1 Introduction

Multiple testing problem occurs when a set of simultaneous statistical tests are considered. In many situations, more than one or even a large number of hypotheses are simultaneously tested, which is referred to as multiple comparisons [12]. For example, in case of high-dimensional data obtained from the field of genetics, medicine, molecular biology, bioinformatics, agricultural science etc. [13]. In statistical hypothesis testing, a pre-defined confidence level generally applies only to each individual test. But, multiple testing considers the same confidence level for the whole family of simultaneous tests. The probability of committing false statistical inferences considerably increase when more than one hypothesis is simultaneously tested at a given confidence level [4, 18]. In general, where all the null hypotheses (\(H_{0}\)) are independent and also supposed to be true, the statistical inference of committing at least one false rejection will become inevitable even if 100 hypotheses are individually tested at confidence level \(\alpha = 0.0{5}.\)Estimation of this error rate is more complex if the hypotheses are correlated and not all of them are true [4]. Hence, the unadjusted or marginal p-values are no longer a useful quantity for this inference process as we are testing many features at the same time.

Therefore, the correction of p-value is obvious for multiple testing to control the type I error [1]. Many correction methods are proposed. However, these corrections are not always easy and straightforward in terms of the selection of proper method suitable for the various experimental properties and study purposes [4]. For example, the genome-wide association studies (GWAS) consider simultaneous testing of individual single-nucleotide polymorphism (SNP) of a gene and picks up positive genes when any one of SNPs in the gene is associated with a trait [3]. The contingency table tests are generally used to analyze the dichotomous case–control traits, where the deviation is measured from independence under \(H_{0}\) of no association between the phenotype and SNP genotype classes. One of the most celebrated forms of this test is the popular chi-square (\(\chi^{2}\)) test [3]. Selection of the largest test statistic (MAX-test) from different genetic models is a powerful approach as it provides safeguard against model uncertainty [7].

Due to rapid advances in genotyping technology and standardized guidelines for reporting statistical evidence, the multitude of comparisons made in a GWAS may result false positives (type I errors). When testing millions of SNPs in a single gene association study will become the standard, consideration of multiple comparisons is an essential part of determining statistical significance [13]. Moreover, correction of multiple testing p-values is also required for the SNPs in linkage disequilibrium (LD) with each other [19]. The closely spaced SNPs frequently yield high correlation because of extensive LD. Therefore, when association studies are conducted with many SNPs, the tests performed on each SNP are usually not independent, depending on the correlation structure among the SNPs. This violation of the independence assumption limits the Šidák and Bonferroni correction’s ability to control the type I error effectively [6]. The permutation test can be applied [11, 20]. But, an enormous number of permutations are required to accurately estimate small p-values, constituting a computational burden [14].

We combined the geometry of multi-way contingency table [21] and the χ2-test [9, 22] to develop a linear algebraic transformation called spherization. Geometrically, the table and tests are defined by their contour hyperplanes in a df-dimensional space. The spherization is used to convert the shape of the space, defined by the contour hyperplanes of the distribution of tables sharing the same marginal counts. Hence, the stochastic distributions of tables are transformed into a standard multivariate normal distribution in df-dimensional space that is able to address tables with smaller p-values because of their symmetricity with respect to distance and direction. If MAX-test is performed for a set of \(H_{0}\), the tables with the same test statistic values are located in the df-dimensional space in the shape of a convex polytope, in which the corrected p-values correspond to the sum of the probability out of the convex polytope of the truncated normal distribution (TND). The TND approach by Botev [2] was applied to estimate the multiple testing corrected p-values.

The method was applied to a real GWAS data and the additive model tests of individual SNPs were repeated for a SNP set. Therefore, the main aim of this paper is to propose a multiple testing correction method for the contingency table tests as well as to study its features using a real data set.

2 Methods

2.1 Spherization

This method combines the idea of algebraic geometry of a contingency table [21] and χ2-statistic [9, 22] to develop the MAX-test from the spherical geometry.

