Rank deficiency and full rank in ANOVA models

ANOVA models are full rank when the factor level combinations in the data make the estimation of all of the terms in the model possible. Rank deficiency occurs if any X variable columns in the design matrix can be written as a linear combination of the other X columns. In practical terms, rank deficiency occurs when the right observations to fit the model are not in the data. When rank deficiency occurs, Minitab removes terms until the model is small enough to fit.

Suppose you try to perform a two-way ANOVA with these factors:

C1 C2 C3
Machine Operator Response
1 Joel 15
1 Joel 18
1 Joel 17
2 Bill 14
2 Bill 15
2 Bill 16

In this example, the machine column has the exact same pattern as the operator column. If you perform ANOVA with this data set, Minitab removes the interaction term and the second factor term from the model in order to perform the analysis.

When you perform ANOVA, rank deficiency can also occur because an interaction term that is in the model does not have at least one observation for each combination of the factor levels. For example, the model is rank deficient when Machine has 3 levels, Operator has 3 levels, you include the Machine*Operator interaction in the model, and one of the possible 9 combinations of factor levels is not in the data:

C1 C2 C3
Machine Operator Response
1 Joel 15
1 Bill 18
1 Robin 17
2 Joel 14
2 Bill 15
2 Robin 16
3 Robin 16
3 Bill 16
3 Robin 17

In this example, Joel does not use Machine 3 so the interaction between Machine and Operator causes a rank deficiency. If you include the interaction term in an ANOVA model, Minitab removes the interaction term in order to perform the analysis.

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