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Classification of Regression models
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Independent variable (exposure)
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| Univariable (single variable)
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Multivariable (multiple variables)
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How to derive coefficients [math]\displaystyle{ b_i }[/math]
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Dependent
variable
(outcome)
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Continuous
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- [math]\displaystyle{ Y = a + bX }[/math]
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- Multivariable† linear regression
- [math]\displaystyle{ Y = a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots }[/math]
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Least squares method
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| Binary
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- 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]
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- 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]
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Maximum likelihood estimation method
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Multinominal
≥ 3
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- Single multinominal logistic regression
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- Multivariable† multinominal logistic regression
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Maximum likelihood estimation method
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| Ordinal
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- Single ordinal logistic regression
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- Multivariable† ordinal logistic regression
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Maximum likelihood estimation method
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| Rate ratio
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- 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]
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Maximum likelihood estimation method
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| Survival time
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- 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]
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Maximum likelihood estimation method
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†'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]
Generalized linear model
Penalized multivariable logistic regression model
Restricted cubic spline
