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I have a good friend who has an experiment where he grows different kind of plants on various soils and measures the protein content and weight each week for 5 weeks. His goal is to determine which plant yields the best protein*weight results for each soil and on which week.

The data is fairly straightforward with columns: Soil, Plant, Time(Week), Weight, Protein/g

Initially this doesn't seem like a statistics related problem as I'd just group all data by soild and find the max value of Weight*Protein from the weeks for each plant. It should give me the best Plant-Time combination for each Soil.

The issue arises when my friend also mentioned that he has to do a One sample, two sample and paired t-test on the data without much context. He has no big knowledge in statistics so he couldn't provide me any more information.

One sample t-test should be executed against a specific mean value to determine whether or not the sample was taken from a population that has a similar mean.

Two sampled and paired t-test compares 2 sample sets and determines if they are from the same population or not. Paired also makes sure to make the comparison pair-wise instead of assuming the values are randomly chosen.

The protein and weight variables are not even normally distributed, therefore a t-test is a no-go in this case.

How can I even apply the t-test to this task so I can use it to find my answers?

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First off, this is a classic statistics problem. Your solution of just picking the maximum from each group ignores a lot of stuff (e.g. the variation in each group).

Second, not having statistics knowledge does not prevent your friend from giving you context. In fact, it may even make it more likely that he can give you context without using statistical terms incorrectly. In my work, I have often found it easier to get the context if my client knows no statistics than if he/she knows a little.

Third, the t-test isn't the best test here. I would use some sort of multilevel model to account for the repeated measures and the resulting lack of independence. A GEE model might also be good.

Fourth, regarding normality and t-tests, see this thread;\

Finally, if your friend needs to use t-tests, there are surely hypotheses that could be found for each type, e.g.

One sample: Does a particular group at a particular time point have a particular mean?

Two sample independent: Do two groups at a particular time point have the same mean?

Two sample dependent: Does a group at time 1 have the same mean as the group at time 2?

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    $\begingroup$ "I have often found it easier to get the context"... yes. I often have to spend time trying to get people to stop attempting to use statistical jargon before it's possible to figure out what they want. $\endgroup$
    – Glen_b
    Jun 27 '15 at 12:54
  • $\begingroup$ Thank you for the excellent answer. I sent you a question because I feel there are insufficient data provided as there are only one entry for each SoilType-PlantType-Week-Protein 'pair'. It seems to me that proving staitstically which SoilType-PlantType combination yields the best protein amount and on which week is impossible with just 1 experiment for each combination. $\endgroup$
    – SLOBY
    Jun 29 '15 at 1:08

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