MarginalModelFit {cmm}

MarginalModelFit

Description

Fits marginal models, by default using maximum likelihood.

Usage

MarginalModelFit(dat, model, ShowSummary = TRUE, MaxSteps = 1000, MaxStepSize = 1, 
    MaxError = 1e-20, StartingPoint = "Automatic", MaxInnerSteps = 2, 
    ShowProgress = TRUE, CoefficientDimensions="Automatic", Labels="Automatic",
    ShowCoefficients = TRUE, ShowParameters = FALSE, ParameterCoding = "Effect",
    ShowCorrelations = FALSE, Method = "ML", Title = "Summary of model fit")

Arguments

dat

vector of frequencies or data frame

model

list specified eg as list(bt,coeff,at)

ShowSummary

Whether or not to execute summary() of the output

MaxSteps

integer: maximum number of steps of the algorithm

MaxStepSize

number greater than 0 and at most 1: step size

MaxError

numeric: maximum error term

StartingPoint

vector of starting frequencies corresponding to all cells in the manifest table

MaxInnerSteps

nonnegative integer: only used for latent variable models, indicates number of steps in M step of EM algorithms

ShowProgress

boolean or integer: FALSE for no progress information, TRUE or 1 for information at every step, an integer k for information at every k-th step

CoefficientDimensions

numeric vector of dimensions of the table in which the coefficient vector is to be arranged

Labels

list of characters or numbers indicating labels for dimensions of table in which the coefficient vector is to be arranged

ShowCoefficients

boolean, indicating whether or not the coefficients are to be displayed

ShowParameters

boolean, indicating whether or not the parameters (computed from the coefficients) are to be displayed

ParameterCoding

Coding to be used for parameters, choice of "Effect", "Dummy" and "Polynomial"

ShowCorrelations

boolean, indicating whether or not to show the correlation matrix for the estimated coefficients

Method

character, choice of "ML" for maximum likelihood or "GSK" for the GSK method

Title

title of computation to appear at top of screen output

Details

The data can be a data frame or vector of frequencies. MarginalModelFit converts a data frame dat to a vector of frequencies using c(t(ftable(dat))).

The model specification is fairly flexible. We first describe the most typical way to specify the model. The model itself should typically first be written in the form of a constraint vector as B'\theta(A'\pi) = 0 where B' is a contrast matrix, A' is matrix, normally of zeroes and ones, such that A'pi gives a vector of marginal probabilities, and the function theta yields a list of (marginal) coefficients. The model is then specified as model=list(bt,coeff,at) where bt is the matrix B', at is the matrix A', and coeff represents the vector of coefficients using the generalized exp-log notation. For most of the models in the book, bt can be obtained directly using ConstraintMatrix, coeff can be obtained directly using SpecifyCoefficient, and at can be obtained directly using MarginalMatrix.

Note that CMM does not permit the C and X matrix in the model C'\theta(A'\pi) = X\beta to be specified for use in the programme. The model has to be rewritten in terms of constraints as above, which is normally straightforward to do with the use of ConstraintMatrix. For many purposes, estimates and standard errors for a beta vector as in the equation above can still be obtained using the optional argument ShowParameters=TRUE.

There are two ways to specify coeff. The first is using the generalized exp-log notation, in which case coeff[[1]] should be a list of matrices, and coeff[[2]] should be a list of predefined functions of the same length. The second is to set coeff equal to a predefined function; for example, marginal loglinear models are obtained by setting coeff="log".

The model can be specified in various other ways: as model=list(coeff,at), model=list(bt,coeff), model=at, or even just model=coeff. Furthermore, the model B'\theta(A'\pi) = d with d a nonzero vector is specified in the form model=list(bt,coeff,at,d).

To specify the simultaneous model B'\theta(A'\pi) = 0\\ \log\pi=X\beta the extended model specification model=list(margmodel,x) should be used, where margmodel has one of the above forms, and x is a design matrix, which can be obtained using DesignMatrix. Fitting is often more efficient by specifying a loglinear model for the joint distribution in this way rather than using constraints.

