Question 9.14: A 2³ factorial experiment was conducted to estimate the effe......
A 2³ factorial experiment was conducted to estimate the effects of three factors on the compressive strength of concrete cylinders. The factors were A: type of sand (fine or coarse), B: type of concrete (standard formulation or new formulation), and C: amount of water (low or high). Three replicates were obtained for each treatment. The strengths, presented in the following table, are measured in MPa. Estimate all effects and interactions.
| Treatment | Yield | Cell Mean | ||
| 1 | 16.157 | 18.777 | 21.351 | 18.7617 |
| a | 18.982 | 14.843 | 18.154 | 17.3263 |
| b | 18.760 | 20.661 | 20.409 | 19.9433 |
| ab | 23.459 | 29.009 | 23.051 | 25.1730 |
| c | 24.209 | 15.112 | 15.833 | 18.3847 |
| ac | 17.427 | 18.606 | 19.143 | 18.3920 |
| bc | 14.571 | 18.383 | 19.120 | 17.3580 |
| abc | 30.510 | 21.710 | 30.745 | 27.655 |
We use the sign table (Table 9.7) to find the appropriate sums and differences of the cell means. We present the calculations for the main effect A, the two-way interaction BC, and the three-way interaction ABC:
A effect estimate = \frac{1}{4} (−18.7617 + 17.3263\ −\ 19.9433 + 25.1730
− 18.3846 + 18.3920 − 17.3580 + 27.6550) = 3.5247
BC interaction estimate = \frac{1}{4}(18.7617 + 17.3263\ −\ 19.9433\ −\ 25.1730
− 18.3846 − 18.3920 + 17.3580 + 27.6550) = −0.1980
ABC interaction estimate = \frac{1}{4} (−18.7617 + 17.3263 + 19.9433\ −\ 25.1730
+ 18.3846 − 18.3920 − 17.3580 + 27.6550) = 0.9062
We present all the estimated effects in the following table, rounded off to the same precision as the data:
| TABLE 9.7 Sign table for a 23 factorial experiment | ||||||||
| Treatment | Cell Mean | A | B | C | AB | AC | BC | ABC |
| 1 | \overline{X}_{1} | − | − | − | + | + | + | − |
| a | \overline{X}_{a} | + | − | − | − | − | + | + |
| b | \overline{X}_{b} | − | + | − | − | + | − | + |
| ab | \overline{X}_{ab} | + | + | − | + | − | − | − |
| c | \overline{X}_{c} | − | − | + | + | − | − | + |
| ac | \overline{X}_{ac} | + | − | + | − | + | − | − |
| bc | \overline{X}_{bc} | − | + | + | − | − | + | − |
| abc | \overline{X}_{abc} | + | + | + | + | + | + | + |
| Term | Effect |
| A | 3.525 |
| B | 4.316 |
| C | 0.146 |
| AB | 4.239 |
| AC | 1.628 |
| BC | −0.198 |
| ABC | 0.906 |