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Let's say part of the mainstream believes that a drug X is more effective than drug Y. Another part of the mainstream believes that drug x is less effective than drug Y. A scientist appears who wants to challenge it and show that the drug X is equivalent to the drug Y, generating the same result and the same side effects, since both act exactly by the same mechanism.

I know the example seems a little forced, but it is just to provoke the classic definition that always aligns the alternative hypothesis with the research hypothesis.

In this case there should be the equality (expressed as a small interval) the alternative hypothesis and therefore the difference (outside of cited small interval) as an null hypothesis?

Previously Gavin Simpson thinks in his answer for that question, that the alternative hypothesis would be possible to use equality, but I would like to see more opinions.

In statistics there are tests of equivalence as well as the more common test the Null and decide if sufficient evidence against it. The equivalence test turn this on its head and posits that effects are different as the Null and we determine if there is sufficient evidence against this Null.

I'm not clear on your drug example. If the response is a value/indicator of the effect, then an effect of 0 would indicate not effective. One would set that as the Null and evaluate the evidence against this. If the effect is sufficiently different from zero we would conclude that the no-effectiveness hypothesis is inconsistent with the data. A two-tailed test would count sufficiently negative values of effect as evidence against the Null. A one tailed test, the effect is positive and sufficiently different from zero, might be a more interesting test.

If you want to test if the effect is 0, then we'd need to flip this around and use an equivalence test where the H0 is the effect is not equal to zero, and the alternative is that H1 = the effect = 0. That would evaluate the evidence against the idea that effect was different from 0.

ChatGPT has agreed with the cited user, and gave me the following answer to the same question (Don't be upset with me. I quote this answer because it raised reflections on me):

Yes, in this case, the alternative hypothesis would be that the two drugs, X and Y, are equivalent in terms of effectiveness and side effects. The null hypothesis would be that there is a significant difference between the two drugs in terms of effectiveness or side effects. The scientist's aim would be to gather evidence that supports the alternative hypothesis and rejects the null hypothesis.

It's worth noting that the null hypothesis is often stated as the opposite of the research hypothesis in order to provide a clear statement of what the scientist is trying to disprove. However, in this case, the null hypothesis does not necessarily contradict the mainstream beliefs that you mentioned. It simply states that there is a difference between the drugs, whereas the mainstream beliefs disagree on the direction of that difference.

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    $\begingroup$ ChatgPT has agreed with the cited user.... I am reading the post till above. $\endgroup$ Commented Apr 16, 2023 at 15:32
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    $\begingroup$ Also it would be appreciable to cite the comment/post where Gavin asserted so for the sake of completion of the post. $\endgroup$ Commented Apr 16, 2023 at 15:38
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    $\begingroup$ Paulo, What I and others would suggest strongly is not to consider even a minuscule of assertions provided by ChatGPT; it's not a stat guru sadly. $\endgroup$ Commented Apr 16, 2023 at 15:39
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    $\begingroup$ @PauloBuchsbaum welcome to the site, Paulo, and congrats on what I view as a very interesting question! But as this thread has demonstrated, a citation of ChatGPT is more likely to sidetrack the debate than to enrich it... $\endgroup$ Commented Apr 16, 2023 at 15:46
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    $\begingroup$ @PauloBuchsbaum Answering is always a learning experience here, quite often people will offer constructive criticism. I find that the majority of my own answers were improved by these interactions $\endgroup$
    – Firebug
    Commented Apr 16, 2023 at 19:39

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I would say that the "alternative hypothesis" is usually NOT a "proposed hypothesis".

You do not define "proposed hypothesis" and it is not a common phrase. Presumably you mean that it is either a statistical hypothesis or it is a scientific hypothesis. They are usually quite different things.

A scientific hypothesis usually concerns a something to do with the true state of the real world, whereas a statistical hypothesis concerns only conditions within a statistical model. It is very common for the real world to be more complicated and less well-defined than a statistical model and so inferences regarding a statistical hypothesis will need to be thoughtfully extrapolated to become relevant to a scientific hypothesis.

For your example a scientific hypothesis concerning the two drugs in question might be something like 'drug x can be substituted for drug y without any noticeable change in results experienced by the patients'. A relevant statistical hypothesis would be much more restricted along the lines of 'drug x and drug y have similar potencies' or that 'drug x and drug y have similar durations of action' or maybe 'doses of drug x and drug y can be found where they have similar effects'. Of course, the required degree of similarity and the assays used for evaluation of the statistical hypothesis will have to be defined. Apart from the enormous differences in scope of the scientific and potential statistical hypotheses, the first may require several or all of the others to be true.

