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What should we consider the null hypothesis?

A test statistic is a statistic used to measure the plausibility of an alternative hypothesis relative to a null hypothesis. (The statistical Sleuth)

From this it can be concluded that the alternate hypothesis should be the scientific hypothesis one is trying to establish.

Now this is from a blog I'm using to understand stats

The null hypothesis, H0 is the commonly accepted fact; it is the opposite of the alternate hypothesis. Researchers work to reject, nullify or disprove the null hypothesis. Researchers come up with an alternate hypothesis, one that they think explains a phenomenon, and then work to reject the null hypothesis.

Even at my college this was taught.

But recently I have started taking help from a Stats teacher who along with a Zoology Practical book by P.K. Banerjee is of the opinion that our scientific hypothesis should be the null!

Infact I came to know that when we are not sure what we should be deducing from our data we test if it is consistent with any known theoretical data. It is then that null becomes central,i.e. the hypothesis the scientist is interested in. I found this also in wiki example.

Is this what makes the two types of assumptions different?

Discussing with a problem

A professor wants to determine whether her department should continue keeping the prerequisite for an introductory stats course that the students applying must [preferably] have college level algebra in previous curriculum. She decided to find out from an year's result of that course.

With algebra = 34 (passed) & 6 (failed)

Without algebra = 12 (passed) & 18 (failed)

Are students who had college level algebra in their previous curriculum were more likely to pass the course?

My answer:

[A chi square contingency]

Ho= The students who had college level algebra in their previous curriculum did not have a greater likelihood to pass the course over those without algebra.

Ha= They did have.

The calculated Chi-square was smaller than critical value and so the null was accepted.

Now the Stats teacher said that it would be the opposite.

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  • $\begingroup$ I think you might be confused about something else. The test in question should reject the null. It isn't clear what is meant by `tabulated Chi-square was smaller than the critical value.' $\endgroup$ – HStamper Mar 30 '17 at 20:28
  • $\begingroup$ Fixed @EricMittman Can you elaborate,The test in question should reject the null. $\endgroup$ – Tyto alba Mar 30 '17 at 20:31
  • $\begingroup$ Running chisq.test in R with the contingency table you gave: X-squared = 13.475, df = 1, p-value = 0.0002418 $\endgroup$ – HStamper Mar 30 '17 at 21:14
  • $\begingroup$ The null hypothesis is usually the hypothesis you try to reject and the alternative hypothesis is usually the scientific hypothesis. $\endgroup$ – Michael R. Chernick Mar 31 '17 at 0:03
  • $\begingroup$ "From this it can be concluded that the alternate hypothesis should be the scientific hypothesis one is trying to establish." -- usually, but not always. Typically you seek to "falsify" some position -- by showing that the position is untenable (inconsistent with data). But sometimes you're not trying to do that. So for example, in some situations you might perhaps be using equivalence tests or noninferiority tests. $\endgroup$ – Glen_b -Reinstate Monica Mar 31 '17 at 9:09
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The purpose of the null is to convert a problem from one of inductive reasoning to one of deductive reasoning. The alternative, and the method that preceded it was the method of inverse probability. That method is now generally called Bayesian statistics.

Imagine that you had three scientific hypotheses, denoted a, b, and c. Imagine that the true model is d, but no one has yet to discover this. The world is still flat, white is still a color, and Mercury follows Newton's laws.

A Bayesian test would create three hypotheses, $H_a$, $H_b$, and $H_c$. For a data set that is large enough, you would end up with the hypothesis or the combination of hypotheses that are most likely true. However, since $H_d$ wasn't tested, you may continue to be fooled by the idea yet to be thought of.

The Frequentist hypothesis testing regime would assume that the alternative hypothesis is $H_A\to{H}_x$, and the null is $H_0\to\neg{H}_x$. The null contains every hypothesis that is not the alternative.

The first example in the academic literature, but not the first null hypothesis, is where R.A. Fisher assumed that Mendel's laws were false as the null. If you discredit the null, then you exclude every explanation, including those not yet considered. The first null hypothesis was that Muriel Bristol (Fisher's boss) could not correctly detect the difference between tea poured into mild from milk poured into tea. That was the very first statistical test.

There is a slight difference between R.A. Fisher's idea of a null and Pearson and Neyman's idea of a null. Fisher felt there was a null, but no alternative hypothesis. If you rejected the null, it told you what was wrong, but was not directive as to what was correct automatically. Pearson and Neyman championed acceptance and rejection regions based on frequencies, and they felt the method directed behavior.

