The new technique involves fitting a main effects model and using the main effects for “fake” factors that are used to create the design but not actual changeable factors (in this case there were two, since the design was based on a six-factor definitive screening design) to provide an estimate the pure error. A more detailed analysis using this method will be saved for another day, since I had one significant effect: heat setting. However, it seemed like there was too much noise based on my past experiments, and the vinegar that was so promising in the last experiment was no longer there. I took a quick look with Graph Builder and wasn’t surprised that there was no main effect for vinegar:
This becomes even more pronounced when looking at the residuals (from fitting the heat setting and block) vs. vinegar:
Typically, when I’m fitting a model from a designed experiment, I follow the principle of effect heredity, which means I won’t add a second-order term unless the main effect component(s) is significant. However, that doesn’t seem to hold for this data. If you look at the rating vs. heat setting, and overlay vinegar, there also appears to be an interaction with heat setting and vinegar amount:
I did some other investigation with other effects, particularly with time. This is because, with the heat setting being so significant (and suggesting high heat), time should be factored in. After all the models I fit, in the final model, time was significant. I also kept the quadratic effects for time and vinegar, as they were marginally significant:
You’ll notice that time shows up as significant in the final model. The estimate itself doesn’t change from the main-effects-only model, since main effects are orthogonal to all main effects and second-order effects. However, the standard error has been reduced because the variation in rating comes from the second-order terms in the model.
Confirming the experiment
Now that I had the final model, I needed to see how it well it works. Using the Maximize Desirability option from the red triangle menu in the Profiler, it looks like 50% vinegar for 23 minutes on heat setting 3 is best.
It turns out I can dye multiple cars within a batch of liquid (hmm … that sounds like the makings of a split-plot design in the future). I used 50% vinegar with heat setting 3 and the lowest dye amount for three cars, taking them out at 10, 20 and 30 minutes. The thought was the 20 minutes should be ideal. From left to right, the cars here are undyed, 30 minutes, 20 minutes and 10 minutes:
I was expecting the 20- and 30-minute cars to look good, and I was happy with the results. In fact, I prefer the 20-minute car, as 30 minutes on high heat made the plastic in the car start to melt.
While the principle of effect heredity is based on empirical studies, sometimes it doesn’t hold, and you have to start investigating if you’ve missed some second-order effects. The definitive screening design worked out nicely in this case. That's because, unlike just using center points, not only could I detect the possibility of quadratic effects, but I could also estimate them.
If you compare the results for the red cars in the first experiment, it’s incredible to see the difference after two follow-up experiments. I have a much better sense of getting other colors as well, especially with the Profiler. For example, if I want medium colors, I can use no vinegar and heat, while the lighter colors involve no heat with some vinegar. (If you missed any of these experiments, check out the whole series on dyeing diecast cars.)
I still want a definitive screening design to try out the new analysis technique. Any suggestions? Thanks for reading!