Case-control studies C&H 16 Bendix Carstensen Steno Diabetes - PDF document

Case-control studies C&H 16 Bendix Carstensen Steno Diabetes Center & Department of Biostatistics, University of Copenhagen bxc@steno.dk http://BendixCarstensen.com PhD-course in Epidemiology, Department of Biostatistics, Tuesday 31 January 2017 Relationship between follow–up studies and case–control studies In a cohort study , the relationship between exposure and disease incidence is investigated by following the entire cohort and 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 (C&H 16) 2/ 59 Case-control study In a case-control study the subjects who develop the disease (the cases) are registered by some other mechanism than follow-up, and a group of healthy subjects (the controls) is used to represent the subjects who do not develop the disease. Case-control studies (C&H 16) 3/ 59

Rationale behind case-control studies ◮ In a follow-up study, rates among exposed and non-exposed are estimated by: D 1 D 0 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 (C&H 16) 4/ 59 ◮ In a case-control study we use the same cases, but select controls to represent the distribution of risk time between exposed and unexposed: H 1 ≈ Y 1 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 (C&H 16) 5/ 59 Choice of controls (I) Failures s Healthy 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 (C&H 16) 6/ 59

What about censoring and late entry? Failures s Healthy 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 (C&H 16) 7/ 59 Choice of controls (II) Failures s Healthy Censored 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 (C&H 16) 8/ 59 Most common way of choosing controls. Case-control probability tree Exposure Failure Selection Probability Case ✟✟✟ 0 . 97 pπ 1 × 0 . 97 ( D 1 ) ❍❍❍ F π 1 ✑✑✑ 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 (C&H 16) 9/ 59

Retrospective analysis of case-control studies Retrospective: Compare the distribution of exposure between cases and controls. ◮ The proportion of cases who smoke compared to controls ◮ The mean age of cases compared to controls Looks at the study backwards. Only works properly for binary explanatory variables. Case-control studies (C&H 16) 10/ 59 The retrospective argument Selection Failure Exposure Probability ✟✟✟✟ p × π 1 × 0 . 97 E 1 (Cases) F ❍❍❍❍ � � E 0 (1 − p ) × π 0 × 0 . 97 � � � ❅ � ❅ ✟✟✟✟ E 1 p × (1 − π 1 ) × 0 . 01 � ❅ � ❅ (Controls) S ❍❍❍❍ � ❅ E 0 (1 − p ) × (1 − π 0 ) × 0 . 01 ❅ ❅ ❅ ❅ Not in study Note: Parameters in the previous tree not on these branches. Case-control studies (C&H 16) 11/ 59 Odds of exposure for cases and controls: p × π 1 × 0 . 97 1 − p × π 1 p Ω cas = (1 − p ) × π 0 × 0 . 97 = π 0 p × (1 − π 1 ) × 0 . 01 1 − p × 1 − π 1 p Ω ctr = (1 − p ) × (1 − π 0 ) × 0 . 01 = 1 − π 0 Odds-ratio for exposure between cases and controls: � 1 − π 1 Ω cas = π 1 = OR( disease ) population Ω ctr π 0 1 − π 0 Case-control studies (C&H 16) 12/ 59

Prospective analysis of case-control studies Compare the case/control ratio between exposed and non-exposed subjects — or more general: How does case-control ratio vary with exposure ? The point is that in the study it varies in the same way as in the population. Case-control studies (C&H 16) 13/ 59 The prospective argument Selection Exposure Failure Probability ✟✟✟✟ p × π 1 × 0 . 97 π 1 F E 1 ❍❍❍❍ p � � p × (1 − π 1 ) × 0 . 01 S 1 − π 1 � � � ❅ � ❅ ✟✟✟✟ π 0 (1 − p ) × π 0 × 0 . 97 F � ❅ � ❅ 1 − p E 0 ❍❍❍❍ � ❅ (1 − p ) × (1 − π 0 ) × 0 . 01 S 1 − π 0 ❅ ❅ ❅ ❅ Not in study Case-control studies (C&H 16) 14/ 59 P { Case given inclusion } Odds of disease = P { Control given inclusion } p × (1 − π 1 ) × 0 . 01 = 0 . 97 p × π 1 × 0 . 97 π 1 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 (C&H 16) 15/ 59

What is the case-control ratio? � s 1 , cas � D 1 = 0 . 97 π 1 π 1 0 . 01 × = × H 1 1 − π 1 s 1 , ctr 1 − π 1 � s 0 , cas � D 0 = 0 . 97 π 0 π 0 0 . 01 × = × H 0 1 − π 0 s 0 , ctr 1 − π 0 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 (C&H 16) 16/ 59 Log-likelihood for case-control studies Log-Likelihood (conditional on being included) is a binomial likelihood with odds-parameters ω 0 and ω 1 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 (C&H 16) 17/ 59 Odds-ratio ( θ ) is the ratio of the odds ω 1 to ω 0 , so: � ω 1 � log( θ ) = log = log( ω 1 ) − log( ω 0 ) ω 0 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 (C&H 16) 18/ 59

The standard errors of the odds are estimated by: � � 1 + 1 1 + 1 and D 1 H 1 D 0 H 0 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 (C&H 16) 19/ 59 Confidence interval for OR First a confidence interval for log(OR) : � 1 + 1 + 1 + 1 log(OR) ± 1 . 96 × 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 (C&H 16) 20/ 59 BCG vaccination and leprosy Does BCG vaccination in early childhood protect against leprosy? New cases of leprosy were examined for presence or absence of the BCG scar. During the same period, a 100% survey of the 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 (C&H 16) 21/ 59

Exercise I BCG scar Leprosy cases Population survey Present 101 46 028 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 (C&H 16) 22/ 59 Solution to I OR = D 1 /H 1 = 101 / 46028 159 / 34594 = 0 . 002194 0 . 004596 = 0 . 48 D 0 /H 0 � 1 1 1 1 s.e.(log[OR]) = D 1 + H 1 + D 0 + 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 (C&H 16) 23/ 59 Exercise II BCG scar Leprosy cases Population controls Present 101 554 Absent 159 446 The table shows the results of a computer-simulated study which picked 1000 controls at random. What is the odds ratio estimate in this study? Give a 95% c.i. for the OR . Use SAS for this: Exercise from the notes. Case-control studies (C&H 16) 24/ 59