fitODBOD (version 1.5.3)

GenerateBOD: Generate Overdispersed Binomial Outcome Data

Description

Using a three step algorithm to generate overdispersed binomial outcome data. When the number of frequencies, binomial random variable, probability of success and overdispersion are given.

Usage

GenerateBOD(N,n,pi,rho)

Value

The output of GenerateBOD gives a vector of overdispersed binomial random variables

Arguments

N

single value for number of total frequencies

n

single value for binomial random variable

pi

single value for probability of success

rho

single value for overdispersion parameter

Details

The generated binomial random variables are overdispersed based on \(rho\) for the probability of success \(pi\).

Step 1: Solve the following equation for a given \(n,pi,rho\), $$phi(z(pi),z(pi),delta)=pi(1-pi)rho + pi^2,$$

For \(delta\) where \(phi(z(pi),z(pi),delta)\) is the cumulative distribution function of the standard bivariate normal random variable with correlation coefficient \(delta\), and \(z(pi)\) denotes the \(pi^{th}\) quantile of the standard normal distribution.

Step 2: Generate $n$-dimensional multivariate normal random variables, \(Z_i=(Z_{i1},Z_{i2},ldots,Z_{in})^T\) with mean \(0\) and constant correlation matrix \(Sigma_i\) for \(i=1,2,\ldots,N,\) where the elements of \((Sigma_i)_{lm}\) are \(delta\) for \(l \ne m\).

Step 3: Now for each \(j=1,2,\ldots,n\) define \(X_{ij} = 1;\) if \(Z_{ij} < z(\pi)\), or \(X_{ij} = 0;\) otherwise. Then, it can be showed that the random variable \(Y_i=\sum_{j=1}^{n} X_{ij}\) is overdispersed relative to the Binomial distribution.

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

References

manoj2013mcdonaldfitODBOD

Examples

N <- 500    # Number of observations
n <- 10      # Dimension of multivariate normal random variables
pi <- 0.5   # Probability threshold
rho <- 0.1  # Dispersion parameter

# Generate overdispersed binomial variables
New_overdispersed_data <- GenerateBOD(N, n, pi, rho)
table(New_overdispersed_data)