Alternative hypothesis

Hi! So, yours is actually a sophisticated question that masquerades as a simple one, so I'll try to answer this in a way that conveys the concept while perhaps alluding to some of its problems. At its heart, the null hypothesis is a sort of "straw man" that is defined by a researcher at the beginning of an experiment that usually represents a state of affairs that would be expected to occur if the researcher's proposal were false. Note that a null hypothesis is entirely imaginary, and it has nothing to do with the actual state of the world. It is contrived, usually to show that the actual state of the world is inconsistent with the null hypothesis. Suppose a researcher is trying to determine whether the heights of men and women are different. A suitable null hypothesis might be that the difference of the two population averages (height of men and height of women) is equal to zero. Then the researcher would conduct his or her experiment by measuring the heights of many men and women. When it comes time to draw a statistical conclusion, he or she will compute the probability that the observed data (the set of heights) could have come from the null hypothesis (i.e., a world where there is no difference). This probability is called a "p-value". Conceptually, this is similar to a "proof by contradiction," in which we assert that, if the probability is very small that the data could have originated from the null hypothesis, it must not be true. This is what is meant by "rejecting the null hypothesis". It is different from a proof by contradiction because rejecting the null proves nothing, except perhaps that the null is unlikely to be the source of the observed data. It doesn't prove that the true difference is 5 inches, or 1 inch, or anything. Because of this, rejecting the null hypothesis is in NO WAY equivalent to accepting an alternative hypothesis. Usually, in the course of an experiment, we observe a result (such as the observed height difference, perhaps it is ~5 inches) that, once we reject, replaces the hypothesized value of 0 under the null. However, we DON'T know anything about the probability that our observed value is "correct", which is why we never say that we have "accepted" an alternative. I actually hesitate to discuss an "alternative" hypothesis because most researchers never state one and it doesn't matter for the purposes of null hypothesis significance testing (NHST). It is just the name given to the conclusion drawn by the researchers after they have rejected their null hypothesis. Philosophically, there is an adage that data can never be used to prove an assertion, only to disprove one. It includes an analogy about a turkey concluding that he is loved by his human family and is proven wrong upon being slaughtered on Thanksgiving. I'll include a link if I can find it. Now, think about this: The concept of rejecting a null hypothesis probably seems very reasonable as long as we are careful not to overinterpret it, and this is how NHST was performed for decades. But consider - what is the probability that the null hypothesis is true in the first place? In other words, how likely is it that the difference between mens' and womens' heights is equal to zero? I propose that the probability is exactly zero, and if you disagree then I will find a ruler small enough to prove me correct. The difference can never be equal to exactly zero (even though this is the "straw man" that our experiment refutes), so we are effectively testing against a hypothesis that can never be true. Rejecting a hypothesis we already know to be false tells us nothing important ("the data are unlikely to have come from this state that cannot be true"). And since every null hypothesis is imaginary, it is suggested that any null hypothesis can be rejected with enough statistical power (read:sample size). Often a "significant" result says more about a study's sample size than it does about the study's findings, even though the language used in papers/media suggests to readers that the findings are more "important" or "likely to be correct". This has, in part, led to a reproducibility crisis in the sciences and, for some, an undermining of subject-matter-experts' trust in the use of applied statistics. More on reddit.com

I think the technical term for my criterion (a) is that a test is "consistent" or "pointwise consistent in power," which means that as you take more and more data, the power gets closer and closer to 1. (I'm not 100% sure of this.) It's a desirable feature of an alternative hypothesis but maybe ... More on stats.stackexchange.com

The alternative determines what test statistics are in your rejection region. Equivalently, what values of the test statistic count as "at least as extreme" for the purpose of computing p-values. In some cases it might impact what the most powerful test is, or whether there even is one (but I don't think in any case you'd be likely to encounter). when the reason of the experiment is just to reject H0? I don't know that I would agree that's what experiments are for Ok, then we prove H0 is wrong. How does it support H1? I mean, the real u could be like u = 2311. You would only tend to use the simple alternative under a few possible circumstances. For example: There were only two realistic possibilities. ("If it's not this value most people in this area think it is, this other theory is the only one that makes any sense at all, because otherwise we'd have seen a, b and c already, which means that "4" would be the value under the alternative"). It doesn't happen much in the social sciences but I have seen this in the 'hard' sciences. I saw one in astronomy just recently, where there were only two competing theories that were seen as having any realistic chance of being right; a more common/ conventional one and a less popular competing theory (that would have overturned a number of other accepted ideas as well and require a lot more new research to figure out what was going on). Each corresponded to a particular, specific value for a parameter. The person I saw talking about it did discuss whether the alternative should be more general but the equality alternative was actually the one considered in the paper that was discussed. 2. There's only one alternative you would care to reject the null for. Again, not common in the social sciences. More on reddit.com

You're close enough for most purposes. In reality (ignoring the Bayesian side of things for now), 'rejecting the null' just means that we can't explain our results as resulting from random statistical variation. The association may be due to the hypothesized causal relationship; it may also be due to some unmeasured confounding factor. The next step would be to try a different experiment to test your hypothesis in a different way, and see if it stands up to that test. Eventually, scientists get tired of this and decide that the hypothesis is true enough for now. More on reddit.com