2.1.1 Geometry of Multi-way Contingency Tables

Let Y be a m-dimensional multi-way contingency table with non-negative elements \(Y_{i}\), where \(i \equiv \left( {i_{1} ,{ }i_{2} , \ldots ,i_{m} } \right)\), \(1 \le i_{j} \le I_{j}\) and \(I_{j}\) is the number of categories of j-th dimension. To consider the geometry of contingency tables, we define the vectorized form of \(Y\) as \(y\) by defining the index. The index of i-th element is

$$ind\left( i \right) \equiv 1 + \mathop \sum \limits_{j = 1}^{m} \left( {i_{j} - 1} \right)\mathop \prod \limits_{k = 1}^{j - 1} I_{k} .$$
(1)

Geometrically, contingency tables can be deployed in \(M\)-dimensional space with \(M \equiv \mathop \prod \limits_{j} I_{j}\). The tables can be located in the same space if the observation structures are the same.

2.1.2 Geometry of the χ2-Test

2.1.2.1 Dimension Reduction of Contingency Tables

For the χ2-test, the tables sharing same marginal counts can be located in the same \({\text{df}} \equiv \mathop \prod \limits_{j} (I_{j} - 1){ }\) dimensional space, which is smaller than \(M\)-dimensional space as the tables are restricted by the marginal counts. The dimension reduction is performed by shifting in parallel and rotating them. First, the center of the distribution of tables is moved to the expected table and the differential vector is defined as

$$d \equiv o - e,$$
(2)

where \(o\) and \(e\) are the observed and expected frequency vectors for arbitrary tables in \(M\)-dimensional space, respectively. Next, a rotation matrix \(R\) is defined depending on \(I_{j}\):

$$R_{M \times M} \equiv \otimes \prod\limits_{j = 1}^{m} {R\left( {I_{j} } \right)}.$$
(3)

Equation (3) defines the Kronecker product of \(m\) rotation matrices of \(m\)-dimensional multi-way contingency table. Each of \(m\) matrices is a rotation matrix in an \(n\)-dimensional space, of which \(n\) is the number of categories in \(I_{j}\) for the \(j\)-th dimension. The structure for one variable is

$$R\left( n \right) \equiv \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {r_{1}^{T} } \\ {r_{2}^{T} } \\ \vdots \\ \end{array} } \\ {r_{n}^{T} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {r_{1,1} } & {r_{1,2} } & \cdots \\ \vdots & \vdots & \ddots \\ {r_{n,1} } & {r_{n,2} } & \ldots \\ \end{array} \begin{array}{*{20}c} { r_{1,n} } \\ \vdots \\ { r_{n,n} } \\ \end{array} } \right),$$
(4)

where \(\left\{ {\begin{array}{*{20}c} {\forall i < n} & {r_{i} = \left( {r_{i,j} } \right) = \left\{ {\begin{array}{*{20}l} 0 \hfill & {j < i} \hfill \\ {\sqrt {\frac{n - i}{{n - i + 1}}} } \hfill & {j = i} \hfill \\ { - \sqrt {\frac{1}{{\left( {n - i} \right)\left( {n - i + 1} \right)}}} } \hfill & {j > i} \hfill \\ \end{array} } \right.} \\ {i = n} & {r_{n} = \left( {r_{n,j} } \right) = \left( {\begin{array}{*{20}c} {\frac{1}{\sqrt n },} & {\frac{1}{\sqrt n },} \\ \end{array} \begin{array}{*{20}c} \cdots & {,\frac{1}{\sqrt n }} \\ \end{array} } \right)} \\ \end{array} } \right.\).

Rotation of the Eq. (2) by (3) is the rotation of a simplex in the \(I_{j}\) dimension so that the vertices of it are placed in a hyperplane, \(\left[ {Rd} \right]_{ji} = \frac{1}{\sqrt n }\), where \(Rd\) is a rotated differential vector. This rotation produces many zero elements (\(\left| {Rd} \right|_{0}\)) that are common for all the tables. Specifically, \(\left| {Rd} \right|_{0} \le {\text{df}}\), where \(\left| {{ } \cdot { }} \right|_{0}\) is the number of non-zero elements. The positions of \(\left| {{ } \cdot { }} \right|_{0}\) for tables sharing same marginal counts are consistent with each other and can be known from \(I_{j}\) s. Then, an index vector \(v\) is defined to specify the positions of \(\left| {Rd} \right|_{0}\) and \(\left| {{ } \cdot { }} \right|_{0}\) elements:

$$v \equiv \otimes \prod\limits_{j = 1}^{m} {u\left( {I_{j} } \right)},$$
(5)

where \(u\) is an \(n\)-dimensional vector, the last element of which is 0, and the others are 1. A matrix \(V\) is defined from \(v\), which removes zero elements from \(Rd\) by specifying the columns for which \({ }\left[ v \right]_{i} = 0\). For example, for a \(2 \times 3\) table,

$$v = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ {\begin{array}{*{20}c} 1 \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \\ 0 \\ \end{array} } \right)\;\;{\text{and}}\;V = \left( {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} { }\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right)$$

