The Three-Way Factorial design has three grouping factors (independent variables A, B, and C) and one observed value (dependent variable), where A, B, and C are the main effects of the three factors. The two-way interactions are represented as A×B, A×C, and B×C, while the three-way interaction is represented as A×B×C.
The Analysis of Variance table reports the sum of squares and resulting F-test for each of the components of the model. To interpret a three-factor analysis, first look at the three-way interaction. If it is not significant, then examine the two-way interactions. If these are not significant, you can analyze the main effects. Differences between groups in main effects of over two levels can be analyzed using multiple comparison procedures. If the three-way interaction is present, the analysis of the two-way interaction terms or the main effects is invalid. In such cases, you must perform comparisons of means by cells or remodel your analysis.
The percentage of hardwood concentration in row pulp, the vat pressure, and the cooking time of the pulp are being investigated for their effects on the strength of paper. Three levels of hardwood concentration, three levels of pressure, and two cooking times are selected. A factorial experiment with two replicates is conducted, and the following data is obtained.
| Hardwood Concentration | Replicates | ||||||
|---|---|---|---|---|---|---|---|
| Cooking time (3 hrs) | Cooking time (4 hrs) | ||||||
| Pressure | Pressure | ||||||
| 400 | 500 | 650 | 400 | 500 | 650 | ||
| 2 | R1 | 196.6 | 197.7 | 199.8 | 198.4 | 199.6 | 200.6 |
| R2 | 196.0 | 196.0 | 199.4 | 198.6 | 200.4 | 200.9 | |
| 4 | R1 | 198.5 | 196.0 | 198.4 | 197.5 | 198.7 | 199.6 |
| R2 | 197.2 | 196.9 | 197.6 | 198.1 | 198.0 | 199.0 | |
| 8 | R1 | 197.5 | 195.6 | 197.4 | 197.6 | 197.0 | 198.5 |
| R2 | 196.6 | 196.2 | 198.1 | 198.4 | 197.8 | 199.8 | |
In the above-mentioned example, Cooking time (Factor A), Pressure (Factor B), and Hardwood Concentration (Factor C) are the factors. Factor A has two levels (A1: 3 hrs, A2: 4 hrs), Factor B has three levels (B1: 400, B2: 500, B3: 650), and Factor C has three levels of concentrations (C1, C2, C3) with two replications. Hence, we have 2 × 3 × 3 = 18 treatment combinations. These treatment combinations should be arranged in the data file or entered in the text area of the webpage in a nested form:
Sequence of Treatment Combinations in Data File
R1 R2
A1B1C1
A1B1C2
A1B1C3
A1B2C1
A1B2C2
A1B2C3
A1B3C1
A1B3C2
A1B3C3
A2B1C1
A2B1C2
A2B1C3
A2B2C1
A2B2C2
A2B2C3
A2B3C1
A2B3C2
A2B3C3
196.6 196.0
198.5 197.2
197.5 196.6
197.7 196.0
196.0 196.9
195.6 196.2
199.8 199.4
198.4 197.6
197.4 198.1
198.4 198.6
197.5 198.1
197.6 198.4
199.6 200.4
198.7 198.0
197.0 197.8
200.6 200.9
199.6 199.0
198.5 199.8
You can enter or paste the data in text area of the webpage. After entering/pasting the data, press the submit button.
Enter the levels for the first factor, levels for the second factor, levels for the third factor, number of replications, and number of sets/characters in the text boxes provided.
Choose the design from the select design group box and select the transformation if needed.
Press the Analyze button to analyze your data.