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Gene Set Analysis (GSA) is a framework for testing the association of a set of genes and the outcome, e.g. disease status or treatment group. The method replies on computing a maxmean statistic and estimating the null distribution of the maxmean statistics via a restandardization procedure. In practice, the pre-determined gene sets have stronger intra-correlation than genes across sets. This may result in biases in the estimated null distribution. We derive an asymptotic null distribution of the maxmean statistics based on sparsity assumption. We propose a flexible two group mixture model for the maxmean statistics. The mixture model allows us to estimate the null parameters empirically via maximum likelihood approach. Our empirical method is compared with the restandardization procedure of GSA in simulations. We show that our method is more accurate in null density estimation when the genes are strongly correlated within gene sets.

A gene pathway commonly refers to a set of genes that share a particular property, carry out a biological function or lead to a certain product in cells/tissues. Performing differential expression (DE) analysis on such gene sets aggregates the signal of individual genes and potentially increases the power of a hypothesis test. Gene set analysis also provides comprehensive understanding of the biological activities associated with the outcome phenotype and may shed light on treatment of disease.

A variety of tools are available for gene set analysis. These methods can be roughly classified into two broad categories, self-contained and competitive [_{S} be the size of S. The maxmean statistic S is defined as,

S + = 1 n S ∑ i ∈ S z i I { z i > 0 } , S − = − 1 n S ∑ i ∈ S z i I { z i < 0 } , S = max ( S + , S − ) , (1.1)

where I { } is the indicator function.

The GSA method estimates the null distribution of S through a restandardization procedure, which is a combination of row (gene) randomization and column (sample label) permutation. It compares the gene set against its permutations and also takes into account the overall distribution of randomly selected null sets. The permutation maintains the correlation structure in the gene set, while row randomization rescales and shifts the permutations to include the competition of genes outside the set.

In practice, the gene sets for analysis are obtained from a public database e.g. Kyoto Encyclopedia of Genes and Genomes (KEGG) [

We propose a two group mixture model for the gene set maxmean statistics. Our model assumes that only a small proportion of the gene sets are truly significantly DE. Based on the mixture model, we apply the maximum likelihood method to estimate the empirical null. The empirical method improves the accuracy of GSA in large scale hypothesis testing. It also reduces the computational burden of the permutation steps in restandardization procedure of GSA. The analysis is demonstrated in simulation studies and a data set from the MSigDB database [

The maxmean statistic in GSA is essentially a maximum statistic of two correlated sample means, S_{+} and S_{−} defined in (1.1), which asymptotically follow bivariate normal distribution for adequately large n_{S},

( S + S − ) ~ N 2 { ( μ + μ − ) , ( σ + 2 ρ σ + σ − ρ σ + σ − σ − 2 ) } , (2.1)

where μ + = E ( S + ) , σ + 2 = Var ( S + ) , μ − = E ( S − ) , σ − 2 = Var ( S − ) , and ρ = corr ( S + , S − ) . Based on the work in [

f 0 ( s ) = 1 σ + ϕ ( − s + μ + σ + ) × Φ ( ρ ( − s + μ + ) σ + 1 − ρ 2 − − s + μ − σ − 1 − ρ 2 ) + 1 σ − ϕ ( − s + μ − σ − ) × Φ ( ρ ( − s + μ − ) σ − 1 − ρ 2 − − s + μ + σ + 1 − ρ 2 ) (2.2)

where ϕ and Φ are the probability density function (pdf) and cumulative density function (cdf) of the standard normal distribution.

We can estimate the parameters in f_{0} by fitting the null gene sets. A special case would be that z i ∼ N ( 0 , σ 2 ) independently. We can easily compute the parameters:

μ + = μ − = 0.40 σ , σ + 2 = σ − 2 = 0.34 σ 2 , ρ = − 0.467.

However, genes in the same set are often correlated. Therefore, the theoretically computed parameters may not match the actual null distribution well. We propose an empirical method to estimate the null distribution of S. Let f be the density function of the maxmean statistics for all the gene sets. Adopting the two group mixture model [_{0}) of the null density f_{0} and a small proportion of the non-null density f_{1},

f ( s ) = p 0 f 0 ( s ) + p 1 f 1 ( s ) . (2.3)

