lm.dibp{bivpois}

R Documentation

General Diagonal Inflated Bivariate Poisson Model

Description

Produces a "list" object which gives details regarding the fit of a bivariate diagonal inflated Poisson regression model of the form

 (Xi,Yi) ~ DIBP( ë1i,  ë2i, ë3i , D(è) )

which is equivalent to

(Xi,Yi) ~

(1-p)BP( xi, yi| ë1i,  ë2i, ë3i )

if xi¹yi

(Xi,Yi) ~

(1-p)BP( xi, yi| ë1i,  ë2i, ë3i )+pD( xi | è )

if xi=yi

 

with

log(ë1) = w1 â

log(ë2) = w2 â

log(ë3) = w3 â3

where

Ø      i =1, 2, … , n; n is the sample size

Ø      ë1 = ( ë11, ë12, … , ë1n )T, ë2 = ( ë21, ë22, … , ë2n )T and ë3 = ( ë31, ë32, … , ë3n )T are vectors of length n with the estimated lambda for each observation

Ø      w1, w2  are n´p data matrices containing explanatory variables for ë1 and ë2 .

Ø      w3        are n´p2 data matrix containing explanatory variables for ë3.

Ø      â          vector of length p; â is common for ë1 and ë2  in order to allow for common effects

Ø      â3         vector of length p2.

Ø      D(è)     is a discrete distribution with parameter vector theta used to inflate the diagonal

Ø      p          is the mixing proportion.

Usage

lm.dibp( x, y,  formula1=y1y2~., formula2=y3~1, data, zeroL3=FALSE, distribution='discrete', jmax=2, maxit=300,  pres=1e-8, print.details=FALSE)

Required Arguments

x, y

names (character objects) of response vectors.

data

Data frame containing the variables in the model.

Optional Arguments

formula1=y1y2~.

Formula type argument specifying the terms used for the linear predictors of  ë1 and ë2 . The default value fits a model with all explanatory variables of the data frame specified in data argument, having different non-equal effects on ë1 and ë2. For details on using formulas see bpformulas.html.

formula2=y3~1

Formula type argument specifying the terms used for the linear predictor of ë3. The default value fits a model with constant covariance term.

zeroL3=FALSE

Logical argument controlling whether ë3 should be set equal to zero (therefore fits a double Poisson model).

Distribution=’discrete’

Specifies the type of inflated distribution;

  1. ='discrete' :  Discrete(J=jmax),
  2. ='poisson'  :  Poisson(è)
  3. ='geometric': Geometric(è).

jmax=2

Number of parameters used in $Discrete$ distribution. This argument is not used for the Poisson or the Geometric distributions are used as for the inflation of the diagonal.

maxit=300

Maximum number of EM steps. Default value is 300 iterations.

Pres =1e-8

Precision used in stopping the EM algorithm. The algorithm stops when the relative log-likelihood difference is lower than the value of  pres.

print.details=FALSE

Argument for controlling the printing details during the iterations of the EM algorithm. The default is to print only the iteration number, the loglikelihood and its relative difference from the previous iteration. If  print.details=TRUE then the model parameters â1, â2 and â3 are additionally printed.

Value

A list object returned with the following variables.

beta

Estimates of the model parameters for â1, â2 and â3 . When a factor is used then its default set of constraints is used.

Beta1,beta2, beta3

Vectors â1, â2 and â3 containing the coefficients involved in the linear  predictors of  ë1 , ë2 and ë3 respectively. When zeroL3=TRUE then this beta3 is not calculated.

lambda1, lambda2

Vectors of length n containing the estimated ë1 and ë2 for each observation

lambda3

vector containing the values of ë3. If zeroL3=TRUE then lambda3 is equal to zero and is not calculated.

fitted1, fitted2

Vectors of length n containing the fitted values for x and y. For the bivariate Poisson model the fitted values are given by ë13 and ë23       respectively.

loglikelihood

Maximized log-likelihood of the fitted model. This is given in a vector form (one value per iteration).With this vector we can monitor the log-likelihood improvement and how EM algorithm works.

