Epidemiological Values of Diagnostic Tests

Diagnostic tests are important aspects in making a diagnosis. Certain statistical information about the accuracy and validity of the tests themselves can help turn the data into usable, applicable information. Some of the most important epidemiological values of diagnostic tests include sensitivity and specificity, false positives and false negatives, positive and negative predictive values, likelihood ratios, and pre-test and post-test probabilities. For example, a test with high sensitivity is useful as a screening test, whereas high specificity is required for an accurate diagnosis. Alternatively, positive and negative predictive values help determine the probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of disease in the case of certain test results.

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Overview of Screening Tests

Screening tests

Screening tests are used to identify people in the early stages of a disease and enable early intervention with the goal of reducing morbidity and mortality.

Screening tests do not provide a definitive diagnosis:

  • Screening tests don’t “prove” that a person has a disease, but only provide suspicion.
  • A positive screening test is followed up by another diagnostic test, which (ideally) can definitively verify the suspicion (e.g., a biopsy).

The usefulness of screening tests requires assessment of:

  • Frequency of overdiagnosis: How often does a test suggest a patient has a disease when in fact they do not?
  • Frequency of misdiagnosis
  • Adverse effects of the test: Is the test painful or otherwise damaging?
  • Screening for diseases for which early intervention has shown no benefit

Contingency tables used in evaluating screening tests

Contingency tables are commonly used in the statistical analysis of multiple variables. To evaluate the epidemiologic value of a screening test, a table similar to that presented below can be used to determine the relative frequencies of individuals with different combinations of screening test results (positive or negative) and true disease state (truly have or do not have the disease).

It is important that the table is set up in a standard fashion in order for standard formulas to be applicable. The standard table is presented below (with screening test results on the left, true disease state on top, and “yes” answers before “no” answers).

Screening tests contingency table

Contingency table for screening tests

Image by Lecturio. License: CC BY-NC-SA 4.0

In this table:

  • A represents true positives (TPs): people with a positive screening test and actually having the disease
  • B represents false positives (FPs): people with a positive screening test but not actually having the disease
  • C represents false negatives (FNs): people with a negative screening test but actually having the disease
  • D represents true negatives (TNs): people with a negative screening test and not having the disease

False Positives and False Negatives

False positive

  • An FP test result indicates that a person has the disease when they do not. 
  • Known as a type I error:
    • An error in which a test result incorrectly indicates the presence of a condition when the condition is not truly present
    • A rejection of a true null hypothesis
  • Effects of FP results:
    • Can lead to unneeded testing and medications
    • Burden on the healthcare system
    • Anxiety for patients

False negative

  • An FN test result indicates a person does not have the disease when, in fact, they do.
  • Known as a type II error:
    • An error where the test result incorrectly fails to detect the presence of a condition when, in fact, the condition is present
    • Nonrejection of a false null hypothesis
  • Effects of FN results:
    • People with the disease are not promptly diagnosed.
    • Leads to a delay in management plan and a possible ↑ in morbidity or mortality
Contingency table for false postives and negatives

Contingency table identifying false positives (B) and false negatives (C)

Image by Lecturio. License: CC BY-NC-SA 4.0

Sensitivity and Specificity

Sensitivity and specificity are measures used to assess the performance of screening and diagnostic tests.

Sensitivity

Definition:

  • Probability of a test in accurately diagnosing a person who has the disease
  • The proportion of diseased people that test positive
  • A measure of the inclusivity of a diagnostic test: Does it identify everyone who actually has the disease?
True positive fraction

True positive fraction:
Diagram illustrating the concept of true positive fractions – the proportion of the population represented by the sensitivity of a test. This figure shows that all patients tested positive; however, the green figures represent individuals who did not have the disease but incorrectly tested positive (false positives). The red figures represent individuals who actually had the disease and also tested positive (true positives).

Image by Lecturio. License: CC BY-NC-SA 4.0

Calculations:

To calculate sensitivity, a 2×2 contingency table should be set up:

Screening tests contingency table

Contingency table for screening tests

Image by Lecturio. License: CC BY-NC-SA 4.0

Sensitivity is the proportion of people who test positive on the screening test and have the disease (TPs, found in square A) divided by all the people who are truly diseased regardless of their screening test results (TPs and FNs, A + C). Sensitivity is represented by the following equation:

$$ Sensitivity = \frac{A}{A + C} $$

Example: A new diagnostic test is evaluated on a group of patients: 100 patients are known to have the disease, and another 100 patients are known to be disease free in a control group. Among them, 90 patients with the disease and 20 individuals in the control group show a positive result. What is the sensitivity of the new test?

