The 2 k Factorial: Calculation of Effects and Analysis of Variance

15.2 The 2 k Factorial: Calculation of Effects and Analysis of Variance

  Consider initially a 2 2 factorial with factors A and B and n experimental obser-

  vations per factor combination. It is useful to use the symbols (1), a, b, and ab to signify the design points, where the presence of a lowercase letter implies that the factor (A or B) is at the high level. Thus, absence of the lower case implies that the factor is at the low level. So ab is the design point (+, +), a is (+, −), b is ( −, +) and (1) is (−, −). There are situations in which the notation also stands

  15.2 The 2 k Factorial: Calculation of Effects and Analysis of Variance

  for the response data at the design point in question. As an introduction to the calculation of important effects that aid in the determination of the influence of the factors and sums of squares that are incorporated into analysis of variance computations, we have Table 15.1.

  Table 15.1: A 2 2 Factorial Experiment

  A Mean

  b ab b+ab 2n

  B

  a (1)+a 2n

  Mean (1)+b

  In this table, (1), a, b, and ab signify totals of the n response values at the

  individual design points. The simplicity of the 2 2 factorial lies in the fact that

  apart from experimental error, important information comes to the analyst in single-degree-of-freedom components, one each for the two main effects A and B and one degree of freedom for interaction AB. The information retrieved on all these takes the form of three contrasts. Let us define the following contrasts among the treatment totals:

  A contrast = ab + a − b − (1),

  B contrast = ab − a + b − (1), AB contrast = ab − a − b + (1).

  The three effects from the experiment involve these contrasts and appeal to com- mon sense and intuition. The two computed main effects are of the form

  effect = ¯ y H − ¯y L ,

  where ¯ y H and ¯ y L are average response at the high, or “+” level and average

  response at the low, or “ −” level, respectively. As a result,

  Calculation of

  ab + a

  − b − (1)

  A contrast

  Main Effects

  B contrast

  The quantity A is seen to be the difference between the mean responses at the low and high levels of factor A. In fact, we call A the main effect of factor A. Similarly, B is the main effect of factor B. Apparent interaction in the data is observed by inspecting the difference between ab − b and a − (1) or between ab − a and b − (1) in Table 15.1. If, for example,

  ab − a ≈ b − (1)

  or

  ab − a − b + (1) ≈ 0,

  Chapter 15 2 k Factorial Experiments and Fractions

  a line connecting the responses for each level of factor A at the high level of factor

  B will be approximately parallel to a line connecting the responses for each level of factor A at the low level of factor B. The nonparallel lines of Figure 15.1 suggest the presence of interaction. To test whether this apparent interaction is significant,

  a third contrast in the treatment totals orthogonal to the main effect contrasts, called the interaction effect, is constructed by evaluating

  ab

  Interaction Effect

  − a − b + (1)

  AB contrast

  High Level of B

  Level of A

  Figure 15.1: Response suggesting apparent interaction.

  Example 15.1: Consider the data in Tables 15.2 and 15.3 with n = 1 for a 2 2 factorial experiment.

  Table 15.2: 2 2 Factorial with No Interaction

  Table 15.3: 2 2 Factorial with Interaction

  The numbers in the cells in Tables 15.2 and 15.3 clearly illustrate how contrasts and the resulting calculation of the two main effects and resulting conclusions can

  be highly influenced by the presence of interaction. In Table 15.2, the effect of A

  is −30 at both the low and high levels of factor B and the effect of B is 20 at both the low and high levels of factor A. This “consistency of effect” (no interaction) can be very important information to the analyst. The main effects are

  2 − 90 = −30,

  while the interaction effect is

  2 −

  AB =

  2 − 75 = 0.

  15.2 The 2 k Factorial: Calculation of Effects and Analysis of Variance

  On the other hand, in Table 15.3 the effect of A is once again −30 at the low level of B but +30 at the high level of B. This “inconsistency of effect” (interaction) also is present for B across levels of A. In these cases, the main effects can be meaningless and, in fact, highly misleading. For example, the effect of A is

  since there is a complete “masking” of the effect as one averages over levels of B. The strong interaction is illustrated by the calculated effect

  Here it is convenient to illustrate the scenarios of Tables 15.2 and 15.3 with inter- action plots. Note the parallelism in the plot of Figure 15.2 and the interaction that is apparent in Figure 15.3.

  Figure 15.2: Interaction plot for data of

  Figure 15.3: Interaction plot for data of

  Table 15.2.

  Table 15.3.

