cohort data of size m is available, and that $m/n={\rho }_{\ast }+o(1)$ for a constant ${\rho }_{\ast }>0$. Let $({\xi }_0^{\mathrm{\top }},{\eta }_0{)}^{\mathrm{\top }}$ be the true value of $({\xi }^{\mathrm{\top }},\eta {)}^{\mathrm{\top }}$. If $W^A$ as defined in Equation (5) is nonsingular, then as n goes to infinity, (a) $n^{1/2}(({\hat{\xi }}^A{)}^{\mathrm{\top }}-{\xi }_0^{\mathrm{\top }},{\hat{\eta }}^A-{\eta }_0{)}^{\mathrm{\top }}\stackrel{d}{\to }N(0,(W^A{)}^{-1})$, where ${\theta }_0>0$ is the true value of θ, and (b) $R^A(0,0)\stackrel{d}{\to }0.5{\chi }_1^2+0.5{\chi }_2^2$, under the null hypothesis $H_0:\gamma =0\mathrm{\&}\theta =0$.

According to Theorem 2.2, when a known distribution of η is available, we recommend testing the non-existence of the genetic effect or $H_0:\gamma =0\mathrm{\&}\theta =0$ using the new likelihood ratio test $R^A(0,0)$ and determine its critical value by solving Equation (4).

3. Simulations

In this section, we report on numerical simulations to study the finite-sample performance of the proposed likelihood ratio test (LRT) for the non-existence of the genetic effect. Specifically, we consider four LRTs: (1) LRT-1, $R(0,0)$ with known disease prevalence η; (2) LRT-2, $R(0,0)$ with η replaced by an estimate $\hat{\eta }$ from an external cohort study; (3) LRT-3, $R(0,0)$ with η replaced by $n_1/(n_0+n_1)$; and (4) LRT-4, $R^A(0,0)$ with a prevalence estimate $\hat{\eta }$ from cohort data of size $m=n/2$. We compare their performances with the following popular tests from the literature: the burden test (Burden) calculated with the R package SKAT, the sequence kernel association test (Wu et al. [15], SKAT), the optimal test in an extended family of SKAT tests (Lee et al. [4], SKAT-O), the mixed effects score test (Sun et al. [13], MiST), and three score tests (SS-MAX, $\mathrm{SS}({\alpha }_p)$ and $\mathrm{SS}(\hat{\alpha })$) under a logistic mixed-effects model from Liu et al. [6].

We simulate case–control data with an equal number of cases ( $n_0=n_1=500$) and controls from a Probit mixed-effects model using

$${\mathrm{\Phi }}^{-1}\{\mathrm{pr}(D=1|x,y,v)\}=\alpha +x^{\mathrm{\top }}\beta +y^{\mathrm{\top }}{\mathit{\gamma }}_0\cdot \gamma +y^{\mathrm{\top }}v\cdot \sqrt{\theta }$$ (6)

or a logistic mixed-effects model using

$$\mathrm{Logit}\{\mathrm{pr}(D=1|x,y,v)\}=\alpha +x^{\mathrm{\top }}\beta +y^{\mathrm{\top }}{\mathit{\gamma }}_0\cdot \gamma +y^{\mathrm{\top }}v\cdot \sqrt{\theta },$$ (7)

where $\mathrm{Logit}(t)$ is the inverse function of $f(t)=e^t/(1+e^t)$. The elements of the random-effect vector v are generated independently from $N(0,2)$. The covariate is $x=(x_1,x_2{)}^{\mathrm{\top }}$, where $x_1$ and $x_2$ are independently generated from a Bernoulli distribution with success probabilities of 0.5 and $N(1,1)$, respectively. The genotype values were simulated under the Hardy–Weinberg and linkage equilibriums. The vector y of genetic variables consists of the frequencies of minor alleles at q loci. We set the MAFs to be $j/(4q+1)$ $(j=1,\dots ,q)$ with q = 10, $\alpha =-1$, and $\beta =(-0.2,0.5{)}^{\mathrm{\top }}$. Examples 3.1–3.3 give three groups of settings for the remaining parameters together with the prevalences. All reported numbers except the prevalences in this section were calculated based on 1000 replicates.

