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I have multiple parameters and want to examine their effect on the response variable, $Y$.

The experimental design is as follows. There are $3$ factors. Two of them: $A_1$ and $A_2$, have two levels (coded as $0$ and $1$). The third factor, $B$, is a blocking parameter and has $5$ levels: $b_1, b_2, \ldots, b_5$. I have a sample of $25$ subjects and plan to measure the response of each subject for each combination of factor values. If we denote the response of subject $i$ for the case when $A_1=j$ and $A_2=k$ in the block $b_r$ with $Y_{b_r,j,k,i}$ ($j,k \in \{0,1\}$, $r=1,\ldots,5$, $i=1,\ldots,25$) the data table for block $b_r$ will look this:

$$ \begin{bmatrix} b_r & 0 & 0 & Y_{r,0,0,1} \\ b_r & 0 & 0 & Y_{r,0,0,2} \\ \vdots \\ b_r & 0 & 0 & Y_{r,0,0,25} \\ \hline b_r & 0 & 1 & Y_{r,0,1,1} \\ b_r & 0 & 1 & Y_{r,0,1,2} \\ \vdots \\ b_r & 0 & 1 & Y_{r,0,1,25} \\ \hline b_r & 1 & 0 & Y_{r,1,0,1} \\ b_r & 1 & 0 & Y_{r,1,0,2} \\ \vdots \\ b_r & 1 & 1 & Y_{r,1,0,25} \\ \hline b_r & 1 & 1 & Y_{r,1,1,1} \\ b_r & 1 & 1 & Y_{r,1,1,2} \\ \vdots \\ b_r & 1 & 1 & Y_{r,1,1,25} \\ \end{bmatrix} $$

I will have $5$ such tables and want to know what are the appropriate ways of determining if $A_1$ and $A_2$ have the effect on $Y$. Let's focus on $A_1$ as the same will hold for $A_2$. I can think of two alternatives:

  1. Pair the responses $Y_{b_r, 0, k, i}$ and $Y_{b_r, 1, k, i}$ (iterating over all $b_r, k, i$) and use paired $t$ test.

  2. Group responses so that those where $A_1=0$ get to one group and those where $A_1=1$ get to another. Use ANOVA to test for the effect of $A_1$.

Are both of these ways ok? My main question is can I pair the responses as I explained for the $t$ test.

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