# Blocking effect in Experiment

Let $T\in\{0,1\}$ be the variable representing the treatment. Let $X\in\{0,1\}$ be the variable representing the attributes of subject. For example, $T$ is a new medicine that will make people grow taller and $X$ is gender.

In order to test the treatment on the response variable $Y$, I could do the following 2 experiment.

1. I do randomized assignment to my sample, and run the following regression $Y = a + bT + cX + e$. As I can observe $X$, so I also add $X$ in the regression.

2. I block the attributes $X$ by first divide my sample into 2 groups based on the value of $X$, then within each subgroup, I do randomized assignment. Then I pool the data together and run the following regression $Y = a + bT + cX + e$

My question is: these 2 regressions look the same, what is the difference of these 2 design? what do I gain by running one extra experiment in the 2nd method with blocking? (I think theoretically speaking, this randomized assignment will make $T$ and any attributes of subjects $X$ independent right?)

For example, in the first method, my treatment group ($T=1$) could end up with $(F_1,F_2,F_3,F_6,M_1,M_5)$, where $F$ means Female and $M$ means Male. My control group ($T=0$) is $(F_4,F_5,M_2,M_3,M_4,M_6)$ (for each subject, I flip a coin to determine treatment or control group). Then I run the regression with both $T$ and $X$

However, if I were using the second blocking method, I would first group Male together and group female together, then I do randomization within group. I could end up with in the Male group $T=1: (M_1,M_5)$ and $T=0: (M_2,M_3,M_4,M_6)$. And within Female Group $T=1: (F_1,F_2,F_3,F_6)$ and $T=0: (F_4,F_5)$. So this is essentially the same as in the Method 1 and this happens with same probability as happened in method 1. And all the data (observation $Y_i$'s) will be the same as collected from method 1.

So there should be no difference between these 2 methods