with 10 factors at two levels each produces 210=1024 combinations. At some point this becomes infeasible due to high cost or insufficient resources. In this case, fractional factorial designs may be used. As with any statistical experiment, the experimental runs in a factorial experiment should be randomized to reduce the impact that bias could have on the experimental results. In practice, this can be a large operational challenge. Factorial experiments can be used when there are more than two levels of each factor. However, the number of experimental runs required for three-level (or more) factorial designs will be considerably greater than for their two-level counterparts. Factorial designs are therefore less attractive if a researcher wishes to consider more than two levels. Main effects and interactions A fundamental concept in experimental design is the contrast. Let be the expected response to treatment combination t, and let be the set of treatment combinations. A contrast in is a linear expression such that . The function is a contrast function, or a contrast vector if we fix an order of the treatment combinations. For example, in a one-factor experiment the expression represents a contrast between level 1 of the factor and the combined impact of levels 2 and 3. The corresponding contrast function is given by , and the contrast vector is . It's easy to see that if and are contrast functions, so is , and so is for any real number . We say that and are orthogonal (denoted ) if . More generally, we say that a contrast function belongs to the main effect of factor i if its value depends only on the value of . For example, in the example above, the contrast represents the main effect of factor . The corresponding contrast function is given by ; as a contrast vector, it is displayed in the column for factor in the table above. The contrast function belongs to the interaction between factors i and j if (a) the value of depends only on and , and (b) it is orthogonal to the contrast functions for the main effects of factors and . These contrasts detect the presence or absence of additivity between the two factors. Similarly, for any subset of having more than two elements, a contrast function belongs the interaction between the factors listed in if (a) its values depend only on the levels and (b)