Introduction
Let's begin by recalling the definition: epistasis is the interaction between alleles at different loci that allows an allele at one locus to alter the effects of alleles at a different locus. Notice that the definition doesn't include any sort of specific alteration, which is what makes epistasis so interesting on the one hand and so frustrating on the other. The interesting part is that epistasis can produce almost any type of genotypephenotype relationship; the frustrating part is that there is no formula or guide to its effects that one can use. In other words, there are almost an infinite number of potential epistatic effects. But don't worry, we will concentrate on just a subset of them in order to make the points we need to appreciate at this level.
Recognizing Epistatic Effects at Two Loci
The key to understanding epistasis for our purposes (and, actually, for any purpose) is to recognize that an epistatic effect on a quantitative trait causes a deviation from an additive effect of alleles across loci. In other words, if we think about a multilocus genotype's phenotypic value, we might expect that the effects of several loci on the phenotype ought to be the same as the individual effects of the loci added together.
Any deviation from such additivity would indicate the effect of epistasis. We will work only with twolocus scenarios in the course, but remember that in the real world, epistasis can involve several loci and be quite complex.
Let's consider a hypothetical example at two loci. Imagine that variation in the diameter of snail shells is controlled by variation in two loci (let's not worry about environmental effects in this context). Snails range in diameter from 2 to 4 cm, and the genotype values are these:

A1A1 
A1A2 
A2A2 
B1B1 
4.0 cm 
3.5 cm 
3.0 cm 
B1B2 
3.5 cm 
3.0 cm 
2.5 cm 
B2B2 
3.0 cm 
2.5 cm 
2.0 cm 
Let's look first at individuals who are homozygous for the B1 allele and examine their genotypic values at the A locus. Notice that the A2A2 homozygote differs from the A1A1 homozygote by 1 cm. We can express this another way by noting that for each exchange of an A2 for an A1 allele (and there are two such exchanges that distinguish the two homozygotes), we subtract 0.5 cm from the genotypic value. Notice that the values for A1A2 heterozygotes are exactly intermediate between those for the two homozygotes. There is therefore no dominance, and the rule of "for each A2 you switch for an A1, subtract 0.5 cm" holds for all genotypes.
Now if we look at individuals who are heterozygous at the B locus, we see that the same rules apply for the A2A1 exchange as they did for individuals who were B1B1. We see the same result if we look at individuals who are homozygous for the B2 allele. In other words, the relationships between genotype and phenotype at the A locus are the same regardless of the genotype of the B locus. When this is so, we have pure additivity across loci and no epistatic interactions.
We would have found the same result if we had exchanged the roles of rows and columns in our example. In other words, if we had said "Let's look at A1A1 individuals and see how exchanging B2 alleles for B1 alleles affects the phenotypes . . . ," we would have come to the same conclusion.
Let's look for additivity or the lack thereof with the method I used in class with the maizeteosinte genotypes. Let's call this the "corner" method because we're looking only at the genotypes at the four corners. If we move across the B1B1 genotype from the A1A1 genotype to the A2A2 genotype, we'd subtract 1 cm from the average snail shell diameter. If we moved down the A1A1 genotype from the B1B1 genotype to the B2B2 genotype, we'd subtract 1 cm from the average diameter. So for each locus, exchanging one homozygote genotype for the other subtracts 1 cm. If the loci acted additively, without any epistasis, then we'd expect the double homozygote for the "2" alleles to be 2 cm smaller than the A1A1B1B1 homozygote. When we examine the A2A2B2B2 homozygote, lo and behold it is indeed 2 cm smaller.
Now suppose that we had found that the phenotypic value for A2A2B2B2 genotypes was not 2.0 cm, but was 1.0 cm. First, if we use the "corner" method, we would immediately see that the loci are not combining additively. In this case we would say that there is "reinforcing" epistasis, that the two loci in combination reduce shell diameter more than then would if acting individually. Had the A2A2B2B2 homozygote been larger than 2 cm, we'd have "diminishing" epistasis, in that the combined effects at two loci were not even as great as we'd expect from the effects of each individual locus.
Second, let's move across the rows to look at the effects of switching A2 alleles for A1 alleles at each of the B genotypes as we did at the beginning of our example (and we'll call this the "rowcolumn" method). Our diagnoses about the effects of allelic exchanges at the A locus would be the same as they were before across the B1B1 row and the B1B2 row. But when we look across at the B2B2 row, we'd see that the exchange of each two A2 alleles for two A1 alleles would alter the genotype by 2 cm (because A2A2B2B2 has 1 cm diameter, and A1A1B2B2 has 3 cm diameter). We immediately see that the effects of exchanging alleles at the A locus are not the same effects for each genotype at the B locus. The loci are therefore not combining additively, so there must be epistasis.
You should always use the "rowcolumn" method to search for epistatic interactions, because the "corner" method can diagnose one type of epistasis but will fail with some other types. Suppose our twolocus array of snail genotypes and their phenotypes looked like this:

A1A1 
A1A2 
A2A2 
B1B1 
4.0 cm 
3.5 cm 
3.0 cm 
B1B2 
3.5 cm 
3.0 cm 
2.5 cm 
B2B2 
3.0 cm 
3.0 cm 
2.0 cm 
If we use the "corner" method, the genotypes appear to be combining additively, and so they are when homozygous. But look at the A1A2 genotype's phenotype for each genotype at the B locus. For B1B1 individuals, the A1A2 heterozygote is exactly intermediate between the two homozygotes at the A locus. For B1B2 individuals, the A1A2 heterozygote is exactly intermediate between the A1A1 and A2A2 homozygotes. But for B2B2 genotypes, the A1A2 genotype is the same as the A1A1 genotype. Thus when an individual has the B2B2 genotype, the A1 allele is dominant to the A2 allele, which does not occur for other Blocus genotypes. This is also an epistatic effect, but not one that would be detected with the "corner" method.
