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CHAPTER 3 Probability Theory

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.

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CHAPTER 3 Probability Theory

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  1. CHAPTER 3Probability Theory 3.1 - Basic Definitions and Properties 3.2 - Conditional Probability and Independence 3.3 - Bayes’ Formula 3.4 - Applications (biomedical)

  2. More on Sensitivity / Specificity 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 Veryhighlysensitive, routinely administered by health insurance companies

  3. Some Additional Biomedical Applications

  4. Overview of Biostatistical Methods Cohort studies Case-Control studies

  5. Overview of Biostatistical Methods Observational study designs that test for a statistically significantassociation 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 cases controls reference group E+ vs. E– ? D+ vs. D– E+ vs. E– D+ vs. D– ? PAST • relatively easy and inexpensive • subject to faulty records, “recall bias” • measures direct effect of E on D • expensive, extremely lengthy… FUTURE Both types of study yield a 22 “contingency table” of data: where a, b, c, d are the numbers of individuals in each cell.

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

  7. ref gp Case-Control studies Cohort studies PRESENT where a, b, c, d are the numbers of individuals in each cell. Example: controls cases E+ vs. E– ? D+ vs. D– E+ vs. E– D+ vs. D– ? PAST Cohort studies Among those exposed, the probability of developing disease is 2.5 times greater than the probability of not developing disease. odds(D | E+) = FUTURE Among those not exposed, the probability of developing disease is 1.333 times greater than the probability of not developing disease. odds(D | E–) = The odds of disease among those exposed are 1.875 times greater than the odds of disease among those not exposed. OR

  8. ref gp Case-Control studies Cohort studies PRESENT where a, b, c, d are the numbers of individuals in each cell. Example: controls cases E+ vs. E– ? D+ vs. D– E+ vs. E– D+ vs. D– ? PAST Cohort studies The odds of disease among exposed are 1.875 times greater than the odds of disease among not exposed. “ODDS RATIO”OR FUTURE Why not just use ???

  9. ref gp Case-Control studies Cohort studies PRESENT where a, b, c, d are the numbers of individuals in each cell. Example: controls cases E+ vs. E– ? D+ vs. D– E+ vs. E– D+ vs. D– ? PAST Cohort studies The odds of disease among exposed are 1.875 times greater than the odds of disease among not exposed. “ODDS RATIO”OR FUTURE The probability of disease among exposed is 1.25 times greater than the probability of disease among not exposed. “RELATIVE RISK”RR Case-Controlstudies (HW problem) The odds of exposure among diseased are 1.875 times greater than the odds of exposure among not diseased. “ODDS RATIO”OR

  10. ref gp Case-Control studies Cohort studies PRESENT where a, b, c, d are the numbers of individuals in each cell. Example: controls cases E+ vs. E– ? D+ vs. D– E+ vs. E– D+ vs. D– ? PAST Cohort 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… “ODDS RATIO”OR FUTURE “RELATIVE RISK”RR if the disease is rare in the popul’n… Case-Controlstudies a is small relative to b, and c is small relative to d… thenRR ≈ OR. “ODDS RATIO”OR

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