If you want to know if a hypothesis is a statistical hypothesis then if it concerns the value of a parameter within a statistical model or can be restated as being about a parameter value, then it is.

Now, the "alternative hypothesis". For the hypothesis testing framework there are two things that are commonly called 'alternative hypotheses'. The first is an arbitrary effect size that is used in the pre-data calculation of test power (usually for sample size determination). That alternative hypothesis is ONLY relevant before the data are in hand. Once you have the data the arbitrarily specified effect size loses its relevance because the observed effect size is known. When you perform the hypothesis test the effective alternative becomes nothing more than 'not the null'.

It is a bad mistake to assume that a rejection of the null hypothesis in a hypothesis test leads to the acceptance of the pre-data alternative hypothesis, and it is just about as bad to assume that it leads to the acceptance of the observed effect size as a true hypothesis.

Of course, the hypothesis test framework is not the only statistical approach, and I would argue, it is not even the most relevant to the majority of scientific endeavours. If you use a likelihood ratio test then you can compare the data support for two specified parameter values within the statistical model and that means that you can do the same within a Bayesian framework.

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  • $\begingroup$ I never say that reject the null hypothesis is equivalent to a proof of alternative hypothesis. $\endgroup$ Commented Apr 17, 2023 at 0:13
  • $\begingroup$ Michel Lew I will change the term "proposed hypothesis" for "research hypothesis". I don't want to dive in tecnical jargon here (Bayes or likehood ratio test), I want a general picture. Here an excerpt of information on the hypothesis test: "Research hypothesis: this is the hypothesis that you propose, also known as the alternative hypothesis" $\endgroup$ Commented Apr 17, 2023 at 0:20
  • $\begingroup$ All my life reading papers it's only hypothesis pairs: null and alternative. Perhaps um paper can includes more than one hypothesis pair, but I always see the alternative hypothesis as complementary to the null hypothesis. $\endgroup$ Commented Apr 17, 2023 at 0:22
  • $\begingroup$ @PauloBuchsbaum Seems to me that to say that likelihood and Bayes approaches are "technical jargon" is more than a bit weird. Also, if you change from "proposed hypothesis" to "research hypothesis"you will need to think about my comments on hypotheses relating to the real world and those that relate to statistical models... $\endgroup$ Commented Apr 17, 2023 at 7:22
  • $\begingroup$ A relevant statistical hypothesis... is followed by reference to the real world rather than a statistical model. So in that sense it is not yet a statistical hypothesis, I think. $\endgroup$ Commented Apr 17, 2023 at 8:32
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The principle of statistical hypothesis tests, by definition, treats the null hypothesis H0 and the alternative H1 asymmetrically. This always needs to be taken into account. A test is able to tell you whether there is evidence against the null hypothesis in the direction of the alternative.

It will never tell you that there is evidence against the alternative.

The choice of the H0 determines what the test can do; it determines what the test can indicate against.

I share @Michael Lew's reservations against a formal use of the term "proposed hypothesis", however let's assume for the following that you can translate your scientific research hypothesis into certain parameter values for a specified statistical model. Let's call this R.

If you choose R as H0, you can find evidence against it, but not in its favour. This may not be what you want - although it isn't out of question. You may well wonder whether certain data contradict your R, in which case you can use it as H0, however this has no potential, even in case of non-rejection, to convince other people that R is correct.

There is however a very reasonable scientific justification for using R as H0, which is that according to Popper in order to corroborate a scientific theory, you should try to falsify it, and the best corroboration comes from repeated attempts to falsify it (in a way in which it seems likely that the theory will be rejected if it is in fact false, which is what Mayo's "severity" concept is about). Apart from statistical error probabilities, this is what testing R as H0 actually allows to do, so there is a good reason for using R as H0.

If you choose R as H1, you can find evidence against the H0, which is not normally quite what you want, unless you interpret evidence against H0 as evidence in favour of your H1, which isn't necessarily granted (model assumptions may be violated for both H0 and H1, so they may both technically be wrong, and rejecting H0 doesn't "statistically prove" H1), although many would interpret a test in this way. It has value only to the extent that somebody who opposes your R argues that H0 might be true (as in "your hypothesised real effect does not exist, it's all just due to random variation"). In this case a test with R as H1 has at least the potential to indicate strongly against that H0. You can even go on and say it'll give you evidence that H0 is violated "in the direction of H1", but as said before there may be other explanations for this than that H1 is actually true. Also, "the direction of H1" is rather imprecise and doesn't amount to any specific parameter value or it's surroundings. It may depend on the application area how important that is. A homeopath may be happy enough to significantly show that homeopathy does something good rather than having its effect explained by random variation only, regardless of how much good it actually does, however precise numerical theories in physics/engineering, say, can hardly be backed up by just rejecting a random variation alternative.

