How to choose the null and alternative hypothesis?

Question

I'm practicing with the hypothesis test and I find myself in trouble with the decision about how to set a null and an alternative hypothesis. My main issue is to determine, in every situation, a "general rule" on how I can decide correctly which is the null and which is the alternative hypothesis.. can someone help me?

Here is an example: As an established scholar, you are requested to evaluate if Customer Relationship Management affects the financial performance of firms. The main issue will be solved by means of a test of hypothesis. Two hypothesis will be tested one against the other: CRM is related to performance, CRM is not related.

Thanks.

Answer 1

The rule for the proper formulation of a hypothesis test is that the alternative or research hypothesis is the statement that, if true, is strongly supported by the evidence furnished by the data.

The null hypothesis is generally the complement of the alternative hypothesis. Frequently, it is (or contains) the assumption that you are making about how the data are distributed in order to calculate the test statistic.

Here are a few examples to help you understand how these are properly chosen.

1. Suppose I am an epidemiologist in public health, and I'm investigating whether the incidence of smoking among a certain ethnic group is greater than the population as a whole, and therefore there is a need to target anti-smoking campaigns for this sub-population through greater community outreach and education. From previous studies that have been published in the literature, I find that the incidence among the general population is $p_0$. I can then go about collecting sample data (that's actually the hard part!) to test $$H_0 : p = p_0 \quad \mathrm{vs.} \quad H_a : p > p_0.$$ This is a one-sided binomial proportion test. $H_a$ is the statement that, if it were true, would need to be strongly supported by the data we collected. It is the statement that carries the burden of proof. This is because any conclusion we draw from the test is conditional upon assuming that the null is true: either $H_a$ is accepted, or the test is inconclusive and there is insufficient evidence from the data to suggest $H_a$ is true. The choice of $H_0$ reflects the underlying assumption that there is no difference in the smoking rates of the sub-population compared to the whole.

2. Now suppose I am a researcher investigating a new drug that I believe to be equally effective to an existing standard of treatment, but with fewer side effects and therefore a more desirable safety profile. I would like to demonstrate the equal efficacy by conducting a bioequivalence test. If $\mu_0$ is the mean existing standard treatment effect, then my hypothesis might look like this: $$H_0 : |\mu - \mu_0| \ge \Delta \quad \mathrm{vs.} \quad H_a : |\mu - \mu_0| < \Delta,$$ for some choice of margin $\Delta$ that I consider to be clinically significant. For example, a clinician might say that two treatments are sufficiently bioequivalent if there is less than a $\Delta = 10\%$ difference in treatment effect. Note again that $H_a$ is the statement that carries the burden of proof: the data we collect must strongly support it, in order for us to accept it; otherwise, it could still be true but we don't have the evidence to support the claim.

3. Now suppose I am doing an analysis for a small business owner who sells three products $A$, $B$, $C$. They suspect that there is a statistically significant preference for these three products. Then my hypothesis is $$H_0 : \mu_A = \mu_B = \mu_C \quad \mathrm{vs.} \quad H_a : \exists i \ne j \text{ such that } \mu_i \ne \mu_j.$$ Really, all that $H_a$ is saying is that there are two means that are not equal to each other, which would then suggest that some difference in preference exists.

Answer 2

The null hypothesis is nearly always "something didn't happen" or "there is no effect" or "there is no relationship" or something similar. But it need not be this.

In your case, the null would be "there is no relationship between CRM and performance"

The usual method is to test the null at some significance level (most often, 0.05). Whether this is a good method is another matter, but it is what is commonly done.

Answer 3

In science proofs, you can never prove anything, you can only demonstrate that your model describes the data better than another model. You want your alternate hypothesis to come from the new model under test, and the null hypothesis to be from a different model.

The null hypothesis should come from a model which others would choose to use when challenging your scientific claims! The most common pattern for a scientific claim is "I think that X is a factor in process Y. If everyone already believes X is a factor in the process, then there is nothing to prove, and everyone can just go out and talk about it over drinks. Scientific arguments with null hypothesis are interesting because, if someone takes the opposing view, "X is not a factor in process Y, then there is a disagreement. This is where science does its thing.

If you believe "X is a factor in process Y" enough to run an experiment, you should generally know what you're looking to see in the results. So now your phrase becomes "X is a factor in process Y, producing visible outcome Z."

This is where you pick your null hypothesis. If someone believes X is not a factor, and your experiment does indeed show Z, then they need an explanation for Z. With your choice of null hypothesis, you are effectively challenging their explanation. The dead simplest explanation is always "Z was caused by random chance because science is based on statistics." Accordingly, most null hypothesis are in the form of "The outcome should be predicted using the previously accepted model plus some random chance to account for statistics.

Both hypothesis should be phrased in terms of the visible outcome, NOT the model you intend to prove.[note] You never start with an alternate hypothesis of "I believe X is a factor." You phrase it "I expect to see this result when I observe Z." The null hypothesis will be phrased similarly, "The status quo predicts that we will see this different result when I observe Z." There is always a statistical phrasing in there such as "I expect to observe a normal distribution on Z when I do this experiment over and over." Once you observe results that defend your alternate hypothesis and reject the null hypothsis, you are THEN in a position to make claims about the validity of your model.

[note] This bolded statement is my opinion, but I feel confident enough in its wording choice to post it. The hypotheses draw a strong line between the intuitive portion of the science, and the data and analysis of the science. If your phrasing is too close to the model, it becomes hard to separate the model from the data, and makes it harder for the next scientist to use your data

In the case of our simple model with process Y and visible outcome Z, the existing belief is that Z will fit a distribution that everyone is already comfortable with, such as "the randomness expected by your particular laboratory equipment setup" or "the purity of the reagents used in the experiment." When you "reject the null hypothesis" what you are saying is most literally, "I have run this experiment, and it is so tremendously unlikely that random chance generated the observed behavior, that everybody should start considering that maybe there's more to this than meets the eye."

The alternative hypothesis is what you offer to the world to replace the null hypothesis. It is one thing to go do experiments to poke at holes in other's models, but that doesn't promote science nearly as well as poking holes in other's models and then replacing them with new models that do a better job.

Summary

With the null and alternate hypothesis, you are trying to challenge the current conventional thinking of the day. Choose the hypotheses so that they effectively declare "Here is a result everybody would expect (null hypothesis). However, I actually went out and did the experiment and gathered data, and it is VERY unlikely that the null hypothesis is true. Here is the result I expected (the alternate hypothesis). Nobody expected this hypothesis to be true but me, but when I gathered the data and did the statistics, it is very likely that my model does a better job of describing reality than the existing model. Accordingly, I reject the null hypothesis, accept my hypothesis, and challenge my fellow scientists to work from this new data."

And the fellow scientists are free to:

• Rejoice and accept your data and model with open arms.
• Ignore your data or model (sorry, it happens... welcome to real life)
• Reject your data, and spend their effort running experiments to show different data (It is very common for the science community to say "We do not trust your sample size of 10. We are going to redo your experiment with a sample size of 1000.)
• Accept your data, but not your model. They then must spend their effort generating a new model which explains the data in a different manner.

The last outcome causes strife and bickering, but is ABSOLUTELY part of the scientific process. By using the scientific method to publish your results, you accept that others are free to use the scientific method to contradict your results. They will do so, and publish their results.

At this point, the scientific community will make a political decision: who has to go out and spend the money to test their model, and whose model do we accept. TYPICALLY, because you published the model and the data first, and they are refuting your data, the onus is on them to run the experiments which proves why their model is better than yours. But this is now WELL beyond the hypothesis that caused the strife in the first place, so I leave you to experience them in your lifetime!