Case-control studies
• Overview of different types of studies • Review of general procedures • Sampling of controls – implications for measures of association – implications for bias • Logistic regression modeling
Learning Objectives • To understand how the type of control sampling relates to the measures of association that can be estimated • To understand the differences between the nested case-control study and the case-cohort design and the advantages and disadvantages of these designs • To understand the basic procedures for logistic regression modeling
Overview of types of case-control studies
No designated cohort, Within a designated cohort but should treat source population as cohort Nested case control Case-cohort
Cases only Cases and controls 1) Type of sampling - incidence density Case-crossover - cumulative “epidemic” 2) Source of controls - population-based - hospital - neighborhood - friends -family
1 Review of General Procedures
Obtaining cases Obtaining controls 1) Define target cases 1) Define target controls 2) Identify potential cases 2) Define mechanism for 3) Confirm diagnosis identifying controls 4) *Obtain physician’s consent 3) Contact control 5) Contact case 4) Confirm control’s eligibility 6) Confirm case’s eligibility 5) *Obtain control’s consent 7) *Obtain case’s consent 6) Obtain exposure data 8) Obtain exposure data
*need to account for nonresponse
Pros/Cons of Different Methods
Collection of information
In-person Telephone Mail - hospital - clinic - home
Case cohort study
T0 Additional cases, Assemble compare all cases to the cohort, sample First case subcohort selected subcohort to be at T0 used in all future analyses
2 Nested case-control Study
T0 Assemble Likewise, select controls cohort again at each separate time point; First case; note: each of these randomly cases was eligible to be a control for the select control first case from remaining cohort
Crossover designs
Period the subject is exposed
Period the subject is unexposed • Case-crossover: variation of crossover; case has a pre-disease period which is used as the control period • Good method for control of confounding • May have limited applications • Assumes that neither exposure nor confounders are changing over time in a systematic way
Source Population
• Think of all case-control studies as nested within a cohort, even when the cohort is not designated • Source population is this underlying cohort • Source population describes the cohort giving rise to the cases; controls are also from this source population • Source population reflects the disease under study, difficulties in diagnosing disease, routine procedures for recording the disease occurrence, and the frequency of disease
3 Classic case-control
Source population
Cases Controls Sampling fraction (f1) Sampling fraction f2
Exposed Non-exposed Exposed Nonexposed (A) (C) (B) (D) Cases Controls Odds ratio = AD/BC Exposed A B If sampling: OR = f1*A*f2*D = AD Unexposed C D f1*C*f2*B BC
Incidence Density Sampling
1) Collect information on each case
2) Collect control at the same time each case is observed; collect control from the underlying source population giving rise to the cases
3) Controls can be cases
Cumulative-based sampling 1) Start after the event has happened
2) Ascertain cases
3) Collect controls from the noncases after event (e.g., epidemic)
4 Measures of Association
• Key concept, sampling fraction, independent of exposure • Incidence density sampling/nested case-control studies – if exposure odds in controls(B/D) approximates person-time ratio for source population, odds ratio will approximate rate ratio – without rare disease assumption
Measures of Association
• Case cohort – if ratio B/D approximates overall prevalence of exposure in source population, odds ratio will approximate risk ratio – without rare disease assumption • Cumulative “Epidemic” case-control studies – odds ratio will approximate rate ratio if proportion diseased in each exposure group is low (< 20%) and remains steady during study period
When is the rare disease assumption needed? • Cumulative-based sampling if want to approximate the relative risk • Otherwise – nested case-control and incidence density sampling will give an estimate of rate ratio without rare disease assumption – case-cohort will give an estimate of risk ratio without rare disease assumption • If disease is rare, all three measures will be very close
5 A closer look Recall: Note: in R&G notation A1 I T A T A1= A 1 1 1 o IR = = = Ao= C I A0 A T 0 0 1 B1= B T0 B0= D A1 A1 (1− A ) B A B OR = 1 = 1 = 1 o A A 0 0 A0 B1 (1− A0 ) B0 If sampling does not lead to bias and the sample can approximate the person-years of distribution, a case- control study is a more efficient design (i.e. for same number of people, more precision). However not as precise as if you use the full cohort.