Example 3.1

We consider the following scenarios: (C1) $\gamma =0$ and $\theta =0.4\times k$ with ${\mathit{\gamma }}_0={\mathbb{1}}_q$ under Model (6); (C2) $\gamma =-0.1$ and $\theta =0.3\times k$ with ${\mathit{\gamma }}_0={\mathbb{1}}_q$ under Model (6); (C3) $\gamma =0.02$ and $\theta =0.3\times k$ with ${\mathit{\gamma }}_0={\mathbb{1}}_q$ under Model (6); (C4) $\gamma =-0.1$ and $\theta =0.3\times k$ with ${\mathit{\gamma }}_0=(-{\mathbb{1}}_{q/2},{\mathbb{1}}_{q/2})$ under Model (6). Scenarios (D1)–(D4) are the same as Scenarios (C1)–(C4), respectively, except that Model (6) is replaced with Model (7).

Example 3.2

We consider the following scenarios: (E1) $\gamma =0.02\times k$ and $\theta =0.04\times k$ with ${\mathit{\gamma }}_0={\mathbb{1}}_q$ under Model (6); (E2) $\gamma =-0.03\times k$ and $\theta =0.05\times k$ with ${\mathit{\gamma }}_0={\mathbb{1}}_q$ under Model (6); (E3) $\gamma =-0.5\times k$ and $\theta =0.2\times k$ with ${\mathit{\gamma }}_0=(-{\mathbb{1}}_{q/2},{\mathbb{1}}_{q/2})$ under Model (6); (F1) $\gamma =0.04\times k$ and $\theta =0.04\times k$ with ${\mathit{\gamma }}_0={\mathbb{1}}_q$ under Model (7); (F2) $\gamma =-0.05\times k$ and $\theta =0.05\times k$ with ${\mathit{\gamma }}_0={\mathbb{1}}_q$ under Model (7); (F3) $\gamma =-0.2\times k$ and $\theta =0.2\times k$ with ${\mathit{\gamma }}_0=(-{\mathbb{1}}_{q/2},{\mathbb{1}}_{q/2})$ under Model (7).

Example 3.3

We consider the following scenarios: (G1) $\gamma =0.02\times k$ and $\theta =0$ with ${\mathit{\gamma }}_0={\mathbb{1}}_q$ under Model (6); (G2) $\gamma =0.02\times k$ and $\theta =0.05$ with ${\mathit{\gamma }}_0={\mathbb{1}}_q$ under Model (6); (G3) $\gamma =0.04\times k$ and $\theta =0.01$ with ${\mathit{\gamma }}_0=(-{\mathbb{1}}_{q/2},{\mathbb{1}}_{q/2})$ under Model (6); (H1) $\gamma =0.04\times k$ and $\theta =0$ with ${\mathit{\gamma }}_0={\mathbb{1}}_q$ under Model (7); (H2) $\gamma =0.04\times k$ and $\theta =0.05$ with ${\mathit{\gamma }}_0={\mathbb{1}}_q$ under Model (7); (H3) $\gamma =0.06\times k$ and $\theta =0.01$ with ${\mathit{\gamma }}_0=(-{\mathbb{1}}_{q/2},{\mathbb{1}}_{q/2})$ under Model (7).

The simulation results under Example 3.1 are reported in Tables 1 and 2. Scenarios (C1)–(C3) from Example 3.1 are designed such that the Probit model in Equation (1), which underlies the proposed tests, holds. In Scenario (C1), there are no fixed effects and only random effect exists. When k = 0, the null hypothesis is true and all 11 tests have good control over their type I errors. When $k\ne 0$, the null hypothesis is violated, and the alternative moves farther from the null as k increases from 1 to 4. We observe that the four proposed LRT tests all outperform the Burden, SKAT, SKAT-O, MiST, and the three score tests by a large margin. For example, the power gain of the LRT tests against score tests from Liu et al. [6] is as large as 20% when k = 4.

Table 1.