Multiplying \(Rd\) by \(V\), the dimension for the vectors of tables are reduced without loss of information.

2.1.2.2 Spherization of the χ2-Statistic

Using vectorized form of \(Y\), the χ2-statistic in \({\text{df}}\)-dimensional space is:

$$\chi_{{{\text{df}}}}^{2} = d^{T} E^{ - 1} d,$$
(6)

where \(E\) is a diagonal matrix having \(e\) in the main diagonals. For dimension reduction, the Eq. (6) can be written as

$$\chi_{{{\text{df}}}}^{2} = d^{T} R^{T} RE^{ - 1} R^{T}Rd ,$$
(7)
$$\chi_{{{\text{df}}}}^{2} = d^{T} R^{T} V^{T} V{ }R{ }E^{ - 1} R^{T} V^{T} VRd,$$
(8)

where \(VRd\) is a \({\text{df}}\)-dimensional vector, and \(V{ }R{ }E^{ - 1} R^{T} V^{T}\) is a \(\left( {{\text{df}} \times {\text{df}}} \right)\)-dimensional symmetric matrix. Using LU decomposition, Eq. (8) can be transformed into

$$\chi_{{{\text{df}}}}^{2} = d^{T} R^{T} V^{T} X^{T} XVR{ }d,$$
(9)
$$\chi_{{{\text{df}}}}^{2} = g^{T} g,$$
(10)

where \(X\) is an LU decomposed matrix of \(V{ }R{ }E^{ - 1} R^{T} V^{T}\) and \(g = XVR{ }d\). Hence, Eq. (10) is a squared Euclidean distance from the center to the rotated hyperplane in \({\text{df}}\)-dimensional space, and \(g\) is assumed to be a sample from \({\text{df}}\)-dimensional multivariate normal distribution.

2.1.2.3 Geometry of the Proportion Trend Test

The geometry of the χ2-statistic can be applied to deploy the proportion trend test in df-dimensional space. Let us consider a 2 × 3 contingency table for the description. To define the proportion trend test, a weight vector has to be defined. For example, (2 1 0) is the weight vector, where the weights are defined for each column. For generalization, the weights are defined for each cell of the contingency table as, \(\left( {\begin{array}{*{20}c} 2 & 1 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right)\). For the single weight vector \(w\) or one test, the test statistic \(\chi_{w}\) is the function of a weighted sum of the differential table:

$$\chi_{w} = f\left( {w^{T} d} \right).$$
(11)

To deploy Eq. (11) in \({\text{df}}\)-dimensional space, we use the deformation of the definition of \(g\) in Eq. (10):

$$g \equiv {\mathcal{P}}d,$$
(12)
$$g = XVRd,$$
(13)
$$VRd = X^{ - 1} g,$$
(14)
$$V^{T} VRd = V^{T} X^{ - 1} g,$$
(15)
$$Rd = V^{T} X^{ - 1} g,$$
(16)
$$d = R^{T} V^{T} X^{ - 1} g,$$
(17)
$$d = {\mathcal{P}}^{\dag } g,$$
(18)

where \({\mathcal{P}}^{\dag } = R^{T} V^{T} X^{ - 1}\) is a sparse version of the pseudo-inverse of \({\mathcal{P}}\). The deformation from Eq. (15) to (16) only holds for cases, in which the dimension reduction to \({\text{df}}\)-dimensional space of spherization is applicable. Then, from Eq. (11):

$$\chi_{w} = { }f\left( {w^{T} {\mathcal{P}}^{\dag } g} \right).$$
(19)

