A DOE Example:  The Effect of Herbicide on Seedling Germination

For a biology/ecology project, a team of students decided to study the effect of herbicide pretreatment of soil on the germination rate of soybean seedlings. One student's father was a farmer who used such herbicides, and she knew that he sometimes complained that they seemed to affect germination, even though the manufacturer said that this should not happen.

The herbicide was supposed to be applied four weeks prior to planting, so the students prepared a test plot in the school greenhouse. Initially. they wanted to try 4 different application rates of the herbicide:

1. no herbicide
2. 50% of recommended treatment rate
3. recommended rate
4. 150% of recommended rate

They decided to plant 100 seeds for each application rate and use the number (=percentage) that germinated as their response measurement. The design space for this experiment would therefore be one-dimensional and consist of the four different herbicide "treatments" that they wanted to try:

Alice, the student whose father was a farmer, then raised an important point. She said that her father felt that the amount of rain that occurred in between the time the herbicide was applied and the seeds were planted seemed to affect what the herbicide did. If not much rain was expected, he would actually reduce the amount of herbicide that he would use; and if a wet spring was forecast, he would apply the herbicide at a somewhat higher than recommended rate. It seemed that more rain would tend to dilute or maybe wash out the herbicide somewhat from the soil.

It was not difficult to set up different watering schedules for the greenhouse plots, so varying the amount of water prior to planting wasn't a problem. At first, the students considered trying three different watering schedules: very dry (one every two weeks), average (once a week), and wet (twice a week). However, they realized that this would require 12 different treatment combinations -- 4 herbicide amounts × 3 watering schedules = 12 combinations. They decided that this would be too complicated to keep track of (and they weren't sure that they could get that many plots in the greenhouse, anyway), so they decided to look at just 5 combinations: 2 herbicide amounts (50% below and 50% above) × 2 watering schedules (dry and wet) + one additional "normal" combination (also called a "centerpoint" -- see the figure following) where the recommended herbicide amount and average watering was used. Because they now had two factors that they simultaneously changed, their design space was two dimensional and consisted of five points (combinations of the factors):

This was the design they ran. One month after applying the herbicide, they planted the soybean seeds. They watered all the plots in the same way after planting, but one of the team members wondered what would happen if they had varied the amount of water during the 2-week germination period, also. In any case, after two more weeks, the soybeans had germinated, and they counted the number of seedlings in each of their five separate plots. To help them visualize and better understand the results, they entered them on the previous design plot:

Just looking at the variables separately, the average germination rates at low and high herbicide amounts were 91.5% and 78.5%, respectively, so it looked like the herbicide definitely had an effect, over this range of applications at least. For the water, the average rate at the dry conditions (the bottom of the square) was 76.5% and for the wet conditions, it was 93.5%, so it looked like the amount of water was also a factor. If we plot the % germination vs. herbicide and water in separate plots, they look like:

However, by looking at the results overlaid on the design space, it is clear that looking at the effects one variable at a time is quite misleading. There is really not much difference (in fact, it could just be random experimental variability) in germination rates among the five combinations tried except for the one combination where herbicide amount was high and it was dry. If the herbicide results are plotted separately, but on the same plot, for the dry and wet conditions, the lines that result are distinctly non-parallel. This is an example of an interaction -- the joint effect of changing herbicide and water together is not what one would expect from what is observed when they are varied one at a time in isolation.

In fact, a better way of visualizing how germination rate jointly depends on herbicide amounts and watering is to turn the previous plot in which the results were overlaid on the design into a 3-d plot in which the rate is graphed as the height on the third axis. This is called a response surface plot of the results.

Note that the entire surface above the 2-d design plane had been shaded to indicate such a surface. Of course, this surface is not known (results were obtained only at the five points shown) and would have to be estimated in some suitable way. However, because the amounts of herbicide and water could take on any values within their respective ranges, it is clear that some sort of surface representing the results at all these combinations does exist. In other cases where the experimental factors are discrete and cannot take on all possible values (perhaps two different brands of herbicide rather than two different amounts to be experimented with), the "surface" may consist of just the five points, or just different curves representing a combination of discrete and continuous factors. However, no matter which case holds, the idea is the same: the response is an "extra" dimension whose value depends on the settings of the experimental variables. After completing this experiment, one of the students on the team commented that they had used only one kind of soil -- the standard greenhouse potting soil -- and she wondered whether varying the type of soil, say from a sandy to a claylike mix, might also affect germination rates. Had they done such an experiment by looking at all combinations of the now three factors each at two different conditions, the design space would consist of the eight corners of an experimental cube. The response (germination rate) would be the fourth dimension!

The ideas of the two factor experiment can be extended to many more factors. However, new methods must be used to reduce the number of experimental combinations, which grow exponentially with the number of factors, and to analyze and visualize the results. Such methods exist and are surprisingly easy to learn and use. If your are interested in learning more about them, many have been incorporated into the DOE modules, which are available for downloading.