Pseudo Rates B B This assumes that the sampling f = 1 = 0 = r 2 T T rates are the same for the 1 0 exposed and unexposed A1 Pseudo-rate1 = B1
A0 if r is unknown : Pseudo-rate0 = B 0 A1 T1 A1 1 A1 A1 * If r is known then B B T T r T 1 = 1 1 = 1 = 1 multiplyby r to get rate A0 A T A0 1 A0 0 0 * B0 T r T A1 A1 B1 B0 T0 o o *r = = I1 B1 B1 T1
Pseudo Risks This assumes that the sampling B1 B0 rates are the same for the exposed f2 = = = f N1 N0 and unexposed A Pseudo- risk = 1 1 If f is unknown : B1 A0 A1 N1 A1 1 A1 A1 Pseudo- risk0 = * B B B N N f N 0 1 = 1 1 = 1 = 1 A0 A N A0 1 A0 If f is known then multiply 0 0 * B0 N f N by f to get incidence proportion B0 N0 0 0
A1 A1 B1 * f = = R1 B1 B1 N1
6 Summary • Odds Ratio • Odds Ratio approximates Rate approximates Risk Ratio Ratio – incidence density sampling – case-cohort design – nested case- control studies – cumulative – cumulative sampling if sampling is disease is rare proportion exposed is steady and relatively low
Source of Controls population-based: not necessarily equal probability, control selection probability is proportional to the individual’s person time at risk neighborhood: need to think about referral patterns (e.g., not good for veteran’s hosp); also need to worry about overmatching hospital: need to be especially concerned that sampling was independent of exposure; try to use a variety of diagnoses for controls. friends: main problem is with overmatching; cedes the control selection to the case rather than to the investigator family:may be worthwhile design to control for certain variable (e.g., spouse control, environment, twin control genetics and environment)
Review of control selection
• Select from source population • Select independent of exposure status • Probability of selection should be proportion to amount s/he would have contributed to person-time in the denominator • Not eligible to be a control, if during same time would not have been eligible to be a case
7 Ille-et-Vilaine study (epidat1.txt) • conducted between January 1972 and April 1974 • French department of Ille-et-Vilaine (Brittany) • Men diagnosed in regional hospitals • Controls sampled from electoral lists in each commune of department
Other methodological points • exposure opportunity – interest is in the fact of exposure – think about cohort design – make same exclusions to cases as to controls • comparability of information – may not want comparability if errors are not independent • number of different types of control groups – may be appropriate in some situations (e.g., spouse, siblings), but generally not recommended • prior disease history
Regression Modeling
Regression model- simpler function used to estimate the true regression function
Benefit of Regression Modeling: overcome the numerical limitations of categorical analysis
Cost of Regression Modeling: assumptions of model; invalid inferences if model is misspecified
E(Y|X = x)
Y = dependent variable, outcome variable, regressand X = set of independent variables, predictors, covariates, regressors
8 Logistic Regression
Y When x = 1, log = α + β x 1−Y eα +β x α +β x α +β x e Y 1+ e α +β x Y = = R(x1) = = e α +β x 1−Y 1+ eα +β x eα +β x 1+ e − When x = 0, 1+ eα +β x 1+ eα +β x eα Y log | x =1 α +β x Y α 1−Y e = 1+ e = eα = log = β 1−Y 1+ eα eα Y eα − log | x = 0 1+ eα 1+ eα 1−Y
R(x1) Logistic risk model, bounded by 0 and 1
Likelihood in logistic regression
α+∑ β x α+∑ β x e i i e i i (1− ) ∏ α+∑ βixi ∏ α+∑ βixi 1+ e 1+ e Cases Controls
Interpreting Coefficients Y log( ) = α + β x 1−Y if X = 1 if exposed, 0 if unexposed Y log( ) | x =1 α + β 1−Y = = β Y log( ) | x = 0 α 1−Y OR = eβ If X is continuous, B represents the impact of a one-unit change in X to Y, the exponential of B will give you the OR for a one unit change Note: both modeling variables as either ordered categorical variables or continuous variables assumes the one unit change in X is the same irrespective of the level of X
9 Logistic Modeling
Linear Model Logistic Model Log ( Y) Y $ (1-Y) $ " "
Not bounded by 0 and 1 Y = Probability of Disease Y = Odds of Disease 1- Y Y log( ) = Log odds of Disease 1- Y
Assessing Linearity
• Create categories of continuous variable; categories should represent the same amount of units (e.g., 0-39 grams, 40-79 grams, 80-119, etc) • Plot Beta coefficients; if pattern is approximately linear, keep variable as a continuous variable in model
Logistic Model
Log ( Y) * * (1-Y) * * * * * *
$1 $2 $3 $4
10 Creating Indicator Variables Also referred to as categorical regressors, dummy variables
Need to pick a reference level
Number of indicator variables created = # categories - 1
The intercept term picks of the effect of the left out category
Note: The precision of the estimate will depend on which category you pick as the referent
Things to keep in mind
Note: For some variables, nonusers/nonconsumers are different than users/consumers, therefore may want to keep separate zero category
Last category may be too heterogeneous
Balance between homogeneity and parsimony
Changing units
One Unit Increase X unit increase (e.g., X=10)
Point esimate eb eb*10 b−1.96*se(b) (b*10)−1.96*se(b)*10 Lower 95% CL e e b+1.96*se(b) (b*10)+1.96*se(b)*10 Upper 95% CL e e
Note: can also rescale, see pg 371 of R&G
11 Examining confounding and mediation 1) Run a model with exposure only
2) Run a model with exposure and potential confounder (and/or mediator)
3) Examine the changes in the estimate with exposure between model 1 and 2 - if different keep in confounder - many use a 10% change in estimate - if estimate is different but B is still associated with outcome -- partial confounder - data cannot distinguish b/w confounder and mediator
Comparing Models y Model B: ln = α + βx 1 − y y Model A: ln = α + βx + γz 1 − y If model B is no worse than model A, we want to use model B. Always favor parsimony if we don’t have to give up anything
One way of comparing models is using the log likelihood ratio test
Likelihood Ratio Tests
likelihood for 'A' (larger model) − 2log LR = −2log likelihood for 'B' (smaller model)
~ Chi Square with df = (total number of parameters in larger model - total number of parameters in smaller model)
12 Comparing models:
Subtract the log likelihoods (part of SAS output)
Can use following code or look up in a Chi Square table: data; p = (1-probchi(diff, df)); put p; run; diff = difference in -2 *log likelihoods df = degrees of freedom see above
13