| Title: | Nonparametric and Semiparametric Mixture Estimation |
| Version: | 2.0-0 |
| Date: | 2025-05-10 |
| Depends: | lsei |
| Imports: | graphics,methods,stats |
| Description: | Mainly for maximum likelihood estimation of nonparametric and semiparametric mixture models, but can also be used for fitting finite mixtures. The algorithms are developed in Wang (2007) <doi:10.1111/j.1467-9868.2007.00583.x> and Wang (2010) <doi:10.1007/s11222-009-9117-z>. |
| Encoding: | UTF-8 |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| URL: | https://www.stat.auckland.ac.nz/~yongwang/ |
| RoxygenNote: | 7.3.2 |
| NeedsCompilation: | no |
| Packaged: | 2025-05-10 18:02:02 UTC; ywan076 |
| Author: | Yong Wang [aut, cre] |
| Maintainer: | Yong Wang <yongwang@auckland.ac.nz> |
| Repository: | CRAN |
| Date/Publication: | 2025-05-10 18:40:02 UTC |
Beta-blockers Data
Description
Contains the data of the 22-center clinical trial of beta-blockers for reducing mortality after myocardial infarction.
Format
A numeric matrix with four columns:
center: center identification code.
deaths: the number of deaths in the center.
total: the number of patients taking beta-blockers in the center.
treatment: 0 for control, and 1 for treatment.
Source
Aitkin, M. (1999). A general maximum likelihood analysis of variance components in generalized linear models. Biometrics, 55, 117-128.
References
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86.
See Also
Examples
data(betablockers)
x = mlogit(betablockers)
cnmms(x)
Z-values of BRCA Data
Description
Contains 3226 z-values computed by Efron (2004) from the data obtained
in a well-known microarray experiment concerning two types of genetic
mutations causing increased breast cancer risk, BRCA1 and BRCA2.
Format
A numeric vector containing 3226 z-values.
References
Efron, B. (2004). Large-scale simultaneous hypothesis testing: the choice of a null hypothesis. Journal of the American Statistical Association, 99, 96-104.
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
Wang, Y. and C.-S. Chee (2012). Density estimation using nonparametric and semiparametric mixtures. Statistical Modelling: An International Journal, 12, 67-92.
See Also
Examples
data(brca)
x = npnorm(brca)
plot(cnm(x), x)
Maximum Likelihood Estimation of a Nonparametric Mixture Model
Description
Function cnm can be used to compute the maximum
likelihood estimate of a nonparametric mixing distribution
(NPMLE) that has a one-dimensional mixing parameter, or simply
the mixing proportions with support points held fixed.
A finite mixture model has a density of the form
f(x; \pi, \theta, \beta) = \sum_{j=1}^k \pi_j f(x; \theta_j,
\beta).
where \pi_j \ge 0 and \sum_{j=1}^k \pi_j
=1.
A nonparametric mixture model has a density of the form
f(x; G) = \int f(x; \theta) d G(\theta),
where G is a mixing distribution
that is completely unspecified. The maximum likelihood estimate of
the nonparametric G, or the NPMLE of G, is known to be
a discrete distribution function.
Function cnm implements the CNM algorithm that is proposed
in Wang (2007) and the hierarchical CNM algorithm of Wang and
Taylor (2013). The implementation is generic using S3
object-oriented programming, in the sense that it works for an
arbitrary family of mixture models defined by the user. The user,
however, needs to supply the implementations of the following
functions for their self-defined family of mixture models, as they
are needed internally by function cnm:
initial(x, beta, mix, kmax)
valid(x, beta, theta)
logd(x, beta, pt, which)
gridpoints(x, beta, grid)
suppspace(x, beta)
length(x)
print(x, ...)
weight(x, ...)
While not needed by the algorithm for finding the solution, one may also implement
plot(x, mix, beta, ...)
so that the fitted model can be shown graphically in a
user-defined way. Inside cnm, it is used when
plot="probability" so that the convergence of the algorithm
can be graphically monitored.
For creating a new class, the user may consult the implementations
of these functions for the families of mixture models included in
the package, e.g., npnorm and nppois.
Usage
cnm(
x,
init = NULL,
model = c("npmle", "proportions"),
maxit = 100,
tol = 1e-06,
grid = 100,
plot = c("null", "gradient", "probability"),
verbose = 0
)
Arguments
x |
a data object of some class that is fully defined by the user. The user needs to supply certain functions as described below. |
init |
list of user-provided initial values for the mixing
distribution |
model |
the type of model that is to estimated: the
non-parametric MLE (if |
maxit |
maximum number of iterations. |
tol |
a tolerance value needed to terminate an
algorithm. Specifically, the algorithm is terminated, if the
increase of the log-likelihood value after an iteration is less
than |
grid |
number of grid points that are used by the algorithm to
locate all the local maxima of the gradient function. A larger
number increases the chance of locating all local maxima, at the
expense of an increased computational cost. The locations of the
grid points are determined by the function |
plot |
whether a plot is produced at each iteration. Useful
for monitoring the convergence of the algorithm. If
|
verbose |
verbosity level for printing intermediate results in each iteration, including none (= 0), the log-likelihood value (= 1), the maximum gradient (= 2), the support points of the mixing distribution (= 3), the mixing proportions (= 4), and if available, the value of the structural parameter beta (= 5). |
Value
family |
the name of the mixture family that is used to fit to the data. |
num.iterations |
number of iterations required by the algorithm |
max.gradient |
maximum value of the gradient function, evaluated at the beginning of the final iteration |
convergence |
convergence code. |
ll |
log-likelihood value at convergence |
mix |
MLE of the mixing distribution, being an object of the
class |
beta |
value of the structural parameter, that is held fixed throughout the computation. |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86
Wang, Y. and Taylor, S. M. (2013). Efficient computation of nonparametric survival functions via a hierarchical mixture formulation. Statistics and Computing, 23, 713-725.
