drmdel (version 1.1)

drmdel: Fit a density ratio model

Description

Fit a semiparametric density ratio model (DRM) to m+1 (m>=1) samples using maximum dual empirical likelihood method.

Denote the population cumulative distribution functions of the m+1 samples as $F_k(x)$'s, $k = 0, \, 1, \, \ldots, \, m$. We pick $F_0(x)$ as a baseline distribution. The DRM assumes that the ratio of the density of each non-baseline distribution to the density of the baseline distribution satisfies $$dF_k(x)/dF_0(x) = \exp(\alpha + \beta^T q(x)), \ k=1, \, \ldots, \, m$$ where $q(x)$ is a pre-specified d-dimensional basis function of data, and $\alpha$, $\beta$ are model parameters. No parametric form for baseline distribution $F_0$ is assumed.

Usage

drmdel(x, n_samples, basis_func, par_init=NULL, g_null=NULL,
       g_null_gr=NULL, par_dim_null=NULL, par_init_null=NULL, ...)

Arguments

x

a vector formed by concatenating multiple samples, $x_0$, $x_1$, ..., $x_m$, in the order of baseline sample ($x_0$), non-baseline sample 1 ($x_1$), ..., non-baseline sample m ($x_m$).

n_samples

a vector of length m+1 specifying the sizes of the multiple samples, in the order of 0, 1, ..., m.

basis_func

basis function q(x) of the DRM; must either be an integer between 1 and 11 or a function of the data, x. The integers represents built-in basis-functions:

1 -- $q(x) = x$.

2 -- $q(x) = \log(|x|)$.

3 -- $q(x) = \

par_init

a vector of length m*(d+1) specifying the initial value of the parameter vector; if not specified, all set to zeors. In fact, it is better to always set initial values to zeros, i.e. default, for all parameters to ensure that, at initial v

g_null

the function specifying the null hypothesis about DRM parameter $\beta$ if there is one; The default is NULL.

g_null_gr

a funciton specifying the gradient of g_null, which must return a matrix of dimension m*(d+1) by dim(par_null), if available. The default is NULL.

par_dim_null

dimension of the parameter vector in null hypothesis if there is one. The default is NULL.

par_init_null

a vector of length par_dim_null specifying the initial value of the parameter vector under null; if not specified (default), all set to zeros (recommended).

...

further arguments to be passed to the R function optim for maximizing the dual empirical likelihood. See help(optim) for details. In the drmdel function, by default, the

Value

• drm_info
  a list of basic information about the DRM:

  m -- number of samples - 1.

  d -- dimension of the basis function.

  n_smaples -- the input vector of length m+1 specifying the size of each sample.

  n_total -- total sample size.

  basis_func -- the input basis function of the DRM.

  rho -- sample proportion: n_samples/n_total.

• mdele
  maximum dual empirical likelihood estimator (MDELE) of the model parameters. The output is a vector organized in the following form: $$(\alpha_1, \, \beta_{1,1}, \, \beta_{1,2}, \, ..., \, \beta_{1,d}, \, \alpha_2, \, \beta_{2,1}, \, \beta_{2,2}, \, ..., \, \beta_{2,d}, \, ..., \, \alpha_m, \, \beta_{m,1}, \, \beta_{m,2}, \, ..., \, \beta_{m,d}).$$
• info_mat
  estimated information matrix.
• negldl
  negative log dual-likelihood evaluated at mdele.
• mdele_null
  mdele of the parameters under the null hypothesis, if available.
• negldl_null
  negative log dual-likelihood evaluated at mdele under the null hypothesis, if available.
• delr
  dual-empirical-likelihood ratio statistic evaluated under the null hypothesis. If no null hypotheis (g_null) is given, this is simply -2*negldl.
• df
  degrees of freedom of the chi-square limiting distribution for DELR statistic under the null.
• p_val
  p-vale of the DELR test.
• p_est
  estimated $dF_k(x)$'s at the observed data points, under the DRM. This is a data frame with the following three columns:

  k -- label for the populations, k = 0, 1, ..., m.

  x -- data points; at which 'x' value $dF_k(x)$ is estimated.

  p_est -- estimated $dF_k(x)$.

  NOTE: To estimate the density of $F_k(x)$, it is recommended to use densityDRM function.

• cdf_est
  estimated CDFs, $F_k(x)$'s, at the observed data points, under the DRM. This is a data frame with the following three columns:

  k -- label for the populations, k = 0, 1, ..., m.

  x -- data points; at which 'x' value $F_k(x)$ is estimated.

  cdf_est -- estimated $F_k(x)$.

  NOTE: To estimate CDF $dF_k(x)$, it is recommended to use cdfDRM function instead of looking at this output.

References

S. Cai, J. Chen and J. V. Zidek (2013), Dual-empirical-likelihood ratio test under density ratio models for multiple samples. Manuscript.

A. Keziou and S. Leoni-Aubin (2008), On empirical likelihood for semiparametric two-sample density ratio models. Journal of Statistical Planning and Inference, 138:915-928.

Examples

# Data generation
set.seed(25)
n_samples <- c(100, 200, 180, 150, 175)  # sample sizes
x0 <- rgamma(n_samples[1], shape=5, rate=1.8)
x1 <- rgamma(n_samples[2], shape=12, rate=1.2)
x2 <- rgamma(n_samples[3], shape=12, rate=1.2)
x3 <- rgamma(n_samples[4], shape=18, rate=5)
x4 <- rgamma(n_samples[5], shape=25, rate=2.6)
x <- c(x0, x1, x2, x3, x4)

# Fit a DRM with the basis function q(x) = (x, log(abs(x))), which
# is the basis function for gamma family.

# There are 11 built-in basis function in drmdel(). And q(x) = (x,
# log(abs(x))) is the 6th basis function, so we can fit the model
# by specifying basis_func=6 in drmdel() as follows:
drmfit <- drmdel(x=x, n_samples=n_samples, basis_func=6)
names(drmfit)

# A brief summary of the DRM fit
summaryDRM(drmfit)

# Another way of specifying basis function for drmdel() is to pass
# a user-specified R function to the basis_func argument of the
# drmdel() function.
# NOTE: If the basis function one wants to use is included in the
# built-in function list, one should use the built-in functions by
# passing an integer between 1 to 11 to the drmdel() function,
# because the computation will be faster with a built-in function
# than with a user-specified function.
basis_gamma <- function(x) return(c(x, log(abs(x))))
drmfit1 <- drmdel(x=x, n_samples=n_samples,
                  basis_func=basis_gamma)

# One can see the summary of this DRM fit is exactly the same as
# that of the previous fit with basis_func=6
summaryDRM(drmfit1)