9.4.6.1 Likelihood Ratios and Predictive Values Comparison

Understanding the Relationship and Differences Between Likelihood Ratios and Predictive Values in Diagnostic Testing

In the context of diagnostic testing, likelihood ratios and predictive values are both important concepts that help assess the performance and utility of a test. However, they serve different purposes and provide different types of information.

Likelihood ratios:

Definition: Likelihood ratios (LRs) indicate how much a test result will change the odds of having a disease. They are used to combine pre-test probability (the likelihood of the disease before knowing the test result) with the test result to estimate the post-test probability (the likelihood of the disease after knowing the test result).

Types:

1. Positive Likelihood Ratio (LR+): This is the ratio of the probability of a positive test result in patients with the disease to the probability of a positive test result in patients without the disease.
   • It is calculated as: LR+ = Sensitivity / (1 − Specificity). An LR+ > 1 indicates that a positive test result is more likely in those with the disease.
2. Negative Likelihood Ratio (LR-): This is the ratio of the probability of a negative test result in patients with the disease to the probability of a negative test result in patients without the disease.
   • It is calculated as: LR− = (1−Sensitivity) / Specificity. ​An LR- < 1 indicates that a negative test result is less likely in those with the disease.

Utility:

• LRs help in adjusting the pre-test probability to get a more accurate post-test probability.
• They are not affected by the prevalence of the disease.

Predictive values:

Definition: Predictive values describe the probability that a patient has or does not have a disease given a certain test result.

Types:

1. Positive Predictive Value (PPV): This is the probability that a person has the disease given that the test result is positive.
   • It is calculated as: PPV = True Positives / (True Positives + False Positives). ​PPV depends on both the sensitivity and specificity of the test and the prevalence of the disease in the population being tested.
2. Negative Predictive Value (NPV): This is the probability that a person does not have the disease given that the test result is negative.
   • It is calculated as: NPV = True Negatives / (True Negatives + False Negatives). PPV, NPV depends on the sensitivity, specificity, and prevalence of the disease.

Utility:

• Predictive values are directly relevant to clinical decision-making because they tell us the likelihood of disease presence or absence based on the test result.
• They are highly influenced by the prevalence of the disease in the population being tested.

Key Differences

1. Dependence on Disease Prevalence:
   • Likelihood Ratios: Independent of disease prevalence.
   • Predictive Values: Dependent on disease prevalence.
2. Purpose:
   • Likelihood Ratios: Used to adjust the pre-test probability to get a post-test probability.
   • Predictive Values: Used to determine the probability of disease presence or absence based on the test result.
3. Interpretation:
   • Likelihood Ratios: Help in quantifying how much a test result will change the odds of having a disease.
   • Predictive Values: Indicate the probability of having or not having the disease given a positive or negative test result.

Example

Imagine a diagnostic test for a disease with a sensitivity of 90% and a specificity of 85%. If the disease prevalence in a 1000 person population is 20%, we can calculate the following:

LR+:

This means a positive test result makes the disease six times more likely.

LR-:

LR- of 0.12 means that a negative result is 0.12 times as likely (or 88% less likely) to occur in someone with the disease compared to someone without the disease.

This suggests that the test is quite good at ruling out the disease. When the LR- is close to 0 (typically <0.1), it indicates that a negative test result is strong evidence that the person does not have the disease.

Worth noting:

• PPV: Using Bayes’ theorem and the given prevalence, sensitivity, and specificity, we can calculate PPV (the actual numerical calculation involves more detailed steps, typically requiring a 2×2 contingency table).
• NPV: Similarly, we can calculate NPV using the same principles.

In summary, likelihood ratios are useful for adjusting probabilities and are not affected by disease prevalence, while predictive values are directly useful for interpreting test results but are affected by disease prevalence.

Detailed calculation of positive predictive value (PPV) in the example:

To calculate the Positive Predictive Value (PPV), we need to understand how sensitivity, specificity, and prevalence interact. Here, we will go through a step-by-step process to calculate the PPV for a diagnostic test given the following parameters:

• Sensitivity (True Positive Rate): 90% (0.90)
• Specificity (True Negative Rate): 85% (0.85)
• Prevalence of the disease: 20% (0.20)

Steps to Calculate PPV

1. Define the Probabilities:
   • Sensitivity (Se): The probability of a positive test result given that the disease is present.
   • Specificity (Sp): The probability of a negative test result given that the disease is not present.
   • Prevalence (P): The probability that any given individual in the population has the disease.
2. Set Up the Probabilities in a 2×2 Table: there is a population of 1000 individuals for this calculation:
   • Number of individuals with the disease = Prevalence x Total population = 0.20 x 1000 = 200
   • Number of individuals without the disease = Total population – Number with the disease = 1000 – 200 = 800
3. Calculate the True Positives, False Positives, True Negatives, and False Negatives:
   • True Positives (TP): Individuals with the disease who test positive, TP = Sensitivity × Number with disease = 0.90 × 200 = 180.
   • False Positives (FP): Individuals without the disease who test positive, FP = (1 − Specificity) × Number without disease = (1 − 0.85) × 800 = 0.15 × 800 = 120.
   • True Negatives (TN): Individuals without the disease who test negative, TN = Specificity × Number without disease = 0.85 × 800 = 680.
   • False Negatives (FN): Individuals with the disease who test negative, FN = (1 − Sensitivity) × Number with disease = (1 − 0.90) × 200 = 0.10 × 200 = 20.
4. Calculate PPV Using the Formula: PPV = True Positives / (True Positives + False Positives).
   • Substituting the values calculated: PPV = 180 / 180 + 120 = 180 / 300 = 0.60. Thus, the PPV is 60%. This means that if the test result is positive, there is a 60% probability that the individual actually has the disease.

Summary of the 2×2 Table for the Example:

	Disease Present	Disease Absent	Total
Test Positive	180 (TP)	120 (FP)	300
Test Negative	20 (FN)	680 (TN)	700
Total	200	800	1000

Concluding points for the example

The Positive Predictive Value (PPV) is crucial in understanding how likely it is that a positive test result indicates the actual presence of the disease. It is affected by the test’s sensitivity, specificity, and the prevalence of the disease in the population. In our example, a test with a sensitivity of 90%, specificity of 85%, and disease prevalence of 20% yields a PPV of 60%, indicating that 60% of positive test results are true positives.

This example of over the top and it is unlikely you will be expected to calculate something with this many steps. It is still useful to understand as you may be expected to calculate an individual step of this.