See Also
Examples
## Simulated data
x = rnppois(200, disc(c(1,4), c(0.7,0.3))) # Poisson mixture
(r = cnm(x))
plot(r, x)
x = rnpnorm(200, disc(c(0,4), c(0.3,0.7)), sd=1) # Normal mixture
plot(cnm(x), x) # sd = 1
plot(cnm(x, init=list(beta=0.5)), x) # sd = 0.5
## Real-world data
data(thai)
plot(cnm(x <- nppois(thai)), x) # Poisson mixture
data(brca)
plot(cnm(x <- npnorm(brca)), x) # Normal mixture
Maximum Likelihood Estimation of a Semiparametric Mixture Model
Description
Functions cnmms, cnmpl and cnmap
can be used to compute the maximum likelihood estimate of a
semiparametric mixture model that has a one-dimensional mixing
parameter. The types of mixture models that can be computed
include finite, nonparametric and semiparametric ones.
Function cnmms can also be used to compute the maximum
likelihood estimate of a finite or nonparametric mixture model.
A finite mixture model has a density of the form
f(x; \pi, \theta, \beta) = \sum_{j=1}^k \pi_j f(x; \theta_j,
\beta).
where pi_j \ge 0 and \sum_{j=1}^k pi_j =1.
A nonparametric mixture model has a density of the form
f(x; G) = \int f(x; \theta) d G(\theta),
where G is a mixing distribution
that is completely unspecified. The maximum likelihood estimate of
the nonparametric G, or the NPMLE of $G, is known to
be a discrete distribution function.
A semiparametric mixture model has a density of the form
f(x; G, \beta) = \int f(x; \theta, \beta) d G(\theta),
where G is a mixing distribution that is completely
unspecified and \beta is the structural parameter.
Of the three functions, cnmms is recommended for most
problems; see Wang (2010).
Functions cnmms, cnmpl and cnmap implement
the algorithms CNM-MS, CNM-PL and CNM-AP that are described in
Wang (2010). Their implementations are generic using S3
object-oriented programming, in the sense that they can work for
an arbitrary family of mixture models that is defined by the
user. The user, however, needs to supply the implementations of
the following functions for their self-defined family of mixture
models, as they are needed internally by the functions above:
initial(x, beta, mix, kmax)
valid(x, beta)
logd(x, beta, pt, which)
gridpoints(x, beta, grid)
suppspace(x, beta)
length(x)
print(x, ...)
weight(x, ...)
While not needed by the algorithms, one may also implement
plot(x, mix, beta, ...)
so that the fitted model can be shown graphically in a way that the user desires.
For creating a new class, the user may consult the implementations
of these functions for the families of mixture models included in
the package, e.g., cvp and mlogit.
Usage
cnmms(x, init=NULL, maxit=1000, model=c("spmle","npmle"), tol=1e-6,
grid=100, kmax=Inf, plot=c("null", "gradient", "probability"),
verbose=0)
cnmpl(x, init=NULL, tol=1e-6, tol.npmle=tol*1e-4, grid=100, maxit=1000,
plot=c("null", "gradient", "probability"), verbose=0)
cnmap(x, init=NULL, maxit=1000, tol=1e-6, grid=100, plot=c("null",
"gradient"), verbose=0)
Arguments
x |
a data object of some class that can be defined fully by the user |
init |
list of user-provided initial values for the mixing
distribution |
maxit |
maximum number of iterations |
model |
the type of model that is to estimated:
non-parametric MLE ( |
tol |
a tolerance value that is used to terminate an
algorithm. Specifically, the algorithm is terminated, if the
relative increase of the log-likelihood value after an iteration
is less than |
grid |
number of grid points that are used by the algorithm
to locate all the local maxima of the gradient function. A
larger number increases the chance of locating all local maxima,
at the expense of an increased computational cost. The locations
of the grid points are determined by the function
|
kmax |
upper bound on the number of support points. This is particularly useful for fitting a finite mixture model. |
plot |
whether a plot is produced at each iteration. Useful
for monitoring the convergence of the algorithm. If |
verbose |
verbosity level for printing intermediate results in each iteration, including none (= 0), the log-likelihood value (= 1), the maximum gradient (= 2), the support points of the mixing distribution (= 3), the mixing proportions (= 4), and if available, the value of the structural parameter beta (= 5). |
tol.npmle |
a tolerance value that is used to terminate the computing of the NPMLE internally. |
Value
family |
the class of the mixture family that is used to fit to the data. |
num.iterations |
Number of iterations required by the algorithm |
grad |
For |
max.gradient |
Maximum value of the gradient function,
evaluated at the beginning of the final iteration. It is only
given by function |
convergence |
convergence code. |
ll |
log-likelihood value at convergence |
mix |
MLE of the mixing distribution, being an object of the
class |
beta |
MLE of the structural parameter |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86
See Also
Examples
## Compute the MLE of a finite mixture
x = rnpnorm(100, disc(c(0,4), c(0.7,0.3)), sd=1)
for(k in 1:6) plot(cnmms(x, kmax=k), x, add=(k>1), comp="null", col=k+1,
main="Finite Normal Mixtures")
legend("topright", 0.3, leg=paste0("k = ",1:6), lty=1, lwd=2, col=2:7)
## Compute a semiparametric MLE
# Common variance problem
x = rcvps(k=50, ni=5:10, mu=c(0,4), pr=c(0.7,0.3), sd=3)
cnmms(x) # CNM-MS algorithm
cnmpl(x) # CNM-PL algorithm
cnmap(x) # CNM-AP algorithm
# Logistic regression with a random intercept
x = rmlogit(k=30, gi=3:5, ni=6:10, pt=c(0,4), pr=c(0.7,0.3),
beta=c(0,3))
cnmms(x)
data(toxo) # k = 136
cnmms(mlogit(toxo))
Class cvps
Description
These functions can be used to study a common variance problem (CVP), where univariate observations fall in known groups. Observations in each group are assumed to have the same mean, but different groups may have different means. All observations are assumed to have a common variance, despite their different means, hence giving the name of the problem. It is a random-effects problem.
