maple.dr.group {mable}

Maximum approximate profile likelihood estimate of the density ratio model for grouped data with given regression coefficients

Description

Select optimal degree of Bernstein polynomial model for grouped data with a given regression coefficients.

Usage

maple.dr.group(
  t,
  n0,
  n1,
  M,
  regr,
  ...,
  interval = c(0, 1),
  alpha = NULL,
  vb = 0,
  controls = mable.ctrl(),
  progress = TRUE,
  message = TRUE
)

Arguments

t

cutpoints of class intervals

n0, n1

frequencies of two sample data grouped by the classes specified by t. coden0:"Control", n1: "Case".

M

a positive integer or a vector (m0, m1).

regr

regressor vector function r(x)=(1,r_1(x),...,r_d(x)) which returns n x (d+1) matrix, n=length(x)

...

additional arguments to be passed to regr

interval

a vector (a,b) containing the endpoints of supporting/truncation interval of x and y.

alpha

a given regression coefficient, missing value is imputed by logistic regression

vb

code for vanishing boundary constraints, -1: f0(a)=0 only, 1: f0(b)=0 only, 2: both, 0: none (default).

controls

Object of class mable.ctrl() specifying iteration limit and the convergence criterion for EM and Newton iterations. Default is mable.ctrl. See Details.

progress

logical: should a text progressbar be displayed

message

logical: should warning messages be displayed

Details

Suppose that n0("control") and n1("case") are frequencies of independent samples grouped by the classes t from f0 and f1 which satisfy f1(x)=f0(x)exp[alpha0+alpha'r(x)] with r(x)=(r1(x),...,r_d(x)). Maximum approximate Bernstein/Beta likelihood estimates of f0 and f1 are calculated with a given regression coefficients which are efficient estimates provided by other semiparametric methods such as logistic regression. If support is (a,b) then replace r(x) by r[a+(b-a)x]. For a fixed m, using the Bernstein polynomial model for baseline f_0, MABLEs of f_0 and parameters alpha can be estimated by EM algorithm and Newton iteration. If estimated lower bound m_b for m based on n1 is smaller that that based on n0, then switch n0 and n1 and use f_1 as baseline. If M=m or m0=m1=m, then m is a preselected degree. If m0<m1 it specifies the set of consective candidate model degrees m0:m1 for searching an optimal degree by the change-point method, where m1-m0>3.

Value

A list with components

• m the given or a selected degree by method of change-point

• p the estimated vector of mixture proportions p = (p_0, \ldots, p_m) with the given or selected degree m

• alpha the given regression coefficients

• mloglik the maximum log-likelihood at degree m

• interval support/truncation interval (a,b)

• baseline ="control" if f_0 is used as baseline, or ="case" if f_1 is used as baseline.

• M the vector (m0, m1), where m1, if greater than m0, is the largest candidate when the search stoped

• lk log-likelihoods evaluated at m \in \{m_0, \ldots, m_1\}

• lr likelihood ratios for change-points evaluated at m \in \{m_0+1, \ldots, m_1\}

• pval the p-values of the change-point tests for choosing optimal model degree

• chpts the change-points chosen with the given candidate model degrees

Author(s)

Zhong Guan <[e-mail]>

References

Guan, Z., Application of Bernstein Polynomial Model to Density and ROC Estimation in a Semiparametric Density Ratio Model