Black Belt · Analyze/Improve phase
Lesson 1 established why a full factorial design, testing every combination of factor levels, is the most direct way to catch an interaction that one-factor-at-a-time testing would miss. This lesson makes that design concrete: the 2ᵏ full factorial, the workhorse design of Black Belt work, where every one of k factors is tested at exactly two levels, "low" and "high." This lesson covers how many runs a 2ᵏ design needs, how its standard-order design matrix is built and coded, how to hand-calculate a main effect and an interaction effect directly from the data, and where a plain 2-level design's linear model runs out of ability to represent a curved response.
A 2ᵏ design tests k factors, each at 2 levels, in every combination, so the number of distinct treatment combinations is 2 raised to the k power. Two factors give 2² = 4 runs; three factors give 2³ = 8 runs; four factors give 2⁴ = 16 runs. If the design is replicated, meaning each treatment combination is independently repeated r times, the total number of runs is r times 2ᵏ: a 2³ design replicated twice needs 2 x 8 = 16 total runs. Run count grows fast with k, which is exactly why Lesson 1's screening step, narrowing a long factor list down first, matters before committing to a full factorial on every candidate factor.
Rather than write a factor's actual units (a specific temperature in degrees, a specific primer brand), a 2ᵏ design codes every factor's low level as -1 and high level as +1. Coding puts every factor on the same scale, which is what makes the sign-based effect calculation in the next section work directly. Runs are listed in "standard order," a fixed convention: for two factors A and B, standard order is (1), a, b, ab, where "(1)" means both factors at their low level, a lowercase letter present means that factor is at its high level, and a letter absent means that factor is at its low level. The standard-order table below is the same design shown as a square in the figure above.
| Run | A (primer) | B (cure temp) | AB | Response |
|---|---|---|---|---|
| (1) | -1 | -1 | +1 | 60 |
| a | +1 | -1 | -1 | 72 |
| b | -1 | +1 | -1 | 68 |
| ab | +1 | +1 | +1 | 90 |
The AB column's sign is simply the product of the A and B signs in that row, a pattern that extends to any interaction column in a larger design.
A main effect is defined as the average response when the factor is at its high level minus the average response when the factor is at its low level. Reading straight from the table: factor A (primer) is at +1 in runs a and ab, and at -1 in runs (1) and b. So:
In plain terms: switching from standard to premium primer raises average adhesion by 17 points; switching from low to high cure temperature raises it by 13 points, each averaged across the other factor's two levels.
The AB interaction effect uses the AB sign column exactly the same way: average the response where AB is +1 minus average the response where AB is -1. From the table, AB is +1 for (1) and ab, and -1 for a and b:
Double-checking with the full-sum contrast method confirms this: multiply each response by its sign in the relevant column and sum, then divide by 2ᵏ⁻¹. For AB, use the AB column's signs (+1, -1, -1, +1) on ((1), a, b, ab): (60) - (72) - (68) + (90) = 10, and 10 / 2²⁻¹ = 10 / 2 = 5, matching the average-difference calculation above. The same check on A: -(60) + (72) - (68) + (90) = 34, and 34 / 2 = 17, matching. On B: -(60) - (72) + (68) + (90) = 26, and 26 / 2 = 13, matching. A positive AB effect of 5 here means primer type's effect is larger at high cure temperature (ab - b = 90 - 68 = 22 points) than at low cure temperature (a - (1) = 72 - 60 = 12 points); half of that difference is 5. With only one run per corner there is no estimate of experimental error, so this design alone cannot say whether 5 is a real interaction or noise.
Why this matters
The exam expects both the average-difference definition and the sign- column contrast method to agree, because they are the same calculation written two ways. If they do not match on a hand calculation, the sign table was built wrong, not the arithmetic.
A 2ᵏ design's results are often written as a fitted regression equation in the coded factors, Y = b0 + b1(A) + b2(B) + b12(AB), and each coefficient in that equation is exactly half of the corresponding effect: coefficient = effect / 2. That factor of two exists because an effect is defined as a difference between two group averages, a distance of two coded units (from -1 to +1), while a regression coefficient describes the change per one coded unit. Using this lesson's numbers: b1 (primer) = 17 / 2 = 8.5, b2 (cure temperature) = 13 / 2 = 6.5, and b12 (interaction) = 5 / 2 = 2.5.
A standard 2ᵏ design, with every factor at only two levels, can only ever fit a linear model in each factor: it has exactly enough information to estimate a straight-line effect and interactions, and no more. It cannot estimate curvature, a quadratic term like A-squared, because fitting a curve requires at least three distinct levels of a factor to see whether the response bends. A common, economical way to check for curvature without running a full three-level design is to add center points: a small number of extra runs with every factor set halfway between its low and high level (coded 0). If the response at the center points falls close to the average of the corner responses, the relationship is adequately linear across that range. If it falls noticeably off that average, curvature is present, and the design needs to be expanded to estimate it properly, which the next lesson takes up.
1. How many runs does an unreplicated 2³ full factorial design require?
2. A 2³ design is replicated 3 times. How many total runs are performed?
3. In coded units for a 2ᵏ design, what does -1 represent?
4. In standard order for a 2-factor design, what does the run labeled "b" mean?
5. Using this lesson's worked example, responses are (1)=60, a=72, b=68, ab=90. What is the main effect of factor A (primer)?
6. Using the same data, what is the main effect of factor B (cure temperature)?
7. Using the same data, what is the AB interaction effect?
8. What is the regression coefficient for factor A, given its effect of 17?
9. Why is a regression coefficient equal to half the corresponding effect?
10. Why can a standard 2ᵏ design not estimate a quadratic (curved) effect of a factor?
11. What is the purpose of adding center points to a 2ᵏ design?
12. If the response measured at the center points falls noticeably above or below the average of the four corner responses, what does this suggest?
13. In the AB interaction column of a 2-factor standard-order table, how is each row's sign determined?
14. A GCHS team studies mailing headline (A) and send day (B) at two levels each on response rate, with a positive AB interaction. What does a positive AB interaction mean in plain terms?