Notation

Let be the response variable with categories . The covariates are denoted by a -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

• is the unit index within cluster of stratum

• denotes the sampling weight

• is a -dimensional column vector whose elements are indicator variables for the first categories for variable . If the response of the th unit of the th cluster in stratum falls in category , the th 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

• denotes the -dimensional row vector of explanatory variables for the th unit of the th cluster in stratum . 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

• is the expected vector of the response variable:

Note that , where 1 is a -dimensional column vector whose elements are .