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

The 1-way ANOVA model and assumptions

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 one-way ANOVA is given by 

where Yij represents the j-th observation (j = 1, 2, on the i-th treatment (i = 1, 2, ..., levels). So, Y23 represents the third observation using level 2 of the factor. m is the common effect for the whole experiment, tirepresents the i-th treatment effect and eij represents the random error present in the j-th observation on the i-th treatment. 

The errors eij  are assumed to be normally and independently (NID) distributed, with mean zero and variance s2e . m is always a fixed parameter and t1, t2, 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 ti = 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 ti's are random variables assumed to be NID(0, s2t). This is the random effects model.

Whether the levels are fixed or random depends on how these levels are chosen in a given experiment 

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