Answer: In this case, there were 100 patients who were known to have the disease. Sensitivity is the proportion of these patients who were correctly identified based on the positive test. Set up a contingency table as follows:

Diseased Control group Total
Positive test 90 20 110
Negative test 10 80 90
Total 100 100 200
Sensitivity = TP / (TP + FN) = 90 / 100 = 90%

Importance of sensitivity:

  • Tests with high sensitivity are important when it is crucial that you miss as few cases as humanly possible.
  • Tests with high sensitivity are good screening tests.
  • Example: HIV screening tests. For the 1st screening test, you want to cast a wide net and catch all positive cases. You will likely end up with a higher number of FPs (which can be identified on later confirmatory testing) by not having missed anyone during your initial screening test.

Specificity

Definition:

  • Probability of a test in correctly rejecting a person who does not have the disease
  • The proportion of healthy people that test negative
  • A measure of the exclusivity of a diagnostic test
True negative fraction

True negative fraction:
Diagram illustrating the concept of true negative fractions – the proportion of the population represented by the specificity of a test. This diagram shows that all patients received a negative test result. The red figures represent individuals who actually had the disease but tested negative (false negatives), whereas the green figures represent individuals who did not have the disease and correctly tested negative (true negatives).

Image by Lecturio. License: CC BY-NC-SA 4.0

Calculation:

Specificity is also calculated using a similar contingency table:

Screening tests contingency table

Contingency table for screening tests

Image by Lecturio. License: CC BY-NC-SA 4.0

Specificity is the proportion of people who are truly negative and have a negative screening test (TNs, found in square D) divided by all people who are truly negative, regardless of their screening test results (TNs and FPs, B + D). Specificity is represented by the following equations:

$$ Specificity = \frac{D}{B + D}\ or \ Specificity = \frac{TN}{(FP + TN)} $$

where TN = true negatives and FP= false positives

Example: A new diagnostic test is tested on a group of patients: 100 patients are known to have the disease, and another 100 patients are known to be disease free in a control group. Among them, 90 patients with the disease and 20 individuals in the control group show a positive result. What is the specificity of the new test?

Answer: In this case, all patients in the control group are known to be disease free. Specificity is the proportion of these patients who were correctly identified based on the negative test. Set up a contingency table as follows:

Diseased Control group Total
Positive test 90 20 110
Negative test 10 80 90
Total 100 100 200
Specificity = TN / (TN + FN) = 80 / 100 = 80%

Importance of specificity:

  • Tests with high specificity are important when it is crucial that you exclude everyone who is truly healthy.
  • Tests with high specificity are good confirmatory/diagnostic tests.
  • Example: HIV-confirmation tests. We did not want to exclude anyone in the initial screening tests; thus, we accepted a high FP rate to make sure no one with HIV was excluded. Before beginning lifelong treatment with antiretrovirals, however, it is important to exclude all FP cases to ensure that only those individuals who are truly HIV positive receive treatment.

Positive and Negative Predictive Values

Predictive values are also called “precision rates.”

Positive predictive value

Definition:

The positive predictive value is the percentage of people with a positive test result who actually have the disease among all people with a positive result (A), regardless of whether or not they have the disease (A+B).

Contingency table highlighting the values needed for the computation of positive predictive value

Contingency table highlighting the values needed for the computation of positive predictive value

Image by Lecturio. License: CC BY-NC-SA 4.0

Calculation:

The positive predictive value is calculated using the equation:

$$ Positive\ predictive\ value = \frac{A}{A + B} $$

where A = true positives and B = false positives

Example: A new diagnostic test is tested on a group of patients: 100 patients are known to have the disease, and another 100 patients are known to be disease free in a control group. Among them, 90 patients with the disease and 20 individuals in the control group show a positive result. What is the positive predictive value of the new test?

Answer: The positive predictive value is asking about the proportion of TP cases out of all positive cases (TP + FP). Set up a contingency table as follows:

Diseased Control group Total
Positive test 90 20 110
Negative test 10 80 90
Total 100 100 200
Positive predictive value = TP / (TP + FP) = 90 / (90 + 20) = 90 / 110 = 81.8%

Difference between positive predictive value and sensitivity:

  • Positive predictive value considers all patients with a positive test, including those who truly have and do not have the disease.
  • Sensitivity considers all patients who truly have the disease, including those with positive and negative tests.

