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CHAPTER 3 Probability Theory • 3.1 - Basic Definitions and Properties • 3.2 - Conditional Probability and Independence • 3.3 - Bayes’ Formula • 3.4 - Applications (biomedical) More on Sensitivity / Specificity Examples of Screening Tests for Early Detection of Colorectal Cancer Highly sensitive and highly Fecal Occult Blood Test (FOBT) specific, but expensive. Cheap, fast, easy, and highly Cost-effective for males 50+. sensitive, but low specificity… not to mention TOTALLY DISGUSTING. “FUITA” Procedure Very highly sensitive, routinely administered by health insurance companies Some Additional Biomedical Applications Case-Control studies Cohort studies 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 Case-Control studies Cohort studies PRESENT PAST FUTURE cases E+ vs. E– ? controls reference group D+ vs. D– E+ vs. E– relatively easy and inexpensive subject to faulty records, “recall bias” D+ vs. D– ? measures direct effect of E on D expensive, extremely lengthy… Both types of study yield a 22 “contingency table” of data: D+ D– E+ a b a+b E– c d c+d a+c b+d n where a, b, c, d are the numbers of individuals in each cell. Case-Control studies PRESENT PAST cases E+ vs. E– ? Cohort studies ref gp controls FUTURE D+ vs. D– E+ vs. E– D+ D– E+ a b a+b E– c d c+d a+c b+d Cohort studies D+ vs. D– ? where a, b, c, d are the numbers of individuals in each cell. n P ( D | E ) a / ( a b) a P ( D | E ) b / ( a b) b P ( D | E ) c / (c d ) c “Odds of Disease, given Not Exposed” = odds(D | E–) = P ( D | E ) d / (c d ) d “Odds of Disease, given Exposed” = odds(D | E+) = odds ( D | E ) a / b a d odds ( D | E –) c / d b c <1 possible protective factor =1 No assoc; D, E stat indep >1 possible risk factor Case-Control studies PRESENT PAST cases E+ vs. E– ? Example: Cohort studies odds(D | E+) = ref gp controls D+ vs. D– E+ vs. E– D+ D– E+ 500 a 200 b a700 +b E– 400 c 300 d c700 +d a900 + c b500 +d P( D | E ) a 500 2.5 200 b P( D | E ) P( D | E ) c 400 1.333 odds(D | E–) = P( D | E ) d 300 Cohort studies FUTURE D+ vs. D– ? where a, b, c, d are the numbers of individuals in each cell. 1400 n 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 1.333 times greater than the probability of not developing disease. odds ( D | E ) a d (500)(300) 1.875 odds ( D | E –) b c (200)(400) The odds of disease among those exposed are 1.875 times greater than the odds of disease among those not exposed. Case-Control studies PRESENT PAST cases E+ vs. E– ? Cohort studies FUTURE ref gp controls D+ vs. D– E+ vs. E– Example: D+ D– E+ a b a+b E– c d c+d a+c b+d Cohort studies Why not just use D+ vs. D– ? n odds ( D | E ) a d 1.875 odds ( D | E –) b c P( D | E ) ??? P ( D | E ) where a, b, c, d are the numbers of individuals in each cell. The odds of disease among exposed are 1.875 times greater than the odds of disease among not exposed. Case-Control studies PRESENT PAST cases E+ vs. E– ? Cohort studies FUTURE ref gp controls D+ vs. D– E+ vs. E– Example: D+ D– E+ 500 a 200 b a700 +b E– 400 c 300 d c700 +d a900 + c b500 +d Cohort studies Case-Control studies D+ vs. D– ? 1400 n odds ( D | E ) a d 1.875 odds ( D | E –) b c P( D | E ) a (c d ) 1.25 P( D | E ) c (a b) (HW problem) where a, b, c, d are the numbers of individuals in each cell. odds ( E | D) a d 1.875 odds ( E | D –) b c The odds of disease among exposed are 1.875 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. The odds of exposure among diseased are 1.875 times greater than the odds of exposure among not diseased. Case-Control studies PRESENT PAST cases E+ vs. E– ? Cohort studies ref gp controls D+ vs. D– E+ vs. E– Example: D+ D– E+ a b a+b E– c d c+d a+c b+d Cohort studies odds ( D | E ) a d odds ( D | E –) b c P( D | E ) a (c d ) P( D | E ) c (a b) Case-Control studies odds ( E | D) a d odds ( E | D –) b c FUTURE D+ vs. D– ? where a, b, c, d are the numbers of individuals in each cell. n 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… a is small relative to b, and c is small relative to d… then RR ≈ OR.