spass (version 1.2)

sandwich2: Generate Time Series with Negative Binomial Distribution and Autoregressive Correlation Structure of Order One: NB-INAR(1)

Description

rnbinom.inar1 generates one or more independent time series following the NB-INAR(1) model. The generated data has negative binomial marginal distribution and an autoregressive covariance structure.

Usage

sandwich2(sigma, rho, theta, k, Time, dropout, Model)

Arguments

sigma

assymptotic standard deviation for Full and subpupulation

rho

correlation coefficient of the underlying autoregressive correlation structure. Must be between 0 and 1.

theta

correlation absorption coefficient if tinepoints are farther appart

k

sample size allocation factor between groups: see 'Details'.

Time

vector of measured timepoints

dropout

vector describing the percentage of dropout in every timepoint

Model

either 1 or 2, describing if 4-regressor or 3-regressor model was used.

Value

rnbinom.inar1 returns a matrix of dimension n x tp with marginal negative binomial distribution with mean mu and dispersion parameter size, and an autoregressive correlation structure between time points.

Details

The generated marginal negative binomial distribution with mean mu = \(\mu\) and size = \(\eta\) has density $$(\mu/(\mu+\eta))^x \Gamma(x + \eta)/(\Gamma(x+1)\Gamma(\eta)) (\eta/(\mu+\eta))^\eta$$ for \(0 < \mu\), \(0 < \eta\) and \(x=0, 1, 2, ...\).

Within the NB-INAR(1) model, the correlation between two time points \(t\) and \(s\) for rho = \(\rho\) is given through $$\rho^|t-s|$$ for \(0 \le \rho \le 1\).

References

McKenzie Ed (1986), Autoregressive Moving-Average Processes with Negative-Binomial and Geometric Marginal Distributions. Advances in Applied Probability Vol. 18, No. 3, pp. 679-705.

Examples

# NOT RUN {
set.seed(8)
random<-rnbinom.inar1(n=1000, size=0.6, mu=2, rho=0.8, tp=6)
cor(random)

#Check the marginal distribution of time point 3
plot(table(random[,3])/1000, xlab="Probability", ylab="Observation")
lines(0:26, dnbinom(0:26, mu=2, size=0.6), col="red")
legend("topright",legend=c("Theoretical Marginal Distribution", "Observed Distribution"), 
col=c("red", "black"), lty=1, lwd=c(1,2))

# }

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