MB: Multivariate multiplicative binomial distribution

In RobinHankin/MM: The Multiplicative Multinomial Distribution

Multivariate multiplicative binomial distribution

Description

\loadmathjax

Various utilities to coerce and manipulate MB objects

Usage

MB(dep, m, pnames=character(0))
## S3 method for class 'MB'
as.array(x, ...)
## S4 method for signature 'MB'
getM(x)
## S3 method for class 'gunter_MB'
print(x, ...)

Arguments

dep	Primary argument to MB(). Typically a matrix with each row being an observation (see ‘details’ section below for an example). If an object of class Oarray, function MB() coerces to an MB object
m	Vector containing the relative sizes of the various marginal binomial distributions
x	Object of class MB to be converted to an Oarray object
...	Further arguments to as.array(), currently ignored
pnames	In function MB(), a character vector of names for the entries

Details

Function MB() returns an object of class MB. This is essentially a matrix with one row corresponding to a single observation; repeated rows indicate identical observations as shown below. Observational data is typically in this form. The idea is that the user can coerce to a gunter_MB object, which is then analyzable by Lindsey().

The multivariate multiplicative binomial distribution is defined by \mjdeqn \prod_i=1^t m_i\choose x_i\, z_ip_i^x_iq_i^z_i\theta_i^x_iz_i \prod_i < j\phi_ij^x_ix_j Equation 20 of the vignette

Thus if \mjeqn\theta=\phi=1theta=phi=1 the system reduces to a product of independent binomial distributions with probability \mjseqnp_i and size \mjseqnm_i for \mjeqni=1,...,t1,...,t.

There follows a short R transcript showing the MB class in use, with annotation.

The first step is to define an m vector:

R> m <- c(2,3,1)
 

This means that \mjeqnm_1=2,m_2=3,m_3=1m1=2,m2=3,m3=1. So \mjeqnm_1=2m1=2 means that \mjseqni=1 corresponds to a binomial distribution with size 2 [that is, the observation is in the set \mjeqn{0,1,2}0,1,2]; and \mjeqnm_2=3m2=3 means that \mjseqni=2 corresponds to a binomial with size 3 [ie the set \mjeqn{0,1,2,3}0,1,2,3].

Now we need some observations:

R> a <- matrix(c(1,0,0, 1,0,0, 1,1,1, 2,3,1, 2,0,1),5,3,byrow=T)
R> a
     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    1    0    0
[3,]    1    1    1
[4,]    2    3    1
[5,]    2    0    1 

In matrix a, the first observation, viz c(1,0,0) is interpreted as \mjeqnx_1=1,x_2=0,x_3=0x1=1,x2=0,x3=0. Thus, because \mjeqnx_i+z_i=m_ixi+zi=mi, we have \mjeqnz_1=1,z_2=3,z_3=1z1=1,z2=3,z3=1. Now we can create an object of class MB, using function MB():

R>  mx <- MB(a, m, letters[1:3])   

The third argument gives names to the observations corresponding to the columns of a. The values of \mjeqnm_1, m_2, m_3m1,m2,m3 may be extracted using getM():

R> getM(mx)
a b c 
2 3 1 
R> 

The getM() function returns a named vector, with names given as the third argument to MB().

Now we illustrate the print method:

R> mx
     a na     b nb     c nc    
[1,] 1 1      0 3      0 1     
[2,] 1 1      0 3      0 1     
[3,] 1 1      1 2      1 0     
[4,] 2 0      3 0      1 0     
[5,] 2 0      0 3      1 0     
R> 

See how the columns are in pairs: the first pair total 2 (because \mjeqnm_1=2m1=2), the second pair total 3 (because \mjeqnm_2=3m2=3), and the third pair total 1 (because \mjeqnm_3=1m3=1). Each pair of columns has only a single degree of freedom, because \mjeqnm_imi is known.

Also observe how the column names are in pairs. The print method puts these in place. Take the first two columns. These are named ‘a’ and ‘na’: this is intented to mean ‘a’ and ‘not a’.

We can now coerce to a gunter_MB:

R> (gx <- gunter(mx))
$tbl
   a b c
1  0 0 0
2  1 0 0
3  2 0 0
[snip]
24 2 3 1

$d
 [1] 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1

$m
a b c 
2 3 1 

Take the second line of the element tbl of gx, as an example. This reads c(1,0,0) corresponding to the observations of a,b,c respectively, and the second line of element d [“d” for “data”], viz 2, shows that this observation occurred twice (and in fact these were the first two lines of a).

Now we can coerce object mx to an array:

R> (ax <- as.array(mx))
, , c = 0

   b
a   0 1 2 3
  0 0 0 0 0
  1 0 0 2 0
  2 0 0 0 0

, , c = 1

   b
a   0 1 2 3
  0 0 1 0 0
  1 0 0 0 0
  2 1 1 0 0
>

(actually, ax is an Oarray object). The location of an element in ax corresponds to an observation of abc, and the entry corresponds to the number of times that observation was made. For example, ax[1,2,0]=2 shows that c(1,2,0) occurred twice (the first two lines of a).

The Lindsey Poisson device is applicable: see help(danaher) for an application to the bivariate case and help(Lindsey) for an example where a table is created from scratch.

Author(s)

Robin K. S. Hankin

See Also

MM, Lindsey, danaher

Examples

a <- matrix(c(1,0,0, 1,0,0, 1,1,1, 2,3,1, 2,0,1),5,3,byrow=TRUE)
m <- c(2,3,1)
mx <- MB(a, m, letters[1:3])   # mx is of class 'MB'; column headings
                   #  mean "a" and "not a".
ax <- as.array(mx)
gx <- gunter(ax)
ax2 <- as.array(gx)

data(danaher)
summary(Lindsey_MB(danaher))

RobinHankin/MM documentation built on May 16, 2024, 5:26 p.m.