Bob, I am not an SME regarding your situation, but I have looked at your table and have the following thoughts. Some of my questions and comments are more Socratic rather than require an answer.
1. First and foremost, is the variation in your response of any practical value? There is no context provided regarding how much of a change in disease is of scientific interest? This is always more important than statistical significance.
2. If I understand your data, it looks like you have one factor (Name) and it is tested at 10 "levels". You have 2 Plant types (crossed with Name). You have 10 plants (Rep) for each level of Name and each plant is measured twice (is this in different locations on the plant or the exact same location on the plant?) and measures of Disease over 3 time periods for "within" plant (you might consider the plants nested within Type and Name). We could debate whether the systematic sampling of time periods is crossed with Name. I have some questions regarding this structure. The numbering scheme for Rep looks questionable. Can the same plant be given a different level of Name? If not, there should be no repeated Rep numbers.How many plants are actually in the study? It also looks like you have 2 measures of disease for each plant? I have attached another version of the data table. It is imperative the data table match how the data was actually collected or any analysis will be suspect.
What is the intent of the study? (e.g., Are you trying to pick the "winner" (e.g., best Name to reduce max disease?) or trying to understand what contributes to disease (growth))? Why 10 plants? Why 3 time periods, each a day apart? Are you interested in the rate of change of disease over time or just maximum disease or what? I guess the question is what do you hypothesize the Name will do to disease? Why?
3. Have you assessed the measurement system? Did you measure the disease multiple times on the same location on same plant at the same time period? As it appears now, the measurement errors are likely confounded with within plant?
4. A graphical look at the data without any summarization shows the time periods (DAI) are the largest source of variation in the study. This may be expected and may be of no interest (hence why you may want to look at the data by DAI), but also notice the variation increases with DAI. There may also be some "outliers" in your data.
Colored by DAI
Colored by Name for DAI=1
"All models are wrong, some are useful" G.E.P. Box