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Case-Control Studies
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Case Control Studies
Readings
Fletcher, chapter 10
Walker, chapter 6 [Case-Control Studies] from
Observation and Inference, 1991 [course pack]
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Objectives
Students will be able to:
1. Define the term case-control study
2. Explain the relationship between case-controland cohort studies
3. Understand the difference between
cumulative incidence and incidence density designs
Case-Control Studies - Slide 1
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Objectives
4. Calculate parameters which may be validly obtainedfrom case-control studies, namely:
a. Odds parameters:
- odds of exposure in cases
- odds of exposure in controls
- odds ratio
b. Risk parameters:
- approximation of relative risk
- attributable fraction
c. Incidence rate parameters:
- incidence rate ratio
- attributable fraction among the exposed
- attributable fraction for the population
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Objectives
5. Indicate situations in which case-control studies
permit estimation of rate differences between
exposure groups
6. Highlight advantages and disadvantages ofcase-control studies, including key biases
7. List possible sources of controls in
case-control studies
8. Identify biases which may result from
different types of control selection
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Case-Control Studies
Fletcher, p. 213:
Patients who have the disease and a group of
otherwise similar people who do not have the
disease are selected. The researchers then look
backward in time to determine the frequency of
exposure in the two groups.
In other words, a study population is first assembled
based on a determination as to whether subjects
have or have not developed an outcome of interest.
Subjects (or person-time) are then classified as to
whether an exposure of interest took place.
Data on other variables (e.g. potential confounders)
is also obtained.
Case Control Studies- Slide 4
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Walker, 1991:
Case-control studies constitute the
major advance in epidemiologic methods
of our time
Classic example:
Doll & Hill, relationship between lung cancer
and cigarette smoking (1950)
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Advantages
Useful for study of conditions that are rareand/or characterized by a long latency
between exposure(s) and outcomes of interest.
May be useful in evaluating the impact of
multiple types of exposure.
Disadvantages
May be particularly vulnerable to biases arising from
selection of subjects (most often of the control group),
and measurement (estimation) of exposure
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In case-control studies, data about exposure status is
calculated after first determining outcome status.
However, subjects may be recruited prospectively
(concurrently), e.g.:
- All persons aged 30-50 who are diagnosed withhypertension on the island of Montreal during 2005,
within 2 weeks of diagnosis.
- Controls recruited among persons of the same age
who are newly diagnosed with appendicitis inMontreal during the same time period.
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Often, outcome status is already available for all subjects
(historical) at the time of initiation, e.g.:
- During 2005, a researcher identifies all women
aged 40-50 who were diagnosed with breast cancer
on the island of Montreal in 2004.
- In 2005, she recruits a control group among
women of the same age who had negative
screening mammograms in Montreal in 2004.
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Note that the terms
prospective and retrospective
are not very useful
with respect to case-control studies,
since data about exposure statusis always retrospective (by definition).
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Cohort and Case-Control Studies
Every case control study corresponds to an underlying cohort study,
which is (ordinarily) hypothetical.
Example (from Doll & Hill, 1950):_____________________________________________________
Women diagnosed with lung cancer vs other diseases
at 20 London hospitals
Smokers Non-Smokers Total
Lung cancer cases 41 19 60No lung cancer (controls) 28 32 60Total 69 51 120
_________________________________________________________
Crudeodds ratio = odds of exposure in cases/odds of exposure in controls
= (a/b)/(c/d)
= ad/bc = (41x32) / (19x28) = 2.5
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In the corresponding cohort study,
women from the same geographic area
would be recruited and classified as to
smoking status, then followed for the
development vs non-development of lung cancer.
