Analysis of matched data HRP 261 02/02/04 Chapter 9 Agresti – read sections 9.1 and 9.2.

Analysis of matched dataAnalysis of matched data

HRP 261 02/02/04HRP 261 02/02/04

Chapter 9 Agresti – read sections 9.1 and 9.2Chapter 9 Agresti – read sections 9.1 and 9.2

Pair Matching: Why match?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.

ExampleExample Johnson and Johnson (NEJM 287: 1122-1125,

1972) selected 85 Hodgkin’s patients who had a sibling of the same sex who was free of the disease and whose age was within 5 years of the patient’s…they presented the data as….

Hodgkin’s

Sib control

Tonsillectomy None

From John A. Rice, “Mathematical Statistics and Data Analysis.

OR=1.47; chi-square=1.53 (NS)

ExampleExample 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 paired…better to analyze data like this:

From John A. Rice, “Mathematical Statistics and Data Analysis.

OR=2.14; chi-square=2.91 (p=.09)

Tonsillectomy

Tonsillectomy None

Control

Pair Matching: Pair Matching: Agresti Agresti exampleexample

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.

Pair Matching: Pair Matching: AgrestiAgresti example example

Which cells are informative?

Just the discordant cells are informative!

Diabetes

No diabetes

25 119

Diabetes No Diabetes

MI cases

MI controls

Pair MatchingPair Matching

Diabetes

No diabetes

25 119

MI cases

MI controls

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 OR>1.

Diabetes

No diabetes

25 119

MI cases

MI controls

P(“favors” case/discordant pair) =

)~/(*)/(~)~/(~*)/(

)~/(~*)/(

DEPDEPDEPDEP

DEPDEP

=the probability of observing a case-control pair with only the control exposed

=the probability of observing a case-control pair with only the case exposed

Diabetes

No diabetes

25 119

MI cases

MI controls

P(“favors” case/discordant pair) =

Diabetes

No diabetes

25 119

MI cases

MI controls

odds(“favors” case/discordant pair) =

Diabetes

No diabetes

25 119

MI cases

MI controls

OR estimate comes only from discordant pairs!!

OR= 37/16 = 2.31

Makes Sense!

Diabetes

No diabetes

MI casesMI controls

McNemar’s TestMcNemar’s Test

...)5(.)5(.39

53)5(.)5(.

53 143915381637

valuep

01.;88.264.3

)5)(.5(.53

Null hypothesis: P(“favors” case / discordant pair) = .5(note: equivalent to OR=1.0 or cell b=cell c)

By normal approximation to binomial:

McNemar’s Test: generallyMcNemar’s Test: generally

22)5)(.5)(.(

By normal approximation to binomial:

Equivalently:

No exp

exp No exp

casescontrols

95% CI for difference in 95% CI for difference in dependent proportionsdependent proportions

Diabetes

No diabetes

25 119

MI cases

MI controls

24.05.)0024.(96.115.17.- 32. : CI %95

0024.144

)11.*26.57.*06(.2)83)(.17(.)68)(.32(.

),(2)1()1(

),(2)()()(

~//~/~///

212121

controlscases

ppCovn

ppCovpVarpVarppVar

Each pair is it’s own “age-Each pair is it’s own “age-gender” stratumgender” stratum

Diabetes

No diabetes

Case (MI) Control

Example: Concordant for

exposure (cell “a” from before)

Diabetes

No diabetes

Case (MI) Control

Diabetes

No diabetes

Case (MI) Control

Diabetes

No diabetes

Case (MI) Control

Diabetes

No diabetes

Case (MI) Control

Mantel-Haenszel for pair-Mantel-Haenszel for pair-matched datamatched data

We want to know the relationship between diabetes and MI controlling for age and gender.

Mantel-Haenszel methods apply.

RECALL: The Mantel-Haenszel RECALL: The Mantel-Haenszel Summary Odds RatioSummary Odds Ratio

Exposed

Not Exposed

Case Control

Diabetes

No diabetes

Case (MI) Control

Diabetes

No diabetes

Case (MI) Control

ad/T = 0

bc/T=0

ad/T=1/2

bc/T=0

Diabetes

No diabetes

Case (MI) Control

Diabetes

No diabetes

Case (MI) Control

ad/T=0

bc/T=1/2

ad/T=0

bc/T=0

Mantel-Haenszel Summary ORMantel-Haenszel Summary OR

Mantel-Haenszel Test StatisticMantel-Haenszel Test Statistic(same as McNemar’s)(same as McNemar’s)

nnnnVar

cellsdisc

cellsdisccon cellsdisccase

21111k

)25)(.(

)](5.)(5[.

]5.5.[

)1)(1)(1)(1()(;

)1)(1(

:cells discordant

0 contribute cells Concordant

)1()(n

)E(n :recall

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.

