# Clarification on Bonferroni in hypothesis testing

Could you help me to clarify how to use Bonferroni correction in hypothesis testing? Suppose I want to test $$H_0: X \perp Y \text{ , } Z \perp Y \text{ , } X \perp Z \hspace{1cm} \text{at level \alpha=5\%}$$ where $\perp$ denotes independence.

One way to do this is to test $$H_0^1: X \perp Y \hspace{1cm} \text{at level \alpha=\frac{5}{3}\%}$$ $$H_0^2: Z \perp Y \hspace{1cm} \text{at level \alpha=\frac{5}{3}\%}$$ $$H_0^3: X \perp Z \hspace{1cm} \text{at level \alpha=\frac{5}{3}\%}$$ What I am not sure about is the following: once I have the results of these three tests, what should I conclude about $H_0$? Should I reject $H_0$ if I reject at least one among $H_0^1, H_0^2, H_0^3$? Should I reject $H_0$ if I reject all of $H_0^1, H_0^2, H_0^3$?

The comma is often used as the replacement for AND ($\wedge$ in logic). Let me write your problem in an equivalent form using truth tables. Recall that the truth table for logical conjunction is $$\begin{array}{|c|c|c|}\hline p & q & p\wedge q\\ \hline T & T & T \\ \hline T & F & F \\ \hline F & T & F \\\hline F & F & F \\\hline \end{array}.$$ The hypothesis $H_0$ will hold if the last column of the below table returns the value true ($T$). We obtain
$$\begin{array}{|c|c|c|c|c|}\hline H_0^1 & H_0^1 & H_0^3 & H_0^1 \wedge H_0^2 & (H_0^1 \wedge H_0^2) \wedge H_0^3 \\ \hline T & T & T & T& T\\ \hline F & T & T & F &F \\ \hline F & F & T & F& F\\ \hline F & F & F & F&F \\ \hline T & F & F & F&F \\ \hline T & T & F & T&F \\ \hline F & T & F & F&F \\ \hline T & F & T & F&F \\ \hline \end{array}$$ Therefore, you can see that $H_0$ holds only in one case. If your comma (AND) was replaced by OR, then you could translate this problem into logic tables and use the one for disjunction (OR).
• Thanks. Hence, just to be sure about my final question: I reject $H_0$ if at least one among $H_0^1, H_0^2, H_0^3$ is false. Correct? – TEX May 10 '17 at 15:21
Usually there are more powerful ways to test a global hypothesis than the Bonferroni test. The advantage of using Bonferroni (and its variants) is that, not only can you reject the global hypotheses $$H_0$$ when there is at least one rejection among the tests at the $$\alpha/3$$ level, you can also reject every individual component hypothesis $$H_0^j$$ that was rejected at that level.
For most tests of global hypotheses (e.g., the $$F$$ test), rejection of the global hypothesis $$H_0$$ does not allow you to make any statements about any of the component hypotheses $$H_0^j$$. More specifically, you can't say anything about the components following a global rejection if you wish you to control the Familywise Error Rate (FWER) over all components, when you use global tests such as the $$F$$ test. The benefit of the Bonferroni test of the global intersection hypothesis (unlike the $$F$$ global test), is that you can make statements about the components of the intersection, with complete FWER control.