Rejection probabilities (%) of the 11 tests compared in Scenarios (C1)–(C4) from Example 3.1.

k	0	1	2	3	4	0	1	2	3	4
	(C1) $\gamma =0$ and $\theta =0.4\times k$					(C2) $\gamma =-0.1$ and $\theta =0.3\times k$
Burden	5.2	10.1	12.6	13.4	11.5	43.9	10.7	4.0	6.7	10.1
SKAT	4.8	5.1	7.1	6.8	6.4	16.5	6.2	4.0	5.8	6.5
SKAT-O	4.9	7.9	10.9	11.6	10.5	37.0	9.2	4.5	7.4	9.2
MiST	4.4	25.7	42.0	37.7	34.1	93.3	17.5	6.9	14.1	21.9
SS-MAX	4.0	50.9	66.7	62.9	59.5	93.2	35.4	37.7	42.9	46.2
$\mathrm{SS}({\alpha }_p)$	4.1	54.7	71.2	66.9	61.3	94.5	37.5	40.1	47.0	47.4
$\mathrm{SS}(\hat{\alpha })$	3.7	53.3	73.0	70.8	68.3	94.6	35.2	37.8	47.7	49.8
LRT-1	4.6	59.6	82.5	87.1	89.3	94.8	41.1	52.0	69.3	77.7
LRT-2	4.5	58.6	82.1	87.1	89.5	94.8	40.8	51.8	69.6	77.4
LRT-3	4.5	56.7	81.3	85.6	88.2	94.1	36.0	47.1	64.8	72.8
LRT-4	4.5	58.7	82.1	87.2	89.6	94.8	40.9	51.9	69.5	77.4
Prevalence	29.6	34.0	38.6	41.3	42.9	22.1	26.0	31.5	35.6	38.1
	(C3) $\gamma =0.02$ and $\theta =0.3\times k$					(C4) $\gamma =-0.1$ and $\theta =0.3\times k$
Burden	6.2	10.9	14.2	13.1	11.9	24.2	33.2	34.5	26.6	24.2
SKAT	5.0	5.2	6.7	7.1	6.7	13.3	15.0	12.7	10.6	10.2
SKAT-O	5.7	9.0	11.0	11.7	10.5	19.2	28.3	27.6	21.3	19.6
MiST	9.3	31.9	47.6	47.7	39.2	66.5	25.9	33.5	29.3	33.8
SS-MAX	12.7	49.9	72.3	71.5	61.1	20.9	16.4	51.3	55.4	51.7
$\mathrm{SS}({\alpha }_p)$	13.9	52.2	77.0	74.6	63.6	25.3	19.1	56.5	60.2	54.9
$\mathrm{SS}(\hat{\alpha })$	13.8	51.2	75.3	75.7	69.6	25.2	18.9	59.4	64.7	61.5
LRT-1	13.7	56.0	81.3	86.3	86.3	23.7	18.5	66.4	77.6	80.8
LRT-2	13.5	56.2	81.5	86.3	85.7	23.9	19.1	66.1	77.5	81.0
LRT-3	13.5	54.6	79.0	84.7	84.7	26.1	21.2	66.1	77.3	79.4
LRT-4	13.6	56.2	81.6	86.2	85.8	23.9	19.1	66.2	77.7	81.1
Prevalence	31.3	34.0	37.6	40.2	41.9	26.2	29.6	34.3	37.6	39.9

Table 2.

Rejection probabilities (%) of the 11 tests compared in Scenarios (D1)–(D4) from Example 3.1.

