# 三种估计区间

## 置信区间

95%的置信区间含义如下：从同一个群体中采样100次，目标是群体的平均数。100个不同的样本，有100个不同的置信区间，95个置信区间中含有群体目标参数（该例中即为平均是）。

## 三个区间的比较

1. 置信区间来源于采样误差。
2. 预测区间来源于采样误差，预测误差。
3. 忍受区间来源于采样误差，群体比例误差。

# 数据显著性

## 三个指标

1. P value
2. significance level（又名α\alpha）
3. confidence level(1-α\alpha)

## P值

### How Do You Interpret P Values?

In technical terms, a P value is the probability of obtaining an effect at least as extreme as the one in your sample data, assuming the truth of the null hypothesis.

For example, suppose that a vaccine study produced a P value of 0.04. This P value indicates that if the vaccine had no effect, you’d obtain the observed difference or more in 4% of studies due to random sampling error.

P values address only one question: how likely are your data, assuming a true null hypothesis? It does not measure support for the alternative hypothesis. This limitation leads us into the next section to cover a very common misinterpretation of P values.

### P Values Are NOT the Probability of Making a Mistake

Incorrect interpretations of P values are very common. The most common mistake is to interpret a P value as the probability of making a mistake by rejecting a true null hypothesis (a Type I error).

There are several reasons why P values can’t be the error rate.

First, P values are calculated based on the assumptions that the null is true for the population and that the difference in the sample is caused entirely by random chance. Consequently, P values can’t tell you the probability that the null is true or false because it is 100% true from the perspective of the calculations.

Second, while a low P value indicates that your data are unlikely assuming a true null, it can’t evaluate which of two competing cases is more likely:

• The null is true but your sample was unusual.
• The null is false.

Determining which case is more likely requires subject area knowledge and replicate studies.

Let’s go back to the vaccine study and compare the correct and incorrect way to interpret the P value of 0.04:

• Correct: Assuming that the vaccine had no effect, you’d obtain the observed difference or more in 4% of studies due to random sampling error.
• Incorrect: If you reject the null hypothesis, there’s a 4% chance that you’re making a mistake.

To see a graphical representation of how hypothesis tests work, see my post: Understanding Hypothesis Tests: Significance Levels and P Values.

### What Is the True Error Rate?

Think that this interpretation difference is simply a matter of semantics, and only important to picky statisticians? Think again. It’s important to you.

If a P value is not the error rate, what the heck is the error rate? (Can you guess which way this is heading now?)

Sellke et al.* have estimated the error rates associated with different P values. While the precise error rate depends on various assumptions (which I discuss here), the table summarizes them for middle-of-the-road assumptions.

P value

Probability of incorrectly rejecting a true null hypothesis

0.05

At least 23% (and typically close to 50%)

0.01

At least 7% (and typically close to 15%)

Do the higher error rates in this table surprise you? Unfortunately, the common misinterpretation of P values as the error rate creates the illusion of substantially more evidence against the null hypothesis than is justified. As you can see, if you base a decision on a single study with a P value near 0.05, the difference observed in the sample may not exist at the population level. That can be costly!

Now that you know how to interpret P values, read my five guidelines for how to use P values and avoid mistakes.

You can also read my rebuttal to an academic journal that actually banned P values!

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