The "equivalence testing" idea would amount to specifying a rather precise R (specific parameter value and small neighbourhood) as H1 and potentially rejecting a much bigger part of the parameter space on both sides of R. This would then be more informative, but has still the same issue with model assumptions, i.e., H0 and H1 may both be wrong. (Obviously model assumption diagnoses may help to some extent. Also even if neither H0 nor H1 is true, arguably some more distributions can be seen as "interpretatively equivalent" with them, e.g., two equal non-normal distributions in a two-sample setup where a normality-based test is applied, and actually may work well due to the Central Limit Theorem even for many non-normal distributions.)

So basically you need to choose what kind of statement you want to allow your test to back up. Choose R as H0 and the data can only reject it. Choose R as H1 and the data can reject the H0, and how valuable that is depends on the situation (particularly on how realistic the H0 looks as a competitor; i.e., how informative it actually is to reject it). The equivalence test setup is special by allowing you to use a rather precise R as H1 and reject a big H0, so the difference between this and rejecting a "random variation/no effect" H0 regards the precise or imprecise nature of the research hypothesis R to be tested.

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In this case there should be the equality as an alternative hypothesis and therefore the difference as an null hypothesis?

Hypothesis testing works well when a particular hypothesis makes a precise prediction. Like the observed value is likely equal or above/below some value. Hypothesis testing is about making predictions based on a theory and observing whether those predictions come true.

If you would have a hypothesis that drug X and Y are unequal, then you would have an untestable theory. Given some differences between the populations of X and Y, whether that difference is zero or not, in nearly any circumstances the observations of samples from X and Y will be different anyway, independent from the hypothesis whether or not X and Y similar distributions.

(and even when samples from X and Y are observed to be equal, then this might not be significant as the difference can be as small as we like, making the observation of equal samples not anything special that falsifies the hypothesis)

Test of equivalence

However what would be possible is a test of equivalence, which relates to a hypothesis that the mean difference between X and Y is between some small range. Then, observing a larger range could falsify that hypothesis.

So the 'alternative hypothesis' can be used as the null hypothesis. But, it needs to be expressed in a form that restricts the observations. The hypothesis 'X ≠ Y' doesn't do that. However a limit on the difference '|X-Y| < a' is a testible theory/hypothesis.

Confidence interval

Another popular alternative to null hypothesis testing is to present confidence intervals. The confidence interval can be seen as the range of hypothetical parameter values that pass a null hypothesis test where the hypothetical value is the null hypothesis.

Related:

Why are standard frequentist hypotheses so uninteresting?

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@Dave gave me a light about the question and told me about the equivalence test, explained here.

The hypothesis test for equivalence can be written as follows:

H0: The difference between the two group means is outside the equivalence interval
H1: The difference between the two group means is inside the equivalence interval

However, I study this question deeply in the last days in many textbooks and other sources. There is no consensus of all statistics on this issue. I found several divergent opinions.

The divergence happens more relative to the debate if the alternative hypothesis is always complementary to the null hypothesis, or is not. I, for my part, find it much more logical to consider the null and alternative hypothesis as perfectly complementary. This, this and this (sources linked to universities) agree with me.

As stated in the referred link, if there is a gray zone, which is neither included in the alternative hypothesis nor in the zero hypothesis.In this case, rejecting the null hypothesis no longer serves conceptually to help highlight the alternative hypothesis, it may merely mean that we are in the gray zone. For me there is no point in thinking that way.

However, almost all experts agrees that the null hypothesis always should contain '=' operator ("<=", "=" or ">="). See, for instance, here, here and here, sources linked to universities, adopt this line of thought.

And I understand why. Always have '=' in the null hypothesis, it creates a certain standardization of statistical methodologies that could reject this equality. It would be very confusing to have to deal with an alternative hypothesis that could contain equality.

If we accept that null and the alternative hypothesis is the complement (">","≠" or "<", respectively) in a exhaustive way.

Sometimes the researcher believes in the alternative hypothesis, sometimes don't.

This applies to the hypothetical situation that I've created, if we suppose the research hypothesis does not necessarily align with the researcher's beliefs.

In this context, the alternative hypothesis it would be the hypothesis that "challenges" the null hypothesis, in the sense of finding out if the study has statistically evidence regarded as sufficient to reject it.

