# If the loglikelyhood function of $Y$ is $\log \binom {N}{Y}+Y \log p+(N-Y)\log (1-p)$ what is the meaning of "$Y$ is linear in $\log \frac {p}{1-p}$"

by MrFranzén   Last Updated October 19, 2019 20:19 PM - source

If the loglikelyhood function of $$Y$$ is $$\log \binom {N}{Y}+Y \log p+(N-Y)\log (1-p)$$ what is the meaning of "$$Y$$ is linear in $$\log \frac {p}{1-p}$$"

The following taken from a book by W. Stroup on generalized linear mixed models, is meant as a motivation for the logistic model. I quote dirictly:

The p.d.f. of the binomal random variable $$y_{i,j }$$ is $$\binom {N_{i,j}} {y_{i,j} } p_{i}^{y_{i,j}}(1-p_{i})^{N_{i,j}-y_{i,j} }$$. The log likelihood is $$\log \binom {N_{i,j}} {y_{i,j} } + y_{i,j} \log p_i + (N_{i,j}-y_{i,j})\log (1-p_i)$$ which we may express as $$y_{i,j} \log (p_i/(1-p_i)) + N_{i,j} \log (1-p_i) + \log \binom {N_{i,j}} {y_{i,j} }$$. The key is the expression $$y_{i,j} \log (p_i/(1-p_i))$$ - it reveals the random variable $$y_{i,j }$$ to be linear in $$\log (p_i/(1-p_i))$$. In statistical modeling, the expression $$\log (p_i/(1-p_i))$$ is known as the logit of $$p_i$$. A possible model is, thus, logit$$(p_i )=\log (p_i/(1-p_i))=\beta_0 + \beta_1 X_i$$.

I cannot understand what is meant with

"it reveals the random variable $$y_{i,j }$$ to be linear in $$\log (p_i/(1-p_i))$$."

How should this be intepreted?

Grateful for any help!

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