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Case-Control Studies- Slide 12
Assuming all cases of lung cancer during the period of interest
were detected,
one possible 2x2 tablewould be
Smokers Non-Smokers Total
Lung cancer 41 19 60
No lung cancer (controls) 859 981 1,840
Total 900 1000 1,900OR = 2.5
but it could also be:
Smokers Non-Smokers TotalLung cancer 41 19 60
No lung cancer (controls) 70 81 151
Total 111 100 211OR = 2.5
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The cases diagnosed and included, and the
controls sampled, relate to the exposure experienceof an underlying source population.
In each scenario, the estimated odds of cigarette
smoking among cases are 2.5 times thoseamong controls.
In each scenario, all cases of lung cancer were
included. The size of the source population
(and hence the number of non-cases) was varied.
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Cumulative incidence case-control studies
Goal is toderive estimate of relative risks
(relative cumulative incidences)
of outcomes among
exposed vs. unexposed
Design:
- Cases are ascertained during a definedobservation period
- Controls are persons who did not become casesduring the period of observation.
- The underlying cohort is a fixed one(not open or dynamic).
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Doll and Hill, 1950
Assume that the source population was as follows:
900 smokers & 1000 non smokers - followed 5 yearsThen the 2x2 table would be:
Smokers Non-Smokers Total
Cancer + 41 19 60
Cancer - 859 981 1,840
Total 900 1,000 2,000
________________________________________________
Case-Control Studies- Slide 15
Risk of cancer in smokers: 41/900 = 0.046
Risk of cancer in non smokers: 19/1000 = 0.019
Risk ratio: 0.046/0.019 = 2.4
Odds of smoking in women with cancer: 41/19 = 2.2
Odds of smoking in women without cancer: 859/981 = 0.88
Odds ratio = 2.5
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In the corresponding case control study we take 100% of cases, but
sample the controls (60/1840 or 3.3% of all potential controls - those
who happened to be admitted to hospital for some other reason).
Hence the new table is:
Smokers Non smokers Total
Cancer + 100% x 41 = 41 100% x 19 = 19 60
Cancer - 3.3% x 859 = 28 3.3% x 981 = 32 60
Total 69 51 120_________________________________________________________
Case-Control Studies- Slide 16
Risk of cancer in smokers: 41/69 = 0.59 INVALID
Risk of cancer in non smokers: 19/51 = 0.37 INVALID
The risk ratio from this 2x2 table is also invalid
Odds of smoking among cases: 41/19 = 2.2 (as before)
Odds of smoking among controls: 28/32 = 0.88 (as before)
Odds ratio: 2.2/0.88 = 2.5 (as before)
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General Form: Cumulative incidence case-control studies
exposure + exposure -
outcome + a b | total cases
outcome - c d | total controls___________ _____________ |
total exposed total unexposed | total subjects
Odds of exposure in cases = a/b
Odds of exposure in controls = c/d
Odds ratio = odds of exposure in cases = a/b = ad______________________ ___ __odds of exposure in controls c/d bc
but:
Odds of disease among exposed = a/c
Odds of disease among unexposed = b/d
Odds ratio = odds of disease among exposed = a/c = ad___________________________ ___ __odds of disease among unexposed b/d bc
Case-Control Studies- Slide 17
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Risk parameter estimationin cumulative incidence case-control studies:
Recall that relative risk = risk of disease in exposed______________________risk of disease in unexposed
From our 2x2 table, this is: a/(a+c) = a(b+d)_______ ______
b/(b+d) b(a+c)
If the disease is rare,
then a
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In a case-control study, it is then possible to estimate
the attributable risk (fraction) among the exposed,
even if the risk for the population is unknown.
In a cohort study, the attributable risk fraction is:
Rexp
- Runexp__________Rexp
= (Rexp/Runexp) - (Runexp/Runexp)_______________________Rexp/Runexp
= RR-1_____RR
In a case-control study, this is estimated by (OR-1)/OR
Case-Control Studies- Slide 19
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Hence, from Doll and Hill (1950),
the estimated fraction of
lung cancer among female smokers
which is attributable to smoking is:
2.5 -1 = 0.6 or 60%______
2.5
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Incidence Density Case-Control Studies
The incidence density case-control study involves theimplicit comparison of the person-timeexperience
of cases and controls with respect to the exposure(s)
of interest.
