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${\displaystyle log({\frac {p_{i}}{1-p_{i}}})=X_{i}\cdot {\mathcal {\beta }}}$

i is the index for each individual observation, or in the other form.

${\displaystyle EY_{i}={\frac {1}{1+e^{-X_{i}\cdot {\mathcal {\beta }}}}}}$

To estimate the parameter ${\displaystyle {\mathcal {\beta }}}$, we maximize the likelihood function:

${\displaystyle L({\mathcal {\beta }})=\Pi _{i=1}^{n}P(Y_{i}|X_{i},{\mathcal {\beta }})=\Pi _{i=1}^{n}\{{\frac {1}{1+e^{-X_{i}\cdot {\mathcal {\beta }}}}}\}^{Y_{i}}\{{\frac {e^{-X_{i}\cdot {\mathcal {\beta }}}}{1+e^{-X_{i}\cdot {\mathcal {\beta }}}}}\}^{1-Y_{i}}}$

where, ${\displaystyle Y_{i}}$ are binary dependent variables, and this maximization procedure is done by numerical method such as Newton Raphson.

Let's set up a simple case: logistic model with one factorial factor ( level, which can be more level):

${\displaystyle logit(p_{0})=\eta }$
${\displaystyle logit(p_{i})=\eta +\alpha _{i}}$

and the Odds Ratio of level-i over the base line level is

 ${\displaystyle OR=e^{\alpha _{i}}}$

The confidence interval of OR is the exponential of confidence interval of ${\displaystyle \alpha _{i}}$.

In more than one factor case, this is this still the same, conditioning on other factors remain fixed at certain level.

To decide a variable to be fix effect or random effect, explanation involve expression in R package lme4 and lattice:

1. continuous variable, there is a observable linear/nonlinear trend/correlation in response;

2. for categorical variable(factor), the level is limited (2 or 3), fixed (not a sample from a larger population), and/or its effect at each level is of major interests.

When a factor have multiple levels and is of less interest to know the effect of each specific level.

1. (1|Factor1): This would give you a random effect, graphical detection should be xyplot(y~x,data), with no systematical trend but oscillation a among different levels;

2. (0+Factor2|Factor1): This would give you random interaction, where Factor1 is random, Factor2 is fixed, graphical detection should be xyplot(y~factor2,groups=factor1,data), and there is observable oscillation for each level Factor2 within Factor1, and the trend is not parallel in groups;

3. (0+Covariate1|Factor1): This would count as random slope, graphical detection is xyplot(y~covariate1|Factor,data,type=c('p','a')), and see the slope at each level of Factor1 is different;

We would also have some doubts about if 1. 2. 3. are correlated, so use model (1+Factor2+Covariate1|Factor1) or smaller model to compare with nested model (using anova).

How to transform data into more normal likely pattern