k	0	1	2	3	4	0	1	2	3	4
	(D1) $\gamma =0$ and $\theta =0.4\times k$					(D2) $\gamma =-0.1$ and $\theta =0.3\times k$
Burden	5.4	6.2	6.1	5.1	6.7	20.6	8.5	6.2	4.9	5.7
SKAT	3.7	4.4	4.9	4.8	5.5	7.8	6.2	5.1	5.6	6.1
SKAT-O	5.2	6.3	5.3	5.0	6.4	16.0	7.7	5.3	5.8	6.4
MiST	5.7	8.4	13.8	15.2	14.3	53.5	10.2	6.9	6.9	7.8
SS-MAX	5.3	12.4	23.0	27.7	22.5	53.7	16.5	17.1	15.2	15.7
$\mathrm{SS}({\alpha }_p)$	6.1	15.3	24.2	29.2	24.0	56.4	17.6	19.0	15.0	15.3
$\mathrm{SS}(\hat{\alpha })$	5.9	14.7	25.6	32.8	32.8	57.6	17.0	17.8	17.5	17.9
LRT-1	6.2	13.9	29.5	44.5	45.6	57.0	18.1	21.5	24.4	32.1
LRT-2	6.3	13.4	29.6	44.6	45.5	57.1	18.0	21.6	24.3	31.8
LRT-3	6.0	14.4	28.3	42.1	45.2	57.7	16.5	19.8	23.8	29.8
LRT-4	6.3	13.5	29.7	44.7	45.7	57.1	18.1	21.7	24.4	32.0
Prevalence	36.1	38.3	40.8	42.5	43.6	30.6	33.4	37.1	39.8	41.3
	(D3) $\gamma =0.02$ and $\theta =0.3\times k$					(D4) $\gamma =-0.1$ and $\theta =0.3\times k$
Burden	4.8	7.6	10.2	7.1	8.3	12.0	13.9	14.0	14.7	15.6
SKAT	4.8	5.5	5.9	4.6	5.5	7.8	7.5	7.4	6.8	9.3
SKAT-O	5.2	6.6	8.3	6.1	7.3	10.0	11.3	12.3	13.6	14.7
MiST	5.1	17.4	20.4	18.1	19.2	20.6	13.7	10.6	15.0	15.9
SS-MAX	6.7	30.9	29.7	27.7	25.8	10.7	8.7	15.6	17.8	26.2
$\mathrm{SS}({\alpha }_p)$	6.3	33.6	33.2	29.2	26.5	12.3	9.4	17.5	19.4	27.6
$\mathrm{SS}(\hat{\alpha })$	5.8	36.1	37.8	37.9	33.8	12.8	10.1	18.3	22.3	31.6
LRT-1	6.9	38.7	52.4	55.6	60.9	12.5	11.3	19.2	28.0	36.1
LRT-2	7.0	38.9	51.9	54.4	60.9	12.4	11.5	19.3	28.5	35.7
LRT-3	6.6	37.5	51.0	54.4	60.2	13.1	12.9	20.0	28.6	36.1
LRT-4	7.0	39.0	52.2	54.6	61.4	12.4	11.7	19.5	28.5	36.1
Prevalence	37.3	41.4	44.3	45.5	46.4	33.7	35.1	37.5	40.0	41.6

The LRT-1, LRT-2, and LRT-4 have nearly the same powers and are slightly more powerful than the LRT-3 in all cases. This indicates that the performances of the proposed LRT tests seem insensitive to the value of η. The power loss of LRT-3 is likely due to the true prevalence η range between 22% and 42%, while it is replaced by $\rho =1/2$ in LRT-3. In Scenarios (C2) and (C3), a non-zero fixed-effect always exists ( $\gamma \ne 0$); therefore, the null hypothesis is always violated, regardless of k. Thus, the proposed LRT tests are again more powerful than the other seven tests. In particular, the power gain of the LRT tests against the score tests from Liu et al. [6]'s are as large as 30% when k = 4.

Scenarios (C4) and (D1)–(D4) are models for the proposed LRT tests in different cases. In Scenarios (C4) and (D1)–(D3), either the Probit model or the project direction of fixed effects is incorrect, while in Scenarios (D4), both are misspecified. In Scenario (D1) with k = 0, as both the random and fixed effects are gone, the gene information does not impact the disease status. The results in Table 2 correspond to Scenario (D1) with $k=0$ and are type I errors. All tests, including the LRT tests, have acceptable type I errors, although Logistic models are misspecified as Probit models. Except for this case, all cases in Scenarios (C4) and (D1)–(D4) have comparable LRT tests (when k = 1) or much more powerful ( $k\ge 2$) than all score tests. It is noted that all tests except the proposed LRT tests were developed under logistic models, which are correct in Scenarios (D1)–(D4). We believe that this advantage of LRT tests is likely due to the use of the likelihood ratio rather than the score.

