Experiments 2D - In-depth case study: analyzing a system with 3 factors by hand

Videos used in the Coursera course: Experimentation for Improvement. Join the course for FREE at https://www.coursera.org/learn/experimentation These videos are also part of the free online book, "Process Improvement using Data", http://yint.org/pid Full script for the video: http://yint.org/scripts/2D -------------------- In today's class, my goal is to show you how we analyze data, from factorial experiments when three factors were used. This example, is based on one from this textbook by Box, Hunter and Hunter, called "Statistics for Experimenters". The experiments described in that example, were run to find the combination of settings that would reduce the amount of pollution discharged from the water treatment facility. This is clearly a case where we would like to minimize the amount of pollutant. So, minimizing our outcome variable would be the objective. Three factors were considered. The first one, factor C, was the chemical compound used. Let's call that compound P, and compound Q. We don't really know their names. Factor T, was the temperature of the treatment. Whether we were treating the water at 72 or 100 Fahrenheit. And factor S, was the stirring speed either a slow speed of 200 revolutions per minute, or a high speed of 400. Notice that every factor has two levels, and going back to that mathematical idea, that two to the power "k", is the total number of experiments; "k" is equal to three in this example, so we get a total of eight possible combinations. Here's a short quiz to test that knowledge. So let's take a look at the results. We will always present our data, and analyze it using what we call "standard order". Standard order, requires that we create a column for each of our factors. So C, S, and T. Note that I could have used A, B, and C for the three factors, but very often we'll switch to letters that actually match of our factor names, but you don't have to. So back to the standard order table. And the rule is, we vary the first factor the fastest: - + - + - + - + The second factor, temperature, is varied the next fastest, between its low and high levels. So two minuses, two pluses, two minuses, two pluses. And then the last factor S, is varied the slowest. So four low levels (-) and four high levels (+). Those make up our entire table. Never run the experiments in the order of this table. The order must be randomly selected. So what we will do, is add a column to our table to keep track of the order in which we actually ran the experiments. Also add a column over here for the outcome variable. In this case, the outcome was the pollutant amount, measured in pounds. One thing that's so great about the standard order table, is that we can get a quick sense of the factor's influence on the outcome variable. Take a look, for example, at how the pollution amounts changes, when we change the chemical compound, factor C. That factor goes low, high, low, high, low, high, low, high. We see that same pattern in the pollution amounts. Take a look at the effect of factor S, the first four experiments, have a very high level of pollution on average, while the last four experiments have a low level of pollution. That also matches with factor S. We can already tell, just from this table, that factor C and factor S are going to be really important to understanding the results. Let's go back to our cube plot. And this time, our cube plot is actually a cube. We can draw it by showing the first factor along the horizontal axis. The next factor on the vertical axis and the final factor, S, is shown in and out of the page in this diagonal way. Next, we transcribe the values onto this cube. This is really easy when we follow the standard order sequence. Take a look: 5, 30, 6, 33. Then 4, 3, 5, and 4. I love this visual representation of the experimental data. It really helps us achieve our objective so quickly. Take a few seconds and answer this question. At what levels, should we set up three factors in order to achieve the lowest pollution amount? That's right. It's very clear we need to use chemical Q, operate at a low temperature, and with high stirring speeds of 400 revolutions per minute. Later on in the course, we're going to start examining what happens when you move outside this cube. And I want you to already start to think along those lines. But let's come back to the data we have right here, and analyze the main effects and the interactions. Start with a first factor C. The choice of either chemical P, or chemical Q at the high level. If we look at the cube, we actually have four estimates of that main effect, along each of the four horizontal edges. At high temperature, and high stirring speed, in other words high T and high S, that effect is equal to 4 - 5. At high temperature and low speed, that's 33 - 6. At T- and high speed, in other words S+, it is 3 - 4. And finally, at low temperature and low speed, it's 30 - 5. So four ...