GenerateBOD function

Generate Overdispersed Binomial Outcome Data

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.

GenerateBOD(N,n,pi,rho)

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

Returns

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

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 $n,pi,rho$,

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.

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)

References

\insertRef manoj2013mcdonaldfitODBOD