it is orthogonal to all contrasts of lower order among those factors. Let denote the set of contrast functions belonging to the main effect of factor , the set of those belonging to the interaction between factors and , and more generally the set of contrast functions belonging to the interaction between the factors listed in for any subset with (here denotes cardinality). We also let denote the set of constant functions on . We have thus defined a set corresponding to each . It is not hard to see that each is a vector space, a subspace of the vector space consisting of all real-valued functions on . We have the following well-known, fundamental facts: If then . , the (orthogonal) sum running over all subsets . For each , dim . These results underpin the usual analysis of variance or ANOVA (see below), in which a total sum of squares is partitioned into the sums of squares for each effect (main effect or interaction), as introduced by Fisher. If contrast functions are written as vectors, then the left-hand side of item 2 is , where , the number of treatment combinations. The dimension of is the degrees of freedom for the corresponding effect. In a two-factor or experiment, the orthogonal sum reads , and the corresponding dimensions are , giving the usual formulas for degrees of freedom for main effects and interaction (the total degrees of freedom is ). Analysis A factorial experiment can be analyzed using ANOVA or regression analysis. To compute the main effect of a factor "A" in a 2-level experiment, subtract the average response of all experimental runs for which A was at its low (or first) level from the average response of all experimental runs for which A was at its high (or second) level. Other useful exploratory analysis tools for factorial experiments include main effects plots, interaction plots, Pareto plots, and a normal probability plot of the estimated effects. When the factors are continuous, two-level factorial designs assume that the effects are linear. If a quadratic effect is expected for a factor, a more complicated experiment should be used, such as a central composite design. Optimization of factors that could have quadratic effects is the primary goal of response surface methodology. Analysis example Montgomery gives the following example of analysis of a factorial experiment:.An engineer would like to increase the filtration rate