Usage
cvps(x)
rcvp(k, ni=2, mu=0, pr=1, sd=1)
rcvps(k, ni=2, mu=0, pr=1, sd=1)
## S3 method for class 'cvps'
print(x, ...)
Arguments
x |
CVP data in the raw form as an argument in |
k |
the number of groups. |
ni |
a numeric vector that gives the sample size in each group. |
mu |
a numeric vector for all the theoretical means. |
pr |
a numeric vector for all the probabilities associated with the theoretical means. |
sd |
a scalar for the standard deviation that is common to all observations. |
... |
arguments passed on to function |
Details
Class cvps is used to store the CVP data in a summarized
form.
Function cvps creates an object of class cvps, given
a matrix that stores the values (column 2) and their grouping
information (column 1).
Function rcvp generates a random sample in the raw form for
a common variance problem, where the means follow a discrete
distribution.
Function rcvps generates a random sample in the summarized
form for a common variance problem, where the means follow a
discrete distribution.
Function print.cvps prints the CVP data given in the
summarized form.
The raw form of the CVP data is a two-column matrix, where each
row represents an observation. The two columns along each row
give, respectively, the group membership (group) and the
value (x) of an observation.
The summarized form of the CVP data is a four-column matrix, where
each row represents the summarized data for all observations in a
group. The four columns along each row give, respectively, the
group number (group), the number of observations in the
group (ni), the sample mean of the observations in the
group (mi), and the residual sum of squares of the
observations in the group (ri).
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Neyman, J. and Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 1-32.
Kiefer, J. and Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. Math. Stat., 27, 886-906.
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86.
See Also
Examples
x = rcvps(k=50, ni=5:10, mu=c(0,4), pr=c(0.7,0.3), sd=3)
cnmms(x) # CNM-MS algorithm
cnmpl(x) # CNM-PL algorithm
cnmap(x) # CNM-AP algorithm
Class disc
Description
Class disc is used to represent an arbitrary univariate
discrete distribution with a finite number of support points.
Usage
disc(pt, pr=1, sort=TRUE, collapse=FALSE)
Arguments
pt |
a numeric vector for support points. |
pr |
a numeric vector for probability values at the support points. |
sort |
=TRUE, by default. If TRUE, support points are sorted (in increasing order). |
collapse |
=TRUE, by default. If TRUE, identical support points are collapsed, with their masses aggregated. |
Details
Function disc creates an object of class disc, given
the support points and probability values at these points.
Function print.disc prints the discrete distribution.
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
See Also
Examples
(d = disc(pt=c(0,4), pr=c(0.3,0.7)))
Density function of a mixture distribution
Description
Computes the density or their logarithmic values of a
mixture distribution, where the component family depends on the
class of x.
x must belong to a mixture family, as specified by its class.
Usage
dmix(x, mix, beta = NULL, log = FALSE)
Arguments
x |
a data object of a mixture model class. |
mix |
a discrete distribution, as defined by class
|
beta |
the structural parameter, if any. |
log |
if |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86
See Also
cnm, cnmms,
npnorm, nppois, disc,
Examples
## Poisson mixture
mix0 = disc(c(1,4), c(0.7,0.3))
x = rnppois(10, mix0)
dmix(x, mix0)
dmix(x, mix0, log=TRUE)
## Normal mixture
x = rnpnorm(10, mix0, sd=1)
dmix(x, mix0, 1)
dmix(x, mix0, 1, log=TRUE)
dmix(x, mix0, 0.5, log=TRUE)
Grid points
Description
A generic method used to return a vector of grid points used for searching local maxima of the gradient funcion.
Usage
gridpoints(x, beta, grid)
Arguments
x |
an object of a class for data. |
beta |
instrumental parameter in a semiparametric mixture. |
grid |
number of grid points to be generated. |
Value
A numeric vector containing grid points.
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
Hierarchical Constrained Newton method
Description
Function hcnm can be used to compute the MLE
of a finite discrete mixing distribution, given the component
density values of each observation. It implements the
hierarchical CNM algorithm of Wang and Taylor (2013).
