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Suppose there are two factors, A and B. There are 7 levels for factor A and 4 levels of factor B. At level 4 of factor A and level 3 of factor B, the yield (y) is 5.

Table 1.

                       A
     B  | 1    2    3    4    5    6    7
   -----|---------------------------------
      1 |12    9    0    17   11   6   13
      2 |8    15    7    12    0   9   20 
      3 |8     6   14     5   10   11  10  
      4 |10   12    4    15   17   18   9

A researcher first want to know that which levels of A are significant for the yield?

To answer the question, I am thinking of doing randomized block design (RBD) where factor B will be treated as a nuisance factor. After confirming that there is really difference at least two levels of A, then is it Tukey hsd test for identifying which levels of A are significant for the yield (y)? As far I know, Tukey hsd tests which pair of treatment means are significantly different, not which treatments are significant.

The 2nd research problem is: Does B influence the efficiency of A significantly?

As soon as this question is posed, it seems the analysis for answering the first question (which level of A is significant for the yield?) is no longer valid since we have conducted RBD taking into account B as a nuisance factor.

I think two-factor factorial design is appropriate to answer the 2nd question because it will allow us to test the interaction effect between factor A and B, hence answering the question whether B influence the efficiency of A significantly.

If so, then I think I don't need to conduct RBD since ANOVA table for the two-factor factorial design will also tell us whether A is significant for the yield (y).

To sum up, there are two research problem in this experiment:

(1) Is A significant for the yield? If so, which levels of A are significant for the yield?

(2) Does B influence the efficiency of A significantly?

Are two-factor factorial design most appropriate to answer the questions? How can I identify which levels of A are significant for the yield?

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