Let Y be the response variable with categories . The p covariates are denoted by a p-dimension row vector .
For a stratified clustered sample design, each observation is represented by a row vector, , where
is the stratum index
is the cluster index within stratum h
is the unit index within cluster i of stratum h
denotes the sampling weight
is a D-dimensional column vector whose elements are indicator variables for the first D categories for variable Y. If the response of the jth unit of the ith cluster in stratum h falls in category d, the dth element of the vector is one, and the remaining elements of the vector are zero, where .
is the indicator variable for the category of variable Y
denotes the k-dimensional row vector of explanatory variables for the jth unit of the ith cluster in stratum h. If there is an intercept, then .
is the total number of clusters in the sample
is the total sample size
The following notations are also used:
denotes the sampling rate for stratum h
is the expected vector of the response variable:
Note that , where 1 is a D-dimensional column vector whose elements are 1.