Download Y-variabeln binär (värden 0 – 1)

Logistic Regression
Statistics Sweden
September 2004
Dan Hedlin
Binary Y variable (0 or 1)
• Contract cancer or not, over or under a poverty
line, response or nonresponse
• Y is not limited in ordinary regression
• Trick:
 p 
    1x1   2 x2
log 
1 p 
• p is probability for cancer, etc.
Alternative expressions
• Common notation
logit  p    1x1   2 x2
• Equivalent:
e  1x1   2 x2 
p
1  e  1x1   2 x2 
Different scales
• Log-odds
(additive effects)
• Odds p/(1-p) (multiplicative effects)
• Probability p
Another difference to ’ordinary’ regression:
• Iterative computation and numerical issues
Interpretation of parameters
• ’Base probability’ for x1  0 and x2  0
e 
p
1  e  
• Maybe most interpretable when x are
interval scaled variables and the zero point
is meaningful
Interpretation of ß
 p 
    x
• One auxiliary variable: log 
1 p 
 p    x 

  e
• So
1 p 
• Hence additive one-step-increment of x

gives multiplicative effect on odds with e
Classical example
• Bliss (1935), also in Agresti (1990)
’Catergorical Data Analysis’, Wiley,
section 4.5.3.
• Beetles, two interval-scaled variables
y = dead/survived,
x = log(dose carbon disulphide)
• There are other models for a binary y that in
some cases may be better. Logistic reg most
common.
Model fitting
1. Table low-high risk vs each variable separately
2. Are there cells with zero observations?
3. First selection with e.g. Forward selection 0.25
significance level
4. Test each remaining variable separately
5. For continuous variables: examine linearity by
dividing the continuous variable in groups and
compute log-odds within group
6. Test interaction effects
7. Consider subject matter knowledge