{"id":813,"date":"2015-08-10T15:25:45","date_gmt":"2015-08-10T14:25:45","guid":{"rendered":"http:\/\/pcool.dyndns.org:8080\/statsbook\/?page_id=813"},"modified":"2025-08-11T08:26:05","modified_gmt":"2025-08-11T07:26:05","slug":"sensitivity-specificity","status":"publish","type":"page","link":"https:\/\/pcool.dyndns.org\/index.php\/sensitivity-specificity\/","title":{"rendered":"Sensitivity \/ Specificity"},"content":{"rendered":"\n

In medicine, diagnostic tests are used to make a diagnosis. For example, an MRI-scan is performed to diagnose a meniscal tear or a CT-scan to see if someone has a tarsal coalition. Some diagnostic tests are better than others. An MRI-scan of the knee, for example, is better in diagnosing a meniscal tear than a CT scan, which in turn is better than a plain radiograph (This of cause does not<\/strong> mean one should not request a radiograph of the knee when a meniscal tear is suspected! The radiograph is very helpful in eliminating other causes of knee pain or locking such as osteochondritis dissecans<\/em>). If one test is better than another in diagnosing a condition, we would like to know how much better<\/em><\/strong> this test is. To do this we can validate<\/em><\/strong> the test against a gold standard<\/em><\/strong>.<\/p>\n\n\n\n

Five measures of a diagnostic test have been described. These are sensitivity<\/strong>, specificity<\/strong>, positive predictive value<\/strong>, negative predictive value<\/strong> and accuracy<\/strong>. These five features allow us to compare<\/em><\/strong> different tests. They will be explained with a fictional example.<\/p>\n\n\n\n

As an example, lets look at the value of MRI in diagnosing a meniscal tear in the knee. The MRI needs to be validated<\/em><\/strong> against a gold standard<\/em><\/strong> (or the \u2018truth\u2019 as it is perceived). In this case, diagnostic arthroscopy is the gold standard. There are 100 patients with a suspected meniscal tear. All patients had an MRI-scan that was reviewed by a radiologist. The radiologist reported the scan as either positive or negative for a meniscal tear. The radiologist was not allowed to be indecisive. After the patient had the  MRI-scan, a diagnostic arthroscopy was performed. The orthopaedic surgeon, who performed the procedure, diagnosed a meniscal tear or not. Other pathology found by the orthopaedic surgeon, such as arthritis, is irrelevant in this context.<\/p>\n\n\n\n

Now the radiological diagnosis is validated against the arthroscopic diagnosis. So, we assume that the arthroscopic diagnosis is always correct (surgeon is always right!). There are 4 possible combinations:<\/p>\n\n\n\n

    \n
  1. The radiologist and orthopaedic surgeon both agree that there is a meniscal tear.<\/li>\n\n\n\n
  2. The radiologist diagnosed a meniscal tear but this was not confirmed at arthroscopy (over diagnosis by radiologist<\/em>).<\/li>\n\n\n\n
  3. The radiologist reported the scan as normal, but there was a meniscal tear at arthroscopy (missed diagnosis by radiologist<\/em>).<\/li>\n\n\n\n
  4. They both agree there is no meniscal tear.<\/li>\n<\/ol>\n\n\n\n\n\n\n\t\n\n\t\n\t
    <\/td>Athroscopy
    \nPositive<\/th>    		Arthroscopy
    \nNegative<\/th>\n<\/tr>\n<\/thead>\n
    MRI
    \nPositive<\/td>    		a<\/td>b<\/td>\n<\/tr>\n
    MRI
    \nNegative<\/td>    		c<\/td>d<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n

    Please note the formulas that follow will be different if the rows \/ columns in the table are changed.<\/p>\n\n\n\n

    In this table, the gold standard (\u2018truth\u2019) is in the columns and the test we validate against this standard is in rows. The result of the MRI-scan is validated against the arthroscopic diagnosis. So, value a<\/strong> is the number of patients who have been correctly diagnosed as having a meniscal tear by MRI-scan. They are called the True Positive <\/strong>scans. Similarly, we can see that value b<\/strong> is the False Positive<\/strong>, value c<\/strong> the False Negative<\/strong>s and value d<\/strong> the True Negative <\/strong>scans. Or:<\/p>\n\n\n\n

    a: True Positive<\/strong><\/p>\n\n\n\n

    b: False Positive<\/strong><\/p>\n\n\n\n

    c: False Negative<\/strong><\/p>\n\n\n\n

    d: True Negative<\/strong><\/p>\n\n\n\n

    The five measures of a diagnostic test that have been described are discussed:<\/p>\n\n\n\n

      \n
    1. Positive Predictive Value<\/li>\n\n\n\n
    2. Negative Predictive Value<\/li>\n\n\n\n
    3. Sensitivity<\/li>\n\n\n\n
    4. Specificity<\/li>\n\n\n\n
    5. Accuracy<\/li>\n<\/ol>\n\n\n\n\n\n\n\t\n\n\t\n\t\n\t
      <\/td>Athroscopy
      \nPositive<\/th>      		Arthroscopy
      \nNegative<\/th>      		<\/td>\n<\/tr>\n<\/thead>\n
      MRI
      \nPositive<\/td>      		49<\/td>5<\/td><\/td>\n<\/tr>\n
      MRI
      \nNegative<\/td>      		1<\/td>45<\/td><\/td>\n<\/tr>\n
      <\/td><\/td><\/td>100<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n

      Positive Predictive Value (ppv, also called precision):<\/strong><\/p>\n\n\n\n

      When the test is positive, what is the probability the person has the condition:<\/p>\n\n\n\n\n\n\n\t\n\n\t\n\t\n\t
      <\/td>Athroscopy
      \nPositive<\/th>      		Arthroscopy
      \nNegative<\/th>      		<\/td>\n<\/tr>\n<\/thead>\n
      MRI
      \nPositive<\/td>      		49<\/strong><\/td>5<\/strong><\/td>54<\/strong><\/td>\n<\/tr>\n
      MRI
      \nNegative<\/td>      		1<\/td>45<\/td><\/td>\n<\/tr>\n
      <\/td><\/td><\/td>100<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n
      \\(PPV = \\frac{49}{54} \\approx 0.907 \\)