7.
Product and Process Comparisons
7.4. Comparisons based on data from more than two processes 7.4.3. Are the means equal?


A model that describes the relationship between the response and the treatment (between the dependent and independent variables)  The mathematical model that describes the relationship between the response and treatment for the oneway ANOVA is given by $$ Y_{ij} = \mu + \tau_i + \epsilon_{ij} \, , $$ where \(Y_{ij}\) represents the \(j\)th observation (\(j = 1, \, 2, \, \ldots, \, n_i\)) on the \(i\)th treatment (\(i = 1, \, 2, \, \ldots, \, k\) levels). So, \(Y_{23}\) represents the third observation using level 2 of the factor. \(\mu\) is the common effect for the whole experiment, \(\tau_i\) represents the \(i\)th treatment effect, and \(\epsilon_{ij}\) represents the random error present in the \(j\)th observation on the \(i\)th treatment.  
Fixed effects model  The errors \(\epsilon_{ij}\) are assumed to be normally and independently (NID) distributed, with mean zero and variance \(\sigma_\epsilon^2\). \(\mu\) is always a fixed parameter, and \(\tau_1, \, \tau_2, \, \ldots, \, \tau_k\) are considered to be fixed parameters if the levels of the treatment are fixed and not a random sample from a population of possible levels. It is also assumed that \(\mu\) is chosen so that $$ \sum \tau_i = 0 \, , \,\,\,\,\, i = 1, \, \ldots, \, k $$ holds. This is the fixed effects model.  
Random effects model  If the \(k\) levels of treatment are chosen at random, the model equation remains the same. However, now the \(\tau_i\) values are random variables assumed to be NID(0, \(\sigma_\tau\)) This is the random effects model.  
Whether the levels are fixed or random depends on how these levels are chosen in a given experiment. 