# getPara.orig: Translate Re-Parameterized Parameters to Original Scale In eLNNpaired: Model-Based Gene Clustering for Genomics Data from Paired/Matched Designs

## Description

Translate re-parameterized parameters to original scale.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 getPara.orig( delta1, xi1, lambda1, nu1, delta2, xi2, lambda2, nu2, lambda3, nu3, c1 = qnorm(0.95), c2 = qnorm(0.05)) 

## Arguments

 delta1 log of the mean of the mean expression levels for gene probes in cluster 1 (over-expressed probes). xi1 the value of the inverse function of the cumulative distribution function of the standard normal distribution at the point that is equal to the scalar in the variance of the mean expression levles for gene probes in cluster 1 (over-expressed probes). lambda1 a parameter related to alpha_1, which is the shape parameter of the distribution of the variance of gene expression levels for gene probes in cluster 1 (over-expressed probes). nu1 log of the rate parameter of the distribution of the variance of gene expression levels for gene probes in cluster 1 (over-expressed probes). delta2 log of the negative mean of the mean expression levels for gene probes in cluster 2 (under-expressed probes). xi2 the value of the inverse function of the cumulative distribution function of the standard normal distribution at the point that is equal to the scalar in the variance of the mean expression levles for gene probes in cluster 2 (under-expressed probes). lambda2 a parameter related to alpha_2, which is the shape parameter of the distribution of the variance of gene expression levels for gene probes in cluster 2 (under-expressed probes). nu2 log of the rate parameter of the distribution of the variance of gene expression levels for gene probes in cluster 2 (under-expressed probes). lambda3 a parameter related to alpha_3, which is the shape parameter of the distribution of the variance of gene expression levels for gene probes in cluster 3 (non-differentially-expressed probes). nu3 log of the rate parameter of the distribution of the variance of gene expression levels for gene probes in cluster 3 (non-differentially-expressed probes). c1 the lower bound for mu_g/sqrt(tau_g^{-1}) for cluster 1 (over-expressed probes). By default c_1=Phi^{-1}(0.95). c2 the upper bound for mu_g/sqrt(tau_g^{-1}) for cluster 2 (under-expressed probes). By default c_2=Phi^{-1}(0.05).

## Details

We assume the following the Bayesian hierarchical models for the 3 clusters of gene probes.

For cluster 1 (over-expressed gene probes):

d_{gl} | (mu_g, tau_g) ~ N(mu_g, tau_g^{-1}),\ mu_g | tau_g ~ N(mu_1, k_1 tau_g^{-1}),\ tau_g ~ Gamma(alpha_1, beta_1).

For cluster 2 (under-expressed gene probes):

d_{gl} | (mu_g, tau_g) ~ N(mu_g, tau_g^{-1}),\ mu_g | tau_g ~ N(mu_2, k_2 tau_g^{-1}),\ tau_g ~ Gamma(alpha_2, beta_2).

For cluster 3 (non-differentially-expressed gene probes):

d_{gl} | (mu_g, tau_g) ~ N(0m, tau_g^{-1}),\ tau_g ~ Gamma(alpha_3, beta_3).

For cluster 1, we add one constraint

alpha_1>1+beta_1( (c_1-Phi^{-1}(0.05)sqrt{k_1})/mu_1 \right)^2

based on

Pr(mu_g/tau_g^{-1} <= c_1 | tau_g^{-1})<0.05,

where c_1=Phi^{-1}(0.05) and Phi is the cumulative distribution function of the standard normal distribution.

For cluster 2, we add one constraint

alpha_2>1+beta_2( (c_2-Phi^{-1}(0.95)sqrt{k_2})/mu_2 \right)^2

based on

Pr(mu_g/tau_g^{-1} >= c_2 | tau_g^{-1})<0.05,

where c_2=Phi^{-1}(0.95) and Phi is the cumulative distribution function of the standard normal distribution.

To do unconstraint numerical optimization, we do parameter reparameterization:

mu_1=exp(delta_1), k_1=Phi(xi_1), beta_1=exp(nv_1),\ alpha_1=exp(lambda_1)+1+beta_1left( frac{c_1-Phi^{-1}(0.05)sqrt{k_1}}{mu_1} right)^2,\ mu_2= -exp(delta_2), k_2=Phi(xi_2), beta_2=exp(nv_2),\ alpha_2=exp(lambda_2)+1+beta_2left( frac{c_2-Phi^{-1}(0.95)sqrt{k_2}}{mu_2} right)^2,\ beta_3=exp(nv_3), alpha_3=exp(lambda_3).

## Value

A 10x1 vector of reparameterized parameters: mu_1, k_1, alpha-1, beta_1, alpha_3, beta_3,

## Author(s)

Yunfeng Li <colinlee1999@gmail.com> and Weiliang Qiu <stwxq@channing.harvard.edu>

## References

Li Y, Morrow J, Raby B, Tantisira K, Weiss ST, Huang W, Qiu W. (2017), <doi:10.1371/journal.pone.0174602>

See Also as getRePara
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 getPara.orig( delta1 = -0.690142787, xi1 = -7.212004793, lambda1 = -13.152520780, nu1 = -2.199687707, delta2 = -0.168584053, xi2 = 0.008683666, lambda2 = -13.582936416, nu2 = -2.671150369, lambda3 = 0.331454152, nu3 = -2.339660241, c1 = qnorm(0.95), c2 = qnorm(0.05) )