Usage
hcnm(
D,
p0 = NULL,
w = 1,
maxit = 1000,
tol = 1e-06,
blockpar = NULL,
recurs.maxit = 2,
compact = TRUE,
depth = 1,
verbose = 0
)
Arguments
D |
A numeric matrix, each row of which stores the component density values of an observation. |
p0 |
Initial mixture component proportions. |
w |
Duplicity of each row in matrix |
maxit |
Maximum number of iterations. |
tol |
A tolerance value to terminate the
algorithm. Specifically, the algorithm is terminated, if the
increase of the log-likelihood value after an iteration is less
than |
blockpar |
Block partitioning parameter. If > 1, the number
of blocks is roughly |
recurs.maxit |
Maximum number of iterations in recursions. |
compact |
Whether iteratively select and use a compact subset (which guarantees convergence), or not (if already done so before calling the function). |
depth |
Depth of recursion/hierarchy. |
verbose |
Verbosity level for printing intermediate results. |
Value
p |
Computed probability vector. |
convergence |
convergence code. |
ll |
log-likelihood value at convergence |
maxgrad |
Maximum gradient value. |
numiter |
number of iterations required by the algorithm |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
Wang, Y. and Taylor, S. M. (2013). Efficient computation of nonparametric survival functions via a hierarchical mixture formulation. Statistics and Computing, 23, 713-725.
See Also
Examples
x = rnppois(1000, disc(0:50)) # Poisson mixture
D = outer(x$v, 0:1000/10, dpois)
(r = hcnm(D, w=x$w))
disc(0:1000/10, r$p, collapse=TRUE)
cnm(x, init=list(mix=disc(0:1000/10)), model="p")
Initialization for a nonparametric/semiparametric mixture
Description
A generic method used to return an initialization for a nonparametric/semiparametric mixture.
Usage
initial(x, beta, mix, kmax)
Arguments
x |
an object of a class for data. |
beta |
instrumental parameter in a semiparametric mixture. |
mix |
an object of class |
kmax |
the maximum allowed number of support points used. |
Value
beta |
an initialized value of beta |
mix |
an initialised or updated object of class |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
Initialisation
Description
The functions examines whether the initial values are proper. If not, proper ones are provided, by employing the function "initial" provided by the class.
Usage
initial0(x, init = NULL, kmax = NULL)
Arguments
x |
an object of a class for data. |
init |
a list with initial values for beta and mix (as in the
output of |
kmax |
the maximum allowed number of support points used. |
Value
beta |
an initialized value of beta |
mix |
an initialised or updated object of class |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
Log-likleihood Extra Term.
Description
Value of possibly an extra term in the log-likleihood function for the instrumental parameter beta
Usage
llex(x, beta, mix)
Arguments
x |
an object of a class for data. |
beta |
instrumental parameter in a semiparametric mixture. |
mix |
an object of class |
Value
a scalar value
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
Derivative of the log-likleihood Extra Term
Description
Derivative of the log-likleihood extra term wrt beta
Usage
llexdb(x, beta, mix)
Arguments
x |
an object of a class for data. |
beta |
instrumental parameter in a semiparametric mixture. |
mix |
an object of class |
Value
a scalar value
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
Log-density and its derivative values
Description
A generic method to compute the log-density values and possibly their first derivatives with respec to theta and beta.
Usage
logd(x, beta, pt, which)
Arguments
x |
an object of a class for data. |
beta |
instrumental parameter in a semiparametric mixture. |
pt |
a vector of values for the mixing variable theta. |
which |
an integer vector of length 3, indicating if,
respectively, the log-density values, the derivatives wrt beta
and the derivatives wrt theta are to be computed and returned if
being 1 ( |
Value
ld |
a matrix, storing the log-density values for each (x[i], beta, pt[j], or NULL if not asked for. |
db |
a matrix, storing the log-density derivatives wrt beta for each (x[i], beta, pt[j], or NULL if not asked for. |
dt |
a matrix, storing the log-density derivatives wrt theta for each (x[i], beta, pt[j], or NULL if not asked for. |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
Log-likelihood value of a mixture
Description
Computes the log-likelihood value
x must belong to a mixture family, as specified by its class.
Usage
loglik(mix, x, beta = NULL, attr = FALSE)
Arguments
mix |
a discrete distribution, as defined by class
|
x |
a data object of a mixture model class. |
beta |
the structural parameter, if any. |
attr |
=FALSE, by default. If TRUE, also returns attributes "dmix" and "logd" |
Value
the log-likelihood value.
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86
See Also
cnm, cnmms,
npnorm, nppois, disc,
Examples
## Poisson mixture
mix0 = disc(c(1,4), c(0.7,0.3))
x = rnppois(10, mix0)
loglik(mix0, x)
## Normal mixture
x = rnpnorm(10, mix0, sd=2)
loglik(mix0, x, 2)
Lung Cancer Data
Description
Contains the data of 14 studies of the effect of smoking on lung cancer.
Format
A numeric matrix with four columns:
study: study identification code.
lungcancer: the number of people diagnosed with lung cancer.
size: the number of people in the study.
smoker: 0 for smoker, and 1 for non-smoker.
Source
Booth, J. G. and Hobert, J. P. (1999). Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. Journal of the Royal Statistical Society, Ser. B, 61, 265-285.
References
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86.
See Also
Examples
data(lungcancer)
x = mlogit(lungcancer)
cnmms(x)
Class mlogit
Description
These functions can be used to fit a binomial logistic regression model that has a random intercept to clustered observations. Observations in each cluster are assumed to have the same intercept, while different clusters may have different intercepts. This is a mixed-effects problem.
