The function `fmrs.gendata`

generates a data set from an FMRs model.
It has the form

fmrs.gendata(nObs, nComp, nCov, coeff, dispersion, mixProp, rho, umax, ...)

where `n`

is the sample size, `nComp`

is the order of FMRs model, `nCov`

is
the number of regression covariates, `coeff`

, `dispersion`

and `mixProp`

are
the parameters of regression models, i.e. regression coefficients,
dispersion (standard deviation) of the errors (sub-distributions) and
mixing proportions,
respectively, and `rho`

is the used in the variance-covariance matrix for
simulating the design matrix `x`

, and `umax`

is the upper bound for
Uniform distribution for generating censoring times.

Depending on the choice of `disFamily`

, the function `fmrs.gendata`

generates
a simulated data from FMRs models. For instance, if we choose
`disFamily = "norm"`

, the function ignores the censoring parameter `umax`

and generates a data set from an FMR model with Normal sub-distributions.
However, if we choose `disFamily = "lnorm"`

or `disFamily = "weibull"`

, the
function generates data under a finite mixture of AFT regression model with
Log-Normal or Weibull sub-distributions.

The `fmrs.gendata`

function returns a list which includes a vector of
responses `$y`

, a matrix of covariates `$x`

and a vector of censoring
indicators `$delta`

as well as the name of sub-distribution of the
mixture model.

The function `fmrs.mle`

in fmrs package provides maximum likelihood
estimation for the parameters of an FMRs model. The function has the
following form,

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

where `y`

, `x`

and `delta`

are observations, covariates and censoring
indicators respectively, and `nComp`

is the order of FMRs, `initCoeff`

,
`initDispersion`

and `initmixProp`

are initial values for EM and NR
algorithms, and the rest of arguments of the function are controlling
parameres. The function returns a fitted FMRs model with estimates of
regression parameters, standard deviations and mixing proportions of the
mixture model.
It also returns the log-likelihood and BIC under the estimated model, the
maximum number of iterations used in EM algorithm and the type of the
fitted model.

Note that one can do ridge regression by setting a value for tuning
parameter of the ridge penalty other than zero in the argument `lambRidge`

.

To do the variable selection we provided the function `fmrs.varsel`

with the form

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

where `penFamily`

is the penalty including `"adplasso"`

, `"lasso"`

, `"mcp"`

,
`"scad"`

, `"sica"`

and `"hard"`

, and `lambPen`

is the set of tuning
parameters for components of penalty. We can run the function `fmrslme`

first and use the parameter estimates as initial values for the
function `fmrs.varsel`

.

There are two approaches for specifying tuning parameters: Common and
Component-wise tuning parameters. If we consider choosing common tuning
parameter, we can use the BIC criteria to search through the a set of
candidate values in the interval $(0,\lambda_M)$, where $\lambda_M$ is a
prespecified number. The BIC is provided by the function `fmrs.varsel`

for
each candidate point and we choose the optimal $\hat\lambda$, the one that
maximizes BIC. This approach will be good for the situations with enough
samples sizes. It also reduces the computational burden.

On the other hand, if we consider choosing component-wise tuning parameters we use the following function to search for optimal $(\lambda_1, \ldots, \lambda_K)$ from the set of candidate values in $(0, \lambda_M)$. The function is

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

It is a good practice run the function `fmrs.mle`

first and use the parameter
estimates as initial values in the function `fmrs.tunsel`

.
The function `fmrs.mle`

then returns a set optimal tuning parameters of
size `nComp`

to be used in variable selection function. Note that this
approach still is under theoretical study and is not proved to give optimal
values in general.

We use a simulated data set to illustrate using `fmrs`

package. We generate
the covariates (design matrix) from a multivariate normal distribution of
dimension `nCov=10`

and sample size 200 with mean vector $\bf 0$ and
variance-covariance matrix $\Sigma=(0.5^{|i-j|})$. We then generate
time-to-event data from a finite mixture of two components AFT regression
models with Log-Normal sub-distributions. The mixing proportions are set
to $\pi=(0.3, 0.7)$. We choose $\boldsymbol\beta_0=(2,-1)$ as the
intercepts, $\boldsymbol\beta_1=(-1, -2, 1, 2, 0 , 0, 0, 0, 0, 0)$ and
$\boldsymbol\beta_2=(1, 2, 0, 0, 0 , 0, -1, 2, -2, 3)$ as the regression
coefficients in first and second component, respectively.

We start by loading necessary libraries and assigning the parameters of model as follows.

library(fmrs) 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

Using the function `fmrs.gendata`

, we generate a data set from a finite
mixture of accelerated failure time regression models with above settings as
follow.

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

Now we assume that the generated data are actually real data. We find MLE of the parameters of the assumed model using following code. Note that almost all mixture of regression models depends on initial values. Here we generate the initial values form a Normal distribution with

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)) coefficients(res.mle) dispersion(res.mle) mixProp(res.mle)

As we see the ML estimates of regression coefficients are not zero.
Therefore MLE cannot provide a sparse solution. In order to provide a sparse
solution, we use the variable selection methods developed by
Shokoohi et. ale. (2016). First we need to find a good set of tuning
parameters.
It can be done by using component-wise tuning parameter selection function
implemented in `fmrs`

as follows. In some settings, however, it is better to
investigate if common tuning parameter performs better.

res.lam <- fmrs.tunsel(y = dat$y, x = dat$x, delta = dat$delta, nComp = nComp, disFamily = "lnorm", initCoeff = c(coefficients(res.mle)), initDispersion = dispersion(res.mle), initmixProp = mixProp(res.mle), penFamily = "adplasso") show(res.lam)

We have used MLE estimates as initial values for estimating the tuning parameters. Now we used the same set of values to do variable selection with adaptive lasso penalty as follows.

res.var <- fmrs.varsel(y = dat$y, x = dat$x, delta = dat$delta, nComp = ncomp(res.mle), disFamily = "lnorm", initCoeff=c(coefficients(res.mle)), initDispersion = dispersion(res.mle), initmixProp = mixProp(res.mle), penFamily = "adplasso", lambPen = slot(res.lam, "lambPen")) coefficients(res.var) dispersion(res.var) mixProp(res.var)

The final variables that are selected using this procedure are those with non-zero coefficients. Note that a switching between components of mixture has happened here.

round(coefficients(res.var)[-1,],3)

Therefore the variable selection and parameter estimation is done simultaneously using the fmrs package.

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