Basic Ideas - Multi-Factor ANOVA
In the previous example (Partitioning of Sums of Squares), it may have occurred to you that we could have simply computed a t-test for independent samples with Basic Statistics and Tables to arrive at the same conclusion. And, indeed, we would get the identical result if we were to compare the two groups using this test. However, ANOVA is a much more flexible and powerful technique that can be applied to much more complex research issues.
Multiple factors
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.
Controlling for factors
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.