The test vector defined by the weight vector and rotation is:

$$\tau^{T} = \frac{{w^{T} {\mathcal{P}}^{\dag } }}{{\left| {w^{T} {\mathcal{P}}^{\dag } } \right|}}.$$
(20)

Therefore, the proportion trend test is projected in \({\text{df}}\)-dimensional space with a direction of the inner relation of the test as:

$$\tau^{T} g = \left| g \right|cos\theta$$
(21)

Equation (21) defines the inner product, which is the relation between observations and weight, where \(\theta\) is the angle between observation and weight. If the observation is in the direction of \(\theta = 0\) so that \(\left( {\tau^{T} g} \right)^{2} = \left| g \right|^{2} = \chi_{{{\text{df}}}}^{2}\), then the χ2-statistic for a vectorized weight is

$$\chi_{w}^{2} = \left| {\tau^{T} g} \right|^{2},$$
(22)

where \(\chi_{w}^{2}\) follows a \(\chi_{1}^{2}\)-distribution, and the p-value is the proportion of tables having χ2-values greater than or equal to the observations per the number of all possible tables.

For one observation and one weight, one test vector is deployed into \({\text{df}}\)-dimensional space. For one observation and multiple weights, the multiple test vectors are deployed into the same \({\text{df}}\)-dimensional space as in the case of the MAX-test. Pairs of truncating hyperplanes can be drawn in \({\text{df}}\)-dimensional space depending on a set of test vectors and \(\chi_{w}\). The hyperplanes are defined as vertical to the test vector, and the difference from the origin to the hyperplane is \(\chi_{w}\). Because, the distribution of tables is transformed into a standard multivariate normal distribution in spherized space, the p-value is the cumulative standard normal probability out of the convex polytope, truncated by the pair of hyperplanes.

3 Geometric Features of Convex Polytopes in Spherized Space

The geometric configuration of the convex polytope defines the multiple testing p-values. Figure 1a shows the case of \({\text{df}} = 2\) with \({\text{df}} = 1\) test for a single test. Tables with the same χ2-values are drawn with a high-dimensional ellipsoid contour for the \({\text{df}} = 2\) test and a pair of truncating parallel hyperplanes for the \({\text{df}} = 1\) test. The ellipsoid contour is not easy to handle from the distributional standpoint, since all the tables on the ellipsoid are not equidistant from the origin (Fig. 1a). These parallel hyperplanes can be transformed into tangent hyperplanes and a high-dimensional ellipse into an \(n\)-sphere (generalization of the ordinary sphere into spaces of \({\text{df}}\)-dimension) by spherization, in which all the tables are placed on the transformed df-dimensional n-sphere. The χ2-value is calculated as the squared Euclidean distance from the expected table to the hyperplane with distance \(\chi\), and the p-value is the proportion of tables outside of the convex polytope (Fig. 1b). The definition of the hyperplane for a single test is

$$- \chi_{w} \le \tau^{T} Z \le \chi_{w} ,$$
(23)
Fig. 1
figure 1

The spherical geometry of tests in df-dimensional space. a The contour hyperplanes of the distribution of tables with the same χ2-values for a single test, where ellipse for the \({\text{df}} = 2\) test and a pair of parallel hyperplanes for the \({\text{df}} = 1\) test, respectively. b, c Spherization in df-dimensional space. b The single test: The transformation of parallel hyperplanes into tangent hyperplanes and ellipse into sphere (circle). The black line crossing the center indicates the direction of the test. The χ2-value is the squared Euclidean distance from the center to the hyperplane having distance \(\chi\). c Multiple testing: Each pair of parallel hyperplanes of the same color presents a contour hyperplane of the tables having the same χ2-values from the individual \({\text{df}} = 1\) test. Multiple pairs of these parallel hyperplanes form a convex polytope (light-yellow-colored area). The spherical space defined by the convex polytope produces the TND. The p-value is the integral of the probability defined by the TND out of the polytope (Color figure online)

Full size image

where \(Z\) is the axis of \({\text{df}}\)-dimensional space.

