Hypothesis Testing II – The logic

A hypothesis is a supposition made as a basis for research or reasoning without regard for its truth. So says the Oxford dictionary. What starts the hypothesis testing process is just such a supposition. In the justice system example we made, there is a supposition on the part of law enforcement that the apprehended person committed the crime.In pharmaceutical testing, there may be a supposition a given medicine will reduce cholesterol values in blood tests.Because of the nature of induction we can never prove theory going forward.

As a prediction we recognize that the samples of interest are not those of the past, but rather those of the future and they have not happened yet. Thus they are not available to be sampled for our study and they may, in some way, be different from the samples that are available for the study.Further, that difference could render the results that are (necessarily) based on the samples in the study inapplicable to the future samples. This problem is unavoidable. So we design a way to test our hypothesis in another way. One aspect will be that the statements we can make will be probabilistic. We are in the land of uncertainty.To do our test we create another hypothesis called the null hypothesis.

The Null hypothesis is essentially an assumption that the research supposition (Research Hypothesis) has no merit. In our justice system case, it is an assumption of innocence. In the pharmaceutical case the null hypothesis would state that the new drug had no effect on lowering cholesterol.So there are two hypotheses. The null hypothesis (usually designated H0 and the research hypothesis which is usually, and somewhat unfortunately, called the alternate hypothesis (usually designated H1 or HA). I say unfortunately because this is the hypothesis that we are most interested in and the word ‘alternate’ makes it seem almost secondary.

Thus the comparison is constructed in such a way as to pose the null hypothesis and the burden of rejecting that (no difference) hypothesis rests with the advocate of the alternate hypothesis. That is, if the research can provide enough evidence to reject the null hypothesis, there is reason to believe that the new drug does have an affect on cholesterol. If the prosecutor can provide enough evidence we will suggest that the apprehended person did commit the crime.

We could be wrong. People sometimes ask a statistician to give a yes or no answer or to provide certain proof. That simply cannot be done and that is a problem fundamental to the inductive nature of this process. It is for this reason that we do not accept the alternate hypothesis in the sense that it is ‘proven’. The logic of hypothesis testing this way leads to either reject the null (no difference) hypothesis or to not reject it.

A standard of proof is given. In the justice example in the United States that standard is that the proof must be ‘beyond a reasonable doubt’. In a research study, such as the pharmaceutical example, we set probability level (more about this later).If that standard is met or exceeded we reject the null hypothesis. If the standard is not met we do not reject it.