Three Factor experiments
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 main effects of the three factors. AXC, AXC and BXC are the two way interactions and AXBXC is the three way interaction.
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, first look at the three way interaction. If it is not significant, then look at the two way interaction. If these are not significant, then you can examine the main effects tests. Differences between groups in main effects of over two levels can be analyzed using multiple comparison procedures. If three way interaction is present, analysis of the two way interaction terms or the main effects is invalid. If there are significant two way interactions, then tests for main effects contained in those interactions are invalid. In these cases, you must perform comparisons of means by cells, or remodel your analysis. OPSTAT provides the analysis for following three factor experiments
1. Three factor CRD experiments
2. Three factor RBD experiments
3. Split-split plot experiments
4. Split plot taking first two factor as main and third as sub factor
5. Split plot taking first factor as main and remianing two factor as sub factors
We will explain the the data arrangements and procedure of analysis for three factor CRD RBD, Split-split plot designs with the help of following example.
Example : 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, the following data are obtained. Three factor designs involve the following
|
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 |
196 |
199.4 |
198.6 |
200.4 |
200.9 |
|
|
4 |
R1 |
198.5 |
196 |
198.4 |
197.5 |
198.7 |
199.6 |
|
R2 |
197.2 |
196.9 |
197.6 |
198.1 |
198 |
199 |
|
|
8 |
R1 |
197.5 |
195.6 |
197.4 |
197.6 |
197 |
198.5 |
|
R2 |
196.6 |
196.2 |
198.1 |
198.4 |
197.8 |
199.8 |
|
In the above mentioned example we have Cooking time (Say factor A), Pressure (Factor B) and Hardwood Concentration (Factor C) the factors. Factor A has two levels (i.e. A1 (3 hrs) and A2 (4hrs), Factor B has three levels [B1 (400), B2 (500), B3 (650)] and Factor C has three levels of concentrations i.e. C1, C2, C3 and two replications. Hence we have 2 x 3 x 3 = 18 treatment combinations. These 18 treatment combinations are arranged in data file or enter the text area of web page in such as way that the replications are arranged within the third factor, third factor within second factor and second factor within first factor in nested form. The arrangement is as under
| Sequence of treatments combinations in data file | ||
|
R1 R2 A1B1C1 A1B1C2 A1B1C3 A1B2C1 A1B2C2 A1B2C3 A1B3C1 A1B3C2 A1B3C3 A2B1C1 A2B1C2 A2B1C3 A2B2C1 A2B2C2 A2B2C3 A2B3C1 A2B3C2 A2B3C3 |
|
Data file look likes
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
Procedure of Analysis