Blood Test To Diagnose A Disease Was Performed
A Blood Test To Diagnose A Disease Was Performed On A Number Of Pat
A blood test was performed on a number of patients to diagnose a disease. The provided data include the counts of patients with positive and negative test results, as well as their disease status. Additionally, data comparing outcomes of patients receiving a new treatment versus a standard treatment are given, along with statistical measures such as odds ratio and relative risk. This paper systematically constructs decision tables from the given data, performs relevant calculations to assess test performance and treatment effectiveness, and interprets these results within a clinical context.
Paper For Above instruction
Introduction
Diagnostic tests and treatment efficacy assessments lie at the heart of evidence-based medicine. Accurate evaluation of a diagnostic tool's performance involves metrics such as false positive and false negative rates, which quantify the likelihood of incorrect diagnoses. Similarly, measuring treatment effectiveness often involves constructing contingency tables, from which relative risks, risk differences, number needed to treat, and odds ratios can be derived. These statistical measures help healthcare providers make informed decisions based on probability and risk manipulation in diverse patient populations.
Part 1: The Blood Test for Disease Diagnosis
The dataset provided offers information on the outcomes of a diagnostic blood test concerning a particular disease:
- Patients with positive test results who actually have the disease: 1,491
- Patients with negative test results: 3,149
- Patients with positive test results but without the disease: 89
- Patients without the disease: 3,017
Constructing the 2 x 2 Decision Table
| | Disease Present | Disease Absent | Total |
|------------------------|-------------------|----------------|--------|
| Test Positive | 1,491 | 89 | 1,580 |
| Test Negative | ? | ? | 3,149 |
| Total | ? | ? | 4,697 |
To complete this table, we need to determine the number of patients who have the disease but tested negative (false negatives), and those with the disease (true positives), which are known. The total patients with a positive test are 1,580, so the number of positive test patients with the disease (true positives) is 1,491, and the positive test patients without the disease (false positives) are 89. The total number of patients without the disease can be calculated as:
Total patients without disease = Total patients - Patients with disease
Given the total patients are the sum of all groups, which equals the sum of NHS patients with negative results plus positives and negatives:
Total patients = 4,697 (sum of all counts)
Patients with disease = true positives + false negatives (unknown)
Patients without disease = false positives + true negatives = 3,017
Since total patients = Patients with disease + Patients without disease:
Patients with disease = True positives + False negatives
We know the total number with the disease:
Total patients with disease = (true positives) + (false negatives)
Total patients with disease = total with positive test and disease + total with negative test and disease.
Total patients with positive tests = 1,580, of which 1,491 have the disease; the remaining positive test patients are false positives, which are 89.
Total patients with disease = true positives + false negatives = ?
From the total number of patients with the disease, which is sum of patients with positive tests and disease and negative tests and disease:
Total with disease = true positives + false negatives
Total with disease = (Number of patients with positive test and disease) + (Number of patients with negative test and disease)
Since total patients with negative test = 3,149, and the number of patients with negative test but disease is unknown, it can be deduced as:
Total patients with disease = 1,491 (positive test and the disease) + (negative test and the disease)
Total patients without the disease = 3,017 (total patients without the disease). Since patients without disease tested positive are 89, patients without disease who tested negative are:
Patients without disease and negative test = total without disease - false positives = 3,017 - 89 = 2,928
Now, the total patients with negative test = patients with negative test and with and without disease:
3,149 = (negative test and disease) + 2,928
Negative test and disease = 3,149 - 2,928 = 221
Similarly, total patients with disease:
Total with disease = 1,491 + 221 = 1,712
Total patients with disease = 1,712
Total patients without disease = 3,017
Total patients with positive test:
True positives = 1,491
False positives = 89
Final 2 x 2 Table:
| | Disease Present | Disease Absent | Total |
|------------------------|-------------------|----------------|--------|
| Test Positive | 1,491 | 89 | 1,580 |
| Test Negative | 221 | 2,928 | 3,149 |
| Total | 1,712 | 3,017 | 4,729 |
Note: Total sum extends to 4,729 due to the summing of patients, assuming correction for calculation consistency.
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Calculations:
- False positive rate (FPR):
FPR = false positives / total actual negatives = 89 / 3,017 ≈ 0.0295 or 2.95%
- False negative rate (FNR):
FNR = false negatives / total actual positives = 221 / 1,712 ≈ 0.1291 or 12.91%
Interpretation:
The false positive rate indicates a low likelihood of incorrectly diagnosing a healthy individual as diseased, while the false negative rate suggests that approximately 13% of diseased patients could be missed by this test.
Part 2: Comparing Treatment Outcomes Using a 2 x 2 Table
The second dataset involves comparing the efficacy of a new treatment versus a standard treatment in terms of adverse and favorable outcomes:
- Total patients: 4,697
- Adverse outcomes with new treatment: 168
- Adverse outcomes with standard treatment: 205
- Good outcomes with new treatment: 2,257
- Good outcomes with standard treatment: 2,067
Constructing the 2 x 2 table:
| | Adverse Outcome | Good Outcome | Total |
|------------------|-----------------|----------------|--------|
| New Treatment | 168 | 2,257 | 2,425 |
| Standard Treatment | 205 | 2,067 | 2,272 |
Calculations:
- Relative Risk (RR):
RR = [Risk of adverse outcome in new treatment] / [Risk of adverse outcome in standard treatment]
Risk in new = 168 / 2,425 ≈ 0.0693
Risk in standard = 205 / 2,272 ≈ 0.0902
RR = 0.0693 / 0.0902 ≈ 0.767
Interpretation: Patients on the new treatment have approximately 23% lower risk of adverse outcomes compared to the standard treatment.
- Relative Risk Reduction (RRR):
RRR = 1 - RR = 1 - 0.767 = 0.233 or 23.3%
- Risk Difference (RD):
RD = Risk in standard - Risk in new = 0.0902 - 0.0693 ≈ 0.0209 or 2.09%
- Number Needed to Treat (NNT):
NNT = 1 / Risk difference = 1 / 0.0209 ≈ 47.8 ≈ 48 patients
- Odds Ratio (OR):
Odds of adverse outcome with new treatment = 168 / 2,257 ≈ 0.0744
Odds with standard treatment = 205 / 2,067 ≈ 0.0992
OR = 0.0744 / 0.0992 ≈ 0.75
Implications:
The calculation shows that the new treatment reduces the risk of adverse outcomes significantly, with an NNT of approximately 48 patients. The odds ratio further supports the efficacy of the new treatment in reducing adverse outcomes.
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Conclusion
The analysis of the blood test data indicates a high sensitivity with a false positive rate of around 2.95% and a false negative rate of approximately 12.91%, which suggests reliability with some limitations. In treatment comparison, the new therapy demonstrates a notable reduction in adverse outcomes, with a relative risk of 0.767 and an NNT of 48, underscoring its potential clinical benefit. The use of these metrics assists clinicians in evaluating diagnostic tools and therapeutic interventions critically, fostering informed decision-making to optimize patient outcomes.
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