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

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!



Related Questions


Updated May 07, 2017 12:19 PM

Updated December 22, 2017 11:19 AM

Updated March 02, 2019 03:19 AM

Updated September 08, 2018 02:19 AM