Epidemiology for PhD students Case-control studies SAS-intro Bendix Carstensen Steno Diabetes Center Copenhagen Gentofte, Denmark http://BendixCarstensen.com/EpiPhD/F2017 Department of Biostatistics, University of Copenhagen, Spring 2017 http://BendixCarstensen.com/EpiPhD/F2017 Monday 29 th January, 2018, 16:34 From /home/bendix/teach/Epi/KU-epi/slides/slides.tex 1/ 43 Case-control studies Tuesday 30 January 2018 Epidemiology for PhD students Department of Biostatistics, University of Copenhagen, Spring 2017 http://BendixCarstensen.com/EpiPhD/F2017 cc-lik Epidemiology Relationship between follow–up studies and for PhD students case–control studies Bendix Carstensen In a cohort study , the relationship between exposure and disease Case-control studies incidence is investigated by following the entire cohort and SAS-intro measuring the rate of occurrence of new cases in the different exposure groups. The follow–up allows the investigator to register those subjects who develop the disease during the study period and to identify those who remain free of the disease. Case-control studies ( cc-lik ) 2/ 43
Epidemiology Case-control study for PhD students Bendix In a case-control study the subjects who develop the disease Carstensen (the cases) are registered by some other mechanism than Case-control studies follow-up, and a group of healthy subjects (the controls) is used SAS-intro to represent the subjects who do not develop the disease. Case-control studies ( cc-lik ) 3/ 43 Epidemiology Rationale behind case-control studies for PhD students Bendix ◮ In a follow-up study, rates among exposed and non-exposed Carstensen are estimated by: Case-control studies D 1 D 0 SAS-intro Y 1 Y 0 ◮ and hence the rate ratio by: � D 0 � Y 1 D 1 = D 1 Y 1 Y 0 D 0 Y 0 Case-control studies ( cc-lik ) 4/ 43 Epidemiology ◮ In a case-control study we use the same cases, but select for PhD students controls to represent the distribution of risk time between Bendix Carstensen exposed and unexposed: Case-control studies H 1 ≈ Y 1 SAS-intro H 0 Y 0 ◮ Therefore the rate ratio is estimated by: � H 1 D 1 D 0 H 0 ◮ Controls represent risk time, not disease-free persons. Case-control studies ( cc-lik ) 5/ 43
Epidemiology Choice of controls (I) for PhD students Bendix Carstensen Failures s Case-control Healthy studies SAS-intro study period The period over which failures are registered as cases is called the study period. A group of subjects who remain healthy over the study period is chosen to represent the healthy part of the source population. — but this is an oversimplification. . . Case-control studies ( cc-lik ) 6/ 43 Epidemiology What about censoring and late entry? for PhD students Bendix Failures Carstensen s Case-control Healthy studies SAS-intro Censored Late entry study period Choosing controls which remains healthy throughout takes no account of censoring or late entry. Instead, choose controls who are in the study and healthy, at the times the cases are registered. Case-control studies ( cc-lik ) 7/ 43 Epidemiology Choice of controls (II) for PhD students Failures Bendix s Carstensen Healthy Case-control studies Censored SAS-intro Late entry study period This is called incidence density sampling . Subjects can be chosen as controls more than once, and a subject who is chosen as a control can later become a case. Equivalent to sampling observation time from vertical bands drawn to enclose each case. Case-control studies ( cc-lik ) 8/ 43
Epidemiology Case-control probability tree for PhD students Exposure Failure Selection Probability Bendix Carstensen Case ✟✟✟ Case-control 0 . 97 p π 1 × 0 . 97 studies ( D 1 ) F ❍❍❍ ✑✑✑ π 1 SAS-intro 0 . 03 E 1 ◗◗◗ Control p ✟✟✟ � 0 . 01 � p (1 − π 1 ) × 0 . 01 ( H 1 ) � 1 − π 1 ❍❍❍ S � 0 . 99 � Case ❅ ✟✟✟ 0 . 97 (1 − p ) π 0 × 0 . 97 ❅ ( D 0 ) F ❍❍❍ ❅ ✑✑✑ π 0 ❅ ❅ 1 − p 0 . 03 E 0 ◗◗◗ Control ✟✟✟ 0 . 01 (1 − p )(1 − π 0 ) × 0 . 01 ( H 0 ) 1 − π 0 S ❍❍❍ 0 . 99 Case-control studies ( cc-lik ) 9/ 43 Epidemiology Prospective analysis of case-control studies for PhD students Bendix Compare the case/control ratio between exposed and Carstensen non-exposed subjects — or more general: Case-control studies How does case-control ratio vary with exposure ? SAS-intro The point is that in the study it varies in the same way as in the population. Case-control studies ( cc-lik ) 10/ 43 Epidemiology The prospective argument for PhD students Selection Exposure Failure Probability Bendix Carstensen ✟✟✟✟ p × π 1 × 0 . 97 F Case-control π 1 studies E 1 ❍❍❍❍ p SAS-intro � � p × (1 − π 1 ) × 0 . 01 S 1 − π 1 � � � ❅ � ❅ ✟✟✟✟ (1 − p ) × π 0 × 0 . 97 F π 0 � ❅ � ❅ 1 − p E 0 ❍❍❍❍ � ❅ (1 − p ) × (1 − π 0 ) × 0 . 01 S 1 − π 0 ❅ ❅ ❅ ❅ Not in study Case-control studies ( cc-lik ) 11/ 43