  Computation of Sums of Squares

  We take advantage of the fact that in the 2 2 factorial, or for that matter in the

  general 2 k factorial experiment, each main effect and interaction effect has an as- sociated single degree of freedom. Therefore, we can write 2 k − 1 orthogonal single-degree-of-freedom contrasts in the treatment combinations, each accounting for variation due to some main or interaction effect. Thus, under the usual in- dependence and normality assumptions in the experimental model, we can make tests to determine if the contrast reflects systematic variation or merely chance or random variation. The sums of squares for each contrast are found by following the procedures given in Section 13.5. Writing

  Y 1.. = b + (1),

  Y 2.. = ab + a,

  c 1 = −1,

  and

  c 2 = 1,

  Chapter 15 2 k Factorial Experiments and Fractions

  where Y 1.. and Y 2.. are the total of 2n observations, we have

  , 2 = − b − (1)] =

  (A contrast) 2

  with 1 degree of freedom. Similarly, we find that

  [ab + b

  − a − (1)] =

  2 (B contrast) 2

  (AB contrast) 2

  SS(AB) =

  Each contrast has l degree of freedom, whereas the error sum of squares, with

  2 (n − 1) degrees of freedom, is obtained by subtraction from the formula

SSE = SST − SSA − SSB − SS(AB).

  In computing the sums of squares for the main effects A and B and the inter- action effect AB, it is convenient to present the total responses of the treatment combinations along with the appropriate algebraic signs for each contrast, as in Table 15.4. The main effects are obtained as simple comparisons between the low and high levels. Therefore, we assign a positive sign to the treatment combination that is at the high level of a given factor and a negative sign to the treatment combination at the low level. The positive and negative signs for the interaction effect are obtained by multiplying the corresponding signs of the contrasts of the interacting factors.

  Table 15.4: Signs for Contrasts in a 2 2 Factorial Experiment

  Treatment

  Factorial Effect

The 2 3 Factorial

  Let us now consider an experiment using three factors, A, B, and C, each with levels

  −1 and +1. This is a 2 3 factorial experiment giving the eight treatment combinations (1), a, b, c, ab, ac, bc, and abc. The treatment combinations and

  the appropriate algebraic signs for each contrast used in computing the sums of squares for the main effects and interaction effects are presented in Table 15.5.

  15.2 The 2 k Factorial: Calculation of Effects and Analysis of Variance

  Table 15.5: Signs for Contrasts in a 2 3 Factorial Experiment

  Treatment

  Factorial Effect (symbolic)

  Figure 15.4: Geometric view of 2 3 .

  It is helpful to discuss and illustrate the geometry of the 2 3 factorial much as we illustrated that of the 2 2 factorial in Figure 15.1. For the 2 3 , the eight design

  points represent the vertices of a cube, as shown in Figure 15.4.

  The columns of Table 15.5 represent the signs that are used for the contrasts and thus computation of seven effects and corresponding sums of squares. These

  columns are analogous to those given in Table 15.4 for the case of the 2 2 . Seven

  effects are available since there are eight design points. For example,

  and so on. The sums of squares are merely given by

  (contrast) 2

  SS(effect) =

  3 n .

  An inspection of Table 15.5 reveals that for the 2 3 experiment all contrasts

  Chapter 15 2 k Factorial Experiments and Fractions

  among the seven are mutually orthogonal, and therefore the seven effects are as- sessed independently.

  Effects and Sums of Squares for the 2 k

  For a 2 k factorial experiment the single-degree-of-freedom sums of squares for the main effects and interaction effects are obtained by squaring the appropriate con- trasts in the treatment totals and dividing by 2 k n, where n is the number of replications of the treatment combinations.

  As before, an effect is always calculated by subtracting the average response at the “low” level from the average response at the “high” level. The high and low for main effects are quite clear. The symbolic high and low for interactions are evident from information as in Table 15.5.

  The orthogonality property has the same importance here as it does for the material on comparisons discussed in Chapter 13. Orthogonality of contrasts im- plies that the estimated effects and thus the sums of squares are independent. This

  independence is readily illustrated in the 2 3 factorial experiment if the responses,

  with factor A at its high level, are increased by an amount x in Table 15.5. Only the A contrast leads to a larger sum of squares, since the x effect cancels out in the formation of the six remaining contrasts as a result of the two positive and two negative signs associated with treatment combinations in which A is at the high level.

  There are additional advantages produced by orthogonality. These are pointed out when we discuss the 2 k factorial experiment in regression situations.