The absolute quantity of person-time sampled - and
hence the sampling fraction - is unknown. This is
analogous to the situation with respect to persons in a
cumulative incidence case-control study.
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Hence the underlying (hypothetical) cohort is an openor dynamic one.
Persons considered controls at one point in timemay then become cases; they can then appear twicein the 2x2 table.
For this cohort, the general form of the 2x2 table is:
Case-Control Studies- Slide 22
exposure + exposure -outcome + a bperson-time Pe Po
Where Pe = person-time among exposed
Po = person-time among unexposed
IRe = a/Pe and IRo= b/Po
IRR = aPo____
bPe
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Suppose that all cases are counted, but the
controls are sampled with respect to person-time,
with sampling fraction f generating the incidencedensity case-control study.
Case-Control Studies- Slide 23
Then the 2x2 table is:
exposure + exposure -outcome + a b
outcome - c = fPe d = fPo
Then OR = ad = afPo
= aPo___ _____ ____
bc bfPe bPe
which is equivalent to the IRR above.
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Note that this formulation does not involve any
assumptions about disease rarity.
It requires that the likelihood of being sampled from the
source population of person-timevaries as
a proportion of the person-time potentially contributed
by each individual.
For example:
A potential control subject who was absent from
the geographic area of interest during most of the
accrual period should have less chance of being selected
than a potential subject who was present throughout.
As with the cumulative incidence design, validity hinges
on the assumption that f (the sampling fraction)
does not vary with exposurestatus.
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An example of an incidence density case-control study:
A researcher wishes to evaluate the associationbetween the use of nonsteroidal anti-inflammatory
drugs (NSAIDS) and ventricular tachycardia (VT)
In an open cohort study lasting 2 years,
subjects are recruited and classified as toexposure status (NSAID use), then followed for
development of VT
In principle, it is possible to document periods
of exposure and non-exposure for individuals,
e.g. months on/off medication, as long as
exposure is somehow reassessed
Case-Control Studies- Slide 25
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Then for the cohort,
incidence rates and an incidence rate ratio can be calculated for
the exposed vs unexposed person-time experience, e.g.
NSAID No NSAID Total
VT, cases 80 40 120
Person-years 800 1200 2000
Incidence 0.1/p-y 0.033/p-y 0.06/p-y
The estimated incidence rate ratio is:
80/800_______
40/1200
= 3
So, assuming no confounding, we estimate that the
incidence of ventricular tachycardia among NSAID users
is 3 times that among non-users
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Suppose we instead devise a case-control study.
Here, cases will be defined by a first diagnosisof VT at Montreal hospitals, and
controls will be recruited among persons whovisit the eye clinics of the same hospitals:
both over a 2-year accrual period.
They will be compared with respect to use of
NSAIDS within the last 24 hours prior to presentation.
If sampling is done correctly (e.g. the probability
of selection is unrelated to NSAID use) thenthe controls should represent theperson-time experience of the source population
Case-Control Studies- Slide 27
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If a possible control spent half the accrual period
on NSAIDS, and half off, he has a 50% chance
of contributing to the exposed group and a50% chance of contributing to the unexposed group
This individual will contribute one or the other,
depending on the date of the visit chosen as control;
but in a larger group of people,
the control days sampled will reflect the proportion
of exposed person-time
A person can be a control early in the accrual period
and a case later
In principle, a single person can also be sampled
repeatedly as a control if the time window for
exposure definition is short (more complicated in
terms of analysis)
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Suppose that the case-control study includes all cases which
would have been detected with the open cohort design.
Two controls are recruited per case. This (unbeknownstto the researchers) corresponds to a sampling fraction
for controls of 0.12 person-day sampled per person-year
of follow-up that would have occurred in the open cohort.
