Problem 15 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

On January 28,1986, the main fuel tank of the space shuttle Challenger exploded shortly after takeoff. Essential in this accident was the leakage of some of the six O-rings of the Challenger. In Section \(1.4\) the probability of failure of an O-ring is given by $$ p(t)=\frac{e^{a+b \cdot t}}{1+e^{a+b \cdot t}} $$ where \(t\) is the temperature at launch in degrees Fahrenheit. In Table \(21.2\) the temperature \(t\) (in \({ }^{\circ} \mathrm{F}\), rounded to the nearest integer) and the number of failures \(N\) for 23 missions are given, ordered according to increasing temperatures. (See also Figure 1.3, where these data are graphically depicted.) Give the likelihood \(L(a, b)\) and the loglikelihood \(\ell(a, b)\). Table 21.2. Space shuttle failure data of pre-Challenger missions. \begin{tabular}{ccccccccc} \hline \hline\(t\) & 53 & 57 & 58 & 63 & 66 & 67 & 67 & 67 \\ \(N\) & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ \hline\(t\) & 68 & 69 & 70 & 70 & 70 & 70 & 72 & 73 \\ \(N\) & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ \hline\(t\) & 75 & 75 & 76 & 76 & 78 & 79 & 81 & \\ \(N\) & 0 & 2 & 0 & 0 & 0 & 0 & 0 & \\ \hline \hline \end{tabular}

Step-by-Step Solution

Answer

Likelihood is \(L(a,b) = \prod_{i} (p(t_i))^{N_i} (1-p(t_i))^{1-N_i}\) and log-likelihood is \( \ell(a, b) = \sum_{i} (N_i \log(p(t_i)) + (1-N_i) \log(1-p(t_i))) \)."

Step 1: Understand the Given Data

The problem provides the temperatures and the number of O-ring failures for several missions. The temperature is denoted as \(t\) and the number of failures as \(N\). Data can be represented as pairs \((t_i, N_i)\) for each mission.

Step 2: Define the Probability of Failure

The probability of an O-ring failure at temperature \(t\) is given by the logistic function: \[ p(t) = \frac{e^{a + b \cdot t}}{1 + e^{a + b \cdot t}} \] where \(a\) and \(b\) are parameters to be estimated and \(t\) is the temperature.

Step 3: Set Up the Likelihood Function

The likelihood function for the given data set is the product of probabilities for the successes and failures across all trials. For each trial \(i\), the likelihood contribution is \[ L_i = (p(t_i))^{N_i} \times (1 - p(t_i))^{1-N_i} \] Thus, the overall likelihood \(L(a, b)\) across all missions is: \[ L(a, b) = \prod_{i}(p(t_i))^{N_i} \times (1 - p(t_i))^{1-N_i} \]

Step 4: Derive the Log-Likelihood Function

To simplify calculations, we use the natural logarithm of the likelihood function, known as the log-likelihood: \[ \ell(a, b) = \sum_{i} \left(N_i \log(p(t_i)) + (1 - N_i) \log(1 - p(t_i))\right) \] Substitute \(p(t_i)\) with its expression from Step 2 to obtain \(\ell(a, b)\) in terms of \(a\) and \(b\).

Key Concepts

Probability of Failure

In logistic regression, determining the probability of a failure occurring under certain conditions is a vital concept. For the case of the space shuttle Challenger, we are interested in the probability of an O-ring failure as a function of the temperature at launch. This is expressed using a logistic function:

• \( p(t) = \frac{e^{a + b \cdot t}}{1 + e^{a + b \cdot t}} \)

Here, \(a\) and \(b\) are parameters that need to be estimated from the data. The variable \(t\) represents the temperature in degrees Fahrenheit.
The nature of the logistic function makes it perfect for modeling probabilities because:

• The function outputs values between 0 and 1, which are suitable for probability measures.
• It scales input temperatures to give a smooth transition between failure likelihoods at different temperatures.

Understanding the relationship between temperature and probability helps scientists predict and mitigate risks related to O-ring failures at various temperatures. This insight is crucial, especially in safety-critical applications like space missions.

Likelihood Function

The likelihood function in statistical modeling measures how well a particular set of parameters explains the observed data.
To construct the likelihood function \(L(a, b)\) for the Challenger missions, we examine the data on O-ring failures across different missions and temperatures. The likelihood is built by considering each mission individually, calculating the probability that the observed outcome occurred.

• For missions with a failure \(N_i = 1\), the probability is \(p(t_i)\).
• For missions without a failure \(N_i = 0\), the probability is \(1 - p(t_i)\).

We then take the product of these probabilities across all missions:

• \( L(a, b) = \prod_{i}(p(t_i))^{N_i} \times (1 - p(t_i))^{1-N_i} \)

This function gives us a comprehensive view of how likely the failure data is, assuming a specific logistic model.
Maximizing this likelihood allows the estimation of parameters \(a\) and \(b\) that best explain the observed data, leading to a reliable predictive model.

Log-Likelihood

The log-likelihood function is a transformation of the likelihood function that simplifies the mathematical treatment. Taking the natural logarithm of the likelihood provides a more manageable form for optimization purposes.
The log-likelihood \(\ell(a, b)\) is expressed as:

• \( \ell(a, b) = \sum_{i} \left(N_i \log(p(t_i)) + (1 - N_i) \log(1 - p(t_i))\right) \)

Here, similar to the likelihood function, each mission contributes to the total sum differently based on whether a failure was observed or not.
Using the log function has several advantages when estimating parameters:

• It turns the product of probabilities in the likelihood function into a sum, making differentiation and finding maxima easier.
• Logarithms ensure numerical stability, especially with very small probability values.

In the context of logistic regression, maximizing the log-likelihood gives us the parameters \(a\) and \(b\) that best fit the failure data, aiding in developing a predictive model that can anticipate future O-ring failures under varying temperature conditions.