Question

We have two possible models in two-way ANOVA: the fixed effects model and the random effects model. What are the differences between these two models and how does the analysis differ?

Answer 1

Here I would like to explain by using the examples instead of theory because we will understanding the fixed effect and Random effect model properly.

In Analysis of Variance and some other methodologies, there are two types of factors: fixed effect and random effect. Which type is appropriate depends on the context of the problem, the questions of interest, and how the data is gathered. Here are the differences:

Fixed effect factor: Data has been gathered from all the levels of the factor that are of interest.
Example: The purpose of an experiment is to compare the effects of three specific dosages of a drug on the response. Dosage is the factor; the three specific dosages in the experiment are the levels; there is no intent to say anything about other dosages.

Random effect factor: The factor has many possible levels, interest is in all possible levels, but only a random sample of levels is included in the data.
Example : A large manufacturer of widgets is interested in studying the effect of machine operator on the quality final product. The researcher selects a random sample of operators from the large number of operators at the various facilities that manufacture the widgets. The factor is "operator". The analysis will not estimate the effect of each of the operators in the sample, but will instead estimate the variability attributable to the factor "operator".

The analysis of the data is different, depending on whether the factor is treated as fixed or as random. Consequently, inferences may be incorrect if the factor is classified inappropriately. Mistakes in classification are most likely to occur when there is more than one factor in the study.

Example: Two surgical procedures are being compared. Patients are randomized to treatment. Five different surgical teams are used. To prevent possible confounding of treatment and surgical team, each team is trained in both procedures, and each team performs equal numbers of surgery of each of the two types. Since the purpose of the experiment is to compare the procedures, the intent is to generalize to other surgical teams. Thus surgical team should be considered as a random factor, not a fixed factor.