1. Please summarize/briefly explain and also explain the difference between (A)

Question

1. Please summarize/briefly explain and also explain the difference between (A) 2x2 contingency table and (B) 2x2 two way ANOVA. For each of the two models, give one example case where the usage of that model is appropriate.

2. Please summarize/briefly explain the difference (as you see it) between (A) ANOVA (the "usual", 1-way), and (B) random effects model. Think at least about these questions: if we have a few batches B1, B2, .., with a number of measurements per batch (all this information is of course considered known and serves as explicit input to a random effects model) - is it possible to use ANOVA in order to assess if batch effect is present? If yes, then what can random effects model give us that ANOVA can not - if anything that is?

Explanation / Answer

The 2-way ANOVA is probably the most popular layout in the Design of Experiments. To begin with, let us define a factorial experiment:
An experiment that utilizes every combination of factor levels as treatments is called a factorial experiment.

In the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups (blocks) that are similar to one another. Typically, a blocking factor is a source of variability that is not of primary interest to the experimenter. An example of a blocking factor might be the sex of a patient; by blocking on sex, this source of variability is controlled for, thus leading to greater accuracy.

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