Negative predictive value (NPV)

Definition:

The NPV is the percentage of people with a negative test result who are actually disease free (D), among all people with a negative result (regardless of whether or not they have the disease, C + D).

Contingency table highlighting the values needed for the computation of negative predictive value

Contingency table highlighting the values needed to compute negative predictive value

Image by Lecturio. License: CC BY-NC-SA 4.0

Calculation:

The NPV is calculated using the following equation:

$$ NPV = \frac{D}{C + D} $$

where D = true negatives and C = false negatives

Example: A new diagnostic test is tested on a group of patients: 100 patients are known to have the disease, and another 100 patients are known to be disease free in a control group. Among them, 90 patients with the disease and 20 individuals in the control group show a positive result. What is the NPV of the new test?

Answer: The NPV is asking about the proportion of TN cases out of all negative cases (TN + FN). Set up a contingency table as follows:

Diseased Control group Total
Positive test 90 20 110
Negative test 10 80 90
Total 100 100 200
NPV = TN / (TN + FN) = 80 / (80 + 10) = 80 / 90 = 88.9%

Difference between NPV and specificity:

  • NPV is looking at all patients with a negative test, including those who truly have and do not have the disease.
  • Specificity is looking at all patients who are truly disease free, including those with both positive and negative tests.

Summary Example: Sensitivity, Specificity, Positive Predictive Value, and NPV

Pregnancy Pregnancy Pregnancy is the time period between fertilization of an oocyte and delivery of a fetus approximately 9 months later. The 1st sign of pregnancy is typically a missed menstrual period, after which, pregnancy should be confirmed clinically based on a positive β-hCG test (typically a qualitative urine test) and pelvic ultrasound. Pregnancy: Diagnosis, Maternal Physiology, and Routine Care example:

In a study, 4,810 women take a home urine pregnancy test. All of them undergo an ultrasound to confirm whether or not they are truly pregnant. Among them, 9 women have a positive urine pregnancy test result and are actually found to be pregnant on ultrasound; 1 woman has a negative urine pregnancy test result but is actually pregnant; 351 women have positive urine pregnancy test results and are found to not be pregnant; 4,449 women have negative urine pregnancy test results and ultrasound results confirm that they are not pregnant. (Note: This is sample data and does not represent real values.)

In this example, the home pregnancy test is the screening test, and “pregnancy” is the “disease” state.

The contingency table is as follows:

Pregnant Not pregnant Total
Positive test 9 351 360
Negative test 1 4,449 4,450
Total 10 4,800 4,810
Table: Summary of sensitivity, specificity, positive predictive value, and NPV using the pregnancy example
Clinical question What is being asked? Equation Answer
If the woman is actually pregnant, what is the probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability that the urine pregnancy test will be positive? Sensitivity = A / (A + C)
= 9 / (10)
90%
If a woman is not actually pregnant, what is the probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability that the urine pregnancy test will correctly show that she is not pregnant? Specificity = B / (B + D)
= 4,449 / 4,800
92.7%
If a woman tests positive in the urine pregnancy test, what is the probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability that she is actually pregnant? positive predictive value = A / (A + B)
= 9 / 360
2.5%
If a woman tests negative in the urine pregnancy test, what is the probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability that she really is not pregnant? NPV = D / (C + D)
= 4,449 / 4,450
99.9%

Likelihood Ratios

Definition

  • Likelihood ratios (LRs) are ORs that indicate the likelihood that a given test result is expected in a patient with the disease compared with the likelihood that that same result would be expected in a patient without the disease.
  • The LRs tell us by how much we should shift our suspicion that a person has a condition based on their test result.
  • Interpretation:
    • LR > 1: 
      • The test is associated with presence of the disease.
      • A high LR (typically and arbitrarily defined as > 5 or > 10) indicates a strong suspicion that a person has the disease if they test positive.
    • LR < 1: 
      • The test is associated with absence of the disease.
      • A low LR indicates a strong suspicion that a person does not have the disease if they test negative.
  • In practice, we usually only use the positive LR (LR+).

