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Statistical significance

From Wikipedia, the free encyclopedia

In statistical hypothesis testing,[1][2] a result has statistical significance when it is very unlikely to have occurred given the null hypothesis.[3] More precisely, a study's defined significance level, denoted α, is the probability of the study rejecting the null hypothesis, given that the null hypothesis were true;[4] and the p-value of a result, p, is the probability of obtaining a result at least as extreme, given that the null hypothesis were true. The result is statistically significant, by the standards of the study, when p < α.[5][6][7][8][9][10][11] The significance level for a study is chosen before data collection, and typically set to 5%[12] or much lower, depending on the field of study.[13]

In any experiment or observation that involves drawing a sample from a population, there is always the possibility that an observed effect would have occurred due to sampling error alone.[14][15] But if the p-value of an observed effect is less than the significance level, an investigator may conclude that the effect reflects the characteristics of the whole population,[1] thereby rejecting the null hypothesis.[16]

This technique for testing the statistical significance of results was developed in the early 20th century. The term significance does not imply importance here, and the term statistical significance is not the same as research, theoretical, or practical significance.[1][2][17] For example, the term clinical significance refers to the practical importance of a treatment effect.

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  • ✪ Statistical significance of experiment | Probability and Statistics | Khan Academy
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  • ✪ Understanding Statistical Significance - Statistics help
  • ✪ Null Hypothesis, p-Value, Statistical Significance, Type 1 Error and Type 2 Error

Transcription

Voiceover:In an experiment aimed at studying the effect of advertising on eating behavior in children, a group of 500 children, seven to 11 years old were randomly assigned to two different groups. After randomization, each child was asked to watch a cartoon in a private room, containing a large bowl of goldfish crackers. The cartoon included two commercial breaks. The first group watched food commercials, mostly snacks while the second group watched non-food commercials, games and entertainment products. Once the child finished watching the cartoon, the conductors of the experiment weighed the cracker bowls to measure how many grams of crackers the child ate. They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. Let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups. You have group one over here and you have group two. Let's say that this right over here is the first group. The first group watched food commercials. This is group number one. They watched food commercials. We could call this the treatment group. We're trying to see what's the effect of watching food commercials and then they tell us. The second group watched non-food commercials, so this is the control group. Number two, this is non-food commercials. This is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here that the kids ate 10 grams greater on average than this group right over here which just looking at that data makes you believe that okay, well something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers but the question that you always have to ask yourself in a situation like this. Well, isn't there some probability that this would have happened by chance that even if you didn't make them watch the commercials. If these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials. There's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here that it looks like the kids ate 10 grams more. How do you figure out, what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here that that could have just happened by chance. Well the way you do it is what they do right over here. Using a simulator, they re-randomize the results into two new groups and measure the difference between the means of the new groups. They repeated the simulation 150 times and plotted the differences given. The resulting difference is as given below. What they did is they said, okay, they have 500 kids and each kid, they had 500 children. Number one, two, three, all the way up to 500. For each child they measured how much was the weight of the crackers that they ate? Maybe child one ate two grams and child two ate four grams and child three ate, I don't know, ate 12 grams all the way to child number 500 ate, I don't know, maybe they didn't eat anything at all, ate zero grams. We already know, let's say the first time around. The first half was in the treatment group when we're just ranking them like this and then the second, they're randomly assigned into these groups and at the second half was in the control group. What they're doing now is they're taking the same results and they're re-randomizing it. Now they're saying, okay, let's maybe put this person in group number two and this person in group number two and this person stays in group number two and this person stays in group number one and this person stays in group number one. Now they're completely mixing up all of the results that they had. It's completely random of whether the student had watched the food commercial or the non-food commercial and then they're testing what's the mean of the new number one group and the new number two group. They're saying well, what is the distribution of the differences in means. They see when they did this way when they're essentially just completely randomly taking these results and putting them into two new buckets. You have a bunch of cases where you get no difference in the means. Out of the 150 times that they repeated the simulation doing this little exercise here. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15. I'm having trouble counting this let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12. It's so small, I'm aging but it looks like there's about, I don't know. High teens about 20 times when there's actually no noticeable difference in the means of the groups where you just randomly allocate the results amongst the two groups. When you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10 gram difference are actually very unlikely. Let's see, is this the difference? The difference between the means of the new groups. It's not clear whether this is group one minus group two or group two minus group one but in either case the situations where you have a 10 gram difference in mean. It's only two out of the 150 times. When you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens two out of the 150 times. There's a 150 dots here. That is on the order of 2% or actually it's less than 2%, it's between one and 2%. Let's say the situation we're talking about. Let's say that this is group one minus group two in terms of how much was eaten and so you're looking at this situation right over here that that's only one out of a 150 times. It happened less frequently than one in a 100 times. It happened only one in a 150 times. If you look at that, you say well, the probability this was just random. The probability of getting the results that you got is less than 1%. To me and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got. The children who watched food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. If you just randomly put 500 kids into two different buckets based on the simulation results it looks like there's only, if you'd run the simulation a 150 times, that only happened one out of the 150 times. It seems like this was very, it's very unlikely that this was purely due to chance. If this was just a chance event, this would only happen roughly one in 150 times but the fact that this happened in your experiment, it makes you feel pretty confident that your experiment is significant. In most studies, in most experiments, the threshold that they think about is the probability of something statistically significant. If the probability of that happening by chance is less than 5%, this is less than 1%. I would definitely say that the experiment is significant.