Case-Control Studies- Slide 29
Then the 2x2 table is:
NSAID No NSAID Total
VT, cases 80 40 120
No VT(controls) 800*0.12 1200*0.12 2000*0.12
= 96 = 144 = 240_____________________________________________
Total 176 184 360
OR = (80x144)/(40x96) = 3.0 same as earlier IRR
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Another example of an incidence density design:
Bronchodilators are used for the treatment of asthma
There is concern that overuse may be associated with
an increased risk of adverse events, including death
Side effects can include arrhythmias, which may lead
to sudden death
Suissa et al conducted a case-control study using
the Saskatchewan health insurance database
They identified 30 persons prescribed anti-asthma
medications who died of cardiovascular events,
rather than of asthma; the date of death was
termed the index date
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4080 control dayswere then sampled randomly
from the 574,103 person-months of follow-upfor the entire asthmatic group; each such day
was also an index date
Cases and controls were then compared as to
use of theophylline and beta-agonists during the
3 months preceding the index date
These were the main exposures of concern
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Questions for discussion:
Why do you think the researchers chose
this study design?
What would have been the corresponding
cohort study?
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With respect to the relationship between theophylline use and
sudden cardiac death, the authors found the following:
Theophylline in last 3 months
Yes No | Total
Cardiac Death Yes 17 13 | 30
No 956 3124 | 4080
Note that numbers in table refer toperson-days (not to persons)
OR (crude) = ad = 17 x 3124 = 4.3 (2.1 - 8.8)__ ________bc 13 x 956
IRR (crude) = 4.3 (2.1 - 8.8)
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The odds of recent theophylline use among persons
aged 5-54 years prescribed anti-asthma drugswho died of cardiovascular events were
4.3 times those among other persons in the same age
range who were also prescribed anti-asthma drugs,
but did not die.
Asthmatics aged 5-54 who are prescribed theophylline
have an estimated 4.3 fold increase in incidence of
fatal cardiovascular events, compared with
asthmatics who are not prescribed theophylline.
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As with the cumulative incidence design, an attributable ratefraction can be estimated for exposed persons:
It is: Ie-Io, where Ie= incidence among exposed and____
Ie Io= incidence among the unexposed
= IRR - 1 = OR - 1______ _____
IRR OR
For the Saskatchewan study, the estimated attributable
rate fraction among asthmatics who were prescribed
theophylline is:4.3 - 1 = 0.77______4.3
Among asthmatics aged 5-54 prescribed theophylline,
an estimated 77% of fatal cardiovascular events
were related to its prescription.
Case-Control Studies- Slide 35
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It is also possible to estimate the attributable rate fraction
for the entire population (PAR%)
In a cohort study, this is simply
It- Io, where It= incidence among the total population_____It Io= incidence among the unexposed
For the corresponding incidence density case-control study,the population attributable ratefraction is
IRR - 1 x proportion of cases who were exposed,____IRR
estimated as OR - 1 x a_____ ____OR a+b
Similar parameters involving riskcan be generated for
the cumulative incidence design
Case-Control Studies- Slide 36
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For the Saskatchewan study, recall the 2 x 2 table
Theophylline in last 3 months
Yes No | Total
Cardiac death Yes 17 13 | 30
No 956 3124 | 4080
OR = 4.3
Pexp |case = 17/30 = 0.57
then PAR fraction = OR -1 x Pexp |case_____OR
= 4.3 - 1 x 0.57 = 0.44______
4.3
Among Saskatchewan asthmatics aged 5-54, an estimated
44% of cardiovascular deaths relate to theophylline prescriptions.
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Attributable rates (rate difference)
The absolute rate difference (i.e., the absolute
rate of disease attributable to exposure) is Ie- Io
Data from a standard case-control study alone
cannot validly be used to estimate
absolute rates of disease.
Even if case ascertainment is complete,
the controls represent an unknown and
arbitrary fraction of the true person-time at risk.
