Statistics 简明教程

Statistics - Type I & II Errors

I 型错误和 II 型错误表示统计假设检验的错误结果。I 型错误表示错误地拒绝了一个有效的零假设，而 II 型错误则表示错误地保留了一个无效的零假设。

Type I and Type II errors signifies the erroneous outcomes of statistical hypothesis tests. Type I error represents the incorrect rejection of a valid null hypothesis whereas Type II error represents the incorrect retention of an invalid null hypothesis.

Null Hypothesis

零假设是指用证据否定了相反说法的陈述。考虑以下示例：

Null Hypothesis refers to a statement which nullifies the contrary with evidence. Consider the following examples:

Example 1

  1. Hypothesis - Water added to a toothpaste protects teeth against cavities.

  2. Null Hypothesis - Water added to a toothpaste has no effect against cavities.

Example 2

  1. Hypothesis - Floride added to a toothpaste protects teeth against cavities.

  2. Null Hypothesis - Floride added to a toothpaste has no effect against cavities.

此处将根据实验数据检验零假设，以否定氟化物和水对牙齿蛀牙的影响。

Here Null hypothesis is to be tested against experimental data to nullify the effect of floride and water on teeth’s cavities.

Type I Error

考虑示例 1。其中零假设为真，即添加到牙膏中的水对蛀牙没有影响。但是，如果我们使用实验数据检测到添加的水对蛀牙有影响，那么我们就否定了真正的零假设。这是一个 I 型错误。它也称为假阳性条件（一种表明存在给定条件但实际上并不存在的状况）。I 型错误率或 I 型显着性水平表示在零假设为真时拒绝零假设的概率。

Consider the Example 1. Here Null hypothesis is true i.e. Water added to a toothpaste has no effect against cavities. But if using experimental data, we detect an effect of water added on cavities then we are rejecting a true null hypothesis. This is a Type I error. It is also called a False Positive condition (a situation which indicates that a given condition is present but it actually is not present). The Type I error rate or significance level of Type I is represented by the probability of rejecting the null hypothesis given that it is true.

I 型错误记为 $ \alpha $，也称为 alpha 水平。一般来说，I 型错误显著性水平为 0.05 或 5% 是可以接受的，这意味着错误拒绝零假设的概率为 5% 是可以接受的。

Type I error is denoted by $ \alpha $ and is also called alpha level. Generally It is acceptable to have Type I error significance level as 0.05 or 5% which means that 5% probability of incorrectly rejecting the null hypothesis is acceptable.

Type II Error

考虑示例 2。在此，零假设为假，即添加到牙膏中的氟化物对龋齿有效。但是，如果使用实验数据，我们没有检测到添加到龋齿中的氟化物的效果，那么我们正在接受一个错误的零假设。这是一个 II 型错误。它也称为假阳性条件（表明给定条件不存在但实际上存在的情况）。

Consider the Example 2. Here Null hypothesis is false i.e. Floride added to a toothpaste has effect against cavities. But if using experimental data, we do not detect an effect of floride added on cavities then we are accepting a false null hypothesis. This is a Type II error. It is also called a False Positive condition (a situation which indicates that a given condition is not present but it actually is present).

II 型错误记为 $ \beta $，也称为 beta 水平。

Type II error is denoted by $ \beta $ and is also called beta level.

统计检验的目标是确定是否可以拒绝零假设。统计检验可以拒绝或无法拒绝零假设。下表说明了以 I 型或 II 型错误为条件的零假设的真或假与检验结果之间的关系。

Goal of a statistical test is to determine that a null hypothesis can be rejected or not. A statistical test can reject or not be able to reject a null hypothesis. Following table illustrates the relationship between truth or falseness of the null hypothesis and outcomes of the test in terms of Type I or Type II error.

Judgment

Null hypothesis ($ H_0 $) is

Error Type

Inference

Reject

Valid

Type I Error (False Positive)

Incorrect

Reject

Invalid

True Positive

Correct

Unable to Reject

Valid

True Negative

Correct

Unable to Reject

Invalid

Type II error(False Negative)

Incorrect