In the world of science, statistics is the basis of any claim. At the end of the day, the numbers do not lie, since they comprise their own and objective reality applicable to all processes, regardless of the culture or geographical distance of whoever applies them. Null hypothesis and p value

Thus, to be able to affirm (or rather, suspect) that we have discovered something, it is necessary that we can present reliable and repeatable data in a numerical language that supports it. In the world of experimentation, there must be **an anchor point that must be refuted from the beginning, that is, the null hypothesis ** .

Statistics and the scientific method seem to be disciplines and methodologies that are too complex for the general population, but nothing could be further from the truth. On this occasion, we open a small window to the world of numerical realities and basal science with the explanation of what the null hypothesis is.

## What is the null hypothesis ?: refuting the assumptions

To be able to move comfortably in the world of hypotheses, it is necessary that we first lay the essential foundations for understanding the subject. We are going **to immerse ourselves, albeit briefly, in the world of the scientific method ** .

### About the scientific method

The scientific method is defined as a research method based on empirical and measurement, which is also subject to the specific principles of reasoning tests. **This concatenation of steps and reasoning is based on two main pillars** :

- Reproducibility: the ability that, if a person proposes it, to repeat any experiment with the necessary means.
- Refutability: any scientific proposition must be susceptible to being falsified or refuted.

In the world of science we never operate in absolute dogmas. **As much as a number supports a hypothesis, it is possible that the hypothesis does not fully represent reality** , that factors extrinsic to the experiment have not been taken into account, or that the sample size is not large enough, for example.

Thus, the scientific method is based on **observation****, measurement, hypothesis, reproducibility, refutability and review by external agents** who have carried out the experiment themselves.

If any reader eager for scientific knowledge finds himself in front of a typical paper from any journal such as Science or Nature, he will be able to observe that it seems that researchers are anything but sure of their discoveries. “Could be”, “could mean”, “this seems to indicate”, “maybe exists” and other phrases dominate the paragraphs.

In addition, any self-respecting research ignores in its last lines that “more experimentation is required to delve into the subject matter.” As we have seen, science, despite what the general population believes, **is based more on discarding falsehoods than on affirming absolute dogmas** .

Now, once we have understood the caution and mistrust that we must have in the face of blunt statements in the world of science, it is time to explain what the null hypothesis is.

## The false claim

According to the Royal Spanish Academy of the language, a hypothesis is defined as an assumption of something possible or impossible in order to draw a consequence from it. If we go to its etymological roots, we will see that the meaning of the word is contained in it, since “hiccup” corresponds to “subordination / below” and “thesis” to “a conclusion that is maintained with reasoning.”

The hypothesis is **an unverified statement that requires a contrast with experience** (that is, an experiment) and after being refuted and tested, in the best of cases, it can become a verified **statement**.

In any case, to affirm that something “is”, we must also rule out that it “is not”, right? Do not despair, because we present this abstraction exercise in a kinder way in the following lines.

Let’s take an example: we want to show that humidity plays an essential role in the spawning of a population of insects of a specific species in an ecosystem. In this case, we have two possible hypotheses:

- That humidity does not influence the number of eggs per spawning, so there will be no differences in the average of this figure depending on the climate and the region. (H0)
- That humidity does influence the number of eggs per spawning. There will be significant differences in the mean depending on the specific parameter that measures humidity. (H1)

The null hypothesis (H0) in this case corresponds to the first of the statements. Thus, we can define the null hypothesis as **a statement about a parameter that holds that two or more events are not correlated with each other** .

This concept is the basis of the approach to scientific hypotheses, because no matter how much you want to demonstrate a relationship between two specific parameters, you have to operate on the fact that if it has not been documented, it is because it does not exist. Furthermore, any credible investigation should do everything possible to test its H1 hypothesis (that the suspected correlation does exist). **It is not about obtaining the desired result “with”, but about reaching it “despite”** .

## P-value

**The p-value, from English, p-value , is the minimum non-arbitrary significance level with which we can reject the null hypothesis (H0) given a distribution function and a test statistic.**

In other words, the p-value is the probability defined by the minimum distribution that can reject the null hypothesis (H0) without defining *a priori* the significance level for the contrast .

If you remember, you will remember that the area under the curve of the distribution function is a probability. So, from this point of view, the p-value will be the probability of observing a test statistic so extreme that the null hypothesis is true.

## Domain

Since the p-value is a probability, this value will be between 0 and 1.

