Regression model

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Odds in statistics and Odds in a horse race
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Regression model
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Missing data and imputation

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'!!

How to convert coefficient of binary logistic regressoin to odds ratio

About a binary logistic regression formula,

[math]\displaystyle{ \begin{align} & \log Y = \\ & \log \left ( \frac{p}{1-p} \right ) = a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots \end{align} }[/math]

adding 1 to [math]\displaystyle{ X_1 }[/math] makes probablity [math]\displaystyle{ p\prime }[/math] as,

[math]\displaystyle{ \log \left ( \frac{p\prime}{1-p\prime} \right ) = a + b_1(X_1 + 1) + b_2X_2 + b_3X_3 + \cdots }[/math]

Subtraction of above 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(X_1 + 1) - b_1X_1 \\ & = b_1 \\ \Leftrightarrow \frac {\left ( \dfrac{p\prime}{1-p\prime} \right )}{ \left ( \dfrac{p}{1-p} \right ) } & = e^{b_1} \end{align} }[/math]

Because [math]\displaystyle{ \frac {\left ( \dfrac{p\prime}{1-p\prime} \right )}{ \left ( \dfrac{p}{1-p} \right ) } }[/math] is the odds ratio of the probablity [math]\displaystyle{ p\prime }[/math] when 1 is added to [math]\displaystyle{ X_1 }[/math] to the probability [math]\displaystyle{ p }[/math],

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

Generalized linear model

Penalized multivariable logistic regression model

Restricted cubic spline