「Regression model」の版間の差分

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!colspan="2" rowspan="2" style="width:150px"|
 
!colspan="2" rowspan="2" style="width:150px"|
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!colspan="2"|Independent variable (exposure)
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!colspan="3"|Independent variable (exposure)
 
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!style="width:300px"|Univariable (single variable)
 
!style="width:300px"|Univariable (single variable)
 
!style="width:300px"|Multivariable (multiple variables)
 
!style="width:300px"|Multivariable (multiple variables)
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!How to derive coefficients <math>b_i</math>
 
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!rowspan="6"|Dependent<br>variable<br>(outcome)
 
!rowspan="6"|Dependent<br>variable<br>(outcome)
 
!Continuous
 
!Continuous
 
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*'''Simple linear regression'''
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*'''Single linear regression'''
 
::<math>Y = a + bX</math>
 
::<math>Y = a + bX</math>
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*'''Multivariable&dagger; linear regression'''
 
*'''Multivariable&dagger; linear regression'''
 
::<math>Y = a + b_1X_1 + b_2X_2 + b_3X_3 + \cdots</math>
 
::<math>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
 
!Binary
 
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*'''Simple binary logistic regression'''
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*'''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>
 
::<math>\log Y = a + bX</math><br>where <math>Y</math> is '''odds''' of outcome <math>\frac{p}{1-p}</math>
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*'''Multivariable&dagger; binary logistic regression'''
 
*'''Multivariable&dagger; 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>
 
::<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>
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Maximum likelihood estimation method
 
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!Multinominal<br>&ge; 3
 
!Multinominal<br>&ge; 3
 
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*'''Simple multinominal logistic regression'''
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*'''Single multinominal logistic regression'''
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*'''Multivariable&dagger; multinominal logistic regression'''
 
*'''Multivariable&dagger; multinominal logistic regression'''
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!Ordinal
 
!Ordinal
 
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*'''Simple ordinal logistic regression'''
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*'''Single ordinal logistic regression'''
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*'''Multivariable&dagger; ordinal logistic regression'''
 
*'''Multivariable&dagger; ordinal logistic regression'''
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!Rate ratio
 
!Rate ratio
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*'''Multivariable Poisson regression'''
 
*'''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 \cdot time}{events_2/person \cdot time}</math>
 
::<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 \cdot time}{events_2/person \cdot time}</math>
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!Survival time
 
!Survival time
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*'''Multivariable proportional hazard regression'''<br>= '''Cox hazard regression'''
 
*'''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>
 
::<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>
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2023年2月4日 (土) 16:50時点における版

Basics & Definition
Epidemiology
Odds in statistics and Odds in a horse race
Collider bias
Data distribution
Statistical test
Regression model
Multivariate analysis
Marginal effects
Prediction and decision
Table-related commands in STATA
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
Ordinal
  • Single ordinal logistic regression
  • Multivariable† ordinal logistic regression
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 \cdot time}{events_2/person \cdot time} }[/math]
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]

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

Penalized multivariable logistic regression model

Restricted cubic spline

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