Contents

History

Statistical significance dates to the 1700s, in the work of John Arbuthnot and Pierre-Simon Laplace, who computed the p-value for the human sex ratio at birth, assuming a null hypothesis of equal probability of male and female births; see p-value § History for details.[18][19][20][21][22][23][24]

In 1925, Ronald Fisher advanced the idea of statistical hypothesis testing, which he called "tests of significance", in his publication Statistical Methods for Research Workers.[25][26][27] Fisher suggested a probability of one in twenty (0.05) as a convenient cutoff level to reject the null hypothesis.[28] In a 1933 paper, Jerzy Neyman and Egon Pearson called this cutoff the significance level, which they named α. They recommended that α be set ahead of time, prior to any data collection.[28][29]

Despite his initial suggestion of 0.05 as a significance level, Fisher did not intend this cutoff value to be fixed. In his 1956 publication Statistical Methods and Scientific Inference, he recommended that significance levels be set according to specific circumstances.[28]

Related concepts

The significance level α is the threshold for p below which the null hypothesis is rejected even though by assumption it were true, and something else is going on. This means α is also the probability of mistakenly rejecting the null hypothesis, if the null hypothesis is true.[4]

Sometimes researchers talk about the confidence level γ = (1 − α) instead. This is the probability of not rejecting the null hypothesis given that it is true.[30][31] Confidence levels and confidence intervals were introduced by Neyman in 1937.[32]

Role in statistical hypothesis testing

In a two-tailed test, the rejection region for a significance level of α = 0.05 is partitioned to both ends of the sampling distribution and makes up 5% of the area under the curve (white areas).
In a two-tailed test, the rejection region for a significance level of α = 0.05 is partitioned to both ends of the sampling distribution and makes up 5% of the area under the curve (white areas).

Statistical significance plays a pivotal role in statistical hypothesis testing. It is used to determine whether the null hypothesis should be rejected or retained. The null hypothesis is the default assumption that nothing happened or changed.[33] For the null hypothesis to be rejected, an observed result has to be statistically significant, i.e. the observed p-value is less than the pre-specified significance level.

To determine whether a result is statistically significant, a researcher calculates a p-value, which is the probability of observing an effect of the same magnitude or more extreme given that the null hypothesis is true.[11] The null hypothesis is rejected if the p-value is less than a predetermined level, α. α is called the significance level, and is the probability of rejecting the null hypothesis given that it is true (a type I error). It is usually set at or below 5%.

For example, when α is set to 5%, the conditional probability of a type I error, given that the null hypothesis is true, is 5%,[34] and a statistically significant result is one where the observed p-value is less than 5%.[35] When drawing data from a sample, this means that the rejection region comprises 5% of the sampling distribution.[36] These 5% can be allocated to one side of the sampling distribution, as in a one-tailed test, or partitioned to both sides of the distribution as in a two-tailed test, with each tail (or rejection region) containing 2.5% of the distribution.