The logic was that the method created a probabilistic version of modus tollens. Modus tollens is "if A then B; and, not B; therefore, not A." Or, if the null is true, then the test will appear in the acceptance region if it does not, then you can reject the null.

The weakness of the methodology was proposed by an author that I cannot currently locate in this somewhat tongue-in-cheek way. There are 535 elected members representing the states in the U.S. Congress. There are 360 million Americans. Therefore since 535/360000000 is less than .05, if you randomly sample a group of Americans and pick a member of Congress, they cannot be an American (p<.05).

While Fisher's no effect hypothesis is the most common version, because of its implication would be that something has an effect in the alternative, it is not a requirement that a parameter equals zero, or a set of parameters all equal zero.

What matters is that the null is the opposite of what you are wanting to assert before seeing the data.

That makes the null hypothesis method a potent tool. Think about this as a rhetorical device. Your opponent opposes $H_d$ that you recently believe you have discovered.

You do not assert $H_d$ is true. You assert your opponent's position of $\neg{H}_d$ is true and build your probabilities on the assumption that you are the only person that is wrong. Everyone is right, and you are wrong.

It is a powerful rhetorical tool to concede the argument from the beginning, but then ask, "what would the world look like if I am the one that is wrong?" That is the null. If you reject the null, then what you are really saying is that "nature rejects all other ideas except mine."

Now as to your question, you want to show that college algebra matters, therefore your null hypothesis is that college algebra does not matter. We will ignore all the other methodological issues that would really be present since people without college algebra may have other self-selection issues as will the people with college algebra.

Your null is that algebra does not matter. The alternative is that it does. If the p-value is less than your $\alpha$ cutoff, chosen before collecting the data, then you can reject the null. If it is not, then you should behave as if it is true until you either do more research or find another way to come to a conclusion.

It would be dubious, ignoring the methodology issues, to assert that college algebra matters as you only have one sample. The method is intended for repetition. Nonetheless, you would only be made a fool of no more than $\alpha$ percent of the time, ignoring the methodological issues by following the results of the test.

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  • $\begingroup$ -1 For the first sentence. $\endgroup$ – Alexis Dec 1 '19 at 19:40
  • $\begingroup$ @alexis how would you change the sentence? $\endgroup$ – Dave Harris Dec 1 '19 at 21:41
  • $\begingroup$ Hypothesis testing, and statistical inference generally, couched as it is in (precisely articulated) uncertainty, seems fundamentally and irreducibly incompatible with deductive reasoning (as I understand it), because deductive reasoning is more or less about arriving at certain consequences derived from first principles. $\endgroup$ – Alexis Dec 2 '19 at 3:54
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It appears you are asking for clarification..

A null, Ho, essentially predicts no effect (no difference between groups, no correlation/association between variables etc), whereas an alternative/experimental, Ha or H1 predicts an effect.

So in your example, you have the gist of Ho and Ha (though the wording could be improved).

Your Chi-square test gives you a chi-square value - you need to either a) compare this with a 'critical' chi-square value b) know the p-value associated with your chi-square value and compare this with an 'alpha' p-value (typically .05 in psychology for example)

These amount to the same kind of thing For this example, if your alpha/cutoff is .05, then your 'critical' chi-square is 3.841.

NHST requires that, if your p-value is LESS than your alpha/cutoff, then you reject the null.

Here's where the confusion might arise: As chi-square values increase, associated p values decrease.

So, if your chi-square value is SMALLER than the critical, your associated p-value would be LARGER than the alpha/cutoff. If p is larger, the null is NOT rejected.

If your chi-square value is LARGER than the critical, your associated p-value would be SMALLER than the alpha/cutoff. If p is smaller, the null IS rejected.

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    $\begingroup$ I have found a number of sources support the fact that a null hypothesis is not always 'not the same' hypothesis. Here are few of them wiki and an old post. $\endgroup$ – Tyto alba Mar 30 '17 at 20:44
  • $\begingroup$ The null $H_{0}:|\theta|\ge \Delta$ is one of an effect at least as big as $\Delta$, with the alternative $H_{\text{A}}:|\theta| < \Delta$, i.e. that the effect is not as big as $\Delta$. Likewise, the null $H_{0}: \theta \ge \Delta$, is a null that an effect is at least $\Delta$, with an alternative $H_{0}:\theta < \Delta$, etc. $\endgroup$ – Alexis Dec 1 '19 at 0:15

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