Tips and tools for modeling counts most precisely
In a previous Stat-Ease blog, my colleague Shari Kraber provided insights into Improving Your Predictive Model via a Response Transformation. She highlighted the most commonly used transformation: the log. As a follow up to this article, let’s delve into another transformation: the square root, which deals nicely with count data such as imperfections. Counts follow the Poisson distribution, where the standard deviation is a function of the mean. This is not normal, which can invalidate ordinary-least-square (OLS) regression analysis. An alternative modeling tool, called Poisson regression (PR) provides a more precise way to deal with count data. However, to keep it simple statistically (KISS), I prefer the better-known methods of OLS with application of the square root transformation as a work-around.
When Stat-Ease software first introduced PR, I gave it a go via a design of experiment (DOE) on making microwave popcorn. In prior DOEs on this tasty treat I worked at reducing the weight of off-putting unpopped kernels (UPKs). However, I became a victim of my own success by reducing UPKs to a point where my kitchen scale could not provide adequate precision.
With the tools of PR in hand, I shifted my focus to a count of the UPKs to test out a new cell-phone app called Popcorn Expert. It listens to the “pops” and via the “latest machine learning achievements” signals users to turn off their microwave at the ideal moment that maximizes yield before they burn their snack. I set up a DOE to compare this app against two optional popcorn settings on my General Electric Spacemaker™ microwave: standard (“GE”) and extended (“GE++”). As an additional factor, I looked at preheating the microwave with a glass of water for 1 minute—widely publicized on the internet to be the secret to success.
Table 1 lays out my results from a replicated full factorial of the six combinations done in random order (shown in parentheses). Due to a few mistakes following the software’s plan (oops!), I added a few more runs along the way, increasing the number from 12 to 14. All of the popcorn produced tasted great, but as you can see, the yield varied severalfold.
| A: | B: | UPKs | ||
|---|---|---|---|---|
| Preheat | Timing | Rep 1 | Rep 2 | Rep 3 |
| No | GE | 41 (2) | 92 (4) | |
| No | GE++ | 23 (6) | 32 (12) | 34 (13) |
| No | App | 28 (1) | 50 (8) | 43 (11) |
| Yes | GE | 70 (5) | 62 (14) | |
| Yes | GE++ | 35 (7) | 51 (10) | |
| Yes | App | 50 (3) | 40 (9) |
I then analyzed the results via OLS with and without a square root transformation, and then advanced to the more sophisticated Poisson regression. In this case, PR prevailed: It revealed an interaction, displayed in Figure 1, that did not emerge from the OLS models.
Figure 1: Interaction of the two factors—preheat and timing method
Going to the extended popcorn timing (GE++) on my Spacemaker makes time-wasting preheating unnecessary—actually producing a significant reduction in UPKs. Good to know!
By the way, the app worked very well, but my results showed that I do not need my cell phone to maximize the yield of tasty popcorn.
To succeed in experiments on counts, they must be:
- discrete whole numbers with no upper bound
- kept with within over a fixed area of opportunity
- not be zero very often—avoid this by setting your area of opportunity (sample size) large enough to gather 20 counts or more per run on average.
For more details on the various approaches I’ve outlined above, view my presentation on Making the Most from Measuring Counts at the Stat-Ease YouTube Channel.