7.4.3.2. The 1-way ANOVA model and assumptions

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

7.4.3.2. 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)

The mathematical model that describes the relationship between the response and treatment for the one-way ANOVA is given by

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

Fixed effects model

The errors

_ij are assumed to be normally and independently (NID) distributed, with mean zero and variance

is always a fixed parameter and

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

is chosen so that

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

_i's are random variables assumed to be NID(0,

). This is the random effects model.

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