{"id":929,"date":"2021-05-29T15:36:34","date_gmt":"2021-05-29T19:36:34","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=929"},"modified":"2024-02-08T14:05:40","modified_gmt":"2024-02-08T19:05:40","slug":"8-2-type-i-and-type-ii-errors","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/8-2-type-i-and-type-ii-errors\/","title":{"raw":"8.2 Type I and Type II Errors","rendered":"8.2 Type I and Type II Errors"},"content":{"raw":"In testing hypotheses, there are only two possible outcomes: either reject [latex]H_0[\/latex] or do not reject [latex]H_0[\/latex]; in reality, there are only two possible scenarios: either [latex]H_0[\/latex] is true or [latex]H_0[\/latex] is false. Hence, regardless of which conclusion we make, we have a chance to make an error. There are two types of errors: Type I and Type II.\r\n\r\nType I error<\/strong>: reject the null [latex]H_0[\/latex]\u00a0when [latex]H_0[\/latex]\u00a0is in\u00a0fact true.\r\n\r\nType II error<\/strong>: do not reject the null [latex]H_0[\/latex]\u00a0when [latex]H_0[\/latex]\u00a0is false.\r\n

Table 8.2<\/strong>: Type I and Type II Error<\/p>\r\n\r\n

\r\n\r\n\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n\r\n
[latex]H_0[\/latex]\u00a0is True<\/strong><\/div><\/th>\r\n
\r\n
[latex]H_0[\/latex]<\/em>\u00a0is False<\/strong><\/div><\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Decision: Do not reject [latex]H_0[\/latex]<\/th>\r\nCorrect decision<\/td>\r\nType II error<\/td>\r\n<\/tr>\r\n
Decision: Reject [latex]H_0[\/latex]<\/th>\r\nType I error<\/td>\r\nCorrect decision<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nThe probability of type I error is denoted as [latex]\\alpha[\/latex], and the probability of type II error is denoted as [latex]\\beta[\/latex]. That is:\r\n

[latex]\\alpha = P(\\text{Type I error}) = P(\\text{Reject }H_0|H_0\u00a0\\text{ is true})[\/latex]<\/p>\r\n

[latex]\\beta = P(\\text{Type II error}) = P(\\text{Do not reject }H_0|H_0\u00a0\\text{ is false})[\/latex]<\/p>\r\nThe type I error rate [latex]\\alpha[\/latex]<\/em>\u00a0is also called the significance level<\/strong> of a hypothesis\u00a0test.\r\n

\r\n

Example: Type I and Type II Errors<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nIn a diabetes blood test, a patient is diagnosed with the disease if the sugar level in their bloodstream is larger than the threshold C=130 mg\/dL. Suppose the distributions of sugar levels for the two populations (diabetes-free and having diabetes) are the two bell-shaped curves shown in the following figure.<\/a>\r\n\r\n[caption id=\"attachment_932\" align=\"aligncenter\" width=\"877\"]\"Two Figure 8.1<\/strong>: Trade-Off Between Type I and Type II Errors. [Image Description (See Appendix D Figure 8.1)<\/a>][\/caption]<\/div>\r\n
\r\n\r\nDefine the hypotheses: [latex]H_0[\/latex] <\/em>(a patient is disease free) vs. [latex]H_a[\/latex]<\/em> (a patient has diabetes).\u00a0What are the type I and type II errors in this example?\r\n\r\nType I error: claim the person has diabetes (reject the null [latex]H_0[\/latex]) but actually the person does not have diabetes ([latex]H_0[\/latex] is in fact true). This is often referred to as a false positive.Type II error: claim the person does not have diabetes(do not reject the null [latex]H_0[\/latex]), but actually the person has diabetes ([latex]H_0[\/latex] is false). This is often referred to as a false negative.\r\n\r\n<\/div>\r\n<\/div>\r\nThe figure in the above example shows the trade-off between type I and type II errors. The gold area gives\u00a0[latex]\\alpha[\/latex]<\/em>, the probability of the type I error; and the blue area gives [latex]\\beta[\/latex]<\/em>, the probability of the type II\u00a0error. If we increase the threshold C<\/em>\u00a0(move the cut-off to the right), the gold area will reduce\u00a0and the blue area will increase. That is the type I error rate [latex]\\alpha[\/latex]<\/em>\u00a0will decrease and the type II\u00a0error rate [latex]\\beta[\/latex]\u00a0will increase. On the other hand, if we reduce the threshold C<\/em>\u00a0(move\u00a0the cut-off to the left), the type I error rate [latex]\\alpha[\/latex]<\/em>\u00a0will increase and the type II error rate will\u00a0decrease. This is the trade-off between the type I and type II errors [latex]\\alpha[\/latex]<\/em>\u00a0and [latex]\\beta[\/latex]. It is not\u00a0a good idea to set either [latex]\\alpha[\/latex]<\/em>\u00a0or [latex]\\beta[\/latex]<\/em> to be too close to 0; otherwise, the other error rate will be huge. For example, if we set the threshold C<\/em>\u00a0very large, few individuals will be diagnosed\u00a0as diabetic; as a result, many diabetic individuals will be misclassified as not having the disease (meaning we\u00a0have a high probability of committing a type I error). On the other hand, if we set the threshold\u00a0C<\/em> very small, most individuals will be diagnosed as diabetic; consequently, many individuals who\u00a0are free of diabetes will be misclassified as diabetic (meaning we have a high probability of committing a type\u00a0II error). In general, we can set [latex]\\alpha[\/latex]<\/em>\u00a0(or [latex]\\beta[\/latex]<\/em>) to be relatively small if the consequence of\u00a0the type I (or type II) error is more serious. The power<\/strong> of a test is defined as\r\n