In multiple testing, multiple pairs of parallel hyperplanes form a convex polytope (Fig. 1c, light-yellow-colored area), in which each pair corresponds to the tables with same χ2-values from the individual \({\text{df}} = 1\) test. The definition of the convex polytope in Eq. (23) is generalized for \(n_{test}\) as:

$$- \chi_{\max } 1_{{n_{{{\text{test}}}} }} \le AZ \le \chi_{\max } 1_{{n_{{{\text{test}}}} }},$$
(24)

where \(\chi_{\max } \equiv \mathop {\max }\limits_{k} \chi_{{w_{k} }}\); \(k = 1,2, \ldots ,n_{{{\text{test}}}}\), \(A\) is a \((n_{{{\text{test}}}} \times {\text{df}})\) matrix with rows that are test vectors, \(\chi_{\max }\) is the largest value of the test statistics among \(n_{{{\text{test}}}}\), and \(1_{{n_{{{\text{test}}}} }}\) is a (\(n_{{{\text{test}}}} \times 1\)) vector, all of the elements of which are 1.

If the maximum value among a set of test statistic values is representative in cases of multiple testing, the tables on the surface of the convex polytope have the same representative χ2-values. Hence, the probability of observing tables with representative χ2-values greater than or equal to the observed table’s χ2-value is the integral of the probability density of a multivariate normal distribution over the space out of the convex polytope. Therefore, the p-value for \(n_{{{\text{test}}}}\) is

$$p = 1 - \int {N\left( {Z|0, I} \right) \delta \left( { - \chi_{\max } 1_{{n_{{{\text{test}}}} }} \le AZ \le \chi_{\max } 1_{{n_{{{\text{test}}}} }} } \right){\text{d}}Z},$$
(25)

where \(\delta\) is the delta function.

Tables in spherized space are considered to be observed in standard normal distributions as having a center corresponding to the expected table in the context of testing \(H_{0}\). Because of the relationship between \(n\)-dimensional normal distributions and \(\chi^{2}\)-distributions, we define \(\chi\) as the distance from the center to the point corresponding to the observed table, and also obtain the spherized space (Fig. 1b, circle) with a multivariate normal distribution demarcated by a convex polytope, producing TND (Fig. 1c).

4 The Spherization and Botev’s (Sph–Btv) Approach

The probability of a df-dimensional vector Z falls outside of a convex polytope can be defined from the linear restriction of d-dimensional multivariate normal law [2] as:

$$1 - \Pr \left( {l \le AZ \le u} \right) = 1 - \int\limits_{l}^{u} {\phi \left( {Z;0, I_{d} } \right){\text{d}}Z},$$
(26)

where \(l\) and \(u\) are the lower and upper truncation limits, respectively; \(A\) is a full rank matrix; \(Z\) ~ \(N\left( {0, I_{d} } \right)\) having a d-dimensional multivariate normal distribution with mean vector 0 and variance–covariance matrix \(I_{d}\); and \(\phi \left( {Z;0, I_{d} } \right)\) is the probability density function of \(N\left( {0,{ }I_{d} } \right)\).

Since, A is a full rank matrix, and \(Z\) ~ \(N\left( {\mu = 0, \Sigma_{Z} = I_{d} } \right)\), we can simply estimate Eq. (26) as [2]:

$$1 - \Pr \left( {l - A\mu \le A{\mathbb{Z}} \le u - A\mu } \right)$$
(27)
$$= 1 - \Pr \left( {l \le A{\mathbb{Z}} \le u} \right)$$
(28)
$$= 1 - \int\limits_{l}^{u} {\phi \left( {{\mathbb{Z}};0, A \Sigma_{Z} A^{T} } \right) d{\mathbb{Z}}},$$
(29)

where \({\mathbb{Z}}\sim N\left( {0, A \Sigma_{Z} A^{T} } \right)\), \(\Delta_{{n_{{{\text{test}}}} \times n_{{{\text{test}}}} }} = A \Sigma_{Z} A^{T} = A A^{T}\) is an inner product matrix. Here, the p-values were estimated by the Botev’s [2] approach after the transformation of the space with spherization. The definition of the convex polytope for \(n_{test}\) can be defined from the Eq. (28):

$$l \le A{\mathbb{Z}} \le u,$$
(30)

where \(A_{{n_{{{\text{test}}}} \times {\text{df}}}}\) is the \(df\)-dimensional matrix, the rows of which are the test vectors defined by the test models (\(W_{{n_{{{\text{test}}}} \times k}}\)) and rotation matrix (\({\mathcal{R}}_{{k \times {\text{df}}}}\)) of the table; \(k\) is the number of column categories of the table. \({\mathbb{Z}}\sim N\left( {0,{ }\Delta} \right)\) is a \({\text{df}}\)-dimensional multivariate normal vector; and \(l\) and \(u\) are two (\(n_{{{\text{test}}}} \times 1\)) vectors of the intercepts of tangent contour hyperplanes of tests.

