差分

{{Epi StatFloating_Menu}}

==Binary logistic regression=====How to convert coefficient Conversion of logit of binary logistic regressoin outcome odds to odds ratiooutcome probability <math>p</math>===About a Equation of binary logistic regression formulacan be converted to outcome probablity <math>p</math> as,

\begin{alignarray}{lrll}& \log Y = \\ & \log \left ( \fracdfrac{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)} } \\\end{alignarray}

adding 1 ===Conversion of coefficient 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,

\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}

Subtraction of above these two equations makes,

\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 \\

\Leftrightarrow \frac {\left ( \dfrac{p\prime}{1-p\prime} \right )}{ \left ( \dfrac{p}{1-p} \right ) } & = \exp (b_1) = e^{b_1}

Here Because <math>\Leftrightarrow dfrac{p\prime}{1-p\prime}</math> and <math>\dfrac{p}{1-p}</math> are odds of <math>p\prime</math> and <math>p</math>, respectively, <math>\frac {\left ( \dfrac{p\prime}{1-p\prime} \right )}{ \left ( \dfrac{p}{1-p} \right ) }</math> is the '''odds ratio ''' of [the probablity probability when <math>p\prime1</math> after 1 is added to <math>X_1</math> ] to [the probability before adding].  Thus, converted <math>pb_1</math>, so to <math>\color{red}{e\^{b_1}}</math> is the conversion of coefficient or <math>\color{red}{\exp (b_1)}</math> to obtain gives the '''odds ratio when of outocome probabilities''' before and after variable <math>X_1</math> gains <math>1</math>. ::<math>\begin{align}\exp (\text{coefficient}) = e^{\text{coefficient}} & = \text{odds ratio} \\\log (\text{odds ratio}) & = \text{coefficient}\end{align}</math>