7.
Product and Process Comparisons
7.4. Comparisons based on data from more than two processes 7.4.2. Are the means equal?


A model
that describes the relationship between the response and the treatment
(between the dependent and independent variables)
Fixed and random effects 
The mathematical model that
describes the relationship between response and treatment for the oneway
ANOVA is given by
where Y_{ij} represents the jth observation (j = 1, 2, ...n_{i}) on the ith treatment (i = 1, 2, ..., k levels). So, Y_{23} represents the third observation using level 2 of the factor. m is the common effect for the whole experiment, t_{i}represents the ith treatment effect and e_{ij }represents the random error present in the jth observation on the ith treatment. The errors e_{ij} are assumed to be normally and independently (NID) distributed, with mean zero and variance s^{2}_{e }. m is always a fixed parameter and t_{1}, t_{2}, ...t_{k} are considered to be fixed parameters if the levels of treatment are fixed,and not a random sample from a population of possible levels. It is also assumed that m is chosen so that S t_{i }= 0 i = 1, ...k holds. This is the fixed effects model. If the k levels of treatment are chosen at random, the model equation remains the same. However, now the t_{i}'s are random variables assumed to be NID(0, s^{2}_{t}). This is the random effects model. Whether the levels are fixed or random depends on how these levels are
chosen in a given experiment
