Epidemiologic design from a sampling perspective

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Epidemiologic design from a sampling perspective

• Epidemiology II Lecture
• April 14, 2005
• David Jacobs

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Why different epidemiologic designs?

• It is generally not possible to observe everyone
  in a population
• New questions arise after data / samples have
  been collected
• Cost and feasibility
• Statistical efficiency and appropriateness to
  study question

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The possibilities

• There are many approaches
• Sampling from the whole population
• Sampling from exposure
• Sampling from caseness
• Haphazard selection

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True Population Configuration Underlying
Epidemiologic Study Designs
Time 0
Time 1
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True Population Configuration Underlying
Epidemiologic Study Designs Approximate numbers
for Minnesota
Time 0
Time 1
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Alternate Format Population Values Exposed and
Diseased
Diseased Not diseased
Exposed A C
Not exposed B D
The numbers A, B, C, D are fixed and known exactly
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Alternate Format Population Values Exposed and
Diseased
Diseased Not diseased
Exposed A 50,000 C 950,000
Not exposed B 50,000 D 2,950,000
The numbers A, B, C, D are fixed and known exactly
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Measures of Risk and Relative Risk Whole
Population
Diseased Not diseased Odds Risk Probability
Exposed A C A/C A/(AC)
Not exposed B D B/D B/(BD)
Exposure Ratio A/(AB) C/(CD)

• Risk Difference A/(AC) B/(BD)
• Risk Ratio, Relative Risk A/(AC) / B/(BD)
• Odds Ratio A/C / B/D AD/BC

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Measures of Risk and Relative Risk Whole
Population
Diseased Not diseased Odds Risk Probability
Exposed 50,000 950,000 0.053 .05
Not exposed 50,000 2,950,000 0.017 .017
Exposure Ratio 0.5 0.244

• Risk Difference 0.033
• Risk Ratio, Relative Risk 3
• Odds Ratio 3.11

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• Epidemiologic studies sample from A, B, C, and D
  to estimate Odds or Risk Risk Difference, Risk
  Ratio, or Relative Risk
• Epidemiologic design is determined by
  investigator control, temporality, sampling
  fraction

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Epidemiologic design level of investigator
control

• Clinical Trial
• Exposure assigned (at random)
• Reflects temporary state
• Observational
• Exposure occurs naturally
• Often reflects long term state

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Epidemiologic design temporality

• Clinical Trial, Cohort, Nested case control,
  Case-cohort
• Exposure assessed at variable times before
  disease
• Cross-sectional
• Exposure assessed simultaneously with disease
• Case-control
• Past exposure assessed simultaneously with disease

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Epidemiologic design sampling fraction

• Cells A, B, C, and D are sampled at random with
  constant probability (called the sampling
  fraction)
• Sample size is a, b, c, d
• If a/A b/B c/C d/D then the sampling
  fraction is equal for all cells

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Sampling fractions
Diseased Not diseased
Exposed a/A fA c/C fC
Not exposed b/B fB d/D fD
The numbers A, B, C, D are fixed and known
exactly. The numbers a,b,c,d are realized in a
given study, determined during the study.
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Expected observations given sampling fractions
Diseased Not diseased
Exposed a 5,000 fA0.1 c 950 fC0.001
Not exposed b 1,250 fB0.025 d 5,900 fD0.002

• Risk naïve (and wrong) 5000/5950 0.84 and
• 1250/7150 0.175 naïve relative risk 4.8
• Correct risk 5000/0.1/ (5000/0.1 950/0.001)
  0.05 and 1250/0.025 / (1250/0.025 5900/0.002)
  0.017 leading to Relative risk 0.05/0.017 3

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Observations given sampling fractions
Diseased Not diseased
Exposed a 5,000 ea fA0.1 c 950 ec fC0.001
Not exposed b 1,250 eb fB0.025 d 5,900 ed fD0.002
All estimates differ from population values by
random amounts (see example in Excel file)
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Epidemiologic design sampling fraction

• Cross-sectional sample equally from everyone fA
  fB fC fD
• Clinical trial, Cohort study sample equally from
  initial exposure groups
• fAC and fBD
• (ac)/(AC) usually differ from (bd)/(BD) in
  clinical trial, usually the same in cohort study

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Cross-sectional Study
Diseased Not diseased
Exposed a/A f c/C f
Not exposed b/B f d/D f

• Sampling fraction is the same in all cells.
• Risk and odds estimates are unbiased, so risk
  differences and ratios are unbiased.