In Example 3.1, only the random effect θ is allowed to vary. Examples 3.2 and 3.3 allow both random and fixed effects but only the fixed effects vary. The corresponding simulation results are reported in Tables 3 and 4, respectively. Our general observation is that the LRT tests all have close performances to those of the score tests from Liu et al. [6]. In certain scenarios, such as (F3) and (H3), the MiST is the most powerful. However, its performance is unstable and is much less powerful in Scenarios (C1)–(C4). Overall, the proposed LRT tests have remarkable power gains against existing score tests when Probit models are correct or moderately misspecified. At the same time, regardless of whether the prevalence is known or estimated, the LRT test has nearly the same type I error and power; thus, its performance is not sensitive to the prevalence.

Table 3.

Rejection probabilities (%) of the 11 tests compared in Scenarios (E1)–(E3) from Example 3.2 and Scenarios (F1)–(F3) from Example 3.2.

k	0	1	2	3	4	0	1	2	3	4
	(E1) $\gamma =0.02\times k$ and $\theta =0.04\times k$					(F1) $\gamma =0.04\times k$ and $\theta =0.04\times k$
Burden	4.40	8.00	9.30	19.30	28.30	5.00	8.50	13.10	22.40	39.40
SKAT	5.30	5.20	6.20	9.40	11.50	5.20	5.90	5.60	7.70	13.90
SKAT-O	3.90	7.00	8.30	16.10	24.10	4.70	6.80	10.50	17.90	33.10
MiST	4.10	10.20	26.60	54.10	76.50	5.10	11.90	35.00	70.90	89.70
SS-MAX	4.90	11.40	30.30	57.10	82.00	4.80	11.60	35.20	72.80	90.60
$\mathrm{SS}({\alpha }_p)$	4.80	11.50	32.20	62.00	83.70	5.30	12.00	39.90	75.70	92.00
$\mathrm{SS}(\hat{\alpha })$	4.90	11.60	32.30	61.50	83.70	5.40	12.30	40.20	76.30	91.80
LRT-1	4.90	10.80	31.50	60.80	83.90	4.90	12.50	40.00	75.90	91.70
LRT-2	4.90	10.90	31.30	60.80	84.10	4.90	12.30	40.20	76.30	91.70
LRT-3	4.80	10.60	31.60	61.50	83.40	4.80	13.10	40.00	76.30	91.70
LRT-4	4.90	10.90	31.40	60.70	84.00	4.90	12.30	40.20	76.40	91.70
Prevalence	29.6	31.36	33.24	35.20	37.21	36.1	38.64	41.03	43.57	46.11
	(E2) $\gamma =-0.03\times k$ and $\theta =0.05\times k$					(F2) $\gamma =-0.05\times k$ and $\theta =0.05\times k$
Burden	4.50	6.50	14.70	22.40	31.30	4.50	7.50	17.30	35.40	47.00
SKAT	5.80	4.60	6.70	10.60	12.90	4.50	4.80	7.10	11.60	16.20
SKAT-O	4.30	5.30	12.10	19.10	24.80	4.90	6.80	13.20	29.30	40.00
MiST	5.90	16.60	37.30	56.20	71.40	5.80	16.10	45.40	79.80	91.50
SS-MAX	5.50	15.00	40.40	63.10	79.70	3.70	16.40	46.60	81.80	93.40
$\mathrm{SS}({\alpha }_p)$	6.00	17.30	41.40	65.00	81.40	3.80	17.90	48.90	83.20	94.90
$\mathrm{SS}(\hat{\alpha })$	5.70	17.00	39.40	64.50	80.60	4.70	17.80	48.70	83.90	94.70
LRT-1	6.30	17.20	42.00	64.80	80.60	5.30	18.20	50.20	83.70	94.80
LRT-2	6.00	17.30	42.90	65.10	80.90	5.10	18.20	50.20	83.80	94.80
LRT-3	5.50	17.30	40.00	63.70	79.90	5.50	18.00	49.00	84.10	94.50
LRT-4	6.00	17.40	42.90	65.10	81.00	5.10	18.20	50.20	83.80	94.80
Prevalence	29.5	27.35	25.48	23.96	22.86	36.1	33.28	30.75	28.63	26.71
	(E3) $\gamma =-0.5\times k$ and $\theta =0.2\times k$					(F3) $\gamma =-0.2\times k$ and $\theta =0.2\times k$
Burden	4.80	6.90	10.80	13.10	9.50	6.10	38.10	83.50	96.20	98.90
SKAT	5.10	5.00	6.70	7.30	5.20	4.60	16.70	60.30	84.80	93.80
SKAT-O	5.50	6.00	8.10	11.30	7.80	4.40	31.20	79.40	94.00	97.90
MiST	5.90	8.60	26.10	38.80	39.70	3.80	71.50	99.80	100.0	100.0
SS-MAX	5.90	16.90	52.80	66.30	69.30	5.30	16.70	29.80	33.10	31.60
$\mathrm{SS}({\alpha }_p)$	5.70	18.10	56.10	71.40	73.10	5.20	19.40	32.40	34.20	31.20
$\mathrm{SS}(\hat{\alpha })$	6.00	16.90	55.00	70.00	74.40	5.60	22.90	39.20	43.00	45.20
LRT-1	6.20	17.90	59.20	78.30	85.60	6.40	19.60	34.40	37.00	34.70
LRT-2	6.00	18.90	59.20	78.60	85.60	6.00	19.70	34.80	36.50	35.80
LRT-3	5.60	15.60	57.20	76.00	83.20	6.10	22.70	41.70	47.10	49.60
LRT-4	6.00	18.90	59.30	78.60	85.60	6.00	19.90	35.10	36.70	35.80
Prevalence	29.5	31.19	34.12	36.73	38.63	36.0	32.13	30.25	29.56	29.36