Hence the rate difference cannot be estimated.
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However, incidence rates can be estimated if there is
additional knowledge about the amount of person-time at risk
Exposure
(+) (-)
Disease (+) a b
Disease (-) c = f x x Pt d = f x (1- ) x Pt
Then Ie = a = a_____ ___________x Pt [c/(c+d)] x Pt
Then Io = b = b_________ ___________(1- ) x P
t
[d/(c+d)] x Pt
and the rate difference is Ie-Io
where = proportion of person-time which is exposed
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Example:
In this nested case-control study,
the researchers knew that in the source cohort
(Saskatchewan asthmatics aged 5-54), there were
47,842 person-years at risk during the study period
The 2x2 table was:Theophylline in last 3 months
Yes No | Total
Cardiac death Yes 17 13 | 30
No 956 3124 | 4080
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Then the estimated incidence of cardiac death in asthmatics
prescribed theophylline (Ie) is:
a = 17 = 0.0015 per person-year___________ ________________[c/(c+d)] x Pt 956/4080 x 47,842
And in asthmatics who were not prescribed theophylline theestimated incidence (I
o) is:
b = 13 = 0.00035 per person-year___________ _________________[d/(c+d)] x Pt 3124/4080 x 47,842
The estimated rate difference is therefore
0.0015-0.00035 = 0.00115 per person-year.
Note that the IRR computed as Ie/Io remains 4.3
Case-Control Studies - Slide 41
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Ieand Io may also be estimated if It is known for the source population
Recall that It = (Iex ) + [Iox (1- )]
But Ie = Iox OR
Then It = Io[(OR x ) + (1- )]
So Io = It = It______________ ________________________
(OR x ) + (1- ) {OR x [c/(c+d)]} + [d/(c+d)]
Then use Ie = Iox OR
Then RD = Ie- Ioas usual [= Io(OR-1)]
Case-Control Studies - Slide 42
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Example:
The total incidence (It) of cardiovascular death
in the Saskatchewan cohort was
30 deaths/47,842 person-years
= 0.00063 per person-year.
Then Io= 0.00063 = 0.00035___________________________
[4.3 x (956/4080)] + (3124/4080)
and Ie= 0.00036 x 4.3 = 0.0015
RD = 0.0015 - 0.00035 = 0.00115
Case-Control Studies - Slide 43
l di l d
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Additional points
Corresponding estimates of attributable risks and
risk differences can be made for cumulative incidence
case-control studies, if the corresponding additional data
is available
Estimates of absolute risks/incidence rates and
risk/rate differences can be made only if thetotal amount of persons/person-time at risk is known,
or at least one absolute risk/incidence rate is known
(i.e. for the total population, the exposed, or
the unexposed)
Nested case-control studies are a special type of study
where cases and controls are explicitly drawn from
a defined larger cohort (as in the Saskatchewan
asthma study)
Case-Control Studies - Slide 44
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Case Control Studies Slide 45
Case-Control Studies: Strengths and Limitations
Advantages of case-control studies:
Efficiency - much less expensive/intensivethan cohort studies.
Very useful for outcomes that are rare
or occur after a long latency period.
Most outcomes are relatively rare overshort-term follow-up.
Permit evaluation of multiple exposures.
Can rapidly accrue person-time experience.
Avoid losses to follow-up inherent in cohort studies.
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Disadvantages
Not useful/efficient for very rare exposures(may not be present in either cases or controls).
Cannot directly compute incidence rates.
Cannot usually evaluate more than one outcome.
Temporality may be lost or distorted.
Potential for considerable bias, i.e. loss of validity.
Bias relates to:
- Measurement of exposure status
- Selection of subjects (usually controls)
Case-Control Studies- Slide 46
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With respect to measurement,
exposure ascertainment must be consistent
for cases and controls.
