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, λM), where
λM is a
prespecified number. The BIC is provided by the function
fmrs.varsel for each candidate point and we choose the
optimal λ̂, 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 (λ1, …, λK) from the set of candidate values in (0, λM). 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 Σ = (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 π = (0.3, 0.7). We choose β0 = (2, −1) as
the intercepts, β1 = (−1, −2, 1, 2, 0, 0, 0, 0, 0, 0)
and β2 = (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.
## BugReports: https://github.com/shokoohi/fmrs/issuesset.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 = 40Using 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$delta,
nComp = nComp, disFamily = "lnorm",
initCoeff = rnorm(nComp*nCov+nComp),
initDispersion = rep(1, nComp),
initmixProp = rep(1/nComp, nComp))
summary(res.mle)## -------------------------------------------
## Fitted Model:
## -------------------------------------------
## Finite Mixture of Accelerated Failure Time
## Regression Models
## Log-Normal Sub-Distributions
## 2 Components; 10 Covariates; 500 samples.
##
## Coefficients:
## Comp.1 Comp.2
## Intercept -0.98813518 1.993430383
## X.1 -1.04289531 2.132445019
## X.2 1.08397203 -1.076622120
## X.3 1.99159549 -2.180934660
## X.4 0.01722128 1.056957193
## X.5 -0.07980287 1.919587713
## X.6 -0.01501062 0.109143832
## X.7 0.08672761 0.115042361
## X.8 -1.15880098 0.127165742
## X.9 2.10200793 -0.006910483
## X.10 -2.12473990 -0.032354752
##
## Dispersion:
## Comp.1 Comp.2
## 0.9544685 0.838036
##
## Mixing Proportions:
## Comp.1 Comp.2
## 0.5861984 0.4138016
##
## LogLik: -759.5507; BIC: 1674.467As we see the ML estimates of regression coefficients are not zero.
Therefore MLE cannot provide a sparse solution. In order to provide a
sparse solution, we use the variable selection methods developed by
Shokoohi et. al. (2016). First we need to find a good set of tuning
parameters. It can be done by using component-wise tuning parameter
selection function implemented in fmrs as follows. In some
settings, however, it is better to investigate if common tuning
parameter performs better.
res.lam <- fmrs.tunsel(y = dat$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")
summary(res.lam)## -------------------------------------------
## Selected Tuning Parameters:
## -------------------------------------------
## Finite Mixture of Accelerated Failure Time
## Regression Models
## Log-Normal Sub-Distributions
## 2 Components; adplasso Penalty;
##
## Component-wise lambda:
## Comp.1 Comp.2
## 0.01 0.0199We 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"))
summary(res.var)## -------------------------------------------
## Fitted Model:
## -------------------------------------------
## Finite Mixture of Accelerated Failure Time
## Regression Models
## Log-Normal Sub-Distributions
## 2 Components; 10 Covariates; 500 samples.
##
## Coefficients:
## Comp.1 Comp.2
## Intercept -1.008742 1.9457714
## X.1 -1.026428 1.9985027
## X.2 1.061368 -0.9324296
## X.3 1.973257 -2.1430407
## X.4 0.000000 0.9920236
## X.5 0.000000 1.9644591
## X.6 0.000000 0.0000000
## X.7 0.000000 0.0000000
## X.8 -1.113816 0.0000000
## X.9 2.087108 0.0000000
## X.10 -2.104553 0.0000000
##
## Selected Set:
## Comp.1 Comp.2
## X.1 1 1
## X.2 1 1
## X.3 1 1
## X.4 0 1
## X.5 0 1
## X.6 0 0
## X.7 0 0
## X.8 1 0
## X.9 1 0
## X.10 1 0
##
## Dispersion:
## Comp.1 Comp.2
## 0.9481664 0.9016874
##
## Mixing Proportions:
## Comp.1 Comp.2
## 0.5807324 0.4192676
##
## LogLik: -764.3914; BIC: 1628.217The 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.
## Comp.1 Comp.2
## X.1 1 1
## X.2 1 1
## X.3 1 1
## X.4 0 1
## X.5 0 1
## X.6 0 0
## X.7 0 0
## X.8 1 0
## X.9 1 0
## X.10 1 0Therefore, the variable selection and parameter estimation is done simultaneously using the fmrs package.
## R version 4.4.1 (2024-06-14)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 24.04.1 LTS
##
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so; LAPACK version 3.12.0
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_US.UTF-8 LC_COLLATE=C
## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
##
## time zone: Etc/UTC
## tzcode source: system (glibc)
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] fmrs_1.17.0 rmarkdown_2.28
##
## loaded via a namespace (and not attached):
## [1] digest_0.6.37 R6_2.5.1 fastmap_1.2.0 Matrix_1.7-1
## [5] xfun_0.48 lattice_0.22-6 splines_4.4.1 maketools_1.3.1
## [9] cachem_1.1.0 knitr_1.48 htmltools_0.5.8.1 buildtools_1.0.0
## [13] lifecycle_1.0.4 cli_3.6.3 grid_4.4.1 sass_0.4.9
## [17] jquerylib_0.1.4 compiler_4.4.1 sys_3.4.3 tools_4.4.1
## [21] evaluate_1.0.1 bslib_0.8.0 survival_3.7-0 yaml_2.3.10
## [25] jsonlite_1.8.9 rlang_1.1.4