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7.
Product and Process Comparisons
7.4. Comparisons based on data from more than two processes 7.4.3. Are the means equal?
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| 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 $$ Y_{ij} = \mu + \tau_i + \epsilon_{ij} \, , $$ 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. | |||