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Expected Cross-sectional Study
Diseased Not diseased
Exposed a 50 fA0.001 c 950 fC0.001
Not exposed b 50 fB0.001 d 2,950 fD0.001
Naïve risks and relative risks are correct!
50/1000 0.05, etc.
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Observed Cross-sectional Study
Diseased Not diseased
Exposed a 50 ea fA0.001 c 950 ec fC0.001
Not exposed b 50 eb fB0.001 d 2,950 eabc fD0.001
All estimates differ from population values by
random amounts
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Clinical Trial or Cohort Study
Diseased Not diseased
Exposed a/A fAC c/C fAC
Not exposed b/B fBD d/D fBD

• Sampling fraction is fixed within exposed and
  within not exposed. Usually fAC not fBD in
  clinical trial, fAC fBD in cohort study
  (which has cross-sectional baseline).
• Risk and odds estimates are unbiased, so risk
  differences and ratios are unbiased.

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Expected Clinical Trial or Cohort Study
Diseased Not diseased
Exposed a 100 fAC0.002 c 1900 fAC0.002
Not exposed b 50 fBD0.001 d 2950 fBD0.001

• fAC usually fBD in a cohort study
• fAC may differ from fBD in a clinical trial (if
  treatment allocation is not 11)

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Expected Measures of Risk and Relative Risk
Clinical Trial or Cohort Study
Diseased Not diseased Odds Risk Probability
Exposed 100 1,900 0.053 .05
Not exposed 50 2,950 0.017 .017
Exposure Ratio 0.67 0.39 ?Differs from total population ?Differs from total population

• Correct Risk Difference 0.033
• Correct Risk Ratio, Relative Risk 3
• Odds Ratio 3.11

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Observed Clinical Trial or Cohort Study
Diseased Not diseased
Exposed a 100 ea fAC0.002 c 1900 - ea fAC0.002
Not exposed b 50 eb fBD0.001 d 2950 - eb fBD0.001
All estimates differ from population values by
random amounts
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Epidemiologic design sampling fraction

• Case control sample differentially within
  diseased and within nondiseased
• fA fB fAB and fC fD fCD
• Usually fAB much greater than fCD

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Case-control Study
Diseased Not diseased
Exposed a/A fAB c/C fCD
Not exposed b/B fAB d/D fCD

• Sampling fraction is fixed with diseased and
  within not diseased.
• Exposure probabilities and odds estimates are
  unbiased, but risk, disease odds, risk
  differences and ratios are biased.
• Odds ratio relative risk when disease is rare.

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Expected Case-control Study
Diseased Not diseased
Exposed a 500 fAB0.01 c 494 fCD0.00052
Not exposed b 500 fAB0.01 d 1534 fCD0.00052
fAB 19.23 fCD
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Expected Measures of Risk and Relative Risk
Case-Control Study
Diseased Not diseased Odds Risk Probability
Exposed 500 494 1.01 0.503
Not exposed 500 1,534 0.33 0.246
Exposure Ratio 0.5 0.24 Differs from total population Differs from total population

• Incorrect Risk Difference 0.257
• Incorrect Risk Ratio, Relative Risk 2.04
• Odds Ratio 3.11 ? correct and approx true Rel
  Risk

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Observed Case-control Study
Diseased Not diseased
Exposed a 500 ea fAB0.01 c 494 ec fCD0.00052
Not exposed b 500 - ea fAB0.01 d 1534 - ec fCD0.00052
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Epidemiologic design sampling fraction

• Nested case control sample differentially within
  diseased and within nondiseased starting with a
  cross-sectional base, so exposure measured prior
  to disease diagnosis
• fA fB fAB and fC fD fCD
• Often fAB 1
• Usually fAB somewhat greater than fCD

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Nested Case-Control Study, 1Observed
Cross-sectional Study
Diseased Not diseased
Exposed a 500 ea fA0.01 c 9500 ec fC0.01
Not exposed b 500 eb fB0.01 d 29500 eabc fD0.01
Previous cross-sectional example with sampling
fractions increased by a factor of 10
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Nested Case-Control Study, 2Sampling from the
cross-section
Diseased Not diseased
Exposed a/A fAB c/C fCD
Not exposed b/B fAB d/D fCD

• Sampling fraction is fixed within diseased and
  within not diseased temporality preserved.
• Exposure probabilities and odds estimates are
  unbiased, but risk, disease odds, risk
  differences and ratios are biased.
• Odds ratio relative risk when disease is rare.