Addendum: 2 textbooks as an example

1) Understandable Statistics - Brase, Brase -10th edition - 2012

...Any hypothesis that differs from the null hypothesis is called an alternate hypothesis...

pg 411

In statistical testing, the null hypothesis H0 always contains the equals symbol.

pg. 412

It agrees with this 2 claims (null hypothesis should contains '=' and null hypothesis and alternative hypothesis are complementary)

2) Statistics - James McClave, Terry Sincich - 13th edition - 2018

...While alternative hypotheses are always specified as strict inequalities, such as μ < 2,400, μ > 2,400, or μ ≠ 2,400, null hypotheses are usually specified as equalities, such as μ = 2,400. Even when the null hypothesis is an inequality, such as μ ≤ 2,400, we specify H0: μ = 2,400, reasoning that if sufficient evidence exists to show that Ha:μ > 2,400 is true when tested against H0: m = 2,400, then surely sufficient evidence exists to reject μ < 2,400 as well...

pg 403

It agrees with that Ho should contain '=' operator, but works with a "hole" in hypothesis space. However, looking at the text, it's clear that it's irrelevant.

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    $\begingroup$ What makes $H_0$ a null hypothesis is that it determines the distribution of the test statistic. That is fundamental; it would be impossible to conduct a classical null hypothesis test otherwise. $\endgroup$
    – whuber
    Commented Apr 16, 2023 at 19:14
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    $\begingroup$ Welcome! Good question and answer. The linked equivalence testing article explains the idea well except that it contains a basic misunderstanding about hypothesis test outcomes in general: "rejection of the null hypothesis (in which case we believe the alternate hypothesis is true at the specified confidence level)". The bracketed section is wrong. Rejecting a null at the 5% level does not imply confidence of 95% in the alternative hypothesis. The author repeats the error a number of times. $\endgroup$ Commented Apr 16, 2023 at 19:37
  • $\begingroup$ Thank you, Graham Bornholt. In fact, the linked article has mixed significance (alpha=5%) with 95% confidence. It's a mess! $\endgroup$ Commented Apr 17, 2023 at 0:04
  • $\begingroup$ I don't see how the word "always" is in any way justified by this answer. The equivalence test specifies a particular situation in which the "hypothesis of research" is the H1, fair enough. But there may be reasons to test it as H0, see my answer. $\endgroup$ Commented Apr 17, 2023 at 20:35
  • $\begingroup$ @whuber - Isn't that the answer, sweet and simple? I'm new to this community, but in some stackexchange sites we'd say "Make that an answer". $\endgroup$
    – JonathanZ
    Commented Apr 18, 2023 at 15:40
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We generally assume as the null hypotheses as an old orthodox belief as true even though we do not have sufficient proof of its truth.

and Alternate Hypothesis H1 as a new radical belief that is challenging our old system of belief.

So we need a great level of effort to reject our old belief H0. We will need a high degree of confidence in H1 to dismantle our old traditional beliefs.

Lets understand this by example.

Aristotle's view of solar system

For nearly 1,000 years, Aristotle’s view of a stationary Earth at the center of a revolving universe dominated natural philosophy

In old times we believed that sun revolves around earth consider this as our old traditional orthodox belief => Null Hypothesis {We will try very hard to stick to this belief}

“We revolve around the Sun like any other planet.” —Nicolaus Copernicus

Our new radical claim is that earth revolves around the sun. => Alternate Hypothesis H1.

So whichever drug company you work in X or Y you will assume your company's drug(H0) is better than the competitor(H1)

Reference NASA

Reference Georgia Tech course EDX.org

Note:- Please correct me if I am wrong I am just a student trying to learn Thanks

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    $\begingroup$ Welcome to CV, Banarasi. I think this answer confuses philosophy with statistical theory and thereby provides a misleading account of the statistical theory. "Old orthodox" and "radical new" are inappropriate (and meaningless) concepts in this context, and insisting that we "believe as true" a null hypothesis is going too far, because a hypothesis is just that: a set of assumptions erected to investigate their consequences in light of the data under analysis. $\endgroup$
    – whuber
    Commented Apr 18, 2023 at 15:35
  • $\begingroup$ Thanks for suggesting correction, I do not have any formal degree so I am not aware about the formal jargons used can you please suggest me some books or blogs to strengthen my knowledge about this . Thanks $\endgroup$ Commented Apr 19, 2023 at 6:13
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    $\begingroup$ You could start by browsing this site for highly upvoted threads. Add keywords to narrow the search, such as hypothesis. $\endgroup$
    – whuber
    Commented Apr 19, 2023 at 13:22

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