2,992
回編集
差分
ナビゲーションに移動
検索に移動
編集の要約なし
{{Epi StatFloating_Menu}}
==Classification of Regression models==
|-
!colspan="2" rowspan="2" style="width:150px"|
!colspan="23"|Independent variable (exposure)
|-
!style="width:300px"|Univariable (single variable)
!style="width:300px"|Multivariable (multiple variables)
!How to derive coefficients <math>b_i</math>
|-
!rowspan="6"|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
|
*'''Simple 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
|
*'''Simple Single multinominal logistic regression'''
|
*'''Multivariable† multinominal logistic regression'''
|Maximum likelihood estimation method
|-
!Ordinal
|
*'''Simple 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>|Maximum likelihood estimation method
|-
!Survival time
*'''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==