The default on-screen output when running fit=MarginalModelFit(...) is given by summary(fit). Important here is the distinction between coefficients and parameters, briefly described above. Standard output gives the coefficients. These are that part of model without the bt matrix, eg if the model is list(bt,coeff,at) then the coefficients are list(coeff,at). If other coefficients are needed, ModelStatistics can be used.

Latent variable models can be specified: if the size of the table for which model is specified is a multiple of the the size of the observed frequencies specified in dat, it is assumed this is due to the presence of latent variables. With respect to vectorization, the latent variables are assumed to change their value fastest.

Convergence may not always be achieved with MaxStepSize=1 and a lower value may need to be used, but not too low or convergence is slow. If the step size is too large, a typical error message is "system is computationally singular: reciprocal condition number = 1.35775e-19"

Value

Most of the following are included in any output. Use summary() to get a summary of output.

FittedFrequencies

Vector of fitted frequencies for the full table (including any latent variables).

Method

Fitting method used (currently maximum likelihood, GSK or minimum discrimination information)

LoglikelihoodRatio

ChiSquare

DiscriminationInformation

WaldStatistic

DegreesOfFreedom

PValue

p-value based on asymptotic chi-square approximation for likelihood ratio test statistic

SampleSize

BIC

Eigenvalues

ManifestFittedFrequencies

For the “coefficients” in the equation bt.coeff(at.pi)=d, the following statistics are available:

ObservedCoefficients

FittedCoefficients

CoefficientStandardErrors

CoefficientZScores

CoefficientAdjustedResiduals

CoefficientCovarianceMatrix

CoefficientCorrelationMatrix

CoefficientAdjustedResidualCovarianceMatrix

CoefficientDimensions

CoefficientTableVariableLabels

CoefficientTableCategoryLabels

The “parameters” are certain linear combinations of the coefficients. For example, if the coefficients are log probabilities, then the parameters are the usual loglinear parameters.

Parameters

For the i-th subset of variables, the parameters are obtained by

Parameters[[i]]$

The following statistics for the parameters belonging to each subset of variable are available.

Parameters[[i]]$ObservedCoefficients

Parameters[[i]]$FittedCoefficients

Parameters[[i]]$CoefficientStandardErrors

Parameters[[i]]$CoefficientZScores

Parameters[[i]]$CoefficientAdjustedResiduals

Parameters[[i]]$CoefficientCovarianceMatrix

Parameters[[i]]$CoefficientCorrelationMatrix

Parameters[[i]]$CoefficientAdjustedResidualCovarianceMatrix

Parameters[[i]]$CoefficientDimensions

Parameters[[i]]$CoefficientTableVariableLabels

Parameters[[i]]$CoefficientTableCategoryLabels

Author(s)

W. P. Bergsma [e-mail]

References

Bergsma, W. P. (1997). Marginal models for categorical data. Tilburg, The Netherlands: Tilburg University Press. http://stats.lse.ac.uk/bergsma/pdf/bergsma_phdthesis.pdf

See Also

SampleStatistics, ModelStatistics

Examples

# see also the built-in data sets

data(NKPS)

# Fit the model asserting Goodman and Kruskal's gamma is zero for
# Child's attitude toward sex role's (NKPS[,3], three categories) and
# parent's attitude toward sex role's (NKPS[,4], three categories).

coeff = SpecifyCoefficient("GoodmanKruskalGamma",c(3,3))
fit = MarginalModelFit(NKPS[,c(3,4)], coeff )


# Marginal homogeneity (MH) in a 3x3 table AB
# Note that MH is equivalent to independence in the 2x3 table of marginals IR, in which 
# the row with I=1 gives the A marginal, and the row with I=2 gives the B marginal 
n <- c(1,2,3,4,5,6,7,8,9)
at <- MarginalMatrix(c("A","B"),list(c("A"),c("B")),c(3,3))
bt <- ConstraintMatrix(c("I","R"),list(c("I"),c("R")),c(2,3))
model <- list( bt, "log", at)
fit <- MarginalModelFit(n,model)

#Output can be tidied up:
fit <- MarginalModelFit(n,model,CoefficientDimensions=c(2,3))