# Random-Effect Model One-Way ANOVA

Video 4 provides an overview of the random effects model, and a comparison analysis between the fixed and random effects models using both MS Excel and Minitab.

Video 4. Fixed vs Random Effect Model Design of Experiments Explained with Examples Using Excel and Minitab

When the levels of a factor are selected randomly from a long list of options for the levels, the factor is called the random factor. The analysis of variance model used for a random factor is called the random effects model. For example, assume that we are interested to check the computing efficiency for the computers in a computer lab. As all the computers come with the same configurations, the mean efficiency is expected to be the same for all of them. Moreover, the computers are not necessarily being treated with something that changes the computing efficiency from computer to computer (fuel ingredients/additives in the fuel economy study for example). Therefore, testing the mean difference between computers will not make sense. Moreover, testing all computers (say 50 of them) in the lab does not make sense either. Therefore, the logical step would be to select a couple of computers randomly, which makes the computer a random factor. The experiment with a random factor is analyzed by a random effects model. For this type of random effects model, the research question would be to check the difference in the variation of computer power rather than the mean value of the computing power.

Note: In some situations, the levels of the factor could be treated in a random effects model, in which natural treatments may exist such as the use of corner computers vs the computers in the middle of the room. Some could get used more than other computers, which may make a significant difference in the long run. However, if the levels are selected at random, rather than a few fixed levels, the factor becomes a random factor, and the experiment will be analyzed using the random effects model. The factor definition that is used to determine whether the experiment is random or fixed. Experiment type (fixed or random) is NOT determined by whether there is a treatment or not.

Equation 3 can be rewritten as Equation 4 for the random effects model with the same notations. However, the treatment effect is now replaced by the random factor. Equation 4

Therefore, the variance of any observation is calculated by Equation 5 for a random effect model. Equation 5

It can be noted that the variance of any observation is only the experimental error variance (σ2) for a fixed effect model in Equation 3.