Epidemiology for PhD P { Case given inclusion } students Odds of disease = Bendix Carstensen P { Control given inclusion } Case-control studies p × (1 − π 1 ) × 0 . 01 = 0 . 97 p × π 1 × 0 . 97 π 1 SAS-intro ω 1 = 0 . 01 × 1 − π 1 (1 − p ) × (1 − π 0 ) × 0 . 01 = 0 . 97 (1 − p ) × π 0 × 0 . 97 π 0 ω 0 = 0 . 01 × 1 − π 0 � OR = ω 1 π 1 π 0 = = OR( disease ) population ω 0 1 − π 1 1 − π 0 Case-control studies ( cc-lik ) 12/ 43 Epidemiology What is the case-control ratio? for PhD students � s 1 , cas � Bendix = 0 . 97 π 1 π 1 D 1 Carstensen 0 . 01 × = × 1 − π 1 1 − π 1 H 1 s 1 , ctr Case-control studies � s 0 , cas � SAS-intro = 0 . 97 π 0 π 0 D 0 0 . 01 × = × 1 − π 0 1 − π 0 H 0 s 0 , ctr D 1 / H 1 = π 1 / (1 − π 1 ) π 0 / (1 − π 0 ) = OR population D 0 / H 0 — but only if the sampling fractions are identical: s 1 , cas = s 0 , cas and s 1 , ctr = s 0 , ctr . Case-control studies ( cc-lik ) 13/ 43 Epidemiology Log-likelihood for case-control studies for PhD students Bendix Log-Likelihood (conditional on being included) is a binomial Carstensen likelihood with odds-parameters ω 0 and ω 1 Case-control studies SAS-intro D 0 log( ω 0 ) − N 0 log(1 + ω 0 ) + D 1 log( ω 1 ) − N 1 log(1 + ω 1 ) where N 0 = D 0 + H 0 and N 1 = D 1 + H 1 . Exposed: D 1 cases, H 1 controls Unexposed: D 0 cases, H 0 controls Case-control studies ( cc-lik ) 14/ 43
Epidemiology Odds-ratio ( θ ) is the ratio of the odds ω 1 to ω 0 , so: for PhD students Bendix � ω 1 � Carstensen log( θ ) = log = log( ω 1 ) − log( ω 0 ) Case-control ω 0 studies SAS-intro Estimates of log( ω 1 ) and log( ω 0 ) are just the empirical odds: � D 1 � � D 0 � log and log H 1 H 0 Case-control studies ( cc-lik ) 15/ 43 Epidemiology The standard errors of the odds are estimated by: for PhD students Bendix � � Carstensen 1 + 1 1 + 1 and Case-control D 1 H 1 D 0 H 0 studies SAS-intro Exposed and unexposed form two independent bodies of data (they are sampled independently), so the estimate of log( θ ) [= log(OR)] is: � D 1 � � D 0 � log − log , H 1 H 0 � 1 + 1 + 1 + 1 � � with s.e. log(OR) = D 1 H 1 D 0 H 0 Case-control studies ( cc-lik ) 16/ 43 Epidemiology Confidence interval for OR for PhD students Bendix First a confidence interval for log(OR) : Carstensen Case-control � studies 1 + 1 + 1 + 1 log(OR) ± 1 . 96 × SAS-intro D 1 H 1 D 0 H 0 Take the exponential: � � � 1 + 1 + 1 + 1 × OR ÷ exp 1 . 96 × D 1 H 1 D 0 H 0 � �� � error factor Case-control studies ( cc-lik ) 17/ 43
Epidemiology BCG vaccination and leprosy for PhD students Bendix Does BCG vaccination in early childhood protect against leprosy? Carstensen New cases of leprosy were examined for presence or absence of Case-control studies the BCG scar. During the same period, a 100% survey of the SAS-intro population of this area, which included examination for BCG scar, had been carried out. The tabulated data refer only to subjects under 35, because vaccination was not widely available when older persons were children. Case-control studies ( cc-lik ) 18/ 43 Epidemiology Exercise I for PhD students Bendix Carstensen BCG scar Leprosy cases Population survey Case-control studies Present 101 46 028 SAS-intro Absent 159 34 594 Estimate the odds of BCG vaccination for leprosy cases and for the controls. Estimate the odds ratio and hence the extent of protection against leprosy afforded by vaccination. Give a 95% c.i. for the OR . Use SAS for this: Exercise from the notes. Case-control studies ( cc-lik ) 19/ 43 Epidemiology Solution to I for PhD students Bendix Carstensen OR = D 1 / H 1 = 101 / 46028 159 / 34594 = 0 . 002194 0 . 004596 = 0 . 48 Case-control studies D 0 / H 0 SAS-intro � D 1 + 1 1 H 1 + 1 D 0 + 1 s.e.(log[OR]) = H 0 � 1 1 1 1 = 101 + 46028 + 159 + 34594 = 0 . 127 The 95% limits for the odds-ratio are: × × OR ÷ exp(1 . 96 × 0 . 127) = 0 . 48 ÷ 1 . 28 = (0 . 37 , 0 . 61) Case-control studies ( cc-lik ) 20/ 43
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