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In the book Applied Longitudinal Analysis, 2nd Edition there is an example in the chapter "Marginal Models: Generalized Estimating Equations (GEE)" in "Muscatine Coronary Risk Factor Study" sub-section. I am illustrating it below :

Let \(Y_{ij}=1\)if the \(i^{\text{th}}\) child is classified as obese at the \(j^{\text{th}}\) occasion, and \(Y_{ij}=0\) otherwise.

The marginal probability of obesity at each occasion follows the logistic model

\(log\frac{\Pr(Y_{ij}=1)}{\Pr(Y_{ij}=0)}= \beta_1+\beta_2\text{gender}_i+\beta_3\text{age}_{ij}+\beta_4\text{age}_{ij}^2+\beta_5\text{gender}_i\text{age}_{ij}+\beta_6\text{gender}_i\text{age}_{ij}^2.\)

If one construct the hypothesis that changes in the log odds of obesity are the same for boys and girls, then $H_0:\beta_5=\beta_6=0$.

To test the hypothesis \(H_0:\beta_5=\beta_6=0\)
\(\Rightarrow\mathbf L\mathbf\beta = 0,\)

where \(\mathbf\beta =
\begin{pmatrix}
\beta_1
&\beta_2
&\beta_3
& \beta_4
&\beta_5
& \beta_6\\
\end{pmatrix}'
\) and \(\mathbf L\) is the contrast matrix.

But I can't write the contrast matrix for the \(H_0:\beta_5=\beta_6=0\).

Because if the \(H_0\) were \(H_0:\beta_5=\beta_6\) (notice that there ISN'T equal to \(0\) at the most right ), then I can construct the contrast matrix easily as :
\(\mathbf L =
\begin{pmatrix}
0& 0&0& 0&1& -1\\
\end{pmatrix}\) so that

\(\mathbf L\mathbf\beta = 0\)
\(\Rightarrow \begin{pmatrix}
0& 0&0& 0&1& -1\\
\end{pmatrix}\begin{pmatrix}
\beta_1\\
\beta_2\\
\beta_3\\
\beta_4\\
\beta_5\\
\beta_6\\
\end{pmatrix}=0\)

\(\Rightarrow \beta_5=\beta_6.\)

But When the \(H_0\) is \(H_0:\beta_5=\beta_6 = 0\) (notice that there IS equal to \(0\) at the most right ), then
\(\mathbf L =
\begin{pmatrix}
0& 0&0& 0&1& 0\\
0& 0&0& 0&0& 1\\
\end{pmatrix}\) so that

\(\mathbf L\mathbf\beta = 0\)
\(\Rightarrow \begin{pmatrix}
0& 0&0& 0&1& 0\\
0& 0&0& 0&0& 1\\
\end{pmatrix}\begin{pmatrix}
\beta_1\\
\beta_2\\
\beta_3\\
\beta_4\\
\beta_5\\
\beta_6\\
\end{pmatrix}=0\)

\(\Rightarrow \beta_5=0 \quad \text{and}\quad \beta_6=0,\)

but necessarily the contrast matrix is NOT correct as the row sum of a contrast matrix is equal to \(0\). How can I define the contrast matrix?
 
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