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Exercise

A new diagnostic test for a certain disease was evaluated using a group of 100 persons at high risk for the disease. Sixty of them found to be positive by the test.

After the 100 people underwent further clinical evaluation, the disease was confirmed in 50 people, including the 40 who had been found positive by the test.

1. Show these results in the form of 2 by 2 table

2. How many brve positive, true negative, false poutive and false negative were there?

3. Calculate the sensitivity and specificity of the test?
Big explanation

1 answer

To evaluate the performance of the new diagnostic test, we can summarize the results in a 2x2 contingency table. Here’s how to represent the findings based on the information provided:

1. 2x2 Contingency Table

We have the following information:

  • Total tested: 100
  • Test positive: 60
  • Confirmed disease: 50
  • Confirmed disease with positive test: 40

Based on this, we can derive the following:

  1. True Positive (TP): Those who tested positive and have the disease. This is given by the number of individuals found positive by the test who were confirmed to have the disease: \( TP = 40 \).

  2. False Positive (FP): Those who tested positive but do not have the disease. We can find this by subtracting the true positives from the total positives:
    \( FP = 60 - TP = 60 - 40 = 20 \).

  3. True Negative (TN): Those who tested negative and do not have the disease. We know that there are 50 confirmed cases, which means the remaining 50 individuals do not have the disease. However, since 60 tested positive, only 40 are true positives, which means:
    \( TN = 100 - (TP + FP + FN) = 100 - (40 + 20 + FN) \).

  4. False Negative (FN): Those who tested negative but actually have the disease. Since there are 50 confirmed cases and 40 tested positive:
    \( FN = 50 - TP = 50 - 40 = 10 \).

Now we can fill in the values in the contingency table:

| | Disease Present (True Positives) | Disease Absent (True Negatives) | Total Positive | Total Negative | |--------------------------|--------------------------|--------------------------|----------------|----------------| | Test Positive | 40 (TP) | 20 (FP) | 60 | 40 | | Test Negative | 10 (FN) | 30 (TN) | 50 | 40 | | Total | 50 | 50 | 60 | 40 |

2. Counts of True Positive, True Negative, False Positive, and False Negative

From the table we just created:

  • True Positives (TP): 40
  • True Negatives (TN): 30
  • False Positives (FP): 20
  • False Negatives (FN): 10

3. Calculate Sensitivity and Specificity of the Test

Sensitivity is the ability of a test to correctly identify those with the disease. It is calculated as:

\[ \text{Sensitivity} = \frac{TP}{TP + FN} \]

Substituting the values:

\[ \text{Sensitivity} = \frac{40}{40 + 10} = \frac{40}{50} = 0.80 \text{ or } 80% \]

Specificity is the ability of a test to correctly identify those without the disease. It is calculated as:

\[ \text{Specificity} = \frac{TN}{TN + FP} \]

Substituting the values:

\[ \text{Specificity} = \frac{30}{30 + 20} = \frac{30}{50} = 0.60 \text{ or } 60% \]

Summary

  • True Positive (TP): 40
  • True Negative (TN): 30
  • False Positive (FP): 20
  • False Negative (FN): 10
  • Sensitivity: 80%
  • Specificity: 60%

This assessment outlines the test's performance, with relatively good sensitivity but moderate specificity, which may have implications for its usage in clinical settings.