Transcript Logistic Regression Pre-Challenger Relation Between Temperature and Field-Joint O-Ring Failure Dalal, Fowlkes, and Hoadley (1989).

Logistic Regression
Pre-Challenger Relation Between
Temperature and Field-Joint O-Ring
Failure
Dalal, Fowlkes, and Hoadley (1989). “Risk Analysis of the Space Shuttle: Pre-Challenger Prediction
of Failure,” Journal of the American Statistical Association, Vol. 84, #408, pp. 945-957
Data Description
• n=23 Space Shuttle Lift-offs prior to Challenger
• Response: Presence/Absence of erosion or
blow-by on at least one O-Ring field joint
 Y=1 if occurred, 0 if not
• Predictor Variable: Temperature at lift-off
 X = Temperature (degrees Fahrenheit)
Data
O-Ring Problem versus Temperature
O-Ring Problem
1
0
50
55
60
65
70
Temperature
Flight#
1
2
3
4
5
6
7
8
9
10
11
12
Temp
66
70
69
68
67
72
73
70
57
63
70
78
O-Ring Problem
0
1
0
0
0
0
0
0
1
1
1
0
Flight#
13
14
15
16
17
18
19
20
21
22
23
Temp
67
53
67
75
70
81
76
79
75
76
58
O-Ring Problem
0
1
0
0
0
0
0
0
1
0
1
75
80
85
Logistic Regression Model
• Distribution of Responses: Binomial
• Link Function: Logit
 ( X )    X   PY  1 X 
  (X ) 
   0  1 X
g (  )  ln
1  ( X ) 
 0  1 X
e
  (X ) 
 0  1 X
1 e
Model Estimation/Inference
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 15.0429
7.3786 2.039
0.0415 *
degrees
-0.2322
0.1082 -2.145
0.0320 *
--Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Null deviance: 28.267 on 22 degrees of freedom
Residual deviance: 20.315 on 21 degrees of freedom
H0: No association between incidence of O-Ring Failure and Temperature (1 = 0)
HA: Association between incidence of O-Ring Failure and Temperature (1 ≠ 0)
Reject H0 (zobs = -2.145, P=0.032), Conclude a negative association exists
15.04  0.23 X
e
 X  
15.04  0.23 X
1 e
^
1
0.9
0.8
0.7
0.6
O-Ring Problem
P(O-Ring Prob)
0.5
0.4
0.3
0.2
0.1
0
50
55
60
65
70
75
80
85
90
Odds Ratio
 e  0  1 X 
 e  0  1 X 
 e  0  1 X
1  e  0  1 X 
1  e  0  1 X 
1  e  0  1 X
 X 





odds X  



1
1  X  
e  0  1 X  1  e  0  1 X  e  0  1 X  
1

 1  e  0  1 X  
 1  e  0  1 X
1  e  0  1 X

 

odds( X  1)  e  0  1 ( X 1)


  e  0  1 X


odds( X  1) e  0  1 ( X 1) e  0  1 X e 1
Odds Rat io : OR 
  0  1 X   0  1 X  e 1
odds( X )
e
e
^
EstimatedOdds Rat io : OR  e
^
1
 ^ 1 1.96 SE  ^ 1  ^ 1 1.96 SE  ^ 1  
 
 
95% CI for P opulationOdds Rat io :  e
,e




ChallengerData :
^
OR  e
^
1
 e 0.23  0.795
 ^ 1 1.96 SE  ^ 1  ^ 1 1.96 SE  ^ 1  
e
 
 
,e
 e 0.231.96 (.1082 ) , e 0.231.96 (.1082 )  e .4421 , e .0179  .6427,.9823




Note: T heodds of failure decreasesas temperatu
re increases(Intervalbelow 1)

 
