fmrs.mle-methods: fmrs.mle method
In fmrs: Variable Selection in Finite Mixture of AFT Regression and FMR

Description Usage Arguments Details Value Author(s) References See Also Examples

Provides MLE for Finite Mixture of Accelerated Failure Time Regression Models or Finite Mixture of Regression Models. It also provides Ridge Regression.

fmrs.mle(y, delta, x, nComp, ...)

## S4 method for signature 'ANY'
fmrs.mle(
  y,
  delta,
  x,
  nComp = 2,
  disFamily = "lnorm",
  initCoeff,
  initDispersion,
  initmixProp,
  lambRidge = 0,
  nIterEM = 400,
  nIterNR = 2,
  conveps = 1e-08,
  convepsEM = 1e-08,
  convepsNR = 1e-08,
  porNR = 2,
  activeset
)

`y`	Responses (observations)
`delta`	Censoring indicator vector
`x`	Design matrix (covariates)
`nComp`	Order (Number of components) of mixture model
`...`	Other possible options
`disFamily`	A sub-distribution family. The options are `'norm'` for `FMR` models, `'lnorm'` for mixture of AFT regression models with Log-Normal sub-distributions,`'weibull'` for mixture of AFT regression models with Weibull sub-distributions
`initCoeff`	Vector of initial values for regression coefficients including intercepts
`initDispersion`	Vector of initial values for standard deviations
`initmixProp`	Vector of initial values for proportion of components
`lambRidge`	A positive value for tuning parameter in Ridge Regression or Elastic Net
`nIterEM`	Maximum number of iterations for EM algorithm
`nIterNR`	Maximum number of iterations for Newton-Raphson algorithm
`conveps`	A positive value for avoiding NaN in computing divisions
`convepsEM`	A positive value for threshold of convergence in EM algorithm
`convepsNR`	A positive value for threshold of convergence in Newton-Raphson algorithm
`porNR`	A positive integer for maximum number of searches in NR algorithm
`activeset`	A matrix of zero-one that shows which intercepts and covariates are active in the fitted fmrs model

Finite mixture of AFT regression models are represented as follows. Let X be the survival time with non-negative values, and \boldsymbol{z} =(z_{1}, …, z_{d})^{\top} be a d-dimensional vector of covariates that may have an effect on X. If the survival time is subject to right censoring, then the observed response time is T=\min \{Y, C\}, where Y=\log X, C is logarithm of the censoring time and δ=I_{\{y<c\}} is the censoring indicator. We say that V=(T,δ,\boldsymbol z) follows a finite mixture of AFT regression models of order K if the conditional density of (T,δ) given \boldsymbol z has the form

f(t,δ;\boldsymbol{z},\boldsymbolΨ) =∑\limits_{k=1}^{K}π_{k}[f_Y(t;θ_{k}(\boldsymbol z), σ_{k})]^{δ}[S_Y(t;θ_{k}(\boldsymbol z) ,σ_{k})]^{1-δ}[f_{C}(t)]^{1-δ}[S_{C}(t)]^{δ}

where f_Y(.) and S_Y(.) are respectively the density and survival functions of Y, f_C(.) and S_C(.) are respectively the density and survival functions of C; and {θ}_{k}(\boldsymbol{z})=h(β_{0k}+\boldsymbol{z}^{\top} \boldsymbolβ_{k}) for a known link function h(.), \boldsymbolΨ=(π_{1},…,π_{K},β_{01},…, β_{0K},\boldsymbolβ_{1}, …,\boldsymbolβ_{K},σ_{1}, …,σ_{K})^{\top} with \boldsymbolβ_{k}= (β_{k1},β_{k2},…,β_{kd})^{\top} and 0<π_{k}<1 with ∑_{k=1}^{K}π_{k}=1. The log-likelihood of a sample of size $n$ is formed as

\ell_{n}(\boldsymbolΨ) = ∑\limits_{i=1}^{n}\log∑\limits_{k=1}^{K}π_{k}≤ft[f_Y(t_{i}, θ_{k}({\boldsymbol z}_{i}),σ_{k}) \right]^{δ_{i}} ≤ft[S_Y(t_{i},θ_{k}({\boldsymbol z}_{i}), σ_{k})\right]^{1-δ_{i}}.

Note that we assume the censoring distribution is non-informative and hence won't play any role in the estimation process. We use EM and Newton-Raphson algorithms in our method to find the maximizer of above Log-Likelihood.

An fmrsfit-class that includes parameter estimates of the specified FMRs model

Farhad Shokoohi <shokoohi@icloud.com>

Shokoohi, F., Khalili, A., Asgharian, M. and Lin, S. (2016 submitted) Variable Selection in Mixture of Survival Models for Biomedical Genomic Studies

Other lnorm, norm, weibull: fmrs.gendata(), fmrs.tunsel(), fmrs.varsel()

set.seed(1980)
nComp = 2
nCov = 10
nObs = 500
dispersion = c(1, 1)
mixProp = c(0.4, 0.6)
rho = 0.5
coeff1 = c( 2,  2, -1, -2, 1, 2, 0, 0,  0, 0,  0)
coeff2 = c(-1, -1,  1,  2, 0, 0, 0, 0, -1, 2, -2)
umax = 40

dat <- fmrs.gendata(nObs = nObs, nComp = nComp, nCov = nCov,
 coeff = c(coeff1, coeff2), dispersion = dispersion,
 mixProp = mixProp, rho = rho, umax = umax,
 disFamily = 'lnorm')

res.mle <- fmrs.mle(y = dat$y, x = dat$x, delta = dat$delta,
     nComp = nComp, disFamily = 'lnorm',
     initCoeff = rnorm(nComp*nCov+nComp),
     initDispersion = rep(1, nComp),
     initmixProp = rep(1/nComp, nComp))
summary(res.mle)