Question: Colorectal cancers can be detected through non-invasive fecal immunochemical tests (FIT) for hemoglobin (Hb). The diagnostic properties of a widely used FIT were assessed in a cohort of 1,822 adults between the ages of 50 and 79 years old. In addition to being screened for colorectal cancer using the FIT, all participants received a definitive diagnosis based on two colonoscopies. The table below shows the results of the study.

1. Suppose the researchers are deciding between two options of cut-off scores to indicate those who have colorectal cancer. For option one, all individuals scoring ≥ 9 screen positive for colorectal cancer, while those scoring < 9 screen negative. For option two, all individuals scoring ≥ 25 screen positive for colorectal cancer, while those scoring < 25 screen negative. Construct separate 2 x 2 tables for both cut-off scores to relate high FIT scores to the correct classification of colorectal cancer.
2. Compute the sensitivity and specificity of colorectal cancer classification corresponding to both cut-off scores.
3. How does the sensitivity and specificity change when the cut-off score changes from 9 to 25?
4. Based on sensitivity and specificity, which cut-off would you recommend to the researchers?
5. Suppose the prevalence of colorectal cancer in the population of interest is 15%. Compute the positive and negative predictive values of the diagnostic FIT with a cut-off of 9.

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