Since, our method’s test vector is a unit vector, and we define the distance from the origin to the hyperplane as \(\chi\), where \(\chi\) is the maximum value of the test statistics among \(n_{test}\) from the proportion trend test. Therefore, the elements of vectors \(l\) and \(u\) are \({-}\chi\) and \({ }\chi\), respectively. The p-value for the Sph-Btv approach is the probability that the \({\text{df}}\)-dimensional vector \({\mathbb{Z}}\) falls outside of a convex polytope defined by the linear inequalities in Eq. (30):

$$p = 1 - \Pr \left( {l_{\chi } \le A{\mathbb{Z}} \le u_{\chi } } \right) = 1 - \int\limits_{{l_{\chi } }}^{{u_{\chi } }} {\phi \left( {{\mathbb{Z}};0, \Delta } \right){\text{d}}{\mathbb{Z}}},$$
(31)

where \(l_{\chi } = - \chi_{max} 1_{{n_{test} }}\) and \(u_{\chi } = \chi_{max} 1_{{n_{test} }}\) are the lower and upper truncation limits, respectively.

The \(\Delta\) in Eq. (31) is a positive definite variance–covariance matrix. The upper or lower triangle of \(\Delta\) consists of \(q\left( { = n_{{{\text{test}}}} \left( {n_{{{\text{test}}}} - 1} \right)/2} \right)\) elements, each of which is the pairwise inner product of two test vectors. These q-elements determine the size of the convex polytope, and they consist of full information of the probability density function of the MAX-test.

5 Application to the Real Data

5.1 Genotype Data

The genotype data from the Nagahama Study [10, 23] of 1813 samples having 996,339 SNPs were considered. The phenotypes were randomly generated using equal probabilities for cases (= 907) and controls (= 906). The real phenotype data were not used in this study because we were interested to extract the features of our method with a GWAS data considering the character of its LD structures per gene, where the case–control-wise null hypotheses throughout. The SNP subsets (\(N\) subsets) located at each of 14,941 gene loci were considered to calculate the gene-based multiple testing corrected p-values.

Multiple cutoff χ2-values were used by selecting various cutoff \(\chi\)-values from (0.01, 8.5) by increasing by 0.1 along with the GWAS cutoff of 5.45 (\(\chi^{2} = 5.45^{2}\), and \(p = 5 \times 10^{ - 08}\)). For each subset, the calculation was repeated 1,000 times, and 1,000 Monte Carlo samples were used for each repetition. More detailed data description is provided as Online Resource in the Supplementary File, ESM_1.pdf.

5.2 Results

The estimated p-values for the two genes with their individual LD structures are presented in Fig. 2. For example, the gene FES in Fig. 2a has a relatively small number of SNPs with quite strong pairwise LD structures, compared to the gene NCS1 in Fig. 2b. These dissimilar physical features of the two genes lead them to produce quite heterogeneous values of \(p\), and strongly motivate the values to move either toward or away from the \({\text{df}} = 1\) testing. The p-values are highly deviated from the \({\text{df}} = 1\) test for the gene NCS1, whereas the gene FES, is closer to the values of \({\text{df}} = 1\) (Fig. 2c). This variation in p-values is an important indicator about the variation in type I error rate per gene at a uniform confidence level, which may affect the false detection of a gene. For example, if we set the GWAS cutoff of \(p = 5 \times 10^{ - 08}\)\(\left( {\chi^{2} = 5.45^{2} } \right){ }\) for all the χ2-values along the horizontal axis of Fig. 2c, the type I error rate for a gene becomes significantly greater as we move from the left-hand corner to the right-hand corner along the horizontal line. Moreover, SNPs per gene and genome-wide gene’s LD structures are not uniform. The genes with more SNPs and weak LD structures tend to produce higher p-values than genes with relatively smaller numbers of SNPs and stronger LD (Fig. 2). So, if we set the uniform cutoff to calculate the type I error rate for all the genes over the genome, then it is very likely that the type I error rate of the gene in Fig. 2a will be higher than that of the gene in Fig. 2b.