[latex]1 - \\beta = 1- P(\\text{Type II error}) [\/latex] [latex]= 1 - P(\\text{Do not reject } H_0 | H_0 \\text{ false}) = P(\\text{Reject } H_0 | H_0 \\text{ false}).[\/latex]<\/p>\r\nThis is the probability that we reject [latex]H_0[\/latex]\u00a0when [latex]H_0[\/latex]\u00a0is false.\u00a0Thus, it is of interest for a statistical test to have a high level of power.\r\n

\"\"<\/div>\r\n
<\/div>\r\n
\r\n
\r\n

Exercise: Type I and Type II Errors<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSuppose you are performing a statistical test to decide whether a nuclear reactor should be approved. The null hypothesis is that the reactor is safe to use, and so failing to reject the null hypothesis corresponds to approval.\r\n
    \r\n \t
  1. Write down the null and alternative hypotheses.<\/li>\r\n \t
  2. What are the type I and type II errors in this example?<\/li>\r\n \t
  3. Which error has a more serious consequence, type I or type II? Which of [latex]\\alpha[\/latex]<\/em>\u00a0or [latex]\\beta[\/latex]\u00a0should be smaller?<\/li>\r\n<\/ol>\r\n
    Show\/Hide Answer<\/summary>Answers:<\/strong>\r\n
      \r\n \t
    1. [latex]H_0[\/latex]: the nuclear reactor is safe versus [latex]H_a[\/latex]: the nuclear reactor is not safe.<\/li>\r\n \t
    2. Type I error: disapprove the nuclear reactor for use given that the nuclear reactor is actually safe.\r\nType II error: approve the nuclear reactor for use given that the nuclear reactor is not safe.<\/li>\r\n \t
    3. The type II error is more serious than the type I. Disapproving a safe reactor would waste time and money, but approving an unsafe reactor could lead to a nuclear meltdown, which is a catastrophic event. For this reason, we should set the type II error rate [latex]\\beta[\/latex] to be relatively small.<\/li>\r\n<\/ol>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>","rendered":"

      In testing hypotheses, there are only two possible outcomes: either reject [latex]H_0[\/latex] or do not reject [latex]H_0[\/latex]; in reality, there are only two possible scenarios: either [latex]H_0[\/latex] is true or [latex]H_0[\/latex] is false. Hence, regardless of which conclusion we make, we have a chance to make an error. There are two types of errors: Type I and Type II.<\/p>\n

      Type I error<\/strong>: reject the null [latex]H_0[\/latex]\u00a0when [latex]H_0[\/latex]\u00a0is in\u00a0fact true.<\/p>\n

      Type II error<\/strong>: do not reject the null [latex]H_0[\/latex]\u00a0when [latex]H_0[\/latex]\u00a0is false.<\/p>\n

      Table 8.2<\/strong>: Type I and Type II Error<\/p>\n

      \n\n\n\n\n\n\n
      <\/td>\n\n
      [latex]H_0[\/latex]\u00a0is True<\/strong><\/div>\n<\/th>\n
      		\n
      [latex]H_0[\/latex]<\/em>\u00a0is False<\/strong><\/div>\n<\/th>\n<\/tr>\n<\/thead>\n
      Decision: Do not reject [latex]H_0[\/latex]<\/th>\nCorrect decision<\/td>\nType II error<\/td>\n<\/tr>\n
      Decision: Reject [latex]H_0[\/latex]<\/th>\nType I error<\/td>\nCorrect decision<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

      The probability of type I error is denoted as [latex]\\alpha[\/latex], and the probability of type II error is denoted as [latex]\\beta[\/latex]. That is:<\/p>\n

      [latex]\\alpha = P(\\text{Type I error}) = P(\\text{Reject }H_0|H_0\u00a0\\text{ is true})[\/latex]<\/p>\n

      [latex]\\beta = P(\\text{Type II error}) = P(\\text{Do not reject }H_0|H_0\u00a0\\text{ is false})[\/latex]<\/p>\n

      The type I error rate [latex]\\alpha[\/latex]<\/em>\u00a0is also called the significance level<\/strong> of a hypothesis\u00a0test.<\/p>\n

      \n
      \n

      Example: Type I and Type II Errors<\/p>\n<\/header>\n

      \n

      In a diabetes blood test, a patient is diagnosed with the disease if the sugar level in their bloodstream is larger than the threshold C=130 mg\/dL. Suppose the distributions of sugar levels for the two populations (diabetes-free and having diabetes) are the two bell-shaped curves shown in the following figure.<\/a><\/p>\n

