Blocking effect in Experiment

Question

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

Answer 1

While these two methods could produce the same result (but not necessarily so), there is an important difference. Your method two, with random assignment separately within both groups, tends to give a better balanced design.

With your method 1, it is possible (although with small probability, with a reasonable sample size), that all the males get treatment 0 and all the females treatment 1. What would you do then? So, it is generally better to use method 2.