Purpose

    These procedures runs Two-way ANOVA or Three-way ANOVA.

    The world is complex and multivariate in nature, and instances when a single variable completely explains a phenomenon are rare. For example, when trying to explore how to grow a bigger tomato, we would need to consider factors that have to do with the plants' genetic makeup, soil conditions, lighting, temperature, etc. Thus, in a typical experiment, many factors are taken into account. One important reason for using ANOVA methods rather than multiple two-group studies analyzed via t-tests is that the former method is more efficient, and with fewer observations we can gain more information. Let us expand on this statement.
    Suppose that in the above two-group example we introduce another grouping factor, for example, Gender. Imagine that in each group we have 3 males and 3 females. We could summarize this design in a 2 by 2 table:
 

    Before performing any computations, it appears that we can partition the total variance into at least 3 sources: (1) error (within-group) variability, (2) variability due to experimental group membership, and (3) variability due to gender. (Note that there is an additional source - interaction - that we will discuss shortly.)
    What would have happened had we not included Gender as a factor in the study but rather computed a simple t-test? If you compute the SS ignoring the Gender factor (use the within-group means ignoring or collapsing across Gender; the result is SS=10+10=20), you will see that the resulting within-group SS is larger than it is when we include Gender (use the within-group, within-gender means to compute those SS; they will be equal to 2 in each group, thus the combined SS-within is equal to 2+2+2+2=8). This difference is due to the fact that the means for Males are systematically lower than those for Females, and this difference in means adds variability if we ignore this factor. Controlling for error variance increases the sensitivity (power) of a test.
    This example demonstrates another principal of ANOVA that makes it preferable over simple two-group t test studies: In ANOVA we can test each factor while controlling for all others; this is actually the reason why ANOVA is more statistically powerful (i.e., we need fewer observations to find a significant effect) than the simple t test.
    For non normal data use Friedman Test.

Preparations   

    Run Statistics→Analysis of Variance(ANOVA)→Two(Three-)-way ANOVA....

Results

   Why Compare Individual Sets of Means? Usually, experimental hypotheses are stated in terms that are more specific than simply main effects or interactions. We may have the specific hypothesis that a particular textbook will improve math skills in males, but not in females, while another book would be about equally effective for both genders, but less effective overall for males. Now generally, we are predicting an interaction here: The effectiveness of the book is modified (qualified) by the student's gender. However, we have a particular prediction concerning the nature of the interaction: we expect a significant difference between genders for one book, but not the other. This type of specific prediction is usually tested via contrast analysis.

Contrast Analysis. Briefly, contrast analysis allows us to test the statistical significance of predicted specific differences in particular parts of our complex design. It is a major and indispensable component of the analysis of every complex ANOVA design. ANOVA/MANOVA has a uniquely flexible contrast analysis facility that allows you to specify and analyze practically any type of desired comparison.

Post-hoc Comparisons. Sometimes we find effects in our experiment that were not expected. Even though in most cases a creative experimenter will be able to explain almost any pattern of means, it would not be appropriate to analyze and evaluate that pattern as if one had predicted it all along. The problem here is one of capitalizing on chance when performing multiple tests post-hoc, that is, without a priori hypotheses. To illustrate this point, let us consider the following "experiment."
    Imagine we were to write down a number between 1 and 10 on 100 pieces of paper. We then put all of those pieces into a hat and draw 20 samples (of pieces of paper) of 5 observations each, and compute the means (from the numbers written on the pieces of paper) for each group. How likely do you think it is that we will find two sample means that are significantly different from each other? It is very likely! Selecting the extreme means obtained from 20 samples is very different from taking only 2 samples from the hat in the first place, which is what the test via the contrast analysis implies. Without going into further detail, there are several so-called post-hoc tests that are explicitly based on the first scenario (taking the extremes from 20 samples), that is, they are based on the assumption that we have chosen for our comparison the most extreme (different) means out of k total means in the design. Those tests apply "corrections" that are designed to offset the advantage of post-hoc selection of the most extreme comparisons. ANOVA/MANOVA offers a wide selection of those tests. Whenever you find unexpected results in an experiment you should use those post-hoc procedures to test their statistical significance.