I am doing text modeling, where each document can belong to multiple topics. I specify the desired topic number to be 3 and the three topics to be
education. So the model will output a triplet representing the topic mixture of each document.
Let $p_1,p_2,p_3$ denote Pr(
education), respectively. So a news article that talks about a football match being canceled due to the bad weather may have a triplet $(p_1,p_2,p_3)=(0.4,0.5,0.1)$. The three elements in the tuple sum to $1$.
Now I wish to explore the relationship between topics and the popularity of the document. I start with multiple linear regression. Since $p_1+p_2+p_3=1$, I should only include two of $p_1,p_2,p_3$ like $$y=\beta_0+\beta_1p_1+\beta_2p_2,$$ where $y$ is the reader-rated popularity of the documents.
Denote $p_3$'s coefficient by $\beta_3$ (not in the current model above). Is it possible to test all of the following three hypotheses with the model above?
Is my existing answer below correct? If not, how can I do that?