CHAPTER 3 Probability Theory Basic Definitions and Properties Conditional Probability and Independence Bayes’ Formula Applications (biomedical)

Examples of Screening Tests for Early Detection of Colorectal Cancer Highly sensitive and highly specific, but expensive. Cost-effective for males 50+. Fecal Occult Blood Test (FOBT) Cheap, fast, easy, and highly sensitive, but low specificity… not to mention TOTALLY DISGUSTING. “FUITA” Procedure Very highly sensitive, routinely administered by health insurance companies More on Sensitivity / Specificity

Some Additional Biomedical Applications

Case-Control studies Cohort studies

E+ vs. E– Observational study designs that test for a statistically significant association between a disease D and exposure E to a potential risk (or protective) factor, measured via “odds ratio,” “relative risk,” etc. Lung cancer / Smoking PRESENT E+ vs. E– ? D+ vs. D– ? Case-Control studies Cohort studies Both types of study yield a 2  2 “contingency table” of data: D+D+D–D– E+E+ aba + b E–E– cdc + d a + cb + dn relatively easy and inexpensive subject to faulty records, “recall bias” D+ vs. D– FUTUREPAST measures direct effect of E on D expensive, extremely lengthy… where a, b, c, d are the numbers of individuals in each cell. cases controlsreference group

D+D+D–D– E+E+ aba + b E–E– cdc + d a + cb + dn where a, b, c, d are the numbers of individuals in each cell. E+ vs. E– PRESENT E+ vs. E– ? D+ vs. D– ? Case-Control studies Cohort studies D+ vs. D– FUTUREPAST cases controls Cohort studies “Odds of Disease, given Exposed” = odds(D | E+) = “Odds of Disease, given Not Exposed” = odds(D | E–) = = 1 No assoc; D, E stat indep < 1 possible protective factor > 1 possible risk factor ref gp

D+D+D–D– E+E+ aba + b E–E– cdc + d a + cb + dn where a, b, c, d are the numbers of individuals in each cell. E+ vs. E– PRESENT E+ vs. E– ? D+ vs. D– ? Case-Control studies Cohort studies D+ vs. D– FUTUREPAST cases controls odds(D | E+) = odds(D | E–) = ref gp D+D+D–D– E+E E–E– Example: Among those exposed, the probability of developing disease is 2.5 times greater than the probability of not developing disease. Among those not exposed, the probability of developing disease is times greater than the probability of not developing disease. The odds of disease among those exposed are times greater than the odds of disease among those not exposed. Cohort studies

D+D+D–D– E+E+ aba + b E–E– cdc + d a + cb + dn where a, b, c, d are the numbers of individuals in each cell. E+ vs. E– PRESENT E+ vs. E– ? D+ vs. D– ? Case-Control studies Cohort studies D+ vs. D– FUTUREPAST cases controls Why not just use ??? ref gp Example: The odds of disease among exposed are times greater than the odds of disease among not exposed. Cohort studies

D+D+D–D– E+E+ aba + b E–E– cdc + d a + cb + dn where a, b, c, d are the numbers of individuals in each cell. E+ vs. E– PRESENT E+ vs. E– ? D+ vs. D– ? Case-Control studies Cohort studies D+ vs. D– FUTUREPAST cases controls ref gp D+D+D–D– E+E E–E– Example: The odds of disease among exposed are times greater than the odds of disease among not exposed. The probability of disease among exposed is 1.25 times greater than the probability of disease among not exposed. Case-Control studies The odds of exposure among diseased are times greater than the odds of exposure among not diseased. (HW problem) Cohort studies

D+D+D–D– E+E+ aba + b E–E– cdc + d a + cb + dn where a, b, c, d are the numbers of individuals in each cell. E+ vs. E– PRESENT E+ vs. E– ? D+ vs. D– ? Case-Control studies Cohort studies D+ vs. D– FUTUREPAST cases controls ref gp Example: Case-Control studies Whereas the Odds Ratio is reliably approximated from either type of study using the same formula, the Relative Risk is not, and is only appropriately defined for cohort studies, except… if the disease is rare in the popul’n… then RR ≈ OR. a is small relative to b, and c is small relative to d … Cohort studies