eLNNpaired (version 0.2.3)

getPara.orig:

Description

Translate re-parameterized parameters to original scale.

Usage

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 \(c_2=\Phi^{-1}(0.05)\).

Value

A \(10\times 1\) vector of reparameterized parameters: \(\mu_1\), \(k_1\), \(\alpha_1\), \(\beta_1\), \(\alpha_3\), \(\beta_3\),

Details

We assume the following the Bayesian hierarchical models for the 3 clusters of gene probes. For cluster 1 (over-expressed gene probes): $$d_{gl}|\left(\mu_g, \tau_g\right)\sim N\left(\mu_g, \tau_g^{-1}\right),\\ \mu_g | \tau_g \sim N\left(\mu_1, k_1 \tau_g^{-1}\right),\\ \tau_g\sim \Gamma\left(\alpha_1, \beta_1\right). $$ For cluster 2 (under-expressed gene probes): $$d_{gl}|\left(\mu_g, \tau_g\right)\sim N\left(\mu_g, \tau_g^{-1}\right),\\ \mu_g | \tau_g \sim N\left(\mu_2, k_2 \tau_g^{-1}\right),\\ \tau_g\sim \Gamma\left(\alpha_2, \beta_2\right). $$ For cluster 3 (non-differentially-expressed gene probes): $$d_{gl}|\left(\tau_g\right)\sim N\left(0, \tau_g^{-1}\right),\\ \tau_g\sim \Gamma\left(\alpha_3, \beta_3\right). $$ For cluster 1, we add one constraint $$ \alpha_1>1+\beta_1\left( \frac{ c_1-\Phi^{-1}(0.05)\sqrt{k_1}}{\mu_1} \right)^2$$ based on $$ Pr\left(\frac{\mu_g}{\tau_g^{-1}}\leq c_1 | \tau_g^{-1}\right)<0.05, $$ 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 $$ \alpha_2>1+\beta_2\left( \frac{ c_2-\Phi^{-1}(0.95)\sqrt{k_2}}{\mu_2} \right)^2$$ based on $$ Pr\left(\frac{\mu_g}{\tau_g^{-1}} \geq c_2 | \tau_g^{-1}\right)<0.05, $$ 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: $$ \mu_1=\exp(\delta_1), k_1=\Phi(\xi_1), \beta_1=\exp(\nu_1),\\ \alpha_1=\exp(\lambda_1)+1+\beta_1\left( \frac{c_1-\Phi^{-1}(0.05)\sqrt{k_1}}{\mu_1} \right)^2,\\ \mu_2= -\exp(\delta_2), k_2=\Phi(\xi_2), \beta_2=\exp(\nu_2),\\ \alpha_2=\exp(\lambda_2)+1+\beta_2\left( \frac{c_2-\Phi^{-1}(0.95)\sqrt{k_2}}{\mu_2} \right)^2,\\ \beta_3=\exp(\nu_3), \alpha_3=\exp(\lambda_3). $$

References

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

See Also

See Also as getRePara

Examples

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)
)