Dear Community,
I struggle about the construction of the Nullhypothesis in a binomial test. Usually I include the equal sign in the Nullhypothesis, e.g. H_0: p0.5. My idea is that the test statistic follows a Binomial distribution with p=0.5 under the Nullhypothesis. Now I had discussion about including the equal sign in the Nilhypothesis, e.g. H_0:p=0.5. On the first sight for me this seems to cause some problems because I don't know the distribution of the test statistic under Nullhypothesis because p can take every value from [0; 0.5[. Somebody told me that in medical science they often put the equal sign in the Nilhypothesis, can somebody confirm this? What do you think about this issue? Do you have any references?
Thank you very much in advance!
Kind regards,
Florian
Dear Pelumi,
thank's for your response. Unfortunately, the document you attached doesn't directly my answer. I'm concerned about the equal sign in the Nilhypothesis and actually I'm not sure if this makes sense. As I stated above, usually the = is included in Nullhypothesis (as in the examples in your document), but does it make sense to include the = in the Nilhypothesis, i.e. H_0: p>p* and H_1:p
Ok, here it goes: the null hypothesis for the two-tailed test is p (pie value) = 0.5, thus the two-tailed hypothesis test will result in rejection if it differs greatly from 0.5 in “either direction” (p ≠0.5). On the other hand, the null hypothesis for one-tailed test is p ≤ 0.5, thus it would be rejected if p>0.5. Now whether you want one-tailed or two-tailed test depends on what you are trying to prove from your data in a particular discipline.
Two tailed tests are appropriate when the directional change is less important than finding whether there is any effect of the independent variable at all. For example, if a drug is tested for a disease x, just to know whether it has any effect (positive or negative) on patients, two tailed tests are appropriate. It will just detect whether any effect is there, good or bad. On the other hand, when direction is known or expected, then one-tailed test would be appropriate. For example, if a drug is known to treat for disease x, then the probability of its positive outcomes can be tested through one-tailed hypothesis.
Summary: The alternative hypothesis in the one-tailed test it is p > 0.5, and the two-tailed test is p ≠ 0.5.
Hey Khandoker,
thanks for your response! Let's consider the one-tailed test, what happens if I include the equal sign in my Nilhypothesis instead of the Nullhypothesis? For your example this would mean H_1:p>=0.5 and H_0:p
Let's put words on the problem that is referred here. Take the classic example of flipping a coin in a large number of trials. In a one-tailed situation, assume that you strongly suspect that the coin is biased toward head, and you want to test it. Now the way researchers define “success” is the number of times “heads” shows up and “failure” is the number of times tails shows up in the trial, right? But that is only a suspicion and not based on any scientific enquiry. The normal expectation (null hypothesis) is that a coin is fair and expected to produce equal number of heads and tails in a large number of trials. Suspicion (research question:) gets a researcher here to test it!
Thus, an "equal" sign (I mean ≤) in the null hypothesis means that the normal expectation is that the coin is not biased towards heads because I get heads “at most” (≤) half the times the number of trials, or I also normally expect that if the times I get heads are below half the times the number of trials, the coin is not biased toward the head. (Is it biased towards the tail? Yes, because prob of success=1-prob of failure, but that is not the hypothesis in one-tailed test here).
Now putting an equal sign in the alternate hypothesis means that the researcher is saying that the coin is biased towards the head even when he/she gets heads exactly half times the number of trials!!! This is theoretically inconsistent. Well, if there is any unique situation in medical science, then I am not aware of it. Please let me know if there is any.
@ Jochen: thank you very much for your detailed answer! I realized that I need to get more knowledge of the two concepts.
@ Khandoker: first thank you for your answer, too. I guess if the question of interest is reformulated you can solve this problem, e.g. Nilhypothesis says the probabilty of success is 50% or higher and the Null says probabilty of success is less than 50%. I would use the equal sign only in the Nilhypothesis.
Jochen mention the same concerns I have because the distribution of the test statistic is determined by the equal sign. I was a little bit confused by the chapter of Casella and Berger about Hypothesis testing (http://people.unica.it/musio/files/2008/10/Casella-Berger.pdf p.399 of the pdf): It's stated: 'H_0:theta element in theta_0 and H_1:theta element in theta_0^c'. On the first sight it was not obvious to me if Null has to include the equal sign.But now I guess the equal sign has to be included in the Nullhypothesis.
Thanks again
Florian
Hey Jochen,
thanks for your response and the references mentioned! I will read them in the near future...
Kind regards,
Florian
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