Why Say "Fail to Reject" in a Hypothesis Test?

Hypotheses Tests
The null and alternative hypotheses can be difficult to distinguish. C.K.Taylor

In statistics, the topic of hypothesis testing or tests of statistical significance is full of new ideas with subtleties that can be difficult for a newcomer. There are Type I and Type II errors. There are one-sided and two-sided tests. There are null and alternative hypotheses. And there is the statement of the conclusion: when the proper conditions are met we either reject the null hypothesis or fail to reject the null hypothesis.

Failing to Reject vs. Accept

One error that is commonly made by people in their first statistics class has to do with wording their conclusions to a test of significance. Tests of significance contain two statements. The first of these is the null hypothesis, which is a statement of no effect or no difference. The second statement, called the alternative hypothesis, is what we are trying to prove with our test. The null hypothesis and alternative hypothesis are constructed in such a way that one and only one of these statements is true.

If the null hypothesis is rejected, then we are correct to say that we accept the alternative hypothesis. However, if the null hypothesis is not rejected, then we do not say that we accept the null hypothesis. Part of this is probably a result of the English language. While the antonym of the word “reject” is the word “accept” we need to be careful that what we know about language does not get in the way of our mathematics and statistics.

Typically in mathematics, negations are formed by simply placing the word “not” in the correct place. Using this convention we see that for our tests of significance we either reject or we do not reject the null hypothesis. It then takes a moment to realize that “not rejecting” is not the same as “accepting.”

What We Are Proving

It helps to keep in mind the statement that we are attempting to provide enough evidence for is the alternative hypothesis. We are not trying to prove that the null hypothesis is true. The null hypothesis is assumed to be an accurate statement until contrary evidence tells us otherwise. As a result, our test of significance does not give any evidence pertaining to the truth of the null hypothesis.

Analogy to a Trial

In many ways, the philosophy behind a test of significance is similar to that of a trial. At the beginning of the proceedings, when the defendant enters a plea of “not guilty,” this is analogous to the statement of the null hypothesis. While the defendant may indeed be innocent there is no plea of “innocent” that is formally made in court. The alternative hypothesis of “guilty” is what the prosecutor attempts to demonstrate.

The presumption at the outset of the trial is that the defendant is innocent. In theory, there is no need for the defendant to prove that he or she is innocent. The burden of proof is on the prosecution. This means that the prosecuting attorney tries to marshal enough evidence to convince a jury that beyond a reasonable doubt, the defendant truly is guilty. There is no proving of innocence.

If there is not enough evidence, then the defendant is declared “not guilty.” Again this is not the same as saying that the defendant is innocent. It only says that the prosecution was not able to provide enough evidence to convince a jury that the defendant was guilty. In a similar way, if we fail to reject the null hypothesis it does not mean that the null hypothesis is true. It only means that we were not able to provide enough evidence to support the alternative hypothesis.

Conclusion

The main thing to remember is that we either "reject" or "fail to reject" the null hypothesis. We do not prove that the null hypothesis is true. In addition to this, we do not accept the null hypothesis.