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