      \"Two
      Figure 8.1<\/strong>: Trade-Off Between Type I and Type II Errors. [Image Description (See Appendix D Figure 8.1)<\/a>]<\/figcaption><\/figure>\n<\/div>\n
      \n

      Define the hypotheses: [latex]H_0[\/latex] <\/em>(a patient is disease free) vs. [latex]H_a[\/latex]<\/em> (a patient has diabetes).\u00a0What are the type I and type II errors in this example?<\/p>\n

      Type I error: claim the person has diabetes (reject the null [latex]H_0[\/latex]) but actually the person does not have diabetes ([latex]H_0[\/latex] is in fact true). This is often referred to as a false positive.Type II error: claim the person does not have diabetes(do not reject the null [latex]H_0[\/latex]), but actually the person has diabetes ([latex]H_0[\/latex] is false). This is often referred to as a false negative.<\/p>\n<\/div>\n<\/div>\n

      The figure in the above example shows the trade-off between type I and type II errors. The gold area gives\u00a0[latex]\\alpha[\/latex]<\/em>, the probability of the type I error; and the blue area gives [latex]\\beta[\/latex]<\/em>, the probability of the type II\u00a0error. If we increase the threshold C<\/em>\u00a0(move the cut-off to the right), the gold area will reduce\u00a0and the blue area will increase. That is the type I error rate [latex]\\alpha[\/latex]<\/em>\u00a0will decrease and the type II\u00a0error rate [latex]\\beta[\/latex]\u00a0will increase. On the other hand, if we reduce the threshold C<\/em>\u00a0(move\u00a0the cut-off to the left), the type I error rate [latex]\\alpha[\/latex]<\/em>\u00a0will increase and the type II error rate will\u00a0decrease. This is the trade-off between the type I and type II errors [latex]\\alpha[\/latex]<\/em>\u00a0and [latex]\\beta[\/latex]. It is not\u00a0a good idea to set either [latex]\\alpha[\/latex]<\/em>\u00a0or [latex]\\beta[\/latex]<\/em> to be too close to 0; otherwise, the other error rate will be huge. For example, if we set the threshold C<\/em>\u00a0very large, few individuals will be diagnosed\u00a0as diabetic; as a result, many diabetic individuals will be misclassified as not having the disease (meaning we\u00a0have a high probability of committing a type I error). On the other hand, if we set the threshold\u00a0C<\/em> very small, most individuals will be diagnosed as diabetic; consequently, many individuals who\u00a0are free of diabetes will be misclassified as diabetic (meaning we have a high probability of committing a type\u00a0II error). In general, we can set [latex]\\alpha[\/latex]<\/em>\u00a0(or [latex]\\beta[\/latex]<\/em>) to be relatively small if the consequence of\u00a0the type I (or type II) error is more serious. The power<\/strong> of a test is defined as<\/p>\n

      [latex]1 - \\beta = 1- P(\\text{Type II error})[\/latex] [latex]= 1 - P(\\text{Do not reject } H_0 | H_0 \\text{ false}) = P(\\text{Reject } H_0 | H_0 \\text{ false}).[\/latex]<\/p>\n

      This is the probability that we reject [latex]H_0[\/latex]\u00a0when [latex]H_0[\/latex]\u00a0is false.\u00a0Thus, it is of interest for a statistical test to have a high level of power.<\/p>\n

      \"\"<\/div>\n
      <\/div>\n
      \n
      \n
      \n

      Exercise: Type I and Type II Errors<\/p>\n<\/header>\n

      \n

      Suppose you are performing a statistical test to decide whether a nuclear reactor should be approved. The null hypothesis is that the reactor is safe to use, and so failing to reject the null hypothesis corresponds to approval.<\/p>\n

        \n
      1. Write down the null and alternative hypotheses.<\/li>\n
      2. What are the type I and type II errors in this example?<\/li>\n
      3. Which error has a more serious consequence, type I or type II? Which of [latex]\\alpha[\/latex]<\/em>\u00a0or [latex]\\beta[\/latex]\u00a0should be smaller?<\/li>\n<\/ol>\n
        \nShow\/Hide Answer<\/summary>\n

        Answers:<\/strong><\/p>\n

          \n
        1. [latex]H_0[\/latex]: the nuclear reactor is safe versus [latex]H_a[\/latex]: the nuclear reactor is not safe.<\/li>\n
        2. Type I error: disapprove the nuclear reactor for use given that the nuclear reactor is actually safe.
          \nType II error: approve the nuclear reactor for use given that the nuclear reactor is not safe.<\/li>\n
        3. The type II error is more serious than the type I. Disapproving a safe reactor would waste time and money, but approving an unsafe reactor could lead to a nuclear meltdown, which is a catastrophic event. For this reason, we should set the type II error rate [latex]\\beta[\/latex] to be relatively small.<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":19,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-929","chapter","type-chapter","status-publish","hentry"],"part":889,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/929","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":30,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/929\/revisions"}],"predecessor-version":[{"id":5298,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/929\/revisions\/5298"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/889"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/929\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=929"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=929"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=929"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=929"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}