Logistic Regression for Binary Responses

 The data below are the launch temperatures (degrees F) and an indicator
of O-ring failures for 24 space shuttle launches prior to the Challenger
disaster of Jan. 27, 1986. The space shuttle has 6 O-rings that can
fail, but the dataset below ignores how many of the 6 rings failed for a
particular launch. As noted on pg. 86 of your text, engineers warned
NASA the night before that the shuttle should not be launched because of
predicted cold temperatures. However, NASA decided to launch the 
Challenger at a temperature of 29F (about -2C).

 QUESTION: Does there seem to be a higher risk of failure at lower
launch temperatures?

########################################################

 # The data. "y" indicates an O-ring failure. 


> shuttle <- data.frame(Temp=c(53,56,57,63,66,67,67,67,
 68,69,70,70,70,70,72,73, 75,75,76,76,78,79,80,81),
 Failure=c("y","y","y","n","n","n","n","n",
 "n","n","n","y","y","y","n","n", "n","y","n","n","n","n","n","n"))

> shuttle[1:3,]
  Temp Failure 
1   53       y
2   56       y
3   57       y

 Let's plot the data, where "Failure" is the response:

 > plot(shuttle$Temp,shuttle$Failure, xlab = "Temp",  ylab = "2 if fail",
        main = "O-Ring failure vs. Temperature")


> shuttle.fit <- glm(Failure ~ Temp, family = binomial, data = shuttle)

> shuttle.fit

Call:  glm(formula = Failure ~ Temp, family = binomial, data = shuttle) 

Coefficients:
(Intercept)         Temp  
    10.8753      -0.1713  

Degrees of Freedom: 23 Total (i.e. Null);  22 Residual
Null Deviance:      28.97 
Residual Deviance: 23.03        AIC: 27.03 





























> summary(shuttle.fit)

Call:
glm(formula = Failure ~ Temp, family = binomial, data = shuttle)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.2125  -0.8253  -0.4705   0.5907   2.0512  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)  
(Intercept) 10.87535    5.70291   1.907   0.0565 .
Temp        -0.17132    0.08344  -2.053   0.0400 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 28.975  on 23  degrees of freedom
Residual deviance: 23.030  on 22  degrees of freedom
AIC: 27.030

Number of Fisher Scoring iterations: 4


 # Predicted probabilities of failures at each temp. in dataset

>  fitted(shuttle.fit)
         1          2          3          4          5          6          7 
0.85758349 0.78268818 0.75214414 0.52052785 0.39369565 0.35362920 0.35362920 
         8          9         10         11         12         13         14 
0.35362920 0.31551837 0.27973723 0.24655222 0.24655222 0.24655222 0.24655222 
        15         16         17         18         19         20         21 
0.18850910 0.16368693 0.12199327 0.12199327 0.10479854 0.10479854 0.07672851 
        22         23         24 
0.06543827 0.05570909 0.04735313 


 # Predict prob. of O-ring failure at 29F.
 # WARNING: THIS IS EXTRAPOLATION OUTSIDE OF THE BOUNDS OF THE 
 # ORIGINAL DATASET 

> predict(shuttle.fit, newdata = data.frame(Temp=29),se.fit=T)
   # Some output deleted
$fit
        1 
 5.907045   # This can't be a probability! It's the log-odds.

 To get prediction of the probability, add another option:

> predict(shuttle.fit, newdata = data.frame(Temp=29),se.fit=T,
           type = "response")

 # Some output deleted

$fit
         1 
 0.9972872

 So P(O-ring failure) is predicted to be over 99% at 29F.