## Not arbitrary

Unlike the significance levels that we are more used to seeing, such as 1%, 5%, and 10%, the p-value depends on the distribution function that the test statistic has. So, the levels of 1%, 5% and 10% are decided at the beginning of the contrast. This selection is called **arbitrary**.

## P-value formula

The p-value is not a single value like the critical value, but will depend on the statistic. For different values of the test statistic, the critical value will be the same. On the other hand, for different values of the test statistic, the p-value will also be different, because the p-value depends on the value taken by the test statistic.

Where,

- D, is a random variable that follows a certain distribution.
- d, is the value of the test statistic.

## Calculation

It is possible to calculate the p-value by hand but you would have to have very precise distribution tables, that is, with many decimals because the p-value tends to be small. Most statistical programs already have the p-value incorporated and it normally appears in the output of the estimation results by Ordinary Least Squares (OLS). It may seem difficult to use but with practice it is a very useful tool.

To calculate the p-value we need:

- Contrast statistic.
- The distribution of the contrast statistic and knowing its parameters.

## Rejection rule

**If p-value < significance level => Rejection H0. **

**If value – p > level of significance => No rejection of H0. **

## Representation

In the case of a Student’s t distribution with 2 degrees of freedom and a contrast statistic equal to 3, the probability of finding such an extreme statistic when the null hypothesis (H0) is true is 4.77%.

In other words , if the null hypothesis (H0) were true, a statistic as large as 3 would only be observed 4.77% of the time.

## Why is it called p-value?

The name of the p-value has its origin in the definition that refers to being the area under the curve of the distribution function outside the confidence interval. So, since that area is the minimum probability of rejecting the null hypothesis, the “p” of p-value refers to the **probability** . And, since the p-value corresponds to a number, and therefore a value, the word “value” of p-value is attributed to the numerical figure. In some books we can find “probability value” referring to the p-value. Maybe saying “minimum probability to reject the null hypothesis” was too long and did not hold any mystery for the students …

## The importance of P-value

The most careful readers will have noticed that, in the example given above of humidity, the hypothesis that shows a correlation between this parameter and the average number of eggs contains **an important term in it: significance** .

This is essential, since different means are observed in the number of insect eggs, no matter how real and observable, it can be a non-significant event, that is, the product of a random sampling beyond correlation.

For example, if an alien came to earth and picked up four 50-year-old men at random and three of them were 1.90 meters tall, it could safely say that 3 out of 4 humans are very tall. These data are not statistically significant, as they are due to chance of the sample. On the other hand, if said alien measured 3 million citizens and recorded the variations in height in all the geographical locations of the world, there perhaps it would observe significant differences in the height of the species according to (x) parameters.

All these conjectures are not based on a mere process of reasoning, since there are numbers that reflect the significance of the data obtained. This is **the case of the “P-value”, a numerical figure that is defined as the probability that a calculated statistical value is possible given a certain null hypothesis** . This figure is a probability that ranges from 0 to 1.

So we want the P-value to be low, very low. In general, it can be said that a hypothesis H0 (remember, the null hypothesis) can be rejected when this number is equal to or less than an arbitrarily established level of significance (generally 0.05). This means that **the probabilities that the results obtained are the product of chance** (that is, that there is no correlation between the parameters, or what is the same, that the null hypothesis is true) are very, very low.

It should be noted that, in any case, hypothesis testing does not allow us to accept a hypothesis in its entirety, but rather to reject it or not. Returning to the example of eggs and insects, if we obtain samples of 300 spawning of 300 different females in 30 different locations and there are significant differences in the means according to the humidity of the ecosystem, we can say that there seems to be a relationship between the size of the cohort and the humidity parameter.

What we cannot, in any case, is affirm it as an immovable dogma. The scientific method is based on repetition and refutability, so **different research teams must repeat the experiment carried out under the same conditions and obtain equally significant results** so that the correlation can be reliable and valid.

Even so, no matter how well established the idea is in the scientific community, an entomologist may arrive and discover that, after dissecting 300 females of that species, it turns out that the red ones have a larger ovipositor apparatus and therefore put a higher average number of eggs. Now what?

## Conclusions

As we have wanted to convey in these lines, science and the scientific method in general are a series of exciting processes, but certainly frustrating, because we do not stop moving in assumptions that can be refuted at any time.

When asked “what is the null hypothesis?” We can affirm that it is the basis of any investigation, since it corresponds to the supposed reality that we want to deny, that is, that there is no correlation between the parameters that we have proposed to investigate.