Question

Suppose you were to conduct a two-factor factorial experiment, factor A at four levels and factor \(\mathrm{B}\) at five levels, with three replications per treatment. a. How many treatments are involved in the experiment? b. How many observations are involved? c. List the sources of variation and their respective degrees of freedom.

Short answer

Answer: In this experiment, there are 20 treatments, 60 observations, and 4 sources of variation (factor A, factor B, interaction (AB), and error). The respective degrees of freedom for these sources of variation are: Factor A (3 degrees of freedom), Factor B (4 degrees of freedom), Interaction (AB) (12 degrees of freedom), and Error (40 degrees of freedom).

Step-by-step solution

Part a: Number of treatments involved in the experiment

In a two-factor factorial experiment, the total number of treatments is the product of the levels of each factor. In this case, factor A has four levels and factor B has five levels. Finding the number of treatments means finding the product of the number of levels of factor A and factor B. Mathematically, Number of treatments = levels of factor A × levels of factor B

Calculate the number of treatments

Number of treatments = 4 levels of factor A × 5 levels of factor B = 20 treatments So, there are 20 treatments involved in the experiment.

Part b: Number of observations involved in the experiment

In this experiment, for each treatment, there are three replications. To find out the total number of observations, we must multiply the total number of treatments by the replications per treatment. Mathematically, Number of observations = Number of treatments × Replications per treatment

Calculate the number of observations

Number of observations = 20 treatments × 3 replications per treatment = 60 observations So, there are 60 observations involved in the experiment.

Part c: Sources of variation and their respective degrees of freedom

In a two-factor factorial experiment, there are three main sources of variation: factor A, factor B, and the interaction (AB). The error is another source of variation. To find out the degrees of freedom for each source, we can use the following formulas: Degrees of freedom for factor A = (levels of factor A - 1) Degrees of freedom for factor B = (levels of factor B - 1) Degrees of freedom for interaction (AB) = (degrees of freedom for factor A) × (degrees of freedom for factor B) Degrees of freedom for error = (Total number of observations) - (Number of treatments)

Calculate the degrees of freedom for each source of variation

Degrees of freedom for factor A = 4 - 1 = 3 Degrees of freedom for factor B = 5 - 1 = 4 Degrees of freedom for interaction (AB) = 3 × 4 = 12 Degrees of freedom for error = 60 - 20 = 40 So, the sources of variation and their respective degrees of freedom are as follows: - Factor A: 3 degrees of freedom - Factor B: 4 degrees of freedom - Interaction (AB): 12 degrees of freedom - Error: 40 degrees of freedom

Key concepts

Sources of Variation in a Two-Factor Factorial Experiment

In a two-factor factorial experiment, understanding the sources of variation helps in determining what contributes to changes in the data. These are the differences we observe in our experimental results.

There are typically four sources of variation:

• Factor A: This is the first variable or factor you're experimenting on. If you're testing four different levels of sunshine, the variation you observe due to these different levels falls under Factor A.
• Factor B: This is your second variable or factor. For example, if you're testing five levels of watering in your experiment, the observed effects stemming from these levels belong here.
• Interaction between Factor A and B (AB): Sometimes, the combined effect of two factors differs from what you would expect by considering each factor alone. This is your interaction effect.
• Error: This refers to unexplained variation. Natural variability or minor factors not of interest in your experiment might influence it. This represents "noise" in the experiment.

Once these sources are identified, we can start to quantify how much each contributes to the overall variation observed in the experiment.

Degrees of Freedom in Factorial Experiments

Degrees of freedom (DF) is a statistical concept that helps quantify how many values in a calculation have the freedom to vary. It provides insight into the sample size's capacity to provide reliable estimates.

When working with a two-factor factorial experiment:

• Degrees of Freedom for Factor A: Calculated as the number of levels minus one. For instance, if Factor A has four levels, then DF for A is 4 - 1 = 3.
• Degrees of Freedom for Factor B: Similar to Factor A, if Factor B has five levels, then the DF for B is 5 - 1 = 4.
• Degrees of Freedom for Interaction (AB): It is a multiplicative combination of the degrees of freedom for both factors: DF for A multiplied by DF for B, which in our case is 3 × 4 = 12.
• Degrees of Freedom for Error: This is calculated as the total number of observations minus the number of treatments. In our example, with 60 observations and 20 treatments, it would be 60 - 20 = 40.

Understanding these degrees of freedom allows scientists to properly allocate variation sources and perform accurate statistical tests.

Understanding Replications per Treatment

Replications amplify the reliability of an experimental result by repeating the same treatment multiple times. In a two-factor factorial experiment, replication ensures the experiment's findings aren't due to random chance.

Why is replication important?

• Improved Accuracy: Repeating a treatment three times provides a more stable and accurate measure of the effect size, reducing the effect that random errors might have on the outcome.
• Statistical Significance: More replications can lead to a better approximation of true variation and enhance the detection of treatment effects.
• Reliability: It substantiates the results, enhancing the credibility of the conclusions drawn from the experiment.

In our example, with three replications per treatment across 20 treatments, we obtain 60 observations. These replications help confirm that any variations observed are genuine and not mere coincidences.