Fitting Saturated Models for the Marginal Probabilities of Categorization via Maximum Likelihood under MAR and MCAR assumption
Description
satMarML
fits saturated models for the marginal probabilities of
categorization as well as missing at random (MAR) or missing completely at random (MCAR)
models for the missingness mechanism by maximum likelihood (ML) methodology.
It is based on input data of a readCatdata
object.
Linear, loglinear and functional linear models may be subsequently fitted,
respectively, using functions linML()
, loglinML()
and
funlinWLS()
.
Usage
1 2 
Arguments
catdataobj 

missing 
the covariance matrix (based on a Fisher information matrix) of the
estimates for the marginal probabilities of categorization may be computed under

method 
the iterative processes available are: 
start 
by default, the function uses the proportions of the complete data as starting
values in the iterative process, but the current argument allows the user to inform an
alternative starting value for all marginal probabilities except the one corresponding to the
last category of each multinomial, i.e., a vector of dimension 
zero 
when there are sampling zeros in the complete data, these frequencies are replaced
by small values just for the computation of the starting values; this avoids the use of
starting values on the boundary of the parameter space and also allows to incorporate
information from other missingness patterns in the EM iterative process; by default, the
function replaces the values by 
maxit 
the maximum number of iterations (the default is 100). 
trace 
the alternatives are: 0 for no printing (default), 1 for showing only the value of the likelihood ratio statistic at each iteration of the iterative process, and 2 for including also the parameter estimates at each iteration. 
epsilon1 
the convergence criterion of the iterative process is attained if the
absolute difference of the values of the likelihood ratio statistic of successive iterations
is less than the value defined in 
epsilon2 
the convergence criterion of the iterative process is attained if the
absolute differences of the values of estimates for all parameters of the marginal
probabilities of categorization in consecutive iterations are less than the value defined
in 
zeroN 
values used to replace null frequencies in the denominator of the Neyman statistic;
by default, the function replaces the values by 
digits 
integer value indicating the number of decimal places to round results when shown
by 
Details
The generic functions print
and summary
are used to print the results and to obtain a
summary thereof.
Value
An object of the class satMarML
is a list containing most of the components of the
readCatdata
source object informed in the argument catdataobj
as well as
the following components:
theta 
vector of ML estimates for all productmultinomial probabilities under the saturated model for the marginal probabilities of categorization and an assumption of an ignorable missingness mechanism; this is the same under MAR and under MCAR. 
Vtheta 
corresponding estimated covariance matrix based on the Fisher information matrix obtained under the assumed missingness mechanism, leading to different results depending whether the assumption is MAR or MCAR). 
QvMCAR 
likelihood ratio statistic for the conditional test of MCAR given a MAR assumption. 
QpMCAR 
Pearson statistic for the conditional test of MCAR given a MAR assumption. 
QnMCAR 
Neyman statistic for the conditional test of MCAR given a MAR assumption. 
glMCAR 
degrees of freedom for the conditional tests of MCAR given a MAR assumption. 
alphast 
ML estimates for the conditional probabilities of missingness under the assumed missingness mechanism (MAR or MCAR). 
yst 
ML estimates for the augmented frequencies under the saturated model for the marginal probabilities and the assumed missingness mechanism (MAR or MCAR). 
References
Paulino, C.D. e Singer, J.M. (2006). Analise de dados categorizados (in Portuguese). Sao Paulo: Edgard Blucher.
Poleto, F.Z. (2006). Analise de dados categorizados com omissao (in Portuguese). Dissertacao de mestrado. IMEUSP. http://www.poleto.com/missing.html.
Poleto, F.Z., Singer, J.M. e Paulino, C.D. (2007). Analyzing categorical data with complete or missing responses using the Catdata package. Unpublished vignette. http://www.poleto.com/missing.html.
Poleto, F.Z., Singer, J.M. e Paulino, C.D. (2012). A productmultinomial framework for categorical data analysis with missing responses. To appear in Brazilian Journal of Probability and Statistics. http://imstat.org/bjps/papers/BJPS198.pdf.
Singer, J. M., Poleto, F. Z. and Paulino, C. D. (2007). Catdata: software for analysis of categorical data with complete or missing responses. Actas de la XII Reunion Cientifica del Grupo Argentino de Biometria y I Encuentro ArgentinoChileno de Biometria. http://www.poleto.com/SingerPoletoPaulino2007GAB.pdf.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35  #Example 13.4 of Paulino and Singer (2006)
e134.TF<c(12,4,5,2, 50,31, 27,12)
e134.Zp<cbind(kronecker(diag(2),rep(1,2)),kronecker(rep(1,2),diag(2)))
e134.Rp<c(2,2)
e134.catdata<readCatdata(TF=e134.TF,Zp=e134.Zp,Rp=e134.Rp)
e134.satmcarml<satMarML(e134.catdata,missing="MCAR")
e134.satmarml<satMarML(e134.catdata,method="FSMCAR")
e134.satmarml2<satMarML(e134.catdata,method="NR/FSMAR")
e134.satmcarml
#Compare the estimates of the probabilities, standard errors,
#number of iterations and augmented frequencies
summary(e134.satmcarml)
summary(e134.satmarml)
summary(e134.satmarml2)
#Example 1 of Poleto et al (2012)
smoking.TF<rbind(c(167,17,19,10,1,3,52,10,11, 176,24,121, 28,10,12),
c(120,22,19, 8,5,1,39,12,12, 103, 3, 80, 31, 8,14))
smoking.Zp < kronecker(t(rep(1,2)),
cbind(kronecker(diag(3),rep(1,3)),
kronecker(rep(1,3),diag(3))))
smoking.Rp<rbind(c(3,3),c(3,3))
smoking.catdata<readCatdata(TF=smoking.TF,Zp=smoking.Zp,Rp=smoking.Rp)
smoking.catdata
smoking.satmcarml<satMarML(smoking.catdata,missing="MCAR")
smoking.satmarml<satMarML(smoking.catdata,method="FSMCAR")
smoking.satmarml2<satMarML(smoking.catdata,method="NR/FSMAR")
smoking.satmarml
summary(smoking.satmcarml)
summary(smoking.satmarml)
summary(smoking.satmarml2)

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