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& \log Y = \\& \log \left ( \frac{p}{1-p} \right ) & = a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots \\\Leftrightarrow \dfrac{p}{1-p} & = \exp (a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots) = e^{(a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots)} \\\Leftrightarrow p & = \dfrac { \exp (a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots) }{ 1 + \exp (a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots) } = \dfrac { e^{(a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots)} }{ 1 + e^{(a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots)} } \\
adding 1 ===How to convert coefficient of binary logistic regressoin to odds ratio===When thinking about outcome probability <math>X_1p</math> makes probablity and the changed outcome probability <math>p\prime</math> asby adding <math>1</math> to explanatory variable <math>X_1</math>, the following two equations are obtained,
Because <math>\frac {\left ( \dfrac{p\prime}{1-p\prime} \right )}{ \left ( \dfrac{p}{1-p} \right ) }</math> is the odds ratio of the probablity <math>p\prime</math> when 1 is added to <math>X_1</math> to the probability <math>p</math>,
→How to convert coefficient of binary logistic regressoin to odds ratio
†'Multivariable' can be rephrased as 'Multiple'; Multivariable is <font color="red">'''NOT equal to 'Multivariate'!!'''</font>
===How to convert coefficient Conversion of binary logistic regressoin regression equation to odds ratiooutcome probability <math>p</math>===About a Equation of binary logistic regression formulacan be converted to outcome probablity <math>p</math> as,
:<math>
\begin{align}
\end{align}
</math>
:<math>
\begin{array}{lcl}\log \left ( \dfrac{p}{1-p} \right ) & = & a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots \\\log \left ( \fracdfrac{p\prime}{1-p\prime} \right ) & = & a + b_1({\color{red}X_1 + 1} ) + b_2X_2 + b_3X_3 + \cdots\end{array}
</math>
Subtraction of above these two equations makes,
:<math>
\begin{align}
\log \left ( \frac{p\prime}{1-p\prime} \right ) - \log \left ( \frac{p}{1-p} \right ) & = b_1({\color{red}X_1 + 1}) - b_1X_1 \\
& = b_1 \\
\Leftrightarrow \frac {\left ( \dfrac{p\prime}{1-p\prime} \right )}{ \left ( \dfrac{p}{1-p} \right ) } & = e^{b_1}
</math>
Because <math>\colorleft ( \dfrac{redp\prime}{e^1-p\prime} \right )</math> and <math>\left ( \dfrac{b_1p}{1-p}\right )</math> or are odds of <math>p\prime</math> and <math>p</math>, respectively, <math>\colorfrac {\left ( \dfrac{p\prime}{red1-p\prime} \right )}{\exp left (b_1\dfrac{p}{1-p} \right )}</math> is the conversion '''odds ratio''' of coefficient [the probability when <math>1</math> is added to <math>X_1</math>] to [the probability before adding]. Thus, converted <math>b_1</math> to obtain <math>\color{red}{e^{b_1}}</math> or <math>\color{red}{\exp (b_1)}</math> gives the '''odds ratio of outocome probabilities''' before and after variable <math>X_1</math> gains 1.
==Generalized linear model==
