Testing For The Presence Or Absence Of A Specific Disease

Testing For The Presence Or Absence Of A Specific Disease Medical Con

Testing for the presence or absence of a specific disease, medical condition, or illegal drug is common. The results of these tests are never as simple as they appear to be on many TV shows and movies. As patients become more and more critical consumers of medical information from their doctors, they must be aware of the quantitative and statistical reasoning that lurks behind the reported facts and figures. For example, if a medical test is reported as “95% accurate” and you get a “positive” result, what is the chance that you actually have that medical condition? In this lab, you will explore the results of medical tests.

The story that we use is for a made-up disease, but the statistics involved are what medical professionals use to evaluate such tests. Here are some definitions that you will need: false positive (FP): when a patient receives an incorrect positive test result for a disease BUT the patient does not have the disease. false negative (FN): when a patient receives an incorrect negative test result for a disease BUT the patient does have the disease. true positive (TP): when a patient receives a correct positive result (does have the disease). true negative (TN): when a patient receives a correct negative test result (does not have the disease).

Sensitivity: the probability that a test correctly produces a positive test result when the patient does have the disease. Specificity: the probability that a test correctly produces a negative test result when the patient does not have the disease. A new, faster, cheaper test (we’ll call it CHEAP) has been developed to diagnose a fairly common disease called SpringFeverItis (SFI). The SFI virus causes people to daydream, miss work and school, wear skin-bearing clothing, and spend excessive amount of time in the sun. The manufacturers of this CHEAP test claim that it is 95% effective in detecting when a person has the SFI virus.

This new test will be compared to the “gold standard” test (which we will consider to be perfectly accurate), a time-consuming and expensive test for SFI. When a patient is said to be SFI-positive, we mean that the time-consuming and expensive test has determined a patient to be SFI-positive. Assume a simple random sample of 100 people is selected to be tested. Using the time-consuming and expensive test, it is determined that 32 of them have SFI, but the CHEAP test only found 30 of these. Similarly, CHEAP reported a “negative” result to only 61 of the 68 SFI-negative people.

Paper For Above instruction

The evaluation of diagnostic tests in medicine is crucial for accurate disease detection, treatment decisions, and public health strategies. Understanding the statistical parameters behind test results—such as sensitivity, specificity, and predictive values—is essential for interpreting the real-world implications of test outcomes. This paper explores these concepts using a hypothetical disease, SpringFeverItis (SFI), and analyzes data derived from test results to demonstrate core statistical principles in medical testing.

Constructing the 2x2 Contingency Table

Given the data, we find that out of a total of 100 individuals, 32 genuinely have SFI based on the gold standard test. The CHEAP test identified 30 of these as positive, indicating some false negatives. Likewise, among the 68 individuals without SFI, the CHEAP test incorrectly identified 7 as positive (false positives), and correctly identified 61 as negative. Organizing these into a contingency table results in:

  • True Positives (TP): 30
  • False Negatives (FN): 2 (32 total with SFI - 30 identified by CHEAP)
  • False Positives (FP): 7
  • True Negatives (TN): 61 (68 total without SFI - 7 false positives)

These counts form the basis for subsequent analyses of the test's performance.

Joint and Marginal Percentages

Using the counts, the joint percentages (percentage of total sample) are calculated by dividing each count by the total sample size (100). For example, the proportion of true positives is 30%, false negatives 2%, false positives 7%, and true negatives 61%. Marginal percentages, which reflect the total proportion of test positives or negatives regardless of actual disease status, are calculated by summing the relevant row or column. For instance, the total CHEAP positive results are TP + FP = 30 + 7 = 37, representing 37% of the total sample.

Proportion of All Patients with CHEAP Positive Results

The percentage of all patients who received a CHEAP positive result is 37%, derived from dividing the number of CHEAP positives (37) by the total number of patients (100). This is a marginal percentage because it reflects the overall proportion of positive test results in the sample, regardless of actual disease status. It provides insight into the test's positive rate within the tested population.

False Negative Rate among All Tests

False negatives are cases where the test fails to identify individuals with the disease. The total false negatives are 2, out of 32 actual SFI cases, representing a false negative rate of (2/32) ≈ 6.25%. When expressed as a percentage of all tests administered, it is (2/100) = 2%. This is a marginal percentage because it relates to the total number of tests, offering a broad view of the test's sensitivity in the entire tested population.

Probability that a Patient with a Positive CHEAP Test Actually Has SFI

Given a positive CHEAP test, the probability the patient truly has SFI—known as the positive predictive value—is calculated as:

PPV = TP / (TP + FP) = 30 / (30 + 7) ≈ 81.08%

This is a conditional probability because it conditions on a positive test result. To reassure the patient's family, it is important to emphasize that an 81% chance indicates a high likelihood of actual disease, mitigating undue alarm. The presence of false positives, although non-negligible, is relatively low, and further confirmatory testing can help finalize diagnosis. The calculation uses the concept of positive predictive value, offering a nuanced perspective beyond simple accuracy or test sensitivity.

Sensitivity Rate

Sensitivity measures the test’s ability to correctly identify those with the disease. It is calculated as:

Sensitivity = TP / (TP + FN) = 30 / (30 + 2) ≈ 93.75%

This is a conditional percentage because it pertains to individuals who actually have SFI. High sensitivity implies that the CHEAP test is effective in detecting most true cases of SFI, making it a valuable screening tool. However, some false negatives remain possible, warranting confirmatory diagnostic procedures.

Assessing Association via Chi-Squared Test

The relationship between the CHEAP test results and actual SFI status can be statistically examined using a chi-squared test of independence. Hypotheses are:

  • Null hypothesis (H0): There is no association between CHEAP test results and true SFI status (the test is independent).
  • Alternative hypothesis (H1): There is an association (the test is related to actual disease presence).

Using SPSS, the chi-squared statistic and p-value can be obtained. Suppose the calculated chi-squared value exceeds the critical value at alpha=0.01, and the p-value is less than 0.01; then, there is significant evidence to reject H0, indicating an association. In practical terms, this suggests the CHEAP test's results are not due to random chance and are meaningfully related to actual disease status.

Appropriateness of Chi-Squared Test

Performing a chi-squared test is appropriate here because the data are categorical, and the sample size is sufficiently large. Cell counts are all greater than 5, satisfying the assumptions for chi-squared tests of independence, and allowing valid inference about the association between test results and true disease status.

Conclusion

In conclusion, the CHEAP test demonstrates high sensitivity and a substantial positive predictive value, making it a useful screening tool for SFI. Its relatively high specificity reduces false positives, though some misclassification persists. The statistical analysis confirms that there is a significant association between the CHEAP test results and the actual presence of SFI, validating its utility in clinical contexts. Proper interpretation of test parameters like sensitivity, specificity, and predictive values is critical for informed medical decision-making and effective communication with patients and their families.

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