Estimation of cells counts using the pseudo-Bayes estimator.
Usage
pBayes(x, method="m.ind", const=NULL)
Arguments
x
A contingency table with observed cell counts. Typically the output of table or xtabs. More in general an R array with the counts.
method
The method for estimating the final cell frequencies. The following options are available:
method = "Jeffreys", consists in adding 0.5 to each cell before estimation of the relative frequencies, in particular the final relative frequencies are derived by means of ep_i = (n_i + 0.5)/(n + c/2), being n the sum of all the counts and c the number of cells in the table. The final estimated cell frequencies are obtained as en_i=ep_i*n.
method = "minimax", consists in adding sqrt(n)/c to each cell before estimation of the relative frequencies, in particular the final relative frequencies are derived by means of ep_i = (n_i + sqrt(n)/c)/(n + sqrt(n)).
method = "invcat", consists in adding 1/c to each cell before estimation of the relative frequencies, in particular the final relative frequencies are derived by means of ep_i = (n_i + 1/c)/(n + 1).
method = "user", consists in adding a used defined constant a (a>0) to each cell before estimation of the relative frequencies with ep_i = (n_i + a)/(n + a*c). The constant a should be passed via the argument const.
method = "m.ind", the prior guess for the unknown cell probabilities is obtained by considering estimated probabilities under the mutual independence hypothesis. In such a case a data driven K is considered (see Details). This option is available when dealing with at least two-way contingency tables (length(dim(x))>=2).
method = "h.assoc", the prior guess for the unknown cell probabilities is obtained by considering estimated probabilities under the homogeneous association hypothesis. In such a case a data drive K is considered (see Details). This option is available when dealing with at least two-way contingency tables (length(dim(x))>=2).
const
Numeric value, a user defined constant a (a>0) added to each cell before estimation of the relative frequencies when method = "user". As a general rule of thumb it is preferable to avoid that the sum of constant over all the cells in table is greater than 0.20*n.
Details
This function estimates the frequencies in a contingency table by using the pseudo-Bayes approach. In practice the estimator being considered is a weighted average of the observed cell counts n_h and a suitable prior guess for cell probabilities gamma_h:
ep_h = n/(n+K)*p_h + K/(n+K)*gamma_h
K depends on the parameters of Dirichilet prior distribution being considered (for major details see Chapter 12 in Bishop et al., 1974).
It is worth noting that with a constant prior guess gamma_h=1/c (h=1,2,...,c), setting K=1 corresponds to adding 1/c to each cell before estimation of the relative frequencies (method = "invcat"); with K=c/2 the constant 0.5 is added to each cell (method = "Jeffreys"); finally when sqrt(n) the quantity sqrt(n)/c is added to each cell (method = "minimax"). All these cases corresponds to adding a flattening constant; the higher is the value of K the more the estimates will be shrinked towards gamma_h=1/c (flattening).
When method = "m.ind" the prior guess gamma_h is estimated under the hypothesis of mutual independence between the variables crossed in the initial contingency table x, supposed to be at least a two-way table. In this case the value of K is estimated via a data driven approach by considering
eK = (1 - sum(p_h^2))/(sum(gamma_h - p_h)^2)
On the contrary, when method = "h.assoc" the prior guess gamma_h is estimated under the hypothesis of homogeneous association between the variables crossed in the initial contingency table x.
Please note that when the input table is estimated from sample data where a weight is assigned to each unit, the weights should e used in estimation of the input table but it is suggested to rescale them so that they sum up to n, the sample size.
Value
A list object with three components.
invo
A vector with the sample size "n", the number of cells ("no.cells") in x, the average cell frequency ("av.cfr"), the number of cells showing frequencies equal to zero ("no.0s"), the const input argument, the chosen/estimated K ("K") and the relative size of K, i.e. K/(n+K) ("rel.K").
prior
A table having the same dimension as x with the considered prior values for the cell frequencies.
pseudoB
A table with having the same dimension as x providing the pseudo-Bayes estimates for the cell frequencies in x.
providing 
.