Table 4.

Rejection probabilities (%) of the 11 tests compared in Scenarios (G1)–(G3) from Example 3.3 and Scenarios (H1)–(H3) from Example 3.3.

k	0	1	2	3	4	0	1	2	3	4
	(G1) $\gamma =0.02\times k$ and $\theta =0$					(H1) $\gamma =0.04\times k$ and $\theta =0$
Burden	6.3	5.1	10.3	15.6	28.5	4.1	7.9	16.5	24.3	36.7
SKAT	6.5	5.8	4.9	7.1	10.3	5.0	7.2	7.3	8.5	12.4
SKAT-O	7.0	5.0	8.3	13.4	22.6	4.2	7.9	13.0	19.8	30.3
MiST	5.2	9.2	23.7	48.4	74.1	4.9	11.2	33.8	64.5	91.6
SS-MAX	5.1	10.3	26.8	49.0	75.4	4.1	13.0	35.1	69.0	92.7
$\mathrm{SS}({\alpha }_p)$	5.4	10.3	27.2	52.3	78.6	4.3	12.4	38.3	72.2	94.2
$\mathrm{SS}(\hat{\alpha })$	4.9	10.0	26.9	51.9	78.8	4.7	12.3	39.2	72.2	94.4
LRT-1	5.2	9.3	27.0	51.2	78.7	4.7	12.5	39.3	71.4	94.0
LRT-2	5.2	9.2	27.2	50.9	78.7	4.8	12.6	39.3	71.3	94.0
LRT-3	5.4	10.0	27.5	51.7	78.5	5.4	12.8	39.1	72.1	94.2
LRT-4	5.2	9.2	27.2	50.9	78.7	4.8	12.6	39.3	71.4	94.0
Prevalence	29.6	31.2	32.9	34.8	36.6	36.2	38.5	41.0	43.3	46.0
	(G2) $\gamma =0.02\times k$ and $\theta =0.05$					(H2) $\gamma =0.04\times k$ and $\theta =0.05$
Burden	4.6	6.9	10.2	18.1	25.5	5.3	9.3	13.3	21.7	42.9
SKAT	5.4	5.0	5.0	8.0	10.0	4.7	5.1	7.5	9.3	12.9
SKAT-O	5.1	7.0	9.4	14.2	21.2	4.9	7.7	11.8	19.3	35.4
MiST	4.6	8.2	26.7	53.2	73.8	4.2	12.2	38.4	65.8	92.0
SS-MAX	5.6	9.4	28.0	53.6	75.2	5.4	13.3	39.4	67.3	92.3
$\mathrm{SS}({\alpha }_p)$	6.0	9.3	29.7	58.8	78.2	6.2	14.6	42.3	69.8	93.7
$\mathrm{SS}(\hat{\alpha })$	6.1	9.2	29.3	59.0	78.0	5.9	14.4	42.7	69.8	93.6
LRT-1	5.4	9.5	30.1	56.9	77.3	6.5	14.7	41.4	69.4	93.7
LRT-2	5.3	9.6	30.1	56.8	77.4	6.4	14.6	41.3	69.2	93.6
LRT-3	6.0	9.2	29.5	57.7	77.7	6.4	14.6	41.5	69.2	93.5
LRT-4	5.3	9.6	30.1	56.8	77.4	6.5	14.6	41.2	69.2	93.6
Prevalence	29.7	31.4	33.1	34.8	36.7	36.2	38.6	41.0	43.6	46.0
	(G3) $\gamma =0.04\times k$ and $\theta =0.01$					(H3) $\gamma =0.06\times k$ and $\theta =0.01$
Burden	4.5	7.7	18.0	36.0	51.9	5.7	8.0	20.3	31.8	48.1
SKAT	4.1	5.6	10.4	18.5	26.7	5.2	6.2	10.9	14.0	25.7
SKAT-O	4.2	6.1	14.7	30.7	43.7	5.4	7.2	17.3	25.6	41.0
MiST	5.1	12.2	37.8	80.0	97.2	5.9	8.5	33.4	75.9	95.9
SS-MAX	6.1	10.3	23.6	44.2	67.5	4.9	7.5	19.0	39.6	60.1
