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( lambda_1i, lambda_2i, lambda_3i , D(theta) ) which is equivalent to (Xi,Yi) ~ (1-p)BP( xi, yi| lambda_1i, lambda_2i, lambda_3i ) if xi not equal to yi (Xi,Yi) ~ (1-p)BP( xi, yi| lambda_1i, lambda_2i, lambda_3i )+pD( xi | theta_ ) if xi=yi with log(lambda_1) = w1 beta_, log(lambda_2) = w2 beta_, log(lambda_3) = w3 beta_3 where *** i =1, 2, … , n; n is the sample size *** lambda_1 = ( lambda_11, lambda_12, … , lambda_1n )T, lambda_2 = ( lambda_21, lambda_22, … , lambda_2n )T and lambda_3 = ( lambda_31, lambda_32, … , lambda_3n )T are vectors of length n with the estimated lambda for each observation *** w1, w2 are n/p data matrices containing explanatory variables for lambda_1 and lambda_2 . *** w3 are n/p2 data matrix containing explanatory variables for lambda_3. *** beta_ vector of length p; beta_ is common for lambda_1 and lambda_2 in order to allow for common effects *** beta_3 vector of length p2. *** D(theta) is a discrete distribution with parameter vector theta used to inflate the diagonal *** 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-08, print.details = FALSE)

Arguments

x,y names (character objects) of response vectors.
data Data frame containing the variables in the model.
formula1 Formula type argument specifying the terms used for the linear predictors of lambda_1 and lambda_2 . The default value fits a model with all explanatory variables of the data frame specified in data argument, having different non-equal effects on lambda_1 and lambda_2. For details on using formulas see bpformulas.
formula2 Formula type argument specifying the terms used for the linear predictor of lambda_3. The default value fits a model with constant covariance term.
zeroL3 Logical argument controlling whether lambda_3 should be set equal to zero (therefore fits a double Poisson model).
distribution Specifies the type of inflated distribution; ='discrete' : Discrete(J=jmax), ='poisson' : Poisson(theta) ='geometric': Geometric(theta).
jmax 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 Maximum number of EM steps. Default value is 300 iterations.
pres 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 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 beta_1, beta_2 and beta_3 are additionally printed.

Details

See bpformulas for help concerning the formulas objects above.

Value

A list object returned with the following variables.

beta Estimates of the model parameters for beta_1, beta_2 and beta_3 . When a factor is used then its default set of constraints is used.
Beta1,beta2, beta3 Vectors beta_1, beta_2 and beta_3 containing the coefficients involved in the linear predictors of lambda_1 , lambda_2 and lambda_3 respectively. When zeroL3=TRUE then this beta3 is not calculated.
lambda1, lambda2 Vectors of length n containing the estimated lambda_1 and lambda_2 for each observation
lambda3 vector containing the values of lambda_3. 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 lambda_1+lambda_3 and lambda_2+lambda_3 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 theta_i=theta[i] and theta_0 =1-sum(theta_i); for distribution=2, theta is the mean of the Poisson; for distribution=3, theta is is the success probability of the Geometric distribution.

Author(s)

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

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

References

1. Karlis, D. and Ntzoufras, I. (2004). Bivariate Poisson and Diagonal Inflated Bivariate Poisson Regression Models in S. (submitted). Technical Report, Department of Statistics, 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.

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

[Package bivpois version 0.43-1 Index]