### NASA's Challenger Crash, as explained by Logistic Regression

On January 28th 1986, NASA's Challenger spacecraft launched skyward from Cape Canaveral Florida. Approximately 73 seconds later, the ship began to break apart in what appeared to be an explosion as seen from land. This chain of events that eventually destroyed the ship initiated when an o-ring in the right rocket booster failed shortly after takeoff.

After each previous NASA launch, the rocket boosters that fall from the ship, typically into the Atlantic Ocean, are retrieved for analysis. The same structural o-rings that proved to be Challenger's downfall were assessed and assigned a damage level. This information can be paired with the respective outside temperature at time of launch. That data set is available today, and we hope it can help us comprehend factors that led to this crash:

We will assess the damage field as binary; any damage level=Failure. No damage=No Failure. That assumption is used throughout this analysis. With failure being either true or false, we get an initial look at the relationship between temperature and o-ring failure:

Next, we can add our original Failure and Non-failure elements back in. I have assigned o-ring failure as red and non-failure as green.

After each previous NASA launch, the rocket boosters that fall from the ship, typically into the Atlantic Ocean, are retrieved for analysis. The same structural o-rings that proved to be Challenger's downfall were assessed and assigned a damage level. This information can be paired with the respective outside temperature at time of launch. That data set is available today, and we hope it can help us comprehend factors that led to this crash:

Put simply, lower temperatures appear to increase the likelihood of o-ring failure. A convenient tool to utilize here is a logistic regression. This regression model converts our binomial output of "Failure" and "No failure" to a probability of failure between 0 and 1 when paired with a continuous variable. In our case the continuous variable used is temperature in degrees Fahrenheit. Essentially a logistic regression will use Euler based logarithms to convert our Failure/Non-failure measure to odds of failure and finally to a probability of failure. When this model is created, our output can be visualized as such:

Each black point is a fitted value from our regression, and the blue line is a visual aid for the model's behavior overall. We can now use this model to extrapolate fitted values outside of our original data set. If we assign this formula to each temperature from 30°F to 80°F at .5° intervals we get the following:

Again, the relationship between temperature and failure is clear. Our model predicts almost certain failure for any temperature below 45°F. This observation makes the next element addition even more striking:

The red triangle is the actual temperature at time of takeoff 1/28/86. The unusually cold weather that morning unfortunately played a critical role in the ultimate failure of the o-ring and the disaster that occurred seconds later.

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