The use of a one-tailed test is dependent on whether the research question or alternative hypothesis specifies a direction such as whether a group of objects is heavier or the performance of students on an assessment is better.[3] A two-tailed test may still be used but it will be less powerful than a one-tailed test because the rejection region for a one-tailed test is concentrated on one end of the null distribution and is twice the size (5% vs. 2.5%) of each rejection region for a two-tailed test. As a result, the null hypothesis can be rejected with a less extreme result if a one-tailed test was used.[37] The one-tailed test is only more powerful than a two-tailed test if the specified direction of the alternative hypothesis is correct. If it is wrong, however, then the one-tailed test has no power.

Stringent significance thresholds in specific fields

In specific fields such as particle physics and manufacturing, statistical significance is often expressed in multiples of the standard deviation or sigma (σ) of a normal distribution, with significance thresholds set at a much stricter level (e.g. 5σ).[38][39] For instance, the certainty of the Higgs boson particle's existence was based on the 5σ criterion, which corresponds to a p-value of about 1 in 3.5 million.[39][40]

In other fields of scientific research such as genome-wide association studies significance levels as low as 5×10−8 are not uncommon,[41][42] because the number of tests performed is extremely large.

Limitations

Researchers focusing solely on whether their results are statistically significant might report findings that are not substantive[43] and not replicable.[44][45] There is also a difference between statistical significance and practical significance. A study that is found to be statistically significant may not necessarily be practically significant.[46]

Effect size

Effect size is a measure of a study's practical significance.[46] A statistically significant result may have a weak effect. To gauge the research significance of their result, researchers are encouraged to always report an effect size along with p-values. An effect size measure quantifies the strength of an effect, such as the distance between two means in units of standard deviation (cf. Cohen's d), the correlation coefficient between two variables or its square, and other measures.[47]

Reproducibility

A statistically significant result may not be easy to reproduce.[45] In particular, some statistically significant results will in fact be false positives. Each failed attempt to reproduce a result increases the likelihood that the result was a false positive.[48]

Challenges

Overuse in some journals

Starting in the 2010s, some journals began questioning whether significance testing, and particularly using a threshold of α=5%, was being relied on too heavily as the primary measure of validity of a hypothesis.[49] Some journals encouraged authors to do more detailed analysis than just a statistical significance test. In social psychology, the journal Basic and Applied Social Psychology banned the use of significance testing altogether from papers it published,[50] requiring authors to use other measures to evaluate hypotheses and impact.[51][52]

Other editors, commenting on this ban have noted: "Banning the reporting of p-values, as Basic and Applied Social Psychology recently did, is not going to solve the problem because it is merely treating a symptom of the problem. There is nothing wrong with hypothesis testing and p-values per se as long as authors, reviewers, and action editors use them correctly."[53] Some statisticians prefer to use alternative measures of evidence, such as likelihood ratios or Bayes factors.[54] Using Bayesian statistics can avoid confidence levels, but also requires making additional assumptions,[54] and may not necessarily improve practice regarding statistical testing.[55]

The widespread abuse of statistical significance represents an important topic of research in metascience.[56]

Redefining significance

In 2016, the American Statistical Association (ASA) published a statement on p-values, saying that "the widespread use of 'statistical significance' (generally interpreted as 'p ≤ 0.05') as a license for making a claim of a scientific finding (or implied truth) leads to considerable distortion of the scientific process".[54] In 2017, a group of 72 authors proposed to enhance reproducibility by changing the p-value threshold for statistical significance from 0.05 to 0.005.[57] Other researchers responded that imposing a more stringent significance threshold would aggravate problems such as data dredging; alternative propositions are thus to select and justify flexible p-value thresholds before collecting data,[58] or to interpret p-values as continuous indices, thereby discarding thresholds and statistical significance.[59] Additionally, the change to 0.005 would increase the likelihood of false negatives, whereby the effect being studied is real, but the test fails to show it.[60]

In 2019 over 800 statisticians and scientists signed a message calling for the abandonment of the term "statistical significance" in science.[61]

See also

References

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Further reading

External links

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