Description Usage Arguments Details Value References See Also Examples

This function test whether a metabolite-set is differential expressed using a stratified kernel-based score test.

1 | ```
sscore(x, y, lower, upper, m)
``` |

`x` |
numeric measurements of metabolite abundance level. |

`y` |
0/1 response indicating whether a subject is a case group or a control group. |

`lower` |
lower bound of the kernel parameter. |

`upper` |
upper bound of the kernel parameter. |

`m` |
number of grid points selected in the interval [lower, upper]. |

Let x be a *p\times n* matrix, where each column is a subject, y be a *n \times 1* 0/1 vector indicating the group label. This function tests whether this *p*-metabolite set is differentially expressed between two groups (more details can be found in Zhan et al. (2015)). It works in the following way.

A score test can be applied when the kernel parameter *ρ* is known. First, fit the null logistic model *logit(pr(y=1))=β_0* to get estimate of *β_0* as *\hat{β_0}*. Let *\hat{μ_0}=invlogit(\hat{β_0})*. Second, The *n\times n* kernel matrix is calculated as *K(ρ)_{ij} = k(x_i,x_j,ρ)*, where *x_i* is *i*th column in x, *k(\cdot)* is the stratified kernel function skernel. Third, the test statistic *Q(ρ)* is calculated as

*Q(ρ)=(y-\hat{μ_0})^T K(ρ) (y-\hat{μ_0}).*

An standardized version *S(ρ)* of *Q(ρ)* can be calculated as *S(ρ)= [Q(ρ)-μ_{Q}]/σ_{Q}*. More details can be found in Liu et al.(2008).

When the kernel parameter *ρ* is not known. Suppose it takes values in [lower, upper]. Davies (1977) and Davies (1987) proposed a test based on the process *\{S(ρ), ρ \in [lower,upper]\}*. This test has rejection region of the form *\{\sup_{L ≤q ρ ≤q U} S(ρ)> c \}*. Using this test, an upper-bound for the p-value is given by:

*Φ(-M)+V \exp(\frac{1}{2}M^2)/√{8π},*

where *Φ(\cdot)* is the cumulative distribution function of standard normal density, *M* is the maximum of *S(ρ)* over the range of *ρ* and *V=|S(ρ_1)-S(lower)|+|S(ρ_2)-S(ρ_1)|+\cdots+|S(upper)-S(ρ_m)|* is the total variation of *S(ρ)* over the interval [lower, upper] and *ρ_1,…,ρ_m* are *m* grid points in the interval [lower, upper].

A p-value indicating whether the metabolite-set is differentially expressed or not.

Davies, R. B. (1977) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 64,247-254.

Davies, R. B. (1987) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 74,33-43.

Liu, D., Ghosh, D., & Lin, X. (2008). Estimation and testing for the effect of a genetic pathway on a disease outcome using logistic kernel machine regression via logistic mixed models. BMC bioinformatics, 9(1), 292.

Zhan, X., Patterson, A. D., & Ghosh, D. (2015). Kernel approaches for differential expression analysis of mass spectrometry-based metabolomics data. BMC Bioinformatics, 16(1), 77.

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