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Observed Nested Case-Control Study
Diseased Not diseased
Exposed a 500 ea fAB1 c 950 ec ec1 fCD0.01
Not exposed b 500 eb fAB1 d 2950 eabc ec1 fCD0.01
ea, eb, ec are ignored if fAB lt 1 then there
is an ea1.
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Expected Measures of Risk and Relative Risk
Nested Case-Control Study
Diseased Not diseased Odds Risk Probability
Exposed 500 950 0.526 0.344
Not exposed 500 2,950 0.169 0.145
Exposure Ratio 0.5 0.24 Differs from total population Differs from total population
Incorrect Risk Difference 0.199 Incorrect Risk
Ratio, Relative Risk 2.38 Odds Ratio 3.11 ?
correct and approx true Rel Risk
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Epidemiologic design sampling fraction

• Case cohort sample differentially within
  diseased and within everyone (diseased
  nondiseased) starting with a cross-sectional
  base, so exposure measured prior to disease
  diagnosis
• fA fB fAB the whole cohort is sampled at
  fABCD
• Usually fAB 1, while fABCD is a sizeable
  fraction like 0.1 or 0.25.

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Case-Cohort Study, 1Observed Cross-sectional
Study
Diseased Not diseased
Exposed a 500 ea fA0.01 c 9500 ec fC0.01
Not exposed b 500 eb fB0.01 d 29500 eabc fD0.01
Previous cross-sectional example with sampling
fractions increased by a factor of 10
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Case-Cohort Study, 2 sampling from the
cross-section
Diseased Cohort (Part of all ppts)
Exposed A/A, fAB1 (ac)/(AC) fABCD
Not exposed B/B, fAB1 (bd)/(BD) fABCD

• Sampling fraction is fixed within diseased and
  within not diseased temporality preserved
  cohort includes cases and noncases.
• Risk and odds estimates are unbiased within
  exposed and within unexposed but differently
  weighted, so risk differences biased
• Risk ratios are unbiased.

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Observed Case-Cohort Study
Diseased Cohort
Exposed a 500 ea fAB1 c 1000ecec1 fABCD0.1
Not exposed b 500 eb fAB1 d 3000eabcec1 fABCD0.1
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Observed Case-Cohort Study
Case, fAB1 Cohort, fABCD0.1 Cohort, fABCD0.1
Diseased Diseased Not diseased
Exposed a 500 ea 50ecec1 950ecec3
Not exposed b 500 eb 50ecec2 2950 eabcec123

• When fAB 1, cohort diseased is a subset of
  case diseased.
• When fAB lt 1, cohort diseased usually overlaps
  case diseased.

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Nested Case-Control vs Case Cohort

• Same cases in both
• For a certain sampling strategy, same noncases in
  both
• Analytic strategy different

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Expected Measures of Risk and Relative Risk
Case-Cohort Study
Diseased Cohort Odds Risk Probability
Exposed 500 1,000 n/a 0.5
Not exposed 500 3,000 n/a 0.167
Exposure Ratio 0.5 0.25 ?Differs from total population ?Differs from total population

• Incorrect Risk Difference 0.333 (true risk
  diff/fABCD)
• Odds ratio Correct Risk Ratio, Relative Risk
  3
• Relative risk would be correct even if the
  disease were rare

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Analysis of Case Cohort Study

• Set up table as if cohort were the control group
• Include the overlapping cases in both cases and
  cohort
• Compute ad/bc
• You have estimated relative risk
• Note
• If you know the cohort sampling fraction, you can
  multiply the cohort up and estimate true risks
• Given additional error in second stage cohort
  sampling, this is less efficient than estimating
  relative risk without upweighting

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Analysis of Case Control and Case Cohort Study

