When an experiment includes one or more binary responses, such as results that
either pass or fail data, logistic regression, named by its use of the logit
function, becomes more appropriate for predictive modeling than linear
regression based on ordinary least squares (OLS). For statistical details, see
“Logistic Regression Models”, Section 13.2 of *Introduction to Linear Regression
Analysis, 5th edition* by Montgomery, Peck and Vining, 2012, John Wiley & Sons,
Inc., Hoboken, New Jersey. Rest assured that, although the mechanics of logistic
regression get a bit complicated (no worries: the software takes care of this),
the results end up being relatively easy to interpret.

Montgomery, et al, present in problem 13.1 the test-firing results for 25 SAMs aimed at targets of varying speeds in a range from 200 to 500 knots. Let’s see how this experiment is set up in Design-Expert (DX) and, after entering the hit (1) or miss (0) results, how they get modeled, diagnosed and interpreted. Imagine you are in an airplane flying over this SAM installation. How fast do you need to fly to reduce your chances of being shot down to an acceptable probability?

Open Design-Expert and go to the **Help** menu, then hover over **Tutorial Data** and click **SAM
Firing Tests** to load your data (shown below).

Run | Factor 1 A:Speed knots | Response 1 Hit or Miss 1 or 0 |

1 | 400 | 0 |

2 | 220 | 1 |

3 | 490 | 0 |

4 | 210 | 1 |

5 | 500 | 0 |

6 | 270 | 0 |

7 | 200 | 1 |

8 | 470 | 0 |

9 | 480 | 0 |

10 | 310 | 1 |

11 | 240 | 1 |

12 | 490 | 0 |

13 | 420 | 0 |

14 | 330 | 1 |

15 | 280 | 1 |

16 | 210 | 1 |

17 | 300 | 1 |

18 | 470 | 1 |

19 | 230 | 0 |

20 | 430 | 0 |

21 | 460 | 0 |

22 | 220 | 1 |

23 | 250 | 1 |

24 | 200 | 1 |

25 | 390 | 0 |

Note

This data was entered via the Historical Data tool. See how by
rebuilding the design via **File, New Design** and clicking **Yes** to “Use
previous design info”. Proceed through the steps back to the design sheet, now
empty. The factor and response values were then entered (Option: paste from a
spreadsheet). Re-open SAM Firing Tests to fill your design back in.

- Click the
**Analysis**branch on the left. Note that DX recognizes the response being binary and defaults to Logistic Regression. Leave the success level at its default of 1, keeping in mind this is from the perspective of those firing the SAMs, not the pilot flying overhead. 😉

- Press ahead to
**Model**—it being defaulted to linear. That is good.

- Continue on to
**ANOVA**, noting at a glance of the multi-view display the significant p-value and decent pseudo R^{2}values. (If you like, go back to Model and upgrade it to Process Order Quadratic. You will then see that the A^{2}term comes out very insignificant. Therefore, the linear model appears adequate.)

- Next view
**Diagnostics**. Nothing looks abnormal in the plots that DX provides for logistic models.

- Finally, click
**Model Graphs**to see how the probability of a hit decreases with speed, as you would expect.

Under the **Optimization** node click **Numerical**. Assuming all else equal,
it will be best to keep speed lower to save fuel and thus increase the distance
that the aircraft can fly for its mission, enter for factor **A** a **Goal** to
**minimize**. Then for the response enter a **Goal** to **minimize** the chance
of a hit (thinking from the perspective of the pilot trying to get by the SAMs)
with **Limits** at **Lower** of **0.1** and **Upper** of **0.2**.

Then click **Solutions**. By the default Ramps view, you will see that a
speed of 460 knots (530 miles per hour) is recommended at a minimum (far, far
faster much better!).

This concludes a speedy flight through the logistic regression tools for Design-Expert software for experiments with binary responses. Consider saving your results and then trying different goals for the optimization, that is, playing with the data. Have fun!