Completely Randomized Single-Factor Experiment

Experimental Design

• A completely randomized single-factor experiment is a design where one variable (factor) is manipulated across several levels (treatments), and the order of applying treatments to experimental units is randomized.
• The role of randomization in this experiment is extremely important, as it helps to balance out the effect of any nuisance variable that may influence the observed results.

Experimental Example: Tensile Strength

• A manufacturer of paper used for making grocery bags wants to improve the tensile strength of the product.
• The experiment involves four levels of hardwood concentration (5%, 10%, 15%, and 20%) and six test specimens at each concentration level.
• The 24 specimens are tested on a laboratory tensile tester in random order.

Randomization and Nuisance Variables

• Randomization helps to balance out the effect of any nuisance variable that may influence the observed results.
• For example, if there is a warm-up effect on the tensile testing machine, randomization helps to eliminate this effect.

Graphical Interpretation and Sample Size

• Graphical interpretation of the data is always useful, and box plots can be used to show the variability of the observations within a treatment and the variability between treatments.
• An important aspect of designing an experiment is to know how many observations are needed to make conclusions of sufficient accuracy and with sufficient confidence.
• Operating characteristic curves (OC curves) can be used as guides in determining the sample size.

Operating Characteristic Curves (OC Curves)

• OC curves are plots of the probability of a Type II error (β) for various sample sizes against values of the parameter under test.
• To determine the sample size (n) using an OC curve, we require the parameters: α, β, σ, and δ.
• The usual process of using these curves is by defining the ratios we wish to detect from the experiment.

Example: Sample Size Determination

• At least n = 6 replicates must be run in order to obtain a test with the required power (0.90).

Fixed vs. Random Factors

• A fixed factor is a statistical model in which the model parameters are fixed or non-random quantities.
• A random factor is a statistical model where the model parameters are random variables.
• If a fixed effects model is used, it means the same people are used in each trial of the study.
• If a random effects model is used, it is more generalizable because different participants are used each time.

Random Effects Model

• The random effects model is similar in appearance to the fixed effects model, but the treatment means are random variables.
• The expected values of the mean squares for treatments and error are somewhat different than in the fixed effects case.
• The variance components are estimated by equating the expected mean square to their observed values in the ANOVA table and solving for the variance components.

Analysis of Variance Method

• The analysis of variance method is a procedure to estimate the variance components (σ² and σ²) in the model.
• It consists of equating the expected mean square to their observed values in the ANOVA table and solving for the variance components.

Example: Textile Manufacturing

• A textile manufacturing company is interested in loom-to-loom variability in tensile strength.
• The variance components are estimated by equating the expected mean square to their observed values in the ANOVA table and solving for the variance components.
• The conclusion is that the looms in the plant differ significantly in their ability to produce fabric of uniform strength.

Randomized Complete Block Design

• In many practical situations, there are “nuisance” factors that influence the results of an experimental program but are not what we are attempting to study.
• Blocking and ANOVA can be used to separate the effect of the nuisance factor from the treatment factor.
• Examples of nuisance factors include multiple equipment setups, different personnel, and bus allocations.
• Blocking essentially removes some of the error that otherwise would be assigned to the treatments and assigns it to a known 'nuisance factor'.
• A blocking factor can be thought of as just another treatment.

Blocking Design

• A classic example of a blocking design is a tire wear evaluation, where tires from different manufacturers are placed on city buses that run different routes.
• The bus designation is a blocking factor, and the tires are assigned to each bus in a randomized complete block design.
• Since all tire types (called "treatment levels") are assigned to each bus (each "block"), this is called a randomized complete block design (RCBD).