- CFA Exams
- 2023 Level I > Topic 1. Quantitative Methods > Reading 4. Common Probability Distributions
- 7. Multiple Tests and Interpreting Significance

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**Subject 7. Multiple Tests and Interpreting Significance**

A type I error is where you incorrectly reject the null hypothesis; In other words, you get a false positive. If we test a hypothesis millions of times, it can result in hundreds of thousands of false positives. The

**false discovery rate**(

**FDR**) is the expected proportion of type I errors.

The FDR is the expected ratio of the number of false positive classifications (false discoveries) to the total number of positive classifications (rejections of the null). The total number of rejections of the null include both the number of false positives (FP) and true positives (TP). Simply put, FDR = FP / (FP + TP).

For a more humorous (an perhaps understandable) look at the problem, take a look at XKCD's "Jelly Bean Problem."(https://xkcd.com/882) The comic shows a scientist finding a link between acne and jelly beans, when a hypothesis was tested at a 5% significance level. Although there is no link between jelly beans and acne, a significant result was found (in this case, a jelly bean caused acne) by testing multiple times. Testing 20 colors of jelly beans, 5% of the time there is 1 jelly bean that is incorrectly fingered as being the acne culprit. The implications for false discovery in hypothesis testing is that if you repeat a test enough times, you're going to find an effect, but that effect may not actually exist.

*Example*

In medical testing, the false discovery rate is when you get a "positive" test result but you don't actually have the disease.

Out of 10,000 people given the test, there are 450 true positive results (box at top right) and 190 false positive results (box at bottom right) for a total of 640 positive results. Of these results, 190/640 are false positives so the false discovery rate is 30%.

#### Adjusting the FDR

If you repeat a test enough times, you will always get a number of false positives. One of the goals of multiple testing is to control the FDR: the proportion of these erroneous results. For example, you might decide that an FDR rate of more than 5% is unacceptable. Note though, that although 5% sounds reasonable, if you're doing a lot of tests, you'll also get a large number of false positives; for 1000 tests, you could expect to get 50 false positives by chance alone. This is called the multiple testing problem, and the FDR approach is one way to control for the number of false positives.

The FDR approach adjusts the p-value for a series of tests. A p-value gives you the probability of a false positive on a single test; If you're running a large number of tests from small samples, you should use q-values instead.

- A p-value of 5% means that 5% of all tests will result in false positives.
- A q-value of 5% means that 5% of significant results will be false positives.

Although controlling for type I errors sound ideal (why not just set the threshold really low and be done with it?), Type I and Type II errors form an inverse of relationship; when one goes down, the other goes up and vice versa. By decreasing the false positives, you increase the number of false negatives - that's where there is a real effect, but you fail to detect it.

#### Practice Question 1

If you run 20 tests at a 5% significance level, the probability of you getting a false positive result (FDR) is ______.

A. 5%B. 64%

C. 95%Correct Answer: B

This figure is obtained by first calculating the odds of having no false discoveries at a 5% significance level for 20 tests. Using the bimomial formula, P(0) = [20!/(20-0)!0!] x (0.05)^{0} x 0.95 ^{20-0} = 0.358. If the probability of having no false conclusions is 35.8%, then the probability of a false conclusion is 64.2%.

#### Practice Question 2

Suppose you run 500 tests at a 5% significance level. You get 50 rejections of null hypothesis. 10 of these rejections are actually false positive. The false discovery rate is:

A. 0.2%B. 5%

C. 20%Correct Answer: C

The FDR is the number of false positives in all of the rejected hypotheses. FDR = Number of false opositive / Number of significant = 10 / 50 = 20%.

### Study notes from a previous year's CFA exam:

7. Multiple Tests and Interpreting Significance