Regression model

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Classification of Regression models

		Independent variable (exposure)
		Univariable (single variable)	Multivariable (multiple variables)	How to derive coefficients [math]\displaystyle{ b_i }[/math]
Dependent variable (outcome)	Continuous	Single linear regression [math]\displaystyle{ Y = a + bX }[/math]	Multivariable† linear regression [math]\displaystyle{ Y = a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots }[/math]	Least squares method
	Binary	Single binary logistic regression [math]\displaystyle{ \log Y = a + bX }[/math] where [math]\displaystyle{ Y }[/math] is odds of outcome [math]\displaystyle{ \frac{p}{1-p} }[/math]	Multivariable† binary logistic regression [math]\displaystyle{ \log Y = a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots }[/math] where [math]\displaystyle{ Y }[/math] is odds of outcome [math]\displaystyle{ \frac{p}{1-p} }[/math]	Maximum likelihood estimation method
	Multinominal ≥ 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]\displaystyle{ \log Y = a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots }[/math] where [math]\displaystyle{ Y }[/math] is rate ratio [math]\displaystyle{ \frac{events_1/person \text{-} time_1}{events_2/person \text{-} time_2} }[/math]	Maximum likelihood estimation method
	Survival time		Multivariable proportional hazard regression = Cox hazard regression [math]\displaystyle{ \log h(T) = \log h_0(T) + b_1X_1 + b_2X_2 + b_3X_3 + \cdots }[/math] where [math]\displaystyle{ h(T) }[/math] is the hazard at time [math]\displaystyle{ T }[/math] and [math]\displaystyle{ h_0(T) }[/math] is the baseline hazard at time [math]\displaystyle{ T }[/math]	Maximum likelihood estimation method

†'Multivariable' can be rephrased as 'Multiple'; Multivariable is NOT equal to 'Multivariate'!!

Binary logistic regression

Conversion of logit of outcome odds to outcome probability [math]\displaystyle{ p }[/math]

Equation of binary logistic regression can be converted to outcome probablity [math]\displaystyle{ p }[/math] as,

[math]\displaystyle{ \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]\displaystyle{ p }[/math] and the changed outcome probability [math]\displaystyle{ p\prime }[/math] by adding [math]\displaystyle{ 1 }[/math] to explanatory variable [math]\displaystyle{ X_1 }[/math], the following two equations are obtained,

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \dfrac{p\prime}{1-p\prime} }[/math] and [math]\displaystyle{ \dfrac{p}{1-p} }[/math] are odds of [math]\displaystyle{ p\prime }[/math] and [math]\displaystyle{ p }[/math], respectively,

[math]\displaystyle{ \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]\displaystyle{ 1 }[/math] is added to [math]\displaystyle{ X_1 }[/math]] to [the probability before adding].

Thus, converted [math]\displaystyle{ b_1 }[/math] to [math]\displaystyle{ \color{red}{e^{b_1}} }[/math] or [math]\displaystyle{ \color{red}{\exp (b_1)} }[/math] gives the odds ratio of outocome probabilities before and after variable [math]\displaystyle{ X_1 }[/math] gains [math]\displaystyle{ 1 }[/math].

[math]\displaystyle{ \begin{align} \exp (\text{coefficient}) = e^{\text{coefficient}} & = \text{odds ratio} \\ \log (\text{odds ratio}) & = \text{coefficient} \end{align} }[/math]

Penalized multivariable logistic regression model

• Penalized Logistic Regression Essentials in R: Ridge, Lasso and Elastic Net
• 罰則付き・正則化回帰モデルについて（About penalized/regularized regression model）

Restricted cubic spline

• Restricted cubic splinesをStataで実行してみる