Usage
mlogit(x)
rmlogit(k, gi=2, ni=2, pt=0, pr=1, beta=1, X)
Arguments
x |
a numeric matrix with four or more columns that stores clustered data. |
k |
the number of groups or clusters. |
gi |
a numeric vector that gives the sample size in each group. |
ni |
a numeric vector for the number of Bernoulli trials for each observation. |
pt |
a numeric vector for all the support points. |
pr |
a numeric vector for all the probabilities associated with the support points. |
beta |
a numeric vector for the fixed coefficients of the covariates of the observation. |
X |
the numeric matrix as the design matrix. If missing, a random matrix is created from a normal distribution. |
Details
Class mlogit is used to store data for fitting the binomial
logistic regression model with a random intercept.
Function mlogit creates an object of class mlogit,
given a matrix with four or more columns that stores,
respectively, the group/cluster membership (column 1), the number
of ones or successes in the Bernoulli trials (column 2), the
number of the Bernoulli trials (column 3), and the covariates
(columns 4+).
Function rmlogit generates a random sample that is saved as
an object of class mlogit.
An object of class mlogit contains a matrix with four or
more columns, that stores, respectively, the group/cluster
membership (column 1), the number of ones or successes in the
Bernoulli trials (column 2), the number of the Bernoulli trials
(column 3), and the covariates (columns 4+).
It also has two additional attributes that facilitate the
computing by function cmmms. The first attribute is
ui, which stores the unique values of group memberships,
and the second is gi, the number of observations in each
unique group.
It is convenient to use function mlogit to create an object
of class mlogit.
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Kiefer, J. and Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. Math. Stat., 27, 886-906.
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86.
See Also
Examples
x = rmlogit(k=30, gi=3:5, ni=6:10, pt=c(0,4), pr=c(0.7,0.3),
beta=c(0,3))
cnmms(x)
### Real-world data
# Random intercept logistic model
data(toxo)
cnmms(mlogit(toxo))
data(betablockers)
cnmms(mlogit(betablockers))
data(lungcancer)
cnmms(mlogit(lungcancer))
Class npgeom
Description
Class npgeom is used to store data that will be
processed as those of a nonparametric geometric mixture.
Function npgeom creates an object of class npgeom,
given values and weights/frequencies.
Function rnpgeom generates a random sample from a geometric
mixture and saves the data as an object of class npgeom.
Function dnpgeom is the density function of a Poisson
mixture.
Function pnpgeom is the distribution function of a Poisson
mixture.
Usage
npgeom(v, w=1, grouping=FALSE)
rnpgeom(n, mix=disc(0.5))
dnpgeom(x, mix=disc(0.5), log=FALSE)
pnpgeom(x, mix=disc(0.5), lower.tail=TRUE, log.p=FALSE)
Arguments
v |
a numeric vector that stores the values of a sample. |
w |
a numeric vector that stores the corresponding weights/frequencies of the observations. |
grouping |
logical, whether or not use frequencies (w) for identical values. |
n |
the sample size. |
x |
an object of class |
mix |
an object of class |
log |
=FALSE, if log-values are to be returned. |
lower.tail |
=FALSE, if lower.tail values are to be returned. |
log.p |
=FALSE, if log probability values are to be returned. |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
See Also
nnls, cnm,
cnmms, plot.nspmix.
Examples
mix = disc(pt=c(0.2,0.5), pr=c(0.3,0.7))
(x = rnpgeom(200, mix))
dnpgeom(x, mix)
pnpgeom(x, mix)
Class npnbinom
Description
Class npnbinom is used to store data that will be
processed as those of a nonparametric negative binomial mixture.
Function npnbinom creates an object of class
npnbinom, given values and weights/frequencies.
Function rnpnbinom generates a random sample from a
negative binomial mixture and saves the data as an object of class
npnbinom.
Usage
npnbinom(v, w=1, size, grouping=TRUE)
rnpnbinom(n, size, mix=disc(0.5))
dnpnbinom(x, mix=disc(0.5), size=NULL, log=FALSE)
pnpnbinom(x, mix=disc(0.5), size=NULL, lower.tail=TRUE, log.p=FALSE)
Arguments
v |
a numeric vector that stores the values of a sample. |
w |
a numeric vector that stores the corresponding weights/frequencies of the observations. |
size |
number of successful trials (ignored if |
grouping |
logical, to use frequencies (w) for identical values |
n |
the sample size. |
x |
an object of class |
mix |
an object of class |
log |
=FALSE, if log-values are to be returned. |
lower.tail |
=FALSE, if lower.tail values are to be returned. |
log.p |
=FALSE, if log probability values are to be returned. |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
See Also
nnls, cnm,
cnmms, plot.nspmix.
Examples
mix = disc(pt=c(0.2,0.5), pr=c(0.3,0.7))
(x = rnpnbinom(200, size=10, mix))
dnpnbinom(x, mix, size=10)
pnpnbinom(x, mix, size=10)
Class npnorm
Description
Class npnorm can be used to store data that
will be processed as those of a nonparametric normal
mixture. There are several functions associated with the class.
Function npnorm creates an object of class npnorm,
given values and weights/frequencies.
Function rnpnorm generates a random sample from a normal
mixture and saves the data as an object of class npnorm.
Function dnpnorm is the density function of a normal
mixture.
Function pnpnorm is the distribution function of a normal
mixture.
