Question

Use the information in Exercises \(2-4\) to construct an ANOVA table showing the sources of variation and their respective degrees of freedom. A two-factor factorial experiment with factor \(\mathrm{A}\) at four levels and factor \(\mathrm{B}\) at five levels, with three replications per treatment.

Short answer

Answer: The degrees of freedom for each source of variation are as follows: - Factor A: 3 - Factor B: 4 - Interaction A x B: 12 - Error (Within): 40

Step-by-step solution

Identify Sources of Variation

The sources of variation in this experiment are factor A, factor B, the interaction between factors A and B (A x B), and the error variation (Within).

Calculate Degrees of Freedom for Factor A

The formula for calculating degrees of freedom for factor A is (number of levels of factor A - 1). In this case, we have 4 levels, so the degrees of freedom for factor A is (4 - 1) = 3.

Calculate Degrees of Freedom for Factor B

The formula for calculating degrees of freedom for factor B is (number of levels of factor B-1). In this case, we have 5 levels of factor B, so the degrees of freedom for factor B is (5 - 1) = 4.

Calculate Degrees of Freedom for Interaction A x B

The formula for calculating degrees of freedom for the interaction A x B is (degrees of freedom for factor A) * (degrees of freedom for factor B). In this case, it is 3 * 4 = 12.

Calculate Degrees of Freedom for Error (Within)

The formula for calculating degrees of freedom for the error (Within) is (number of levels of factor A) * (number of levels of factor B) * (number of replications per treatment - 1). In this case, it is 4 * 5 * (3 - 1) = 40. Now that we have calculated the degrees of freedom for all sources of variation, we can create the ANOVA table.

ANOVA Table

| Source of Variation | Degrees of Freedom | |---------------------|--------------------| | Factor A | 3 | | Factor B | 4 | | Interaction A x B | 12 | | Error (Within) | 40 |

Key concepts

Degrees of Freedom

In statistical analysis, the concept of 'degrees of freedom' is vital to understand the precision and reliability of various calculations. In the context of an ANOVA table, degrees of freedom refer to the number of independent values that are free to vary in the data set, once certain restrictions have been applied.

For example, when you calculate the degrees of freedom for a factor in an experiment, you subtract one from the number of levels that factor has. This subtraction accounts for the fact that the total sum of deviations from the mean must equal zero, thereby imposing a constraint and reducing the number of free values.

When constructing an ANOVA table, calculate each source of variation’s degrees of freedom carefully. For a factor with four levels, such as Factor A in the given factorial experiment, there are three independent opportunities to deviate from the overall mean ((4 - 1) = 3). Similarly, for Factor B with five levels, there are four independent deviations ((5 - 1) = 4). These must be calculated correctly, as they influence subsequent calculations in the ANOVA, such as the sum of squares and mean squares.

Factorial Experiment

A factorial experiment is a sophisticated design used in experiments aiming to investigate the effects of several factors simultaneously. This type of experiment can reveal not only the individual impact of each factor but also how factors interact with each other to influence the response variable.

In a factorial experiment, such as the one described with Factor A at four levels and Factor B at five levels, each combination of factor levels is considered a treatment. When these treatments are replicated, it gives more data points, providing a clearer picture of variability and interactions. Factorial designs are powerful tools in research because they are efficient in terms of the amount of information they can deliver relative to the number of experimental runs conducted.

Source of Variation

The 'source of variation' in an ANOVA table identifies where variability in the data comes from. These sources typically include the main effects, which are the individual factors of the experiment, and may also include interactions among the factors, as well as error or within-group variation.

Understanding each source is integral in interpreting the ANOVA results. Main effects will show if there’s a significant difference due to the levels of a single factor, while the interaction effect reveals whether the effect of one factor varies across the levels of another factor. Finally, error, or the ‘within’ variation, accounts for the randomness or unexplained variance that does not fit into systematic effects from the factors or their interactions.

Interaction Effect

The interaction effect in an ANOVA is crucial in experiments where two or more factors are studied simultaneously. It represents the combined effect of factors on the response variable that is not simply additive. This means that the impact of one factor could depend on the level of another factor.

For instance, in the given experiment with factors A and B, an interaction effect would imply that the response to a specific level of Factor A is different depending on the level of Factor B. Calculating the degrees of freedom for the interaction, which is the product of the degrees of freedom for the individual factors (3 * 4 = 12), allows you to include this complex source of variability in the ANOVA table. The presence of a significant interaction effect can reveal synergies or conflicts between factors, which is essential for a deep understanding of the experimental data.