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).