• Case Control
• Logistic regression
• eb is an odds ratio
• Temporal bias?
• Nested Case Control
• Logistic regression
• eb is an odds ratio
• No temporal bias
• Case Cohort
• Logistic regression or Linear regression
• eb is a relative risk, not an odds ratio
• No Temporal bias
• Variance somewhat high unless robust variance
  estimate is used (e.g. PROC GENMOD with GEE
  option)

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Disadvantage of Case Control vs. Case Cohort Study

• Case Control and Nested Case Control
• Inflexible Outcome is fixed
• Even in nested case control study the sampling
  structure is usually unknown
• Case Cohort
• Ideal for the intended outcome or for multiple
  outcomes
• If cohort is large enough, multiple outcomes can
  be analyzed
• Cases can be included in analysis of alternate
  dependent variable because sampling structure is
  known

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Person years of risk, 1

• The foregoing assumes that all cases occur at the
  same time (or can safely be treated as such)
• In many, even most studies, this assumption is
  reasonable
• Person years is length of followup number of
  participants when events are rare and/or all
  participants start followup at nearly the same
  time

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Person years of risk, 2

• Even if there are 50 events and they occur on
  average somewhat after the midpoint of followup,
  person years gt ¾ length of followup number of
  participants
• Incidence density rates are somewhat higher than
  correspondingly scaled cumulative incidence
  rates, but relative risks are probably not much
  affected by computation of incidence density vs
  cumulative incidence

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Person years of risk, 3

• Proportional hazards models do not allow time
  dependency in prediction, so most analyses are
  not considering followup time in this way.
• The timing of events vs. censoring and competing
  risk may cause differences in findings for
  incidence density vs. cumulative incidence, but
  this would be rare
• Subgroups with very different followup times
  could create problems, but this is rare

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• In prospective studies, events are accumulated
  over time, so incidence density methods can be
  applied
• Nested case control
• Case cohort

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Nested Case-Control Study (1)
Consider the following hypothetical cohort
X
X lung cancer case O loss to follow-up
X
O
O
X
t1
t2
t3
Time
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Nested Case-Control Study (2)

• At time t1 the first case occurs for which 8
  eligible controls are identified
• Similarly, there are 5 eligible controls for the
  case at time t2, and 4 eligible controls for the
  case occurring at time t3
• A control can become a case at a later time
  (e.g., cases at t2 and t3 serve as control for
  case at t1)
• Controls can be selected randomly from all
  eligible controls (i.e., 1 or more controls for
  each case)
• Number of eligible controls decreases with
  increasing number of matching factors

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Case-Cohort Study (1)
Consider the following hypothetical cohort
X
X lung cancer case O loss to follow-up
X
O
O
X
t0
Time
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Case-Cohort Study (2)

• In closed cohort (in this case, when everybody
  enters cohort at t0), a sample of all subjects
  (sub-cohort) is randomly selected from cohort
  members at start of follow-up t0
• In open cohort (i.e., when time of entry into
  cohort is variable), a sample of all subjects
  (sub-cohort) is randomly selected from members
  of cohort as it is followed over time (i.e.,
  regardless of when subjects entered the cohort)

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Merits of time-based selection

• From a black and white theoretical perspective,
  time-based selection makes sense a person is a
  noncase until a certain time, then becomes a
  case.
• In the life table approach, consistent with this
  thought, the risk set at any time point is cases
  and noncases at that time point.
• However, much chronic disease develops slowly and
  the black-white, case-noncase formulation does
  not apply very well.
• I am unenthusiastic generally about time-based
  selection of controls or noncases because I would
  like to maintain maximum separation between cases
  and noncases.

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Analysis of events that evolve in time

• Nevertheless, taking person years into account
  (incidence density analysis) is more precise than
  is analysis of cumulative incidence.
• Analysis is therefore by Cox proportional hazards
  life table regression methods, or some similar
  technique.
• The nested case-control method is not easily
  adapted to this type of analysis
• The case-cohort method is easily analyzed this
  way (PROC PHREG or GENMOD with Poisson
  regression), but the variances of the slopes tend
  to be too large.
• Robust variance estimation is possible in GENMOD
  with GEE option Barlow provides a SAS macro for
  use of PHREG.