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Thread: Defining Contrast Matrix

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    Defining Contrast Matrix




    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}=1if 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?
    Last edited by Cynderella; 05-18-2016 at 07:11 AM.

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