The null density f_{0} is assumed to have the form in (2.2), in which the parameters (µ_{+}, µ_{−}, σ_{+}, σ_{−}, ρ) need to be estimated. For identifiability of f_{0} we further assume the non-null density f 1 ( s ) ≈ 0 ∀ s ∈ A 0 for some interval A_{0}. Under this assumption, S follows a truncated distribution f_{T}(s) for s ∈ A_{0},

f T ( s ) ≈ p 0 f 0 ( s ) ∫ A 0 p 0 f 0 ( s ) = f 0 ( s ) ∫ A 0 f 0 ( s ) . (2.4)

Fitting the maxmean statistics in A_{0} to (2.4) by maximum likelihood yields f ^ 0 . In addition, p_{0} can be estimated by

p ^ 0 = # { s i ∈ A 0 } n ∫ A 0 f ^ 0 ( s ) . (2.5)

In this paper we let A_{0} be the interval (0, q_{S}) where q_{S} is 90% quantile of the maxmean statistics of all gene sets.

We simulate 2000 gene sets, 5% of them are DE sets and the other 95% are null sets. All sets contain n_{S} = 50 genes. Let C_{1} and C_{2} be the sample indices of two conditions and each condition has m = 10 samples. For gene i in sample j, the expression data x_{ij} is generated in a hierarchical fashion,

x 0 k j ~ N ( δ k I { j ∈ C 1 } , τ k 2 ) , x i j ~ N ( α i x 0 k j , σ i 2 ) ∀ i ∈ S k ,

where τ k = σ i = 1 , δ k = 1 for DE sets and δ_{k} = 0 for null nets. x_{0kj} is the expression of the hub gene [_{k} and all genes in the same set are correlated with the hub gene. The DE genes of set S_{k} are jointly controlled by δ_{k} and α_{i}. In particular, δ_{k} controls the differential expression of the hub gene between two conditions. The parameter α_{i} controls the inter-correlation within the set. When α_{i} =0 genes are independent within gene sets. For null sets we let α_{i} = 0 (independent) or ±0.2 (correlated). For DE sets α_{i} = ±1 so that the correlation is stronger than the null sets. _{0} density estimate by our empirical method and the restandardization procedure in GSA.

When genes within the gene sets are independent, the null distribution obtained by the two methods are very close (

We evaluate our empirical method and GSA with the 4722 curated gene sets in MSigDB database using the gender dataset included in the GSEA package [

cedure [

GSA is a representative tool for gene set DE analysis. Compared with the state- of-the-art method GSEA, the maxmean statistic used in GSA is more powerful than the KS statistic in GSEA [

In studying the GSA method, we propose a new method to estimate the null distribution of the gene set maxmean statistics. Unlike the permutation test of GSA in which every gene set is compared against its own permutations, the fundamental difference of our method is that it estimates an overall null distribution f_{0} parametrically and all gene sets are compared against f_{0}. The possibility of parametric estimation of f_{0} is rendered by the large number of gene sets in the hypothesis testing. A similar idea is proposed in [

Our method is based on the sparsity assumption that only a small proportion of the gene sets are truly DE. Further, we adopt a two group parametric mixture distribution to model the gene set maxmean statistics. The parameters of the null distribution is estimated under the sparsity assumption. We show that our new method provides more accurate estimation of the null distribution. It also avoids the computational intensity in the permutation steps of GSA.

In the simulations, we compare the two methods under independence and correlation. When the intra-set correlation is greater than the cross-set correlation, the GSA shows a mismatch to the true null. The reason is that the randomization step samples genes in the entire dataset, which has a different correlation structure from gene sets. As a result the mean and standard deviation obtained in the randomization step is inaccurate.

In application to the gender data set, our method has fewer gene sets with unadjusted p-value < 0.05, but identifies more gene sets after controlling for FDR. A reason is that in GSA, gene-set p-values are limited by the number of permutations. Due to the large number of genes and gene sets, with 10,000 permutations, GSA takes 64 minutes on an Intel i7 3770K 3.5 GHz CPU. In contrast, our empirical method takes less than 1 minute.

An important aspect of our empirical estimation method is specifying an appropriate range for the zero assumption region A_{0}. In this paper we arbitrarily choose A_{0} to be the interval (0, q_{S}), where q_{S} is some quantile of the observed maxmean statistics. Determination of q_{S} is a trade-off between variance and bias. With small q_{S}, the bias of the null estimate is small but the variance is large and vice-versa. How to determine an optimal A_{0} interval requires further exploration.

Ren, X., Wang, J.M., Liu, S. and Miecznikowski, J.C. (2017) Estimating the Empirical Null Distribution of Maxmean Statistics in Gene Set Analysis. Open Journal of Statistics, 7, 761-767. https://doi.org/10.4236/ojs.2017.75053