$\mathrm{SS}({\alpha }_p)$	5.8	8.7	19.0	38.4	59.1	5.7	5.9	16.1	35.7	56.2
$\mathrm{SS}(\hat{\alpha })$	5.8	8.9	18.8	37.7	58.9	5.6	5.4	15.3	33.5	54.0
LRT-1	6.8	9.6	20.3	40.3	60.5	6.5	6.2	16.1	35.3	55.8
LRT-2	7.0	9.5	20.3	40.5	60.8	6.3	6.1	16.2	34.6	55.5
LRT-3	6.8	9.3	19.9	38.1	58.0	6.4	5.8	15.3	33.6	54.2
LRT-4	7.0	9.5	20.3	40.5	60.8	6.2	6.1	16.2	34.7	55.6
Prevalence	29.7	31.2	32.7	34.4	36.1	36.2	37.7	39.5	41.2	42.9

4. Specific application

This section analyzes a malaria data set from Kenya from the Malaria Genomic Epidemiology Network (MalariaGEN) to illustrate the usefulness of the proposed tests. MalariaGEN investigates how genetic variations affect the biology and epidemiology of malaria and uses this knowledge to develop new tools to control the disease. Nearly 80% of global deaths caused by malaria occurred in Africa in 2015. Kenya is one of the primary countries where malaria occurs, making malaria a major health problem in the country. The data set includes the gender, ethnic group, and genetic information of 3595 individuals (including 1859 cases and 1736 controls) from Kenya. The sampled individuals can be divided into two gender groups (including 1813 males and 1782 females) and four ethnic groups (Chonyi, 1124 individuals; Giriama, 1853 individuals; Kauma, 339 individuals; Other, 279 individuals). In addition, a cross-sectional country representative survey from the 2015 Kenya Malaria Indicator Survey (KMIS), indicates that the prevalence of malaria in Kenya is approximately 8%.

Ndila et al. [7] investigated 121 polymorphisms in 70 candidate severe malaria-associated genes. They found significant associations between risks for severe malaria overall and polymorphisms in 15 genes or positions, most of which were related to red blood cells: ABO, ATP2B4, ARL14, CD40LG, FREM3, INPP4B, G6PD, HBA (both HBA1 and HBA2), HBB, IL10, LPHN2 (also known as ADGRL2), LOC727982, RPS6KL1, CAND1, and GNAS. As noted by Kariuki and Williams [2], malaria-protective variants exist only in the ABO, ATP2B4, G6PD, HBA1, and HBB genes.