Fig. 2
figure 2

The effect of gene-based heterogeneity in p-values for multiple cutoffs. The two LD triangles present two heterogeneous gene structures in terms of LD and SNPs per gene. The strength of LD is shown by colors with greater strength indicated by blacker color. a The LD triangle of the FES gene has 4 SNPs and the average pairwise \(r^{2} = 0.8\). b The LD triangle of the NCS1 gene has 50 SNPs and the average pairwise \(r^{2} = 0.098\). c The plot of the \(log_{10} \left( p \right)\) of two genes FES and NCS1, along with the \({\text{df}} = 1\) test for multiple χ2-values (Color figure online)

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Figure 3a is presenting the results from the evaluation of all the gene-based SNP subsets (\(N\) subsets) for multiple cutoff χ2-values, including the GWAS cutoff. The eight colored subpanels in this figure depict the distributions of \(log_{10} \left( p \right)\)-values for eight different cutoffs. From here, it was observable that there was a shift of p-values for every change in \(\chi^{2}\). The shape of the distributions and their modality at the left tails for each subpanel are changing along with the changes in cutoffs. For example, the shape of the distribution for \(\chi^{2} = 5.45^{2}\) (magenta) is very dissimilar from that of \(\chi^{2} = 8.31^{2}\) (purple). The evenly spaced distributions were observed along the horizontal log scale, indicating that the distributions are shifting with respect to \(p\), keeping the width of the spaces almost the same. Figure 3a demonstrates how the distribution function shapes vary with multiple cutoff χ2-values and the changes in the shape is not simple. Therefore, it is advisable to apply our method for individual gene sets by considering the corresponding LD structures and for target χ2-values individually.

Fig. 3
figure 3

The overall heterogeneity of p-values for multiple cutoffs. a The distributions of \(log_{10} \left( p \right)\) for multiple cutoffs, each different colored subpanel of which present the distribution of every different \(\chi^{2}\)-cutoff. b The distributions of \(log_{10} \left( p \right)\)-values for two cutoff χ2-values, \(2.51^{2}\) and \(3.91^{2}\), respectively. The horizontal and vertical axes of (a) and (b) present \(log_{10} \left( p \right)\) and corresponding density values, respectively. The scatter plots of \(log_{10} \left( p \right)\) for \(\chi^{2}\)-cutoffs, c\(2.51^{2}\) and \(3.91^{2}\). d\(2.51^{2}\) and \(7.51^{2}\) (Color figure online)

Full size image

To visualize the variations in p-values more obviously, only the two cutoff χ2-values of \(2.51^{2}\) and \(3.91^{2}\) were considered. The two distributions of \(log_{10} \left( p \right)\)-values of SNP subsets are presented in Fig. 3b. The shapes of the distributions are almost identical, but a modality change was observed in the left tail. The two evenly spaced distributions in the horizontal axis with an almost total shift in p-values indicate the fold changes in \(p\) with a change in the cutoff. For a typical investigation of heterogeneity in p-values, we considered the evaluation of three cutoff χ2-values: \(2.51^{2}\), \(3.91^{2}\) and \(7.51^{2}\), respectively. The scatter plot between \(log_{10} \left( p \right)\)-values for \(\chi^{2} = 2.51^{2}\) with the other two cutoffs is presented in Fig. 3c and d, respectively. From both figures, it was observed that the p-values are positively correlated having quite different strength and patterns for a change in the cutoffs. The values in Fig. 3c are linearly related to a quite strong, positive correlation \(\left( {r^{2} = 0.98} \right)\), whereas the strength of the relation is quite weak \(\left( {r^{2} = 0.07} \right)\) in Fig. 3d.

These results show that the p-values are heterogeneous for the gene-based SNP subsets over the genome for multiple cutoff χ2-values. Also, they are not uniform for a fixed cutoff. For example, the shape of each colored subpanel of Fig. 3a is far from the shape of uniform distribution, indicating that some genes tend to produce type I error more frequently than others based on their individual characteristics, such as LD pattern and number of SNPs per gene.