Usage
npnorm(v, w = 1)
rnpnorm(n, mix=disc(0), sd=1)
dnpnorm(x, mix=disc(0), sd=1, log=FALSE)
pnpnorm(x, mix=disc(0), sd=1, lower.tail=TRUE, log.p=FALSE)
Arguments
v |
a numeric vector that stores the values of a sample. |
w |
a numeric vector that stores the corresponding weights/frequencies of the observations. |
n |
the sample size. |
mix |
an object of class |
sd |
a scalar for the component standard deviation that is common to all components. |
x |
a numeric vector or an object of class |
log, log.p |
logical, for computing the log-values or not. |
lower.tail |
logical, for computing the lower tail value or not. |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
See Also
nnls, cnm,
cnmms, plot.nspmix.
Examples
mix = disc(pt=c(0,4), pr=c(0.3,0.7)) # a discrete distribution
x = rnpnorm(200, mix, sd=1)
dnpnorm(-2:6, mix, sd=1)
pnpnorm(-2:6, mix, sd=1)
dnpnorm(npnorm(-2:6), mix, sd=1)
pnpnorm(npnorm(-2:6), mix, sd=1)
Class nppois
Description
Class nppois is used to store data that will
be processed as those of a nonparametric Poisson mixture.
Function nppois creates an object of class nppois,
given values and weights/frequencies.
Function rnppois generates a random sample from a Poisson
mixture and saves the data as an object of class nppois.
Function dnppois is the density function of a Poisson
mixture.
Function pnppois is the distribution function of a Poisson
mixture.
Usage
nppois(v, w=1, grouping=TRUE)
rnppois(n, mix=disc(1), ...)
dnppois(x, mix=disc(1), log=FALSE)
pnppois(x, mix=disc(1), lower.tail=TRUE, log.p=FALSE)
Arguments
v |
a numeric vector that stores the values of a sample. |
w |
a numeric vector that stores the corresponding weights/frequencies of the observations. |
grouping |
logical, to use frequencies (w) for identical values |
n |
the sample size. |
x |
an object of class |
mix |
an object of class |
log |
logical, to compute the log-values or not. |
lower.tail |
=FALSE, if lower.tail values are to be returned. |
log.p |
=FALSE, if log probability values are to be returned. |
... |
arguments passed on to function |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
See Also
nnls, cnm,
cnmms, plot.nspmix.
Examples
mix = disc(pt=c(0,4), pr=c(0.3,0.7)) # a discrete distribution
x = rnppois(200, mix)
dnppois(0:10, mix)
pnppois(0:10, mix)
dnppois(nppois(0:10), mix)
pnppois(nppois(0:10), mix)
Plot a discrete distribution function
Description
Class disc is used to represent an arbitrary
univariate discrete distribution with a finite number of support
points.
Function disc creates an object of class disc, given
the support points and probability values at these points.
Function plot.disc plots the discrete distribution.
Usage
## S3 method for class 'disc'
plot(
x,
type = c("pdf", "cdf"),
add = FALSE,
col = 4,
lwd = 1,
ylim,
xlab = "",
ylab = "Probability",
...
)
Arguments
x |
an object of class |
type |
plot its pdf or cdf. |
add |
add the plot or not. |
col |
colour to be used. |
lwd, ylim, xlab, ylab |
graphical parameters. |
... |
arguments passed on to function |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
See Also
Examples
plot(disc(pt=c(0,4), pr=c(0.3,0.7)))
plot(disc(rnorm(5), 1:5))
for(i in 1:5)
plot(disc(rnorm(5), 1:5), type="cdf", add=(i>1), xlim=c(-3,3))
Plotting a nonparametric geometric mixture
Description
Function plot.npgeom plots a geometric mixture,
along with data.
Usage
## S3 method for class 'npgeom'
plot(
x,
mix,
beta,
col = "red",
add = FALSE,
components = TRUE,
main = "npgeom",
lwd = 1,
lty = 1,
xlab = "Data",
ylab = "Density",
...
)
Arguments
x |
an object of class |
mix |
an object of class |
beta |
the structural parameter (not used for a geometric mixture). |
col |
the color of the density curve to be plotted. |
add |
if |
components |
if |
main, lwd, lty, xlab, ylab |
arguments for graphical parameters
(see |
... |
arguments passed on to function |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
See Also
nnls, cnm, cnmms,
plot.nspmix.
Examples
mix = disc(pt=c(0.2,0.6), pr=c(0.3,0.7)) # a discrete distribution
x = rnpgeom(200, mix)
plot(x, mix)
Plotting a nonparametric negative binomial mixture
Description
Function plot.npnbinom plots a negative
binomial mixture, along with data.
Usage
## S3 method for class 'npnbinom'
plot(
x,
mix,
beta,
col = "red",
add = FALSE,
components = TRUE,
main = "npnbinom",
lwd = 1,
lty = 1,
xlab = "Data",
ylab = "Density",
...
)
Arguments
x |
an object of class |
mix |
an object of class |
beta |
the structural parameter (not used for a negative binomial mixture). |
col |
the color of the density curve to be plotted. |
add |
if |
components |
if |
main, lwd, lty, xlab, ylab |
arguments for graphical parameters
(see |
... |
arguments passed on to function |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
See Also
nnls, cnm,
cnmms, plot.nspmix.
Examples
mix = disc(pt=c(0.2,0.5), pr=c(0.3,0.7)) # a discrete distribution
x = rnpnbinom(200, 10, mix)
plot(x, mix)
Plotting a Nonparametric or Semiparametric Normal Mixture
Description
Function plot.npnorm plots the normal mixture.
