# What statistical test should be used to accomplish the objectives of the experiment

I'd like to have your feedback to analyze my data from the following experiment.

Experiment Information

1. A total of 30 (15 from one city and other 15 from another city) slaughtered animals were sampled.
2. Three tissues (Muscle, Kidney and Liver) were collected from each sampled slaughtered animal.
3. Each sample was analyzed for the presence of three drugs and each drug amount was measured.

Objectives

The objectives of the experiment are to compare the level of drugs between

1. Types of drugs
2. Cities
3. Tissues.

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I think that 'biostatistics' is not a very good tag for this, but I'm not sure what is. I added tags related to issues I discuss in my answer, but I don't know that they are ideal either. –  gung May 14 '12 at 18:10

Firstly, I think what you have is not properly called an experiment. For example, it appears there was no random assignment or independent manipulation of any variable. Thus, we should avoid referring to 'independent' and 'dependent' variables, and not infer causality. Instead, this is observational research. There is nothing wrong or lesser about that; indeed, it is perfectly appropriate for comparing levels of a variable between factors, which is your stated goal here.

Next, I think that your response variable(s) are best understood as the levels of the different types of drugs (#1). The level of drugs itself cannot properly be thought of as a unitary response variable that is distinct from the type of drug. Rather, the level of drugs is bound up with type of drug in a way that is not true of the other variables listed. This means that you have more than one response variable.

Furthermore, the levels of the different types of drugs cannot be normally distributed, as it cannot go below 0. It is less vital for response variables to be normally distributed than people typically seem to believe, but it would be best if they are at least approximately normal. I suspect that some form of transformation will be in order. I would inspect the univariate distribution of each type of drug, and see if they look normal-ish, and possibly chose a transformation that yields a more normal result. If this is done first, it is perfectly valid--all that needs to be avoided is testing your data and searching for transformations that give you results you like better.

You will also want to assess that the transformations chosen, when taken together, yield bivariate relationships between the various drugs that are approximately linear (or perfectly uncorrelated). That is, I suspect the distributions of levels of drugs will be related to each other. This is more appropriately thought of in terms of the residuals from the appropriate model, but at this stage you can simply look at the marginal distributions.

If they were all completely uncorrelated, then you could simply run several ANOVA's (one for each type of drug), but realistically, that's not going to happen. As a result, you should conduct a 2 (city) X 3 (tissue) MANOVA.

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