The p-value correction by our method for gene-subsets is reasonably simple and straightforward. But, the biological interpretation is not so simple, because the number of truly associated genetic variants in one particular gene is not always one, and its association structure between multiple genetic variants and phenotypes is believed complicated [15]. In this study, the features of the proposed method were extracted only for the gene-based SNP subsets of a real GWAS data. This is the limitation of our study to apply these gene specific extracted features for the other genetic studies as the gene-coding regions comprise a small proportion of the human genome.

6 Discussion

This is a geometric approach of multiple testing for contingency tables such as MAX-test of multiple genetic models (additive, recessive, dominant). It seems one of the realistic approaches to know the variations of the type I error among subsets, each of which is considered of multiple tests but their dependency is heterogeneous. Our results suggested that the type I error may vary based on the individual structure of genes. Therefore, our proposed method offers the estimates of the probability density of MAX-test p-values. Because, there can be the functional relation among variants in each gene, it is advisable not to correct simple correction based on our method estimate without considering other potential factors. However, still our method will give meaningful information for GWAS interpretation.

We illustrated an example from the genetics using a real GWAS data for the SNP subsets per gene. However, because of the big data era, the simple positive signal detection by multiple testing seems to exist in various fields. This method can be applicable to other arbitrary fields, as far as the positive signal detection is similar. For example, for large categorical data as well as non-categorical data set, where multiple \({\text{df}} = 1\) tests are repeated according to the procedure of the MAX-based test.

Our spherization transformation is based on the algebraic geometry of correspondence analysis (CA) [5, 8] with the squared Euclidean distance as a measure of \(\chi^{2}\) in \({\text{df}}\)-dimensional space. This \(\chi^{2}\)-distance is a standardized form of the Mahalanobis distance [17]. A detailed illustration of the relationship among the CA, Mahalanobis distance and Euclidean distance.is provided as Online Resource in the Supplementary File, ESM_2.pdf.

The sum-of-chi-squares (S-O-C-S) from multiple \(df = 1\) tests can also be used as the gene-wise test statistic for calculating multiple testing p-value (Supplementary File, ESM_3.pdf) [15, 16] instead of the maximum-of-chi-squares (M-O-C-S). The geometric features of multiple testing can also be evaluated by this S-O-C-S approach. However, this method is not suitable for our Sph–Btv approach because the Botev method [2] considers the computational problems that are based on the d-dimensional multivariate normal law under linear restrictions. Also, this method is applicable when the number of linear inequalities is less than the dimension of the space. But, the S-O-C-S approach generates the non-linear contour hyperplanes for the tests that are not linear (Supplementary File, ESM_4.eps). In ESM_4.eps, Fig. (a) is presenting the contour hyperplanes of the tables for a \(2 \times 3\) table test from the S-O-C-S approach, where all possible tables sharing the same marginal counts with the observed table are evaluated using three genetic models (additive, dominant and recessive). From this figure, it was observed that the each contour hyperplane is consisted of a linear summation of χ2-values, which means that the contour hyperplane contains many straight segments. Since, there are many segments, overall they look like “curved”. Also, in ESM_4.eps, Fig. (b) is showing the non-linearity among the relations of the two co-ordinate values of the contour hyperplanes and the S-O-C-S values. So, this non-linear and too complicated contour hyperplanes are not feasible for Botev’s approach.

There are multiple potential directions to extend our approach. One example is to handle the combinations of variables. In this paper, we demonstrated the multiple tests that were consisted of one particular test for multiple items (SNPs). However, the combinatorial effect of multiple SNPs is also the active research target, and our method can take the combinations along with single SNPs, as far as the combinatorial effect is expressed as linear that corresponds to the hyperplane in our geometric approach.

Another example is to apply our method to estimate type II errors rather than type I errors that we demonstrated in this paper. To estimate type II errors in the context of MAX-test, only one difference is to be introduced to our method. The difference is the parallel shift of normal distribution without moving the convex polytope of MAX-test, to the location in the space where an alternative hypothesis indicates.

In summary, we demonstrated a geometric testing procedure of contingency table in the context of MAX-test that enables to estimate the multiple testing corrected p-value having multiple extendibility.