4.1 Definition of the Multinomial Model

• Let \(K\) be the number of categories of the variable of interest \(𝑌\sim multinomial\left(\boldsymbol{\theta}\right)\), with \(\boldsymbol{\theta}=\left(p_{1},p_{2},\dots ,p_{k}\right)\) and \(\sum_{k=1}^{K}p_{k}=1\).

• Let \(N_i\) be the number of elements in the i-th domain and \(N_{ik}\) be the number of elements in the k-th category. Note that \(\sum_{k=1}^{K}N_{ik}=N_{i}\) and \(p_{ik}=\frac{N_{ik}}{N_{i}}\).

• Let \(\hat{p}_{ik}\) be the direct estimation of \(p_{ik}\) and \(v_{ik}=Var\left(\hat{p}_{ik}\right)\), and denote the estimator of the variance by \(\hat{v}_{ik}=\widehat{Var}\left(\hat{p}_{ik}\right)\)

Note that the design effect changes between categories; therefore, the first step is to define the effective sample size per category. This is:

The estimation of \(\tilde{n}\) is given by \(\tilde{n}_{ik} = \frac{(\tilde{p}_{ik}\times(1-\tilde{p}_{ik}))}{\hat{v}_{ik}},\)

\(\tilde{y}_{ik}=\tilde{n}_{ik}\times\hat{p}_{ik}\)

Then, \(\hat{n}_{i} = \sum_{k=1}^{K}\tilde{y}_{ik}\)

From where it follows that \(\hat{y}_{ik} = \hat{n}_i\times \hat{p}_{ik}\)

Let \(\boldsymbol{\theta}=\left(p_{1},p_{2}, p_{3}\right)^{T}=\left(\frac{N_{i1}}{N_{i}},\frac{N_{i2}}{N_{i}}\frac{N_{i3}}{N_{i}}\right)^{T}\), then the multinomial model for the i-th domain would be:

\[ \left(\tilde{y}_{i1},\tilde{y}_{i2},\tilde{y}_{i3}\right)\mid\hat{n}_{i},\boldsymbol{\theta}_{i}\sim multinomial\left(\hat{n}_{i},\boldsymbol{\theta}_{i}\right) \] Now, you can write \(p_{ik}\) as follows:

\(\ln\left(\frac{p_{i2}}{p_{i1}}\right)=\boldsymbol{X}_{i}^{T}\beta_{2} + u_{i2}\) and \(\ln\left(\frac{p_{i3}}{p_{i1}}\right)=\boldsymbol{X}_{i}^{T}\beta_{3}+ u_{i3}\)

Given the restriction \(1 = p_{i1} + p_{i2} + p_{i3}\) then \[p_{i1} + p_{i1}(e^{\boldsymbol{X}_{i}^{T}\boldsymbol{\beta_{2}}}+ u_{i2})+p_{i1}(e^{\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{3}} + u_{i3})\] from where it follows that

\[ p_{i1}=\frac{1}{1+e^{\boldsymbol{X}_{i}^{T}\boldsymbol{\beta_{2}}}+ u_{i2}+e^{\boldsymbol{X_{i}}^{T}\boldsymbol{\beta_{3}}}+ u_{i3}} \]

The expressions for \(p_{i2}\) and \(p_{i3}\) would be:

\[ p_{i2}=\frac{e^{\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{2}} + u_{i2}}{1+e^{\boldsymbol{X}_{i}^{T}\boldsymbol{\beta_{2}}}+ u_{i2}+e^{\boldsymbol{X_{i}}^{T}\boldsymbol{\beta_{3}}}+ u_{i3}} \]

\[ p_{i3}=\frac{e^{\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{3}}+ u_{i3}}{1+e^{\boldsymbol{X}_{i}^{T}\boldsymbol{\beta_{2}}}+ u_{i2}+e^{\boldsymbol{X_{i}}^{T}\boldsymbol{\beta_{3}}}+ u_{i3}} \]