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I have the logistic regression model
$$\text{logit}(\pi_i)=\beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_3+\beta_4x_4+\beta_5x_5$$ when $x_5=x_3\times x_4$.
I set $H_0:\beta_3=\beta_4=\beta_5=0$ and the alternative to be $\beta_3,\beta_4,\beta_5\neq0$.

As the title suggests, I would use Analysis of Deviance, but my (maybe silly) question is: when I determine the models (the full and the sub models), do I include the rest of the parameters $x_1,x_2,x_3$, or do I create a model containing only the predictors $x_3,x_4,x_5$ for the alternative "full" model and the $1$ for the "sub" model?

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You need nested models to get a useful statistic in analysis of deviance (see this short lecture). So, to test $H_0:\beta_3=\beta_4=\beta_5=0$ vs $H_1:\beta_3,\beta_4,\beta_5\neq0$, we have:

\begin{align} \text{full model: }&\text{logit}(\pi_i)=\beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_3+\beta_4x_4+\beta_5x_5\\ \text{sub model: }&\text{logit}(\pi_i)=\beta_0+\beta_1x_1+\beta_2x_2 \end{align}

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