We take the corresponding dummy variables of the gender and ethnic groups as covariates and test the association between the above 15 genes and the status of severe malaria. Table 5 presents the number of single nucleotide polymorphisms (SNPs) along with the p-values for the Burden, SKAT, SKAT-O, MiST, SS-MAX, SS( $\hat{\alpha }$), and LRT-2 for each gene. At the 5% significance, our LRT-2 test identifies seven genes (ABO, ATP2B4, INPP4B, G6PD, HBA1, HBB and GNAS) that include all five genes with malaria-protective variants. In contrast, the remaining six tests fail to identify at least one of the five genes, although they may identify more significant genes. For example, the p-values corresponding to the ABO gene are all much greater than 5%, while that for the LRT-1 is 1.82%, which is much smaller than 5%. This indicates that to some extent, the proposed LRT test has priority against existing competitors .

Table 5.

P-values of Burden, SKAT, SKAT-O, MiST, SS-MAX, $\mathrm{SS}(\hat{\alpha })$, and LRT-2 when applied to malaria data.

Gene name	No. of SNPs	Burden	SKAT	SKAT-O	MiST	SS-MAX	$\mathrm{SS}(\hat{\alpha })$	LRT-2
ABO	20	0.2248	0.2636	0.2530	0.4129	0.3263	0.3273	0.0182
ATP2B4	117	0.1386	0.0237	0.0344	0.0004	$4.4\times {10}^{-5}$	0.0014	0.0008
ARL14	3	0.9092	0.0287	0.0433	0.1174	0.9760	0.9671	0.9592
CD40LG	3	0.0863	0.0415	0.0470	0.3229	0.2398	0.4752	0.4190
FREM3	39	0.8405	0.0690	0.0966	0.2289	0.6379	0.6578	0.3410
INPP4B	230	0.8089	0.0010	0.0012	$1.0\times {10}^{-7}$	0.7101	0.9497	0.0031
G6PD	6	0.3679	0.0041	0.0046	$1.0\times {10}^{-7}$	0.0095	0.0586	$2.0\times {10}^{-6}$
HBA1	4	0.4385	0.0052	0.0054	0	0.0050	0.0475	0
HBB	6	0.2234	0.0073	0.0090	0.0024	0.0159	0.0147	0.0048
IL10	7	0.3150	0.0587	0.0782	0.0875	0.5920	0.5127	0.5003
CAND1	36	0.2959	0.0207	0.0248	0.1664	0.5556	0.4770	0.4865
ADGRL2	119	0.4465	0.0016	0.0018	0.2581	0.5458	0.4638	0.2188
GNAS	30	0.6446	$6.5\times {10}^{-5}$	0.0001	$6.2\times {10}^{-6}$	0.1322	0.6320	0.0009
RPS6KL1	3	0.2622	0.1989	0.2392	0.217	0.1536	0.1532	0.1541
LOC727982	8	0.2439	0.0800	0.1081	0.061	0.5009	0.4972	0.5315

5. Discussion

Under mixed-effects models based on case–control data, score tests are the most popular for genetic association studies and likelihood ratio tests have rarely been used. The main reason is that the latter involves formidable high-dimensional integration while the former successfully avoid such numerical challenges. Fortunately, under Probit mixed-effects models with normal random effects, the likelihood has a closed-form and automatically avoids high-dimensional integration. This makes it feasible to test the gene-disease association using the likelihood ratio test, which depends on the true prevalence. We systematically investigate the large-sample properties of the likelihood ratio test for genetic association with the disease based on the availability of the prevalence information.

In our test development, we fix the direction of the fixed effect to a pre-specified vector ${\mathit{\gamma }}_0$, which may be artificial or subjective. To reduce the risk of misspecifying ${\mathit{\gamma }}_0$, we consider a few candidate directions and take the maximum of the corresponding LRT statistic as the final test statistic. However, the null limiting distribution of this test will become too complicated for general use.

Funding Statement

Dr Liu's research was supported by the National Natural Science Foundation of China [grant numbers 11771144, 71931004, 32030063], and the 111 project [grant number B14019]. Dr Li's research was supported by Natural Sciences and Engineering Research Council of Canada [grant number RGPIN-2020-04964].

Disclosure statement

No potential conflict of interest was reported by the author(s).