Logistic Regression (Proportions)

  We will take a 2nd look at some of the data from the Challenger
disaster of Jan. 20, 1986. The explosion was caused by combustion of
leaking gas through damaged O-rings.

   Reminder of background:

     - Booster rockets have joints that are sealed with rubber O-rings.

     - 6 primary O-rings (3 on each rocket).

     - O-rings sensitive to temperature

	 - if it's cold, can't recover their shape

	 - joints then won't be sealed ... gas leak

     - O-ring damage information was available from 23 previous flights.
       We previously studied simply if any of the 6 rings failed or not.
       Now we'll incorporate how many of the 6 failed on a flight.

  Day of Launch: Temperature of 31F (-0.5C)

    - Previous coldest launch: 53F (12C)

  Analysis done before flight:

    - 7 previous flights had O-ring damage

           Temp (F):  53 | 57 | 58 | 63 | 70 | 70 | 75
    --------------------------------------------------
    Number of Rings:   6 |  6 |  6 |  6 |  6 |  6 |  6
    --------------------------------------------------
      Rings Damaged:   2 |  1 |  1 |  1 |  1 |  1 |  2 


     - No apparent relationship between temperature and proportion of
        rings damaged.

     - Any problems with this analysis?


> chal.dat[1,]
  Temp Breaks Total 
1   53      2     6


> chal.fit <- glm(cbind(Breaks, Total - Breaks) ~ Temp,
                    family = binomial, data = chal.dat)























> summary(chal.fit)

Call:
glm(formula = cbind(Breaks, Total - Breaks) ~ Temp, family = binomial, 
    data = chal.dat)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-0.95227  -0.78299  -0.54117  -0.04379   2.65152  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)  
(Intercept)  5.08498    3.05247   1.666   0.0957 .
Temp        -0.11560    0.04702  -2.458   0.0140 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 24.230  on 22  degrees of freedom
Residual deviance: 18.086  on 21  degrees of freedom
AIC: 35.647

Number of Fisher Scoring iterations: 5

# Predict Prob. of O-ring failure at 31F.

>  predict(chal.fit, newdata = data.frame(Temp=31),
                  type="response",se.fit=T)
$fit:
         1 
 0.8177744     = P(O-ring failure)

$se.fit:
       1 
 0.2404526

$residual.scale:
[1] 1

> 6 * 0.8177744 = 4.907. Predict around 5 of 6 O-rings would fail