hmmm.model {hmmm}

define a hmm model

Description

Function to define a hierarchical multinomial marginal model.

Usage

hmmm.model(marg = NULL, dismarg = 0, lev, cocacontr = NULL, strata = 1,
Z = NULL, ZF = Z, X = NULL, D = NULL, E = NULL, 
names = NULL, formula = NULL, sel = NULL)

Arguments

marg

A list of the marginal sets and their marginal interactions as described in Bartolucci et al. (2007). See below

dismarg

Similar to marg but used to define inequalities Kln(Am)>0. Default 0 if there are no inequalities

lev

Number of categories of the variables

cocacontr

A list of zero-one matrices to build "r" logits created by the function ‘recursive’

strata

Number of strata defined by the combination of the categories of the covariates

Z

Zero-one matrix describing the strata

ZF

Zero-one matrix for strata with fixed number of observations

X

Design matrix for Cln(Mm)=Xbeta. Identity matrix if not declared. It can be defined later or changed only by using the function ‘create.XMAT’

D

If the matrix D is declared, the inequalities are expressed as DKln(Am)>0. Useful for changing the sign of inequalities or for selecting a subset of inequalities

E

If E is a matrix, then E defines the equality contrasts as ECln(Mm)=0

names

A character vector whose elements are the names of the variables

formula

Formula of the reference log-linear model

sel

Vector reporting the positions of the interactions constrained to be zero

Details

Variables are denoted by integers, the lower the number identifying the variable the faster its category subscript changes in the vectorized contingency table. Suppose that the variables are 1 and 2 with categories k_1, k_2, the joint frequencies yij, where i=1,...,k_1, j=1,...,k_2, are arranged in a vector so that the subscript i changes faster than j. If strata is greater than one, the vectorized contingency tables must be entered strata by strata. So that, for example, if the variables are distinguished in responses and covariates, the categories of the covariates determine the strata and the data are arranged in such a way that the categories of the response variable changes faster than the categories of covariate. The names of the variables in names must be declared according to the order of the variables.

The list marg of the marginal sets of a complete hierarchical marginal parameterization, together with the types of logits for the variables, must be created by the function ‘marg.list’. See the help of this function for more details. If marg is not specified the multivariate logit model by Glonek and McCullagh (1995) with interactions of type local is used. The list marg is used to create the link function Cln(Mm) and its derivative (m is the vector of expected frequencies).

If the model is defined in the form Cln(Mm) = Xbeta, the matrix X has to be declared (see the function ‘create.XMAT’). If there are only nullity constraints on parameters, the model is in the form ECln(Mm)=0 and X is ignored. In such a case, E can be declared as matrix or it is automatically constructed if sel is declared. If sel is not NULL, then the model is defined under equality constraints, i.e. ECln(Mm)=0. When X, E and sel are left at default level, a saturated model is defined.

For models with inequality constraints on marginal parameters, the input argument dismarg is declared as a list whose components are of type: list(marg=c(1,2),int=list(c(1),c(1,2)), types=c("g","l" )), with elements marg: the marginal set, int: the list of the interaction subject to inequality constraint, and types: the logit used for every variable ("g"=global, "l"=local, "c"=continuation, "rc"=reverse continuation, "r"=recursive, "b"=baseline, "marg" is assigned to each variable not belonging to the marginal set). This list is used to create the link function Cln(Mm) and its derivative for the inequality constraints.

The matrix Z is of dimension c x s, where c is the number of counts and s is the number of strata or populations. Thus, the rows correspond to the number of observations and the columns correspond to the strata. A 1 in row i and column j means that the ith count comes from the jth stratum. Note that Z has exactly one 1 in each row, and at least one 1 in each column. When the population matrix Z is a column vector of 1 indicates that all the counts come from the same and only stratum. For hmm models, it is assumed that all the strata have the same number of response levels. If Z is not given, a population Z matrix corresponding to data entered by strata is defined and ZF=Z. For non-zero ZF, the columns are a subset of the columns in Z. If the jth column of Z is included in ZF, then the sample size of the jth stratum is considered fixed, otherwise if the jth column of Z is NOT included in ZF, the jth stratum sample size is taken to be a realization of a Poisson random variable. As ZF=Z the sample size in every stratum is fixed; this is the (product-)multinomial setting.

The formula of the reference log-linear model must be defined using the names of the variables declared in names, for example names<-c("A","B","C","D"), formula=~A*C*D+B*C*D+A:B. The interactions not involved in formula cannot be further constrained in the marginal model. The default formula = NULL indicates the saturated log-linear model as reference model. The likelihood function of the reference model is maximized by ‘hmmm.mlfit’ under the constraints ECln(Mm)=0 on the marginal parameters.

The arguments dismarg and formula can be used only if strata=1.

Value

An object of the class hmmmmod; it describes a marginal model that can be estimated by ‘hmmm.mlfit’.

References

Bartolucci F, Colombi R, Forcina A (2007) An extended class of marginal link functions for modelling contingency tables by equality and inequality constraints. Statistica Sinica, 17, 691-711.

Bergsma WP, Rudas T (2002) Marginal models for categorical data. The Annals of Statistics, 30, 140-159.

Cazzaro M, Colombi R (2009) Multinomial-Poisson models subject to inequality constraints. Statistical Modelling, 9(3), 215-233.

Colombi R, Giordano S, Cazzaro M (2014) hmmm: An R Package for hierarchical multinomial marginal models. Journal of Statistical Software, 59(11), 1-25, URL http://www.jstatsoft.org/v59/i11/.

Glonek GFV, McCullagh P (1995) Multivariate logistic models for contingency tables. Journal of the Royal Statistical Society, B, 57, 533-546.

See Also

hmmm.model.X, create.XMAT, summary.hmmmmod, print.hmmmmod, marg.list, recursive, hmmm.mlfit

Examples

data(madsen)
# 1 = Influence; 2 = Satisfaction;  3 = Contact; 4 = Housing 
names<-c("Inf","Sat","Co","Ho")
y<-getnames(madsen,st=6)

# hmm model -- marginal sets: {3,4} {1,3,4} {2,3,4} {1,2,3,4}
margi<-c("m-m-l-l","l-m-l-l","m-l-l-l","l-l-l-l")
marginals<-marg.list(margi,mflag="m")
model<-hmmm.model(marg=marginals,lev=c(3,3,2,4),names=names)
summary(model)

# hmm model with equality constraints
# independencies 1_||_4|3 and 2_||_3|4 impose equality constraints 
sel<-c(12:23,26:27,34:39) # positions of the zero-constrained interactions 
model_eq<-hmmm.model(marg=marginals,lev=c(3,3,2,4),sel=sel,names=names)
summary(model_eq)

# hmm model with inequality constraints
# the distribution of 1 given 4 is stochastically decreasing wrt the categories of 3;
# the distribution of 2 given 3 is stochastically decreasing wrt the categories of 4:
marg134ineq<-list(marg=c(1,3,4),int=list(c(1,3)),types=c("l","marg","l","l"))
marg234ineq<-list(marg=c(2,3,4),int=list(c(2,4)),types=c("marg","l","l","l"))
ineq<-list(marg134ineq,marg234ineq)
model_ineq<-hmmm.model(marg=marginals,lev=c(3,3,2,4),dismarg=ineq,D=diag(-1,8),names=names)
summary(model_ineq)
# The argument D is used to turn the 8 inequalities from 
# non-negative (default) into non-positive constraints