Example: Salmonella Example: Salmonella Outbreak in France, 1996Outbreak in France, 1996

Epidemic CurveEpidemic Curve

Matched Case Control StudyMatched Case Control Study

Case = Salmonella gastroenteritis.

Community controls (1:1) matched for: age group (< 1, 1-4, 5-14, 15-34, 35-44,

45-54, 55-64, or >= 65 years) gender city of residence

ResultsResults

In 2x2 table form: any goat’s In 2x2 table form: any goat’s cheesecheese

Goat’s cheese

Goat’ cheese None

Controls

In 2x2 table form: Brand B In 2x2 table form: Brand B Goat’s cheeseGoat’s cheese

Goat’s cheese B

Goat’ cheese B None

Controls

Brand B

Case (MI) Control

Brand B

Case (MI) Control

Brand B

Case (MI) Control

Brand B

Case (MI) Control

0)12(4

1*0*1*2

)1()(n

011)n(

1*2)E(n :exposed concordant 8

11k11k

1111k11k

nnnnVar

Observed

0)12(4

1*2*1*0

)1()(n

000)n(

1*0)E(n :unexposed concordant 25

11k11k

1111k11k

nnnnVar

Observed

Summary: 8 concordant-exposed pairs (=strata) contribute nothing to the numerator (observed-expected=0) and nothing to the denominator (variance=0).

Summary: 25 concordant-unexposed pairs contribute nothing to the numerator (observed-expected=0) and nothing to the denominator (variance=0).

Summary: 2 discordant “control-exposed” pairs contribute -.5 each to the numerator (observed-expected= -.5) and .25 each to the denominator (variance= .25).

1*1*1*1

)1()(n

5.5.1)n(2

)1)(1( :casefavor cells discordant 24

11k11k

nnnnVar

Observed

1*1*1*1

)1()(n

5.5.0)n(2

)1)(1(:controlfavor cells discordant 2

11k11k

nnnnVar

Observed

Summary: 24 discordant “case-exposed” pairs contribute +.5 each to the numerator (observed-expected= +.5) and .25 each to the denominator (variance= .25).

)25(.26

)25(.22

)25(.2)25(.2400

)]5(.2)5(.24)0(25)0(8[

M:1 matched studiesM:1 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…

2:1 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.

M:1 matched studiesM:1 matched studies Prediagnostic Levels of Carcinoembryonic Antigen and CA Prediagnostic Levels of Carcinoembryonic Antigen and CA

242 in Colorectal Cancer: A Matched Case-Control Study242 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 it’s too costly to test everyone’s blood; so only test stored blood of cases and matched controls from the cohort.

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 (<4 hours, 4–8 hours, >8 hours).

2:1 matching:2:1 matching:

•stratum=matching groupstratum=matching group

•3 subjects per stratum3 subjects per stratum

•6 possible 2x2 tables…6 possible 2x2 tables…

Case (CRC) Controls

Everyone exposed; non-informative

Case exposed; 1 control unexposed

Case (CRC) Controls

0 2Case exposed; both controls unexposed

Case (CRC) Controls

Case unexposed; both controls exposed

Case unexposed; 1 control exposed

Case (CRC) Controls

Everyone unexposed; non-informative

Case (CRC) Controls

1 2102

Case (CRC) Controls

2 Tables with 2 exposed

Case (CRC) Controls

Represents all possible

discordant tables (either 2 or 1 total exposed)

Case (CRC) Controls

)1()() tablesecond(

)1()1()efirst tabl(

DEDEDE

)1()()1(

)exposed total2exposed/ case(

~/~//~//

DEDEDEDEDE

DEDEDE

)1(2)()1(

)1(2)(

)1()()1(

~/~/~//

~//~//

~/~//~//

DEDEDEDE

DEDEDEDEDE

DEDEDE

Case (CRC) Controls

)1()1() tablesecond(

)1()efirst tabl(

DEDEDE

)1()1()1(

)exposed total1exposed/ case(

~/~//~//

DEDEDEDEDE

)1()1()1(

~//~//

~/~//~//

DEDEDEDE

DEDEDEDEDE

SummarySummary

P(case exposed/2 total exposed)=2OR/(2OR+1) P(case unexposed/2 total exposed)=1-2OR/(2OR+1) P(case exposed/1 total exposed) = OR/(OR+2) P(case unexposed/1 total exposed)= 1-OR/(OR+2)

Therefore, we can make a likelihood equation for our data that is a function of the OR, and use MLE to solve for OR

Applying to example dataApplying to example data

ORORdataP

A little complicated to solve further…

Applying to example dataApplying to example data

BD give a more simple robust estimate of OR for 2:1 matching:

0.26)1(1)0(2

)12(2)2(1

exposed) control & exposed total1 where1(#exposed) controls 2 & exposed total2 where#(2

exposed) case & exposed total1 where2(#exposed) case & exposed total2 where#(1

Analysis of matched data HRP 261 02/02/04 Chapter 9 Agresti – read sections 9.1 and 9.2.

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