There may be potential for misclassification of
exposure in relation to disease status
Case-Control Studies- Slide 47
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Example 1
Differential recall of exposures among casesvs controls
e.g. medication use and congenital malformations
- particularly if mothers attuned to
study hypothesis.
If cases more likely to recall exposure,
results will be biased toward a
positive association between exposure and outcome.
The more objective the source of exposure data,
the better.
Case-Control Studies- Slide 48
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Example 2
Different sources of information about exposure
e.g. family members asked about
alcohol consumption of persons
who died of gastric cancer,
vs Direct questioning of control subjects.
If family members tend to underestimate cases
alcohol consumption, results will be biased
against finding a positive association between
alcohol and gastric cancer.
Case-Control Studies- Slide 49
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Example 3
Exposure status changes as a consequence ofthe outcome
e.g. patients with symptoms of lung cancer
stop smoking
If patients with newly diagnosed lung cancer arecompared to controls with respect to current
or recent smoking, results may be biased, i.e.,
the association between smoking and lung cancer
will be underestimated.
Data collection must reflect relevant person-time
experience and temporality of exposure and outcome.
Case-Control Studies- Slide 50
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Association may also be missed
if the exposure of interest is poorly documented
(an example of non-differential misclassification)
Example: mesothelioma
It can be caused by brief, intense exposures
to asbestos, with a very long latency period(>30 years).
In a case control study,
both cases and controls may recall such exposures
very poorly, thereby leading to an underestimate
of the true association.
Case-Control Studies- Slide 51
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Control selection in case-control studies
Recall that the validity of case-control studies
hinges on the assumption that the
sampling fraction for cases (which may be 100%)
and that for controls (usually unknown)
does not vary by exposure status.
In other words, controls should represent the
source population from which the cases arose,
with respect to exposure experience.
Case-Control Studies- Slide 52
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Example 1
A researcher wishes to test the hypothesis thatuse of nonsteroidal anti-inflammatory drugs (NSAIDs)
is associated with development of gastric cancer.
She plans a case-control study comparing gastric cancer
patients (cases) with patients seen at the same hospitalfor peptic ulcer disease (controls).
- NSAID use is a known risk factor for ulcers.
What will be the effect on her findings:
a) if NSAID use is truly a risk factor for gastric cancer?
b) if NSAID use is truly unassociated with gastric cancer?
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Hence, controls should not differ systematicallyfrom the population of interest
with respect to exposure experience.
Sometimes the bias may be less obvious,
i.e. unrelated to explicit criteria for
control selection.
Case-Control Studies- Slide 54
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Example 2
A researcher wishes to evaluate the association between
cellular phone use and brain tumoursusing a case-control design.
Cases are recruited from the brain tumour clinic at theRoyal General Hospital, a neurosurgery referral centre.
Controls are recruited from the family medicine clinicat the same hospital. This clinic primarily serves alow-income population from the area adjacent tothe hospital.
This control group is less likely than the general population
to own cellular phones.
Result:
The study will be biased toward detecting anassociation between brain tumours and cell phone use.
Case-Control Studies- Slide 55
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Controls should be at risk for developing the outcome of interest
- otherwise they do not contribute useful data to the study
(inefficient)
- inclusion of individuals not at risk may
also distort the results if the reason they are not at risk
relates to the exposure under study. This may not be obvious.
Example:
Sleep apnea (exposure) and risk of traffic accidents (outcome)
Cases: Drivers involved in car accidents.
Including non-drivers in the control group would be
a waste of time
- it could bias the results if
persons with severe apnea have chosen not to drive
and are over-represented in the control group.
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Controls should be persons who,
had they developed the outcome of interest,
would have had the same opportunity asthe actual cases to be included as such.
Similarly, cases should have
had the same opportunity as actual controls
to be included, had they
not developed the outcome of interest.
If this is not the case, controls may not properly
represent the source population.
e.g., study of brain tumours and cell phone use
discussed above
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Types of controls in case-control studies
1. Population Controls
Suitable if cases are a representative sample
(or all cases) arising from a well-defined
source population.
