1. Introduction to Logistic Regression）
武汉大学公共卫生学院流行病与卫生统计学系
2011,5,31
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2. New Words
LPM 　　　　　　线性概率模型 Odds Ratio　　　 优势比 Nominal Variables 名义变量 Dummy Variable 哑变量 Multiple Logistic Regression 多重Logistic回归
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3. CONTENTS
1. Review the Type of Variables 2. Variables In Logistic Regression 3. Why cannot we use a Linear Regression for Categorical Response? 4. Logistic Regression Model 5. What Is an Odds Ratio? 6. Multiple Logistic Regression
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4. 1. Review the Type of Variables
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5. Choosing the Scale of Measurement
Before analyzing, select the measurement scale for each variable.
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6. 分类（定性）变量
数值（定量）变量
名义变量
有序变量
离散变量
连续变量
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7. Nominal Variables
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8. Ordinal Variables
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9. Binomial Variables
Weather Good or Bad ？
Male or Female ？
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10. Continuous Variables
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11. 2. Variables In Logistic Regression
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12. Predicted ,Outcome ,Dependent variable
应变量
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13. Types of Logistic Regression
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3. 有序分类logistic回归

14. What Does Logistic Regression Do?
自变量
to predict the probability of specific outcomes.
二分类应变量
Predictor variables Predicted variable
Explanatory variables Response variable
Covariables Outcome variable
Independent variables Dependent variable
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15. Independent variables of Logistic Regression
Continuous variables
Dummy Variable for Nominal
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16. 3. Why cannot we use a Linear Regression for Categorical Response?
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17. Example: Failing or Passing an Exam
Let us define a variable ‘Outcome’
Outcome = 0 if the individual fails the exam
= 1 if the individual passes the exam
Predictor variable：the quantity of hours we use to study
Linear Probability Model’ (LPM) ：
Prob (Outcome=1) = α + β*Quantity of hours of study
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18. Linear Probability Models (LPM)
Student id
Outcome
Quantity of Study Hours 1 3 2 34 17 4 6 5 12 15 7 26 8 29 9 14 10 58 11 31 13 ？
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19. 4. Logistic Regression Model
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20. Logistic Regression Curve
1.0
0.9
0.8
0.7
Probability
0.6
0.5
0.4
0.3
0.2
0.1
0.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 x
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21. Logit Transformation
Logistic regression models transform probabilities called logits.
where
i indexes all cases (observations).
is the probability the event (a sale, for example) occurs in the ith case.
ln is the natural log (to the base e).
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22. Assumption 1
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23. Logistic Regression Model
logit ( ) = b0 + b1X1
where
logit( ) logit transformation of the probability of the event
b0 intercept of the regression line
b1 slope of the regression line.
线性关系
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24. LOGISTIC Procedure SAS SPSS Analyze Regression Binary Logistic…
PROC LOGISTIC DATA=SAS-data-set <options>;
CLASS variables </option>;
MODEL response=predictors </options>;
OUTPUT OUT=SAS-data-set keyword=name </option>;
RUN;
SPSS
Analyze
Regression
Binary Logistic…
Dependent: y
Covariates: x
Method: Forward Ward
Save…——
Predicted Values
 Probabilities
 Group membership
Option——
 CI for exp 95%
Probability for Stepwise
Entry: Removal
Maximum Likelihood Estimation is a statistical method for estimating the coefficients of a model.
The likelihood function
L = Prob (p1* p2* … * pn)
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25. SPSS Output result
Odds Ratio
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26. LPM and Logistic Regression Models
Student id
Outcome
Quantity of Study Hours 1 3 2 34 17 4 6 5 12 15 7 26 8 29 9 14 10 58 11 31 13
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27. Comparing LPM and the Logistic Curve
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28. 5. What Is an Odds Ratio?
An odds ratio indicates how much more likely, with respect to odds, a certain event occurs in one group relative to its occurrence in another group.
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29. Probabilities from odds
The odds, calculated as
Can be rearranged to express the probability of an event in terms of the odds:
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30. Probabilities and Odds
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31. Probability of Outcome
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32. Odds
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33. Odds Ratio
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34. Properties of the Odds Ratio
No Association
Odds Ratio
Group B
More
Likely
Group A
More
Likely
Regression Coefficient b
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-∞ ∞

35. Odds Ratio from a Logistic Regression Model
Estimated logistic regression model:
Estimated odds ratio (each more 1 Study Hours):
odds ratio = (e (a+1))/(e (a))
odds ratio = eb=e.495 = 1.640
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36. 6. Multiple Logistic Regression
logit ( ) = b0 + b1X1 + b2X2 + b3X3
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37. Backward Elimination Method
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38. Adjusted Odds Ratio
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39. Interaction in Multiple Logistic Regression
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40. Interaction Plot Predicted Logit Income Level Females Males Low Medium
High
Income Level
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41. Backward Elimination Method. . .
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42. Multicollinearity in Multiple Logistic Regression
The presence of multicollinearity will not lead to biased coefficients.
But the standard errors of the coefficients will be inflated.
If a variable which you think should be statistically significant is not, consult the correlation coefficients.
If two variables are correlated at a rate greater than .6, .7, .8, etc. then try dropping the least theoretically important of the two.
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43. =15~20 times number of variables
Sample Sizes
=15~20 times number of variables
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44. Thanks for your
attention！
Thanks for your attention
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