In this section, I'm going to show you a couple of additional examples of single-factor experiments. These occur in lots of areas and I've chosen two. There's another one in the book, but I've chosen two that I think you'd find particularly interesting. The first one has to do with an experiment investigating the effect of consuming chart chocolate on your cardiovascular health. This study was described in an article in Nature back in 2003. The experiment consisted of using three different types of chocolate: 100 grams of dark chocolate, 100 grams of dark chocolate with 200 milliliters of full-fat milk, and 200 grams of milk chocolate, and then 12 subjects were used. There were seven men and five women. Average age range of 32.2 plus or minus one year, average weight of 65.8 plus or minus 3.1 kilograms, and a body mass index of 21.8 plus or minus 0.4 kilograms per square meter. On different days, a subject consumed one of these factor level combinations, and an hour later, the total antioxidant capacity of their blood plasma was measured in an assay. Here's the data from this experiment. Now, it says that data similar to that summarized in the article or shown in table 3.1 too. I couldn't get the individual observations, so I had to basically construct them from the summary data that they supplied. So there's the data, and you'll recognize this as a single-factor experiment with three levels of the factor, the chocolate consumption type, and 12 replicates. This has been done in random order with one subject taking one of the treatments on each day. Here are the box plots. Wow, this is interesting. Look at that. Dark chocolate seems to have a much higher antioxidant capacity than either of the other two factor levels. So that seems to be the pretty evident. But when you run the analysis of variance, it's highly statistically significant. Notice that the P-value for comparing these factor level means mini tab reports it to be zero. Well, that's a little bit of a glitch I think in mini tab. These P-values can't be zero. The F-distributions continuous all the way out to plus infinity. So it's less than some very small value like 0.00001. It's not actually zero. Notice that mini tab also reports something called the R-square for this experiment. The R-square is the proportion of variability in the data explained by the treatment factor. Well, the way it's computed is by dividing the factor sum of squares by the total sum of squares. So this experiment explains about 85 percent of the variability in the data. That's pretty good. Generally, we were comfortable when our model explains most of the variability in the data. If the model only explains a relatively small proportion of the data, then that's often a clue or a signal or indicator that you may have left out important factors, or that there may be other factors varying out of control during the course of the experiment. So now we know that there's a difference in means. Which means are different? Well, mini tab does this by using the Fisher LSD. But instead of doing a T-ratio, it does a confidence interval on the difference in means. So let's look at the bottom part of this display. In this display, we are looking at a confidence interval on difference in means where dark chocolate is subtracted from the other two. So this is comparing dark chocolate with dark chocolate plus milk, and this is comparing dark chocolate with milk chocolate. Well, you notice that both of these confidence intervals do not include zero. So that's a strong indication that dark chocolate is significantly different from the other two treatments. Then at the bottom, dark chocolate is being compared to milk chocolate, and you notice that that confidence interval does indeed overlap zero. So there's an indication that dark chocolate and milk chocolate produce the same level of antioxidant capacity in your blood plasma. Now, here's a second example. Our first example had to do with a study on cardiovascular health, and designed experiments have had a great impact on the health sciences and also in manufacturing industries, design of new products, improvement of existing products, development of new manufacturing processes, there had been really beautiful applications of this methodology in these areas. But now we're seeing examples that are coming from areas that are outside this traditional environment, such as financial services, telecommunications, e-commerce, legal services, marketing, transportation, all places. This is really interesting because these businesses are often thought of as what we call the real economy, and the real economy probably accounts for 70 or 80 percent of the total US economy. So applications of experimental design in these areas are really important. Here's a very simple example. We've got a soft drink distributor that knows that end-all displays are an effective way to increase sales of the product. There are a lot of ways to design these displays. They can vary the amount of display itself, the size, the text that's displayed, the colors, the visual images. There are lots of ways to do this. Well, this marketing group has designed three new end-all displays and they want to test their effectiveness. They've identified 15 stores of similar size and similar type, and the stores have agreed to participate in the study. So each store will test one of the display periods for a period of a month. The response variable is going to be the percentage increase in sales activity over the typical sales for that store when they don't use the end-all display. So the data from this experiment is shown in the table below, and so this is the result that we get. Single factor experiment, three levels of the factor, and five replicates. So here's the results from Jump. Now, you'll notice that the R-square is about 86 percent. So this experiment explained a pretty sizable portion of the variability in the data. The F-ratio, 35, P-value less than 0.0001. So there's a strong indication here that there really is a difference in these end-all displays. So what difference do we really see? What difference is really there? Well, here's the results of the Fisher LSD test as presented by Jump. They also use confidence intervals to do this. What you see in this table is a display where in each cell, we have the difference in means, then the standard error of the difference in means, and then the lower and upper confidence interval. When you see a pair of means, where the lower and upper confidence interval spans zero, that says that that pair of means are not different. For example, one and two do not differ because that confidence interval spans zero. On the other hand, one and three are different because that confidence interval does not span zero. Well, when we summarize all of the results from this experiment, this is what we get. End-all display 3 is different from end-all displays 2 and 1, and 2 and 1 are the same. So there's a statistically significant difference here between end-all display design 3 and the other two. Here's a plot of residuals plot predicted for this experiment, again, displayed in Jump. You may notice a slight funnel shaped appearance to this plot. That's really not enough of a funnel shaped appearance to cause me any significant difficulties. So I think there are really no serious problems with assumptions, and so we could probably proceed to conclude that one of these displays is definitely better than the other.