Controls are then randomly sampledfrom the same population.
With the incidence-density design,the probability of being sampled should
vary with an individuals person-time at risk.
Often, it is not easy to define the
precise source population.
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2. Neighbourhood Controls
May match controls to individual cases
with respect to neighbourhood of residence.
If cases are from a hospital, their neighbours
may or may not be equally likely to betreated at the same hospital should
they develop the disease in question.
Example:
A hospital which caters to a particular group
within society.
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3. Family members or friends as controls
May share exposure characteristics with cases
as opposed to broader source population
(e.g. tobacco and alcohol use, dietary intake,
use of household products).
This can obscure relevant associations.
Depends on information provided by cases;
investigator loses control over factors leading
to selection.
Cases friends may overlap, leading to
disproportionate probabilities of selection
of certain individuals as controls.
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4. Hospital/clinic based controls
Often used when cases accrued at specific
hospital(s)/clinic(s).
Controls are recruited among persons seen
at the same hospitals/clinics forother reasons or conditions.
To avoid bias, the basis for control selection
cannot be related to the exposure under study.
The incidence of the control condition(s)
determines the sampling fraction.
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Example:A researcher wishes to examine the relationship
between anti-hypertensive medication use
and car accidents.
What will happen if controls are recruited
in the cardiology clinic?
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The best hospital controls are
persons with acute conditions that
consistently require hospital care but
are not related to the exposure of interest.
Example:
In a case control study of smoking as a
risk factor for colon cancer, a researcher
recruits controls who undergo appendectomy,
prostatectomy, or hysterectomy atthe same hospital as the cases.
Case-Control Studies- Slide 63
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Derivation of formula - Part 1
For the cohort study, the 2 x2 table is:
exposed unexposed total
Cases a b a + b
Person-time Pe Po Pe+ Po= Pt
IRR = Ie = a/Pe = aPo___ _____ ____Io b/Po bPe
= a(Pt - Pe) = a (1 - Pe/Pt)_______ _________
bPe b (Pe/Pt)
= a x (1-)_ ____b
Where = Pe/Pt = the proportion of person-years withexposure among total person-yearsin the source population
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Furthermore,
a = a/(a+b) = Pexp|case_ _______ __________
b b/(a+b) = 1- Pexp|case
where Pexp|case = proportion of cases exposed
Then IRR = Pexp|case (1- ) Equation 1_____________
(1-Pexp|case)
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Derivation of formula - Part 2
if = proportion of person-years with exposure
then 1-= proportion of person-years without exposure
and It = Ie
+ Io(1-
)
i.e. a weighted average of incidence rates
among exposed and unexposed persons
Supplemental Material- Slide 3
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Then the PAR fraction is:
It- Io = (Ie) + [(Io(1- )] - Io_____ ____________________
It (Ie) + [Io(1- )]
= (Ie/Io) + (1- ) (Io/Io) - Io/Io______________________________________
(Ie/Io) + (Io/Io) (1- )
= (IRR) + 1 - - 1________________
(IRR) + 1 -
= (IRR - 1)____________
(IRR - 1) + 1
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Derivation - Part 3
= IRR - 1_____________IRR + (1/ ) - 1
= IRR - 1____________IRR + ( 1- )______
= IRR - 1______________IRR + IRR (1- )_________
IRR ()
pp
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Substituting equation 1 for IRR, this is
IRR - 1____________________________IRR + IRR (1- ) () (1 - Pexp |case)_______________________
() (Pexp |case) (1- )
= IRR - 1__________________
IRR + IRR (1-Pexp |case)_____________
Pexp case
= IRR - 1______________________________
IRR (Pexp |case) + IRR - IRR (Pexp |case)______________________________
Pexp case
= IRR - 1 x Pexp |case = OR -1 x Pexp |case______ ____
IRR OR
Supplemental Material- Slide 6