AIC, BIC

AIC and BIC of the model. Values are also provided for the double Poisson model and the saturated model.                    

parameters

Number of parameters

iterations

Number of iterations

diagonal.distribution

label for the diagonal inflated distribution used

P

mixing proportion

theta

Parameter vector of the diagonal distribution. For distribution=1 then theta has length equal to jmax with èi=theta[i] and è0 =1-sum(èi); for distribution=2, theta is the mean of the Poisson; for distribution=3, theta is is the success probability of the Geometric distribution.

Details

See bpformulas.html for help concerning the formulas objects above.

References

1.      Karlis, D. and Ntzoufras, I. (2004). Bivariate Poisson and Diagonal Inflated Bivariate Poisson Regression Models in S. (submitted). Technical Report, Athens University of Economics and Business, Athens, Greece.

2.      Karlis, D. and Ntzoufras, I. (2003). Analysis of Sports Data Using Bivariate Poisson Models. Journal of the Royal Statistical Society, D, (Statistician), 52, 381 – 393.

Authors Information

1.      Dimitris Karlis, Department of Statistics, Athens University of Economics and Business, Athens, Greece, e-mail: karlis@aueb.gr .

2.      Ioannis Ntzoufras, Department of Statistics, Athens University of Economics and Business, Athens, Greece, e-mail: ntzoufras@aueb.gr  .

See Also

pbivpois,  simple.bp , lm.bp .

Examples

#
# formula for lambda1 and lamba2
form1 <- y1y2~noncommon + z1:noncommon + z3 + I(l2*z5)
# formula for lambda3 
form2 <- y3~z1+z2+z3+z4
#
# Model  1: BivPois
ex2.m1 <-lm.bp  ( 'x', 'y', form1, form2, data=ex2.sim)
# Model  2: Zero Inflated BivPois 
ex2.m2 <-lm.dibp( 'x', 'y', form1, form2, data=ex2.sim, distribution='discrete', jmax=0 )
# Model  3: Diagonal Inflated BivPois with DISCRETE(1) diagonal inflation distribution
ex2.m3 <-lm.dibp( 'x', 'y', form1, form2, data=ex2.sim, distribution='discrete', jmax=1 )
# Model  4: Diagonal Inflated BivPois with DISCRETE(2) diagonal inflation distribution
ex2.m4 <-lm.dibp( 'x', 'y', form1, form2, data=ex2.sim, distribution='discrete', jmax=2 )
# Model  5: Diagonal Inflated BivPois with DISCRETE(3) diagonal inflation distribution
ex2.m5 <-lm.dibp( 'x', 'y', form1, form2, data=ex2.sim, distribution='discrete', jmax=3 )
# Model  6: Diagonal Inflated BivPois with DISCRETE(4) diagonal inflation distribution
ex2.m6 <-lm.dibp( 'x', 'y', form1, form2, data=ex2.sim, distribution='discrete', jmax=4 )
# Model  7: Diagonal Inflated BivPois with DISCRETE(5) diagonal inflation distribution
ex2.m7 <-lm.dibp( 'x', 'y', form1, form2, data=ex2.sim, distribution='discrete', jmax=5 )
# Model  8: Diagonal Inflated BivPois with DISCRETE(6) diagonal inflation distribution
ex2.m8 <-lm.dibp( 'x', 'y', form1, form2, data=ex2.sim, distribution='discrete', jmax=6 )
# Model  9: Diagonal Inflated BivPois with POISSON diagonal inflation distribution
ex2.m9 <-lm.dibp( 'x', 'y', form1, form2, data=ex2.sim, distribution='poisson' )
# Model 10: Diagonal Inflated BivPois with GEOMETRIC diagonal inflation distribution
ex2.m10<-lm.dibp( 'x', 'y', form1, form2, data=ex2.sim, distribution='geometric' )
#
# printing parameters of model 7
ex2.m7$beta1
ex2.m7$beta2
ex2.m7$beta3
ex2.m7$p
ex2.m7$theta
#
# printing parameters of model 9
ex2.m9$beta1
ex2.m9$beta2
ex2.m9$beta3
ex2.m9$p
ex2.m9$theta

 

 


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