# One-Way ANOVA

# Fixed-Effect Model Analysis Basics

This easy to understand few words with practical examples will make a whole lot of sense while design and analysis of an actual experiment (in the next section of this module). Curious and interested readers can consult the Oscar Kempthorne textbooks for more real sugar (Kempthorne 1952; Hinkelmann and Kempthorne 2005; Hinkelmann and Kempthorne 2008)!

Although the identical twins are used for the experimental units with the exact same experimental conditions, there will be natural variations in mostly anything in this world. Therefore, the response is the function of the factors or the explanatory variables with some experimental error as in Equation 1.

Equation 1

# Means Model

In the fuel economy study, the research question is “whether there is a difference in fuel types with respect to the mean miles per gallon, mpg.” Therefore, Equation 1 can be written for testing the mean difference in response (mpg) from treatment to treatment (different fuel types in this case) as in Equation 2. As the response is written with respect to the mean treatment effect, this model is known as the **means model. **More precisely, the **population means model **as the model is intended for the generalization of the sample results for the population. It can also be noted that the model is linear as the power of the mean term (*µ _{i}*) is one.

Equation 2

# Effects Model

Assume that the three fuel types are made by mixing three different ingredients that improve fuel economy when added with a common base fuel. The base fuel without the ingredients will run the vehicle just fine with some mileage. If the ingredient effects are separated from the base fuel economy, the model Equation 2 can be written as Equation 3.

Equation 3

As the model in Equation 3 is designed to test the effects from the treatment (fuel ingredient in this case), the model is known as the **effects model** or **population effects model.**

# Error

The error could occur from many different sources, including the following (Hinkelmann and Kempthorne 2008).

## Experimental Error

- treatment error - error due to our inability to replicate a treatment from one application to the next
- state error - error due to random changes in the physical state of an EU
- selection error - error due to the random selection of EUs for the experiment

## Observational Error

- measurement error - error due to imprecision in our measurement or scoring procedure
- sampling error - error due to the random selection of observational units (OUs) for the investigation.

# Fixed vs Random Effect Model

In this fuel type study, only three fixed level, type 1, type 2, and type 3 fuel are tested as the level of the fuel factor. When fixed levels are used, the model is known as **fixed model**. Therefore, the complete name for this fuel type study will be **fixed effect model**. **Random effect model** will be discussed later in this module. The mixed effect models that include both fixed and random factors will be discussed in the advanced design of experiment section.