# The identity link not used for binary response

Questions: The identity link is the standard one with normal responses but is not often used with binary or count responses. Why do you think this is?

My idea: The range for a linear predictor, and the range of the identity link applied to a binomial probability or to a Poisson mean. I am kinda confused

• It’s to map the real line to a range of plausible values of the parameter of the response distribution. Instead of $Poisson(-1)$, the $log$ link gives us $Poisson(e^{-1})$. – Dave Feb 14 '20 at 3:47
• @Dave What do mean by Poisson(-1)? – Simpson's Paradox Feb 14 '20 at 3:51

If you use the identity link for the binary data, then your regression equation would be $${\rm Pr}(Y_i|x_i)\equiv p_i = \beta_0 + \beta_1 x_i.$$ Without any restriction on the coefficients, RHS can range in real-line (for example, a negative value). Since this does not make any sense for probability $$p_i$$, we transform it so that $$g(p_i)$$ takes its value in $$[-\infty, \infty]$$.