What this lesson covers
Design and analysis of experiments is one topic of Quantitative Methods and Tools, the largest section of the ASQ CQE exam at 34 of the 160 scored questions. Factorial questions are dependable marks: once you can build the sign table, every main effect and interaction is the same short calculation. This free lesson works one three-factor experiment from raw results to a decision, then lists the traps that cost people the mark.
The idea in one line
A main effect is the average response at a factor's high level minus the average at its low level.
In a two-level factorial, every factor is run at a low setting (written -) and a high setting (written +), in every combination with the other factors. Because each factor's high runs and low runs contain the same mix of the other factors, those other factors cancel out when you compare the two averages. That balance is what makes the arithmetic so simple, and it is why changing one factor at a time is a poor substitute: it cannot see interactions and needs more runs for the same precision.
The worked example (invented)
A team wants a stronger bonded joint. It tests three factors at two levels each, one run per combination (a 2^3 design, 8 runs), in random order:
- A: cure temperature (low, high)
- B: clamp pressure (low, high)
- C: surface preparation (wipe, abrade)
The response is peel strength in newtons per centimeter. Runs are labeled by the factors at their high level, so "ac" means A and C high, B low, and "(1)" means everything low.
| Run | A | B | C | AB | Strength |
|---|---|---|---|---|---|
| (1) | - | - | - | + | 22.4 |
| a | + | - | - | - | 27.0 |
| b | - | + | - | - | 23.6 |
| ab | + | + | - | + | 31.4 |
| c | - | - | + | + | 24.0 |
| ac | + | - | + | - | 28.2 |
| bc | - | + | + | - | 25.4 |
| abc | + | + | + | + | 33.6 |
The AB column is the product of the A and B signs: plus when A and B are both low or both high.
Step 1: main effect of A by level averages
- A high runs: a, ab, ac, abc = 27.0 + 31.4 + 28.2 + 33.6 = 120.2, average 30.05.
- A low runs: (1), b, c, bc = 22.4 + 23.6 + 24.0 + 25.4 = 95.4, average 23.85.
- A effect = 30.05 - 23.85 = 6.2.
Check by contrast. Multiply each response by its sign in the A column and add: -22.4 + 27.0 - 23.6 + 31.4 - 24.0 + 28.2 - 25.4 + 33.6 = 24.8. Divide by half the number of runs (4): 24.8 / 4 = 6.2. Both routes agree.
Step 2: the other main effects
- B: high runs b, ab, bc, abc = 23.6 + 31.4 + 25.4 + 33.6 = 114.0 (average 28.50); low runs (1), a, c, ac = 22.4 + 27.0 + 24.0 + 28.2 = 101.6 (average 25.40). B effect = 3.1. Check: (114.0 - 101.6) / 4 = 12.4 / 4 = 3.1.
- C: high runs c, ac, bc, abc = 24.0 + 28.2 + 25.4 + 33.6 = 111.2 (average 27.80); low runs (1), a, b, ab = 22.4 + 27.0 + 23.6 + 31.4 = 104.4 (average 26.10). C effect = 1.7. Check: (111.2 - 104.4) / 4 = 6.8 / 4 = 1.7.
Step 3: the AB interaction
- AB plus runs (1), ab, c, abc = 22.4 + 31.4 + 24.0 + 33.6 = 111.4 (average 27.85); minus runs a, b, ac, bc = 27.0 + 23.6 + 28.2 + 25.4 = 104.2 (average 26.05). AB effect = 1.8.
- Check another way. With B low, raising A adds (27.0 - 22.4) = 4.6 and (28.2 - 24.0) = 4.2, an average of 4.4. With B high, it adds (31.4 - 23.6) = 7.8 and (33.6 - 25.4) = 8.2, an average of 8.0. The interaction is half the difference: (8.0 - 4.4) / 2 = 1.8.
So cure temperature helps more when clamp pressure is high. Worked the same way, AC = 0.0, BC = 0.3 and ABC = 0.2.
Step 4: read the result
Ranked by size: A 6.2, B 3.1, AB 1.8, C 1.7, then BC 0.3, ABC 0.2 and AC 0.0. With one run per combination there is no pure error to test against, so a common approach is to treat the cluster of very small effects (0.0 to 0.3) as a rough picture of noise and plot all the effects on a normal probability plot or a Pareto chart of effects. Here A and B stand well clear; AB and C are worth confirming.
Prediction check. The grand mean is 215.6 / 8 = 26.95. A model with A, B, C and AB adds half of each effect at the chosen levels. At all high: 26.95 + 3.1 + 1.55 + 0.85 + 0.9 = 33.35, against the observed 33.6 for run abc. Close agreement says the small effects left out really are small. The next step would be confirmation runs at the chosen settings.
Common traps
- Dividing the contrast by 8. For a 2^3 design, divide by 4 (half the runs). Dividing by 8 halves every effect.
- Mixing up the sign of an interaction column. Multiply the signs; never read them off a factor column.
- Calling an effect important because it is not zero. Without replicates, judge it against the other small effects, and confirm with new runs.
- Reading a main effect without its interaction. If AB is large, the best level of A depends on B; report A's effect at each level of B.
- Running in standard order. Randomize the run order, or a drift over the day can hide in a factor's effect.
Try it
The free practice set has 10 questions on this topic: counting runs, main effects in 2^2 and 2^3 designs, an interaction, sign columns, aliases and resolution in a half fraction, and why run order is randomized. Each answer explanation works the calculation twice.
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