Positive likelihood ratio

  • Definition: the probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of a positive test result for a person who really has the disease (TPs) divided by the probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of a positive test result for someone who does not really have the disease (FPs)
  • Equations:
    • LR+ = probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of TPs / probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of FPs
    • LR+ = P(TP) / P(FP) 
    • Can be expressed as a function of sensitivity and specificity:
$$ LR+ = \frac{Sensitivity}{1 – specificity} $$

Negative likelihood ratio (LR‒)

  • Definition: the probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of a negative test result for a person who really is healthy (TNs) divided by the probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of a negative test result for someone who actually has the disease (FNs)
  • Equations:
    • LR‒ = probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of TNs / probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of FNs
    • LR‒ = P(TN) / P(FN)
    • Can be expressed as a function of sensitivity and specificity:
$$ LR- = \frac{1 – sensitivity}{Specificity} $$

Example and interpretation of LRs

Using the same pregnancy example in the section above, and knowing that the sensitivity was 90% and the specificity was 92.7%, the LR+ and LR‒ can be computed as follows:

LR+ = 0.9 / (1 ‒ 0.927) = 12.3 = 1,230%

LR‒ = (1 ‒ 0.9) / 0.927 = 0.11 = 11%

Interpretation: There is a 12-fold greater likelihood that a woman who tests positive is truly pregnant. A negative test result reduces the odds of being pregnant by 89%.

Pre-test and Post-test Probabilities

Pre-test probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability

  • Probability of a screened person of having the disease
  • Determining the pre-test probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability:
    • Can be determined using published epidemiological data: typically the prevalence of a disease in the population
    • Clinical criteria scales can also be used to calculate the pre-test probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability (e.g., Well’s deep vein thrombosis Deep vein thrombosis Deep vein thrombosis (DVT) usually occurs in the deep veins of the lower extremities. The affected veins include the femoral, popliteal, iliofemoral, and pelvic veins. Proximal DVT is more likely to cause a pulmonary embolism (PE) and is generally considered more serious. Deep Vein Thrombosis ( DVT DVT Deep vein thrombosis (DVT) usually occurs in the deep veins of the lower extremities. The affected veins include the femoral, popliteal, iliofemoral, and pelvic veins. Proximal DVT is more likely to cause a pulmonary embolism (PE) and is generally considered more serious. Deep Vein Thrombosis) criteria for clinically determining the pre-test probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of DVT DVT Deep vein thrombosis (DVT) usually occurs in the deep veins of the lower extremities. The affected veins include the femoral, popliteal, iliofemoral, and pelvic veins. Proximal DVT is more likely to cause a pulmonary embolism (PE) and is generally considered more serious. Deep Vein Thrombosis))
  • Clinical uses: 
    • To calculate post-test probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability (see below)
    • If high enough, can be used to validate at the beginning of treatment without testing
    • If low enough, can be used to reject the diagnosis as unlikely

Post-test probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability

  • Probability of a person having the disease after getting the results of a test
  • Calculations:
    • Typically calculated using online calculators in clinical studies
    • Involves the following variables:
      • Pre-test probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability
      • Sensitivity of the test
      • Specificity of the test
  • Post-test probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of a positive result: probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability that the disease is present when the test result is positive
  • Post-test probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability of a negative result: probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability that the disease is present when the test result is negative

References

  1. Greenberg, R.S. (2014). Diagnostic testing. In R.S. Greenberg (Ed.), Medical Epidemiology: Population health and effective health care, 5e. New York, NY: McGraw-Hill Education. 
  2. Garibaldi, B.T., Olson, A.P.J. (2018). The hypothesis-driven physical examination. Medical Clinics of North America, 102(3), 433-442. 
  3. Safari, S., Baratloo, A., Elfil, M., Negida, A. (2016). Evidence-based emergency medicine; Part 4: Pre-test and post-test probabilities and Fagan’s nomogram. Emergency (Tehran, Iran), 4(1), 48–51.
  4. Colquhoun, D. (2017). The reproducibility of research and the misinterpretation of p-values. Royal Society Open Science. 4 (12), 171085.
  5. Colquhoun, D. (2018). The false-positive risk: A proposal concerning what to do about p values. The American Statistician. 73, 192–201.
  6. Mahutte, N.G., Duleba, A.J. (2021). Evaluating diagnostic tests. In Armsby, C. (Ed.), UpToDate. Retrieved July 1, 2021, from https://www.uptodate.com/contents/evaluating-diagnostic-tests 
  7. Calculator: Post-test probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability from pre-test probability Probability Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. Basics of Probability, sensitivity, and specificity. UpToDate. Retrieved July 1, 2021, from https://www.uptodate.com/contents/calculator-post-test-probability-from-pre-test-probability-sensitivity-and-specificity

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