Select personalised ads. Apply market research to generate audience insights. Measure content performance. Develop and improve products. List of Partners vendors. The Bonferroni test is a type of multiple comparison test used in statistical analysis. When performing a hypothesis test with multiple comparisons, eventually a result could occur that appears to demonstrate statistical significance in the dependent variable, even when there is none. The Bonferroni test attempts to prevent data from incorrectly appearing to be statistically significant like this by making an adjustment during comparison testing.
The Bonferroni test, also known as "Bonferroni correction" or "Bonferroni adjustment" suggests that the p-value for each test must be equal to its alpha divided by the number of tests performed. The test is named for the Italian mathematician who developed it, Carlo Emilio Bonferroni — A criticism of the Bonferroni test is that it is too conservative and may fail to catch some significant findings.
In statistics, a null hypothesis is essentially the belief that there's no statistical difference between two data sets being compared. Hypothesis testing involves testing a statistical sample to confirm or reject a null hypothesis. The test is performed by taking a random sample of a population or group. While the null hypothesis is tested, the alternative hypothesis is also tested, whereby the two results are mutually exclusive.
However, with any testing of a null hypothesis, there's the expectation that a false positive result could occur. This is formally called a Type I error , and as a result, an error rate that reflects the likelihood of a Type I error is assigned to the test. In other words, a certain percentage of the results will likely yield a false positive. However, when many comparisons are being made in an analysis, the error rate for each comparison can impact the other results, creating multiple false positives.
Bonferroni designed his method of correcting for the increased error rates in hypothesis testing that had multiple comparisons. Bonferroni's adjustment is calculated by taking the number of tests and dividing it into the alpha value. The most common way to control the familywise error rate is with the Bonferroni correction. You find the critical value alpha for an individual test by dividing the familywise error rate usually 0.
Thus if you are doing statistical tests, the critical value for an individual test would be 0. They found the following results:. Under that criterion, only the test for total calories is significant. The Bonferroni correction is appropriate when a single false positive in a set of tests would be a problem. It is mainly useful when there are a fairly small number of multiple comparisons and you're looking for one or two that might be significant. However, if you have a large number of multiple comparisons and you're looking for many that might be significant, the Bonferroni correction may lead to a very high rate of false negatives.
For example, let's say you're comparing the expression level of 20, genes between liver cancer tissue and normal liver tissue. Based on previous studies, you are hoping to find dozens or hundreds of genes with different expression levels. If you use the Bonferroni correction, a P value would have to be less than 0. Only genes with huge differences in expression will have a P value that low, and could miss out on a lot of important differences just because you wanted to be sure that your results did not include a single false positive.
An important issue with the Bonferroni correction is deciding what a "family" of statistical tests is. But they also measured 13 non-dietary variables such as age, education, and socioeconomic status; should they be included in the family of tests, making the critical P value 0. There is no firm rule on this; you'll have to use your judgment, based on just how bad a false positive would be. Obviously, you should make this decision before you look at the results, otherwise it would be too easy to subconsciously rationalize a family size that gives you the results you want.
An alternative approach is to control the false discovery rate. This is the proportion of "discoveries" significant results that are actually false positives. For example, let's say you're using microarrays to compare expression levels for 20, genes between liver tumors and normal liver cells.
One good technique for controlling the false discovery rate was briefly mentioned by Simes and developed in detail by Benjamini and Hochberg Put the individual P values in order, from smallest to largest. Thus the first five tests would be significant. Note that whole milk and white meat are significant, even though their P values are not less than their Benjamini-Hochberg critical values; they are significant because they have P values less than that of proteins.
When you use the Benjamini-Hochberg procedure with a false discovery rate greater than 0. Then with a false discovery rate of 0. This may seem wrong, but if all 25 null hypotheses were true, you'd expect the largest P value to be well over 0.
You would only expect the largest P value to be less than 0. You should carefully choose your false discovery rate before collecting your data. Usually, when you're doing a large number of statistical tests, your experiment is just the first, exploratory step, and you're going to follow up with more experiments on the interesting individual results.
If the cost of additional experiments is low and the cost of a false negative missing a potentially important discovery is high, you should probably use a fairly high false discovery rate, like 0. Sometimes people use a false discovery rate of 0. The Benjamini-Hochberg procedure is less sensitive than the Bonferroni procedure to your decision about what is a "family" of tests.
If you increase the number of tests, and the distribution of P values is the same in the newly added tests as in the original tests, the Benjamini-Hochberg procedure will yield the same proportion of significant results. This doesn't mean you can completely ignore the question of what constitutes a family; if you mix two sets of tests, one with some low P values and a second set without low P values, you will reduce the number of significant results compared to just analyzing the first set by itself.
Sometimes you will see a "Benjamini-Hochberg adjusted P value. If the adjusted P value is smaller than the false discovery rate, the test is significant. For example, the adjusted P value for proteins in the example data set is 0. In my opinion "adjusted P values" are a little confusing, since they're not really estimates of the probability P of anything.
I think it's better to give the raw P values and say which are significant using the Benjamini-Hochberg procedure with your false discovery rate, but if Benjamini-Hochberg adjusted P values are common in the literature of your field, you might have to use them. The Bonferroni correction and Benjamini-Hochberg procedure assume that the individual tests are independent of each other, as when you are comparing sample A vs. D, E vs. F, etc. If you are comparing sample A vs.
C, A vs. D, etc. One place this occurs is when you're doing unplanned comparisons of means in anova, for which a variety of other techniques have been developed, such as the Tukey-Kramer test. Another experimental design with multiple, non-independent comparisons is when you compare multiple variables between groups, and the variables are correlated with each other within groups. Purpose: The Bonferroni correction adjusts probability p values because of the increased risk of a type I error when making multiple statistical tests.
The routine use of this test has been criticised as deleterious to sound statistical judgment, testing the wrong hypothesis, and reducing the chance of a type I error but at the expense of a type II error; yet it remains popular in ophthalmic research.
The purpose of this article was to survey the use of the Bonferroni correction in research articles published in three optometric journals, viz.
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