Usage
## S3 method for class 'npnorm'
plot(
x,
mix,
beta,
breaks = NULL,
col = 2,
len = 100,
add = FALSE,
border.col = NULL,
border.lwd = 1,
fill = "lightgrey",
main,
lwd = 2,
lty = 1,
xlab = "Data",
ylab = "Density",
components = c("proportions", "curves", "null"),
lty.components = 2,
lwd.components = 2,
...
)
Arguments
x |
an object of class |
mix |
an object of class |
beta |
the structural parameter. |
breaks |
the rough number bins used for plotting the histogram. |
col |
the color of the density curve to be plotted. |
len |
the number of points roughly used to plot the density curve over the interval of length 8 times the component standard deviation around each component mean. |
add |
if |
border.col |
color for the border of histogram boxes. |
border.lwd |
line width for the border of histogram boxes. |
fill |
color to fill in the histogram boxes. |
main, lwd, lty, xlab, ylab |
arguments for graphical parameters
(see |
components |
if |
lty.components, lwd.components |
line type and width for the component curves. |
... |
arguments passed on to function |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
See Also
nnls, cnm,
cnmms, plot.nspmix.
Examples
mix = disc(pt=c(0,4), pr=c(0.3,0.7)) # a discrete distribution
x = rnpnorm(200, mix, sd=1)
plot(x, mix, beta=1)
Plotting a nonparametric Poisson mixture
Description
Function plot.nppois plots a Poisson mixture,
along with data.
Usage
## S3 method for class 'nppois'
plot(
x,
mix,
beta,
col = 2,
add = FALSE,
components = c("proportions", "curves", "null"),
main = "nppois",
lwd = 1,
lty = 1,
xlab = "Data",
ylab = "Density",
xlim = NULL,
...
)
Arguments
x |
an object of class |
mix |
an object of class |
beta |
the structural parameter (not used for a Poisson mixture). |
col |
the color of the density curve to be plotted. |
add |
if |
components |
if |
main, lwd, lty, xlab, ylab, xlim |
arguments for graphical
parameters (see |
... |
arguments passed on to function |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
See Also
nnls, cnm,
cnmms, plot.nspmix.
Examples
mix = disc(pt=c(0,4), pr=c(0.3,0.7)) # a discrete distribution
x = rnppois(200, mix)
plot(x, mix)
Plots a function for an object of class nspmix
Description
Plots a function for the object of class
nspmix, currently either using the plot function of the
class or plotting the gradient curve (or its first derivative)
data must belong to a mixture family, as specified by its class.
Class nspmix is an object returned by function cnm,
cnmms, cnmpl or cnmap.
Usage
## S3 method for class 'nspmix'
plot(x, data, type = c("probability", "gradient"), ...)
## S3 method for class 'nspmix'
plot(x, data, type=c("probability","gradient"), ...)
Arguments
x |
an object of a mixture model class |
data |
a data set from the mixture model |
type |
the type of function to be plotted: the probability model of the
mixture family ( |
... |
arguments passed on to the |
Details
Function plot.nspmix plots either the mixture model, if the family of
the mixture provides an implementation of the generic plot function,
or the gradient function.
data must belong to a mixture family, as specified by its class.
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86
See Also
plot.nspmix, nnls,
cnm, cnmms, npnorm,
nppois.
nnls, cnm, cnmms,
cnmpl, cnmap, npnorm,
nppois.
Examples
## Poisson mixture
x = rnppois(200, disc(c(1,4), c(0.7,0.3)))
plot(cnm(x), x)
## Normal mixture
x = rnpnorm(200, disc(c(0,4), c(0.3,0.7)), sd=1)
r = cnm(x, init=list(beta=0.5)) # sd = 0.5
plot(r, x)
plot(r, x, type="g")
plot(r, x, type="g", order=1)
## Poisson mixture
x = rnppois(200, disc(c(1,4), c(0.7,0.3)))
r = cnm(x)
plot(r, x, "p")
plot(r, x, "g")
## Normal mixture
x = rnpnorm(200, mix=disc(c(0,4), c(0.3,0.7)), sd=1)
r = cnm(x, init=list(beta=0.5)) # sd = 0.5
plot(r, x, "p")
plot(r, x, "g")
Plot the Gradient Function
Description
Function plotgrad plots the gradient function or its first
derivative of a nonparametric mixture.
Usage
plotgrad(
x,
mix,
beta,
len = 500,
order = 0,
col = 4,
col2 = 2,
add = FALSE,
main = paste0("Class: ", class(x)),
xlab = expression(theta),
ylab = paste0("Gradient (order = ", order, ")"),
cex = 1,
pch = 1,
lwd = 1,
xlim,
ylim,
...
)
Arguments
x |
a data object of a mixture model class. |
mix |
an object of class 'disc', for a discrete mixing distribution. |
beta |
the structural parameter. |
len |
number of points used to plot the smooth curve. |
order |
the order of the derivative of the gradient function to be plotted. If 0, it is the gradient function itself. |
col |
color for the curve. |
col2 |
color for the support points. |
add |
if |
main, xlab, ylab, cex, pch, lwd, xlim, ylim |
arguments for
graphical parameters (see |
... |
arguments passed on to function |
Details
data must belong to a mixture family, as specified by its class.
The support points are shown on the horizontal line of gradient 0. The vertical lines going downwards at the support points are proportional to the mixing proportions at these points.
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86
See Also
plot.nspmix, nnls,
cnm, cnmms, npnorm,
nppois.
