When two factors, A and B, are studied at low and high levels, four combinations result. These four points are not merely a list: they are the four vertices of a square. That symmetric structure separates main effects from interactions.
This unit has one question.
When every combination of factor levels is run, which effects can be viewed without mixing them together?
A 2² full factorial design includes all four vertices
When low is coded −1 and high +1, the design table has these columns.
| A | B | AB |
|---|---|---|
| −1 | −1 | +1 |
| +1 | −1 | −1 |
| −1 | +1 | −1 |
| +1 | +1 | +1 |
The AB column is A×B. Each of A, B, and AB contains equal numbers of +1 and −1 values, so the columns are balanced. The sum of elementwise products for any two different columns is zero. That is orthogonality in this setting.
Because of orthogonality, the +/− values of B and AB cancel in a balanced way when the A effect is calculated. An estimate of one effect is not pulled by the true value of another. Missing runs, unequal weights, or execution failures can break this perfect orthogonal structure.
A main effect is the difference between the high-level and low-level means
A effect = mean(Y | A=+1) − mean(Y | A=−1)
Because this averages over both B levels, it is the average change associated with A. The B effect is defined the same way. The AB effect is the mean at the two vertices where AB is +1 minus the mean at the two vertices where AB is −1.
If the ±1-coded regression equation is ŷ=b₀+bAA+bBB+bABAB, each regression coefficient is half of its factorial effect. Moving A from −1 to +1 spans two coded units.
When AB is large, the average main effect of A merely averages effects under two different B conditions. Interpret an interaction plot and condition-specific simple effects together. Removing related main effects from a model with an interaction breaks hierarchy.
Nonparallel lines show dependence on conditions
Connect the mean at A low and A high separately for B=low and B=high.
- parallel lines: AB effect is 0
- different slopes: interaction
- crossing lines: the direction of the A effect can reverse by B level
Slightly nonparallel lines do not by themselves establish a practically important interaction. Estimate pure error through independent replication, then inspect effect CIs, ANOVA, and residuals. In a saturated 2² run once at every combination, fitting all effects leaves no residual degrees of freedom.
A 2³ design has eight vertices of a cube
Three two-level factors produce 8 runs and effects A, B, C, AB, AC, BC, and ABC. When all combinations are observed, these seven effect columns are orthogonal. With k factors, the number of runs grows quickly as 2^k. That growth motivates fractional factorial designs in U20, but we must first understand what the full design separates in order to understand the cost of saving runs.
In-Silico Lab: match hand calculations to regression coefficients
The Lab creates all four vertices of Y=70+8A+5B+γAB.
- Verify that the A effect is 16 and the B effect is 10.
- With γ=6, see why the AB effect is 12 and the regression coefficient is 6.
- Calculate the +/− sums in each column and the cross-products between columns to confirm balance and orthogonality.
- Consider what problem occurs if one of the four responses is removed.
2² 설계표에서 효과를 직접 복원하세요
네 조합의 반응으로 A·B·AB 효과와 ±1 coded 회귀계수를 비교합니다.
처음이라면: 무엇을 눌러야 하나요?
- 1. 질문을 먼저 읽기Lab 제목에서 이번에 비교할 한 가지를 확인합니다.
- 2. 조건 하나만 바꾸기처음에는 n, 효과, 산포 같은 입력 중 하나만 바꾸십시오.
- 3. 새 합성 표본 누르기새 합성 데이터가 만들어집니다. 같은 조건도 표본에 따라 달라질 수 있습니다.
- 4. 그림과 계산 결과 비교하기바꾸기 전후 무엇이 움직이고 무엇이 그대로인지 한 문장으로 적어보십시오.
막히면 초기화로 돌아가 기본 결과를 본 뒤 조건 하나만 바꾸십시오. 이 Lab은 정답 판정기가 아니라 패턴 관찰 도구입니다.
같은 설정의 합성 관측
| run | A | B | AB | Y |
|---|---|---|---|---|
| 1 | -1 | -1 | 1 | 63.0 |
| 2 | -1 | 1 | -1 | 61.0 |
| 3 | 1 | -1 | -1 | 67.0 |
| 4 | 1 | 1 | 1 | 89.0 |
계산 결과
효과의 부호는 high−low와 반응의 좋은 방향을 함께 적어야 해석할 수 있습니다.
교육용 synthetic model · bjs-factorial-sequence-v1. 실제 연구 판단에는 실험단위, 결측, 분포, 다중성, 사전계획과 도메인 기준을 별도로 반영해야 합니다.
JMP output connects the design table and effect plots
−1/+1 조합, 균형과 실행 수를 확인합니다.
고수준 평균과 저수준 평균의 차이를 효과로 읽습니다.
꼭짓점 반응과 비평행성을 공간에서 확인합니다.
Do not confuse coded levels in a Design Table with physical units. Effect Estimates show direction according to the coding convention. Cube Plots and Interaction Plots are mean summaries of raw data, so also inspect the spread of replicate points and residuals.
Example result statement
A 2² full factorial design of temperature (A: 30/37°C) and pH (B: 6.8/7.4) was independently replicated in two batches. Using high−low contrasts, the temperature effect was +16.2, the pH effect +9.8, and the AB interaction effect +11.5 units. The positive AB effect indicated that the pH effect was larger at high temperature. The coded and natural-unit equations, run order, and residual diagnostics were retained together.
At the end of this unit
- A full factorial design includes every combination of factor levels.
- ±1 coding makes balance and orthogonality explicit.
- A main effect is the high−low difference averaged over the other factors.
- A coded regression coefficient is half of a two-level factorial effect.
- When an interaction is large, interpret effects by condition.
- A saturated design without replication has no residual degrees of freedom.
The next unit explains why the quality of evidence can differ with run order, replication, and blocks even when the same four combinations are run.
Official supplementary resources
The design and Lab in this article use synthetic educational data.