Mixed- and Random-Effects Models

A Simple Random-Effects Model

Here is a simple example, taken from Douglas C. Montgomery's book on experimental designs. A manufacturer wishes to investigate the research question of whether batches of raw materials furnished by a supplier differ significantly in their calcium content. Suppose data on the calcium content will be obtained on five batches that were received in 1 day. Furthermore, suppose six determinations of the calcium content will be made on each batch. Here batch is an input variable, and the response is the calcium content. The input variable is also called a factor. This is an example of a single-factor experiment, the factor being batch. The different possible categories of the factor are referred to as levels of the factor. Thus the factor batch has five levels. Note that on each level of the factor (i.e., on each batch), we obtain the same number of observations, namely six. In such a case, the data are called balanced. In some applications, it can happen that the numbers of observations obtained on each level of the factor are not the same; that is, we have unbalanced data.

In the above example, suppose the purpose of the data analysis is to test whether there is any difference in the calcium content among five given batches of raw materials obtained in one day. In experimental design terminology, we want to test whether the five batches have the same effects. This example, as stated, involves a factor (namely batch) having fixed effects. The reason is that there is nothing random about the batches themselves; the manufacturer has five batches given to him on a single day, and he wishes to make a comparison of the calcium content among the five given batches. However, there are many practical problems in which the factor could have random effects. In the context of the same example, suppose a large number of batches of raw materials are available in the warehouse, and the manufacturer does not have the resources to obtain data on all the batches regarding their calcium content. A natural option in this case is to collect data on a sample of batches, randomly selected from the population of available batches. Random selection is done to ensure that we have a representative sample of batches. Note that if another random selection is made, a different set of five batches could have been selected. If five batches are selected randomly, we then have a factor having random effects. Note that the purpose of our data analysis is not to draw conclusions regarding the calcium content of the five batches randomly selected; rather, we would like to use the random sample of five batches to draw conclusions regarding the population of all batches. The difference between fixed effects and random effects should now be clear. In the fixed-effects case, we have a given number of levels of a factor, and the purpose of the data analysis is to make comparisons among these given levels only, based on the responses that have been obtained. In the random-effects case, we make a random selection of a few levels of the factor (from a population of levels), and the responses are obtained on the randomly selected levels only. However, the purpose of the analysis is to make inferences concerning the population of all levels.

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