Two Factor Experiments

In research, a common challenge is to investigate the effects of multiple variables, or factors, on a response variable, denoted as 'Y'. Traditionally, factors were studied one at a time through separate experiments. However, R.A. Fisher demonstrated the advantages of studying several factors simultaneously in a single factorial experiment. This approach compares all treatments formed by combining the levels of different factors.

Factorial experiments are highly efficient because each observation provides information about all the factors involved. Moreover, they offer a systematic way to explore relationships between factors and their interactions.

Terminology

• Factors: The individual treatments.
• Levels: The different treatment levels within a factor.

Factorial Experiment Notation

A factorial experiment with two factors, each having two levels, is called a 2x2 factorial experiment. In general, a factorial experiment with 'f' factors at 't' levels is denoted as an 'ft' factorial experiment. If the levels differ among treatments, the notation changes to tA x tB. For instance, if factor A has 3 levels and factor B has 5, the experiment is denoted as a 3x5 factorial experiment.

OPSTAT provides analysis for commonly used two-factor experiments such as two-factor CRD, two-factor RBD, and Split-Plot designs. Below is an example demonstrating the analysis process:

Example: ANOVA for a 2x2 Factorial (CRD)

Data for the RCBD analysis of a 2 x 2 factorial design

Replicates
Character 1

Treatments combinations    1    2    3    4
a0b0                      12   15   14   13
a0b1                      19   22   23   21
a1b0                      29   27   33   30
a1b1                      32   35   38   37

Character 2

Treatments combinations
a0b0                      12.5 15.7 13.4 14.2
a0b1                      16.5 14.5 16.3 17.2
a1b0                      14.6 15.8 13.6 11.6
a1b1                      12.8 13.5 13.2 15.1
        

Data Arrangement

In this example, we have two factors, A and B, each with two levels (0 and 1), resulting in 2x2 = 4 treatment combinations. The data for these combinations should be arranged as follows:

• Enter all replications of the first level of the first factor and the first level of the second factor (a0b0) in one line. Use spaces or tabs to separate the observations.
• Next, enter the data for the first level of the second factor corresponding to the first level of the first factor (a0b1).
• Repeat this process for the second level of the first factor (a1), including the corresponding levels of the second factor (b0 and b1).
• If you have data for multiple characters, enter the data for the next character immediately after the first one, with no line gaps.

Example Data Entry

12  15  14  13
19  22  23  21
29  27  33  30
32  35  38  37

12.5  15.7  13.4  14.2
16.5  14.5  16.3  17.2
14.6  15.8  13.6  11.6
12.8  13.5  13.2  15.1
        

Procedure of Analysis

1. Enter or paste your data directly in the text area following the structure described above.
2. Specify the levels of the first factor, the levels of the second factor, the number of replications, and the number of sets in the provided text boxes.
3. Select the appropriate design (Two-Factor CRD, Two-Factor RBD, or Split-Plot) from the design options.
4. If needed, choose a transformation option.
5. Click the "Analyze" button to process your data. The results will be displayed on a separate web page.

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