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*'''Multiple logistic regression'''::<math>\log Y = a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots</math><br>where <math>Y</math> is odds of outcomeMaximum likelihood estimation method
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==Classification of Regression models==
|-
!colspan="2" rowspan="2" style="width:150px"|
!colspan="23"|Independent variable (exposure)
|-
!style="width:250px300px"|Monovariable Univariable (single variable)!style="width:250px300px"|Multivariable (multiple variables)!How to derive coefficients <math>b_i</math>
|-
!rowspan="36"|Dependent <br>variable<br>(outcome)
!Continuous
|
*'''Simple Single linear regression'''
::<math>Y = a + bX</math>
|
*'''Multivariable † linear regression'''
::<math>Y = a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots</math>
|Least squares method
|-
!Binary
|
*'''Logistic Single binary logistic regression'''::<math>\log Y = a + bX</math><br>where <math>Y</math> is '''odds''' of outcome <math>\frac{p}{1-p}</math>|*'''Multivariable† binary logistic regression'''::<math>\log Y = a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots</math><br>where <math>Y</math> is '''odds ''' of outcome<math>\frac{p}{1-p}</math>|Maximum likelihood estimation method|-!Multinominal<br>≥ 3|*'''Single multinominal logistic regression'''|*'''Multivariable† multinominal logistic regression'''|Maximum likelihood estimation method|-!Ordinal|*'''Single ordinal logistic regression'''|*'''Multivariable† ordinal logistic regression'''|Maximum likelihood estimation method|-!Rate ratio||*'''Multivariable Poisson regression'''::<math>\log Y = a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots</math><br>where <math>Y</math> is '''rate ratio''' <math>\frac{events_1/person \text{-} time_1}{events_2/person \text{-} time_2}</math>
|
|-
!Survival time
|
|
*'''Multiple Multivariable proportional hazard regression'''<br>= '''Cox hazard regression'''::<math>\log h(T) = \log h_0(T) + b_1X_1 + b_2X_2 + b_3X_3 + \cdots</math><br>where <math>h(T)</math> is the hazard at time <math>T</math><br>and <math>h_0(T)</math> is the baseline hazard at time <math>T</math>|Maximum likelihood estimation method
|}
†'Multivariable' can be rephrased as 'Multiple'; Multivariable is <font color="red">'''NOT equal to 'Multivariate'!!'''</font>
==Binary logistic regression==
===Conversion of logit of outcome odds to outcome probability <math>p</math>===
Equation of binary logistic regression can be converted to outcome probablity <math>p</math> as,
:<math>
\begin{array}{lrll}
\log Y = & \log \left ( \dfrac{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{array}
</math>
===Conversion of coefficient to odds ratio===
When thinking about outcome probability <math>p</math> and the changed outcome probability <math>p\prime</math> by adding <math>1</math> to explanatory variable <math>X_1</math>, the following two equations are obtained,
:<math>
\begin{array}{lcl}
\log \left ( \dfrac{p}{1-p} \right ) & = & a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots \\
\log \left ( \dfrac{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 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 ) } & = \exp (b_1) = e^{b_1}
\end{align}
</math>
Because <math>\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 probability when <math>1</math> is added to <math>X_1</math>] to [the probability before adding].
Thus, converted <math>b_1</math> to <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 <math>1</math>.
::<math>
\begin{align}
\exp (\text{coefficient}) = e^{\text{coefficient}} & = \text{odds ratio} \\
\log (\text{odds ratio}) & = \text{coefficient}
\end{align}
</math>
==Generalized linear model==
==Penalized multivariable logistic regression model==
*[http://www.sthda.com/english/articles/36-classification-methods-essentials/149-penalized-logistic-regression-essentials-in-r-ridge-lasso-and-elastic-net/ Penalized Logistic Regression Essentials in R: Ridge, Lasso and Elastic Net]
*[https://jojoshin.hatenablog.com/entry/2016/07/06/180923 罰則付き・正則化回帰モデルについて（About penalized/regularized regression model）]
==Restricted cubic spline==
*[https://statakahiro.com/restricted-cubic-splines%E3%82%92stata%E3%81%A7%E5%AE%9F%E8%A1%8C%E3%81%97%E3%81%A6%E3%81%BF%E3%82%8B Restricted cubic splinesをStataで実行してみる]
