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 \(H_0:\beta_5=\beta_6=0\).
But I am not understanding that why it is not \(H_0:\beta_2=0\) to indicate the hypothesis that changes in the log odds of obesity are the same for boys and girls? Since \(\beta_2\) indicates changes in log odds of obesity for male than that of female (assuming female is reference category).
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 \(H_0:\beta_5=\beta_6=0\).
But I am not understanding that why it is not \(H_0:\beta_2=0\) to indicate the hypothesis that changes in the log odds of obesity are the same for boys and girls? Since \(\beta_2\) indicates changes in log odds of obesity for male than that of female (assuming female is reference category).