Slide 1Slide 2Slide 3Overview of Biostatistical MethodsOverview of Biostatistical MethodsSlide 6Slide 7Slide 8Slide 9Slide 10CHAPTER 3Probability Theory•3.1 - Basic Definitions and Properties•3.2 - Conditional Probability and Independence•3.3 - Bayes’ Formula•3.4 - Applications (biomedical)Examples of Screening Tests for Early Detection of Colorectal CancerHighly 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” ProcedureVery highly sensitive, routinely administered by health insurance companiesMore on Sensitivity / SpecificitySome Additional Biomedical ApplicationsOverview of Biostatistical MethodsCase-Control studiesCase-Control studiesCohort studiesCohort studiesE+ vs. E– Overview of Biostatistical MethodsObservational 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 / SmokingPRESENTE+ vs. E– ? D+ vs. D– ? Case-Control studiesCase-Control studiesCohort studiesCohort studiesBoth types of study yield a 22 “contingency table” of data: D+ D–E+a b a + bE–c d c + da + c b + d n 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. casescontrols reference groupD+ D–E+a b a + bE–c d c + da + c b + d nwhere a, b, c, d are the numbers of individuals in each cell. E+ vs. E– PRESENTE+ vs. E– ? D+ vs. D– ? Case-Control studiesCase-Control studiesCohort studiesCohort studiesD+ vs. D– FUTUREPASTcasescontrolsCohort studies Cohort studies “Odds of Disease, given Exposed” = odds(D | E+) =( | )( | )P D EP D E+ +- +/ ( )/ ( )a a bb a b+�+ab=“Odds of Disease, given Not Exposed” = odds(D | E–) =( | )( | )P D EP D E+ -- -/ ( )/ ( )c c dd c d+�+cd=“ODDS RATIO” OR ( | )( | –)D ED E+=oddsodds//a bc d� =a db c= 1 No assoc; D, E stat indep< 1 possible protective factor> 1 possible risk factorref gpD+ D–E+a b a + bE–c d c + da + c b + d nwhere a, b, c, d are the numbers of individuals in each cell. E+ vs. E– PRESENTE+ vs. E– ? D+ vs. D– ? Case-Control studiesCase-Control studiesCohort studiesCohort studiesD+ vs. D– FUTUREPASTcasescontrolsodds(D | E+) =( | )( | )P D EP D E+ +- +ab�odds(D | E–) =( | )( | )P D EP D E+ -- -cd�OR ( | )( | –)D ED E+=oddsoddsa db c�ref gpD+ D–E+500 200 700E–400 300 700900 500 1400Example:500200= =2.5400300= =1.333(500)(300)(200)(400)= =1.875Among 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.The odds of disease among those exposed are 1.875 times greater than the odds of disease among those not exposed.Cohort studies Cohort studiesD+ D–E+a b a + bE–c d c + da + c b + d n“ODDS RATIO” OR where a, b, c, d are the numbers of individuals in each cell. E+ vs. E– PRESENTE+ vs. E– ? D+ vs. D– ? Case-Control studiesCase-Control studiesCohort studiesCohort studiesD+ vs. D– FUTUREPASTcasescontrolsWhy not just use ???( | )( | )P D EP D E+ ++ -ref gpExample:( | )( | –)D ED E+=oddsoddsa db c�=1.875The odds of disease among exposed are 1.875 times greater than the odds of disease among not exposed.Cohort studies Cohort studiesD+ D–E+a b a + bE–c d c + da + c b + d nwhere a, b, c, d are the numbers of individuals in each cell. E+ vs. E– PRESENTE+ vs. E– ? D+ vs. D– ? Case-Control studiesCase-Control studiesCohort studiesCohort studiesD+ vs. D– FUTUREPASTcasescontrols( | )( | )P D EP D E+ ++ -ref gpD+ D–E+500 200 700E–400 300 700900 500 1400Example:The odds of disease among exposed are 1.875 times greater than the odds of disease among not exposed.( | )( | –)D ED E+=oddsoddsa db c�=1.875“ODDS RATIO” OR ( )( )a c dc a b+�+=1.25“RELATIVE RISK” RR The probability of disease among exposed is 1.25 times greater than the probability of disease among not exposed.Case-Control studies Case-Control studies ( | )( | –)E DE D+=oddsoddsa db c�=1.875“ODDS RATIO” OR The odds of exposure among diseased are 1.875 times greater than the odds of exposure among not diseased.(HW problem)Cohort studies Cohort studiesD+ D–E+a b a + bE–c d c + da + c b + d nwhere a, b, c, d are the numbers of individuals in each cell. E+ vs. E– PRESENTE+ vs. E– ? D+ vs. D– ? Case-Control studiesCase-Control studiesCohort studiesCohort studiesD+ vs. D– FUTUREPASTcasescontrols( | )( | )P D EP D E+ ++ -ref gpExample:( | )( | –)D ED E+=oddsoddsa db c�“ODDS RATIO” OR ( )( )a c dc a b+�+“RELATIVE RISK” RR Case-Control studies Case-Control studies ( | )( | –)E DE D+=oddsoddsa db c�“ODDS RATIO” OR 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…( )( )a c dc a b+/�+/Cohort studies Cohort