Examples
## Poisson mixture
x = rnppois(200, disc(c(1,4), c(0.7,0.3)))
r = cnm(x)
plotgrad(x, r$mix)
## Normal mixture
x = rnpnorm(200, disc(c(0,4), c(0.3,0.7)), sd=1)
r = cnm(x, init=list(beta=0.5)) # sd = 0.5
plotgrad(x, r$mix, r$beta)
Prints a discrete distribution function
Description
Class disc is used to represent an arbitrary
univariate discrete distribution with a finite number of support
points.
Function disc creates an object of class disc, given
the support points and probability values at these points.
Function print.disc prints the discrete distribution.
Usage
## S3 method for class 'disc'
print(x, ...)
Arguments
x |
an object of class |
... |
arguments passed on to function |
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
See Also
Examples
(d = disc(pt=c(0,4), pr=c(0.3,0.7)))
Sorting of an Object of Class npnorm
Description
Function sort.npnorm sorts an object of class
npnorm in the order of the obsersed values.
Usage
## S3 method for class 'npnorm'
sort(x, decreasing = FALSE, ...)
Arguments
x |
an object of class |
decreasing |
logical, in the decreasing (default) or increasing order. |
... |
arguments passed to function |
Examples
mix = disc(pt=c(0,4), pr=c(0.3,0.7)) # a discrete distribution
x = rnpnorm(20, mix, sd=1)
sort(x)
Sorting of an Object of Class nppois
Description
Function sort.nppois sorts an object of class
nppois in the order of the obsersed values.
Usage
## S3 method for class 'nppois'
sort(x, decreasing = FALSE, ...)
Arguments
x |
an object of class |
decreasing |
logical, in the decreasing (default) or increasing order. |
... |
arguments passed to function |
Examples
mix = disc(pt=c(0,4), pr=c(0.3,0.7)) # a discrete distribution
x = rnppois(20, mix)
sort(x)
Support space
Description
Range of the mixing variable (theta).
Usage
suppspace(x, beta)
Arguments
x |
an object of a class for data. |
beta |
instrumental parameter in a semiparametric mixture. |
Value
A vector of length 2.
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
Illness Spells and Frequencies of Thai Preschool Children
Description
Contains the results of a cohort study in north-east Thailand in which 602
preschool children participated. For each child, the number of illness
spells x, such as fever, cough or running nose, is recorded for all
2-week periods from June 1982 to September 1985. The frequency for each
value of x is saved in the data set.
Format
A data frame with 24 rows and 2 variables:
x: values of x.
freq: frequencies for each value of x.
Source
Bohning, D. (2000). Computer-assisted Analysis of Mixtures and Applications: Meta-analysis, Disease Mapping, and Others. Boca Raton: Chapman and Hall-CRC.
References
Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.
See Also
Examples
data(thai)
x = nppois(thai)
plot(cnm(x), x)
Toxoplasmosis Data
Description
Contains the number of subjects testing positively for toxoplasmosis in 34 cities of El Salvador, with various rainfalls.
Format
A numeric matrix with four columns:
city: city identification code.
y: the number of subjects testing positively for toxoplasmosis.
n: the number of subjects tested.
rainfall: the annual rainfall of the city, in meters.
References
Efron, B. (1986). Double exponential families and their use in generalized linear regression. Journal of the American Statistical Association, 81, 709-721.
Aitkin, M. (1996). A general maximum likelihood analysis of overdispersion in generalised linear models. Statistics and Computing, 6, 251-262.
Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86.
See Also
Examples
data(toxo)
x = mlogit(toxo)
cnmms(x)
Valid parameter values
Description
A generic method used to return TRUE if the
values of the paramters use for a nonparametric/semiparametric
mixture are valid, or FALSE if otherwise.
Usage
valid(x, beta, theta)
Arguments
x |
an object of a class for data. |
beta |
instrumental parameter in a semiparametric mixture. |
theta |
values of the mixing variable. |
Value
A logical value.
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
Weights
Description
Weights or frequencis of observations.
Usage
weight(x, beta)
Arguments
x |
an object of a class for data. |
beta |
instrumental parameter in a semiparametric mixture. |
Value
a numeric vector of the weights.
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
Weighted Histograms
Plots or computes the histogram with observations with
multiplicities/weights.
Just like hist, whist can either plot the histogram
or compute the values that define the histogram, by setting
plot to TRUE or FALSE.
The histogram can either be the one for frequencies or density, by
setting freq to TRUE or FALSE.
Description
Weighted Histograms
Plots or computes the histogram with observations with multiplicities/weights.
Just like hist, whist can either plot the histogram
or compute the values that define the histogram, by setting
plot to TRUE or FALSE.
The histogram can either be the one for frequencies or density, by
setting freq to TRUE or FALSE.
Usage
whist(
x,
w = 1,
breaks = "Sturges",
plot = TRUE,
freq = NULL,
xlim = NULL,
ylim = NULL,
xlab = "Data",
ylab = NULL,
main = NULL,
add = FALSE,
col = "lightgray",
border = NULL,
lwd = 1,
...
)
Arguments
x |
a vector of values for which the histogram is desired. |
w |
a vector of multiplicities/weights for the values in
|
breaks, plot, freq, xlim, ylim, xlab, ylab, main, add, col, border, lwd |
These arguments have similar functionalities to their namesakes
in function |
... |
arguments passed on to function |
Value
breaks |
the break points. |
counts |
weighted counts over the intervals determined by
|
density |
density values over the intervals determined by
|
mids |
midpoints of the intervals determined by
|
Author(s)
Yong Wang <yongwang@auckland.ac.nz>
See Also
hist.