Analysis of matched data

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Analysis of matched data
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Pair Matching Why match?

• Pairing can control for extraneous sources of
  variability and increase the power of a
  statistical test.
• Match 1 control to 1 case based on potential
  confounders, such as age, gender, and smoking.

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Example

• Johnson and Johnson (NEJM 287 1122-1125, 1972)
  selected 85 Hodgkins patients who had a sibling
  of the same sex who was free of the disease and
  whose age was within 5 years of the
  patientsthey presented the data as.

OR1.47 chi-square1.53 (NS)
From John A. Rice, Mathematical Statistics and
Data Analysis.
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Example

• But several letters to the editor pointed out
  that those investigators had made an error by
  ignoring the pairings. These are not independent
  samples because the sibs are pairedbetter to
  analyze data like this

OR2.14 chi-square2.91 (p.09)
From John A. Rice, Mathematical Statistics and
Data Analysis.
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Pair Matching Agresti example

• Match each MI case to an MI control based on age
  and gender.
• Ask about history of diabetes to find out if
  diabetes increases your risk for MI.

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Pair Matching Agresti example
Which cells are informative?
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Pair Matching
OR estimate comes only from discordant pairs! The
question is among the discordant pairs, what
proportion are discordant in the direction of the
case vs. the direction of the control. If more
discordant pairs favor the case, this indicates
ORgt1.
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P(favors case/discordant pair)
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odds(favors case/discordant pair)
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OR estimate comes only from discordant
pairs!! OR 37/16 2.31 Makes Sense!
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McNemars Test
Null hypothesis P(favors case / discordant
pair) .5 (note equivalent to OR1.0 or cell
bcell c)
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McNemars Test
Null hypothesis P(favors case / discordant
pair) .5 (note equivalent to OR1.0 or cell
bcell c)
By normal approximation to binomial
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McNemars Test generally
By normal approximation to binomial
Equivalently
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McNemars Test
McNemars Test
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RECALL 95 confidence interval for a difference
in INDEPENDENT proportions
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95 CI for difference in dependent proportions
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95 CI for difference in dependent proportions
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The connection between McNemar and
Cochran-Mantel-Haenszel Tests
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View each pair is its own age-gender stratum
Example Concordant for exposure (cell a from
before)
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x 9
x 37
x 16
x 82
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Mantel-Haenszel for pair-matched data
We want to know the relationship between diabetes
and MI controlling for age and gender (the
matching variables). Mantel-Haenszel methods
apply.
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RECALL The Mantel-Haenszel Summary Odds Ratio
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ad/T 0 bc/T0
ad/T1/2 bc/T0
ad/T0 bc/T1/2
ad/T0 bc/T0
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Mantel-Haenszel Summary OR
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Mantel-Haenszel Test Statistic(same as McNemars)
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Concordant cells contribute nothing to
Mantel-Haenszel statistic (observedexpected)
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Discordant cells
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Example Salmonella Outbreak in France, 1996
From Large outbreak of Salmonella enterica
serotype paratyphi B infection caused by a goats'
milk cheese, France, 1993 a case finding and
epidemiological study BMJ 312 91-94 Jan 1996.
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Epidemic Curve
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Matched Case Control Study

• Case Salmonella gastroenteritis.
• Community controls (11) matched for
• age group (lt 1, 1-4, 5-14, 15-34, 35-44, 45-54,
  55-64, or gt 65 years)
• gender
• city of residence

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Results
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In 2x2 table form any goats cheese
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In 2x2 table form Brand A Goats cheese
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x8
x24
x2
x25
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Summary 8 concordant-exposed pairs (strata)
contribute nothing to the numerator
(observed-expected0) and nothing to the
denominator (variance0).
Summary 25 concordant-unexposed pairs contribute
nothing to the numerator (observed-expected0)
and nothing to the denominator (variance0).
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Summary 2 discordant control-exposed pairs
contribute -.5 each to the numerator
(observed-expected -.5) and .25 each to the
denominator (variance .25).
Summary 24 discordant case-exposed pairs
contribute .5 each to the numerator
(observed-expected .5) and .25 each to the
denominator (variance .25).
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ExtensionM1 matched studies

• This is just a thought problem! I will not test
  you on the material that follows, although its
  very good practice in thinking like a
  statistician
• We will soon learn conditional logistic
  regression, which can handle all of the data we
  are discussing today with much more ease
• You can see that as M increases, so does
  complexity!

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M1 matched studies

• One-to-one pair matching provides the most
  cost-effective design when cases and controls are
  equally scarce.
• But when cases are the limiting factor, as with
  rare diseases, statistical power may be increased
  by selecting more than 1 control matched to each
  case.
• But with diminishing returns

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M1 matched studies

• 21 matched study of colorectal cancer.
• Background Carcinoembryonic antigen (CEA) is the
  classical tumor marker for colorectal cancer.
  This study investigated whether the plasma levels
  of carcinoembryonic antigen and/or CA 242 were
  elevated BEFORE clinical diagnosis of colorectal
  cancer.
• From Palmqvist R et al. Prediagnostic Levels of
  Carcinoembryonic Antigen and CA 242 in Colorectal
  Cancer A Matched Case-Control Study. Diseases of
  the Colon Rectum. 46(11)1538-1544, November
  2003.

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M1 matched studies Prediagnostic Levels of
Carcinoembryonic Antigen and CA 242 in Colorectal
Cancer A Matched Case-Control Study

• Study design A so-called nested case-control
  study.
• Idea Study subjects who were members of an
  ongoing prospective cohort study in Sweden had
  given blood at baseline, when they had no
  disease. Years later, blood can be thawed and
  tested for the presence of prediagnostic
  antigens.
• Key innovation The cohort is large, the disease
  is rare, and its too costly to test everyones
  blood so only test stored blood of cases and
  matched controls from the cohort.

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M1 matched studies

• Two cancer-free controls were randomly selected
  to each case from the corresponding cohort at the
  time of diagnosis of the matched case.
• Matched for
• Gender
• age at recruitment (12 months)
• date of blood sampling 2 months
• fasting time (lt4 hours, 48 hours, gt8 hours).

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21 matching

• stratummatching group
• 3 subjects per stratum
• 6 possible 2x2 tables

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Everyone exposed non-informative
Case exposed 1 control unexposed
Case exposed both controls unexposed
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Case unexposed both controls exposed
Case unexposed 1 control exposed
Everyone unexposed non-informative
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RESULTS
0
2
12
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0
1
102
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2 Tables with 2 exposed (CEA)
2
x0
2
x2
Represents all possible discordant tables (either
2 or 1 total exposed)
13 Tables with 1 exposed (CEA)
1
x12
1
x1
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2 Tables with 2 exposed
2
2
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Summary

• P(case exposed/2 total exposed)2OR/(2OR1)
• P(case unexposed/2 total exposed)1-2OR/(2OR1)
• P(case exposed/1 total exposed) OR/(OR2)
• P(case unexposed/1 total exposed) 1-OR/(OR2)
• Therefore, we can make a likelihood equation for
  our data that is a function of the OR, and use
  Maximum Likelihood Estimation to solve for OR

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Applying to example data

• Well talk (briefly!) about MLE estimation next
  week, but for now
• This is the probability of our data as a function
  of the unknown OR.
• To find the value of the OR that maximizes the
  function (and therefore the likelihood of our
  data)?Take derivative set equal to 0 solve for
  OR.

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Applying to example data
Breslow-Day give a more simple robust estimate of
OR for 21 matching