Random levels are chosen at random from a large or infinite set of levels.
 
 

An example
of random factors
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

ANOVA table for example
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Computation of the components of variance
 

In the previous example, the levels of the factor temperature were considered as fixed, that is, the three temperatures were the only ones that we were interested in (this may sound somewhat unlikely, but let us accept it without opposition).  The model employed for fixed levels is called a fixed model. However, when the levels of a factor are random, such as operators, days, lots or batches, where the levels in the experiment might have been chosen at random from a large number of possible levels, the model is called a random model, and inferences are to be extended to all levels of the population. 

In a random model the experimenter is often  interested in estimating components of variance. Let us run an example that analyzes and interprets a component of variance or random model.

Components of Variance Example

A company supplies a customer with a larger number of batches of raw materials. The customer makes three sample determinations from each of 5 randomly selected batches to control the quality of the incoming material. The model is 

and the k levels (e.g. the batches) are chosen at random from a population with 0 mean and variance s_t². The data are shown below 

The test-statistic from the ANOVA table is, F = 36.94 / 1.80 = 20.5. 

If we had chosen an a value of .01, then the tabled F value from the F distribution critical value table in Chapter 1 for a df of 4 in the numerator and 10 in the denominator, is 14.5. 

Since the test-statistic is larger than the critical value, we reject the hypothesis of equal variances.  Since these batches were chosen via a random selection proces, it may be of interest to find out how much of the variance in the experiment might be attributed to batch diferences and how much to random error. In order to answer these questions, we can use the EMS column.  The estimate of s²_e is 1.80 and  the computed treatment mean square of 36.94 is an estimate of  s²_e + 3s²_t. .Setting the MSS estimates equal to the EMS values (this is called the Method of Moments), we obtain 

In terms of percentages we see that 11.71/13.51or 86.7 percent of the total variance is attributable to batch differences  and 13.3 percent to error within the batches.