An Introduction to Matched Pairs Designs

Wang, W. and Liu, X.

February 25, 2013

Outline

1 Related BackgroundMatchingMatched Pairs Designs

2 Theoretical ResultsTheory of Matched StudyMarginal Homogeneity TestMcNemar’s TestLogit Model

3 ImplementationX-rays and Leukemia Example

4 Limitation

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 1 / 19

Related Background Matching

Outline

1 Related BackgroundMatchingMatched Pairs Designs

2 Theoretical ResultsTheory of Matched StudyMarginal Homogeneity TestMcNemar’s TestLogit Model

3 ImplementationX-rays and Leukemia Example

4 Limitation

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 1 / 19

Related Background Matching

What is Matching?

Matching is a strategy to remove bias in the comparison of groupsby ensuring equality of distributions of the matching covariatesemployed.

It changes the sample frame in the analysis from the individual inan unmatched study to the matched set in matched study.

Matching is a common way to control for the effects of a covariate,such as frequency/within-class matching. A common approach tomatching is to construct matched pairs.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 2 / 19

Related Background Matching

What is Matching?

Matching is a strategy to remove bias in the comparison of groupsby ensuring equality of distributions of the matching covariatesemployed.

It changes the sample frame in the analysis from the individual inan unmatched study to the matched set in matched study.

Matching is a common way to control for the effects of a covariate,such as frequency/within-class matching. A common approach tomatching is to construct matched pairs.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 2 / 19

Related Background Matching

What is Matching?

Matching is a strategy to remove bias in the comparison of groupsby ensuring equality of distributions of the matching covariatesemployed.

It changes the sample frame in the analysis from the individual inan unmatched study to the matched set in matched study.

Matching is a common way to control for the effects of a covariate,such as frequency/within-class matching. A common approach tomatching is to construct matched pairs.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 2 / 19

Related Background Matched Pairs Designs

Outline

1 Related BackgroundMatchingMatched Pairs Designs

2 Theoretical ResultsTheory of Matched StudyMarginal Homogeneity TestMcNemar’s TestLogit Model

3 ImplementationX-rays and Leukemia Example

4 Limitation

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 2 / 19

Related Background Matched Pairs Designs

Matched Pairs Designs in Experiment

It is a special case of a randomized block design.

It is used when the experiment has only two treatment conditions.

Subjects can be grouped into pairs, based on some blockingvariable.

In each pair, subjects are randomly assigned to differenttreatments.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 3 / 19

Related Background Matched Pairs Designs

Matched Pairs Designs in Experiment

It is a special case of a randomized block design.

It is used when the experiment has only two treatment conditions.

Subjects can be grouped into pairs, based on some blockingvariable.

In each pair, subjects are randomly assigned to differenttreatments.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 3 / 19

Related Background Matched Pairs Designs

Matched Pairs Designs in Experiment

It is a special case of a randomized block design.

It is used when the experiment has only two treatment conditions.

Subjects can be grouped into pairs, based on some blockingvariable.

In each pair, subjects are randomly assigned to differenttreatments.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 3 / 19

Related Background Matched Pairs Designs

Matched Pairs Designs in Experiment

It is a special case of a randomized block design.

It is used when the experiment has only two treatment conditions.

Subjects can be grouped into pairs, based on some blockingvariable.

In each pair, subjects are randomly assigned to differenttreatments.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 3 / 19

Related Background Matched Pairs Designs

Why Does Matching Reduce Bias?

The matched pairs design is an improvement over a completelyrandomized design.

Like the completely randomized design, the matched pairs designuses randomization to control for confounding.

But not as the other design, the matched pairs design explicitlycontrols for two potential lurking variables.

That means matched pair designs can eliminate the bias!

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 4 / 19

Related Background Matched Pairs Designs

Why Does Matching Reduce Bias?

The matched pairs design is an improvement over a completelyrandomized design.

Like the completely randomized design, the matched pairs designuses randomization to control for confounding.

But not as the other design, the matched pairs design explicitlycontrols for two potential lurking variables.

That means matched pair designs can eliminate the bias!

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 4 / 19

Related Background Matched Pairs Designs

Why Does Matching Reduce Bias?

The matched pairs design is an improvement over a completelyrandomized design.

Like the completely randomized design, the matched pairs designuses randomization to control for confounding.

But not as the other design, the matched pairs design explicitlycontrols for two potential lurking variables.

That means matched pair designs can eliminate the bias!

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 4 / 19

Related Background Matched Pairs Designs

Why Does Matching Reduce Bias?

The matched pairs design is an improvement over a completelyrandomized design.

Like the completely randomized design, the matched pairs designuses randomization to control for confounding.

But not as the other design, the matched pairs design explicitlycontrols for two potential lurking variables.

That means matched pair designs can eliminate the bias!

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 4 / 19

Related Background Matched Pairs Designs

Two typical Matched Pairs Designs

Compare two treatments on one experimental unit. The blockconsists of the one experimental unit.

Compare two treatments on two very similar experimental units.The block consists of the two experimental units.

We usually use the second form.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 5 / 19

Related Background Matched Pairs Designs

Two typical Matched Pairs Designs

Compare two treatments on one experimental unit. The blockconsists of the one experimental unit.

Compare two treatments on two very similar experimental units.The block consists of the two experimental units.

We usually use the second form.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 5 / 19

Related Background Matched Pairs Designs

Two typical Matched Pairs Designs

Compare two treatments on one experimental unit. The blockconsists of the one experimental unit.

Compare two treatments on two very similar experimental units.The block consists of the two experimental units.

We usually use the second form.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 5 / 19

Theoretical Results Theory of Matched Study

Outline

1 Related BackgroundMatchingMatched Pairs Designs

2 Theoretical ResultsTheory of Matched StudyMarginal Homogeneity TestMcNemar’s TestLogit Model

3 ImplementationX-rays and Leukemia Example

4 Limitation

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 5 / 19

Theoretical Results Theory of Matched Study

Theory of Matched Study

A 2 × 2 frequency table:

The likelihood of the sample is

P(e, f ,g | N) =N!

e!f !g!h!πe

11πf12π

g21π

h22.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 6 / 19

Theoretical Results Theory of Matched Study

What We Care About the Study

To test the null hypothesis of marginal homogeneity under

H0 : π•1 = π1• ,

which is equivalent to test the null of symmetry

H0 : π21 = π12 .

To test the hypothesis of no association between exposure anddisease, where the large sample McNemar test can be employed.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 7 / 19

Theoretical Results Theory of Matched Study

What We Care About the Study

To test the null hypothesis of marginal homogeneity under

H0 : π•1 = π1• ,

which is equivalent to test the null of symmetry

H0 : π21 = π12 .

To test the hypothesis of no association between exposure anddisease, where the large sample McNemar test can be employed.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 7 / 19

Theoretical Results Marginal Homogeneity Test

Outline

1 Related BackgroundMatchingMatched Pairs Designs

2 Theoretical ResultsTheory of Matched StudyMarginal Homogeneity TestMcNemar’s TestLogit Model

3 ImplementationX-rays and Leukemia Example

4 Limitation

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 7 / 19

Theoretical Results Marginal Homogeneity Test

Association Test: Marginal Homogeneity Test

Marginal homogeneity test under matching

H0 : P(D|E) = Pm(D|E), or π1• = π•1,

and this is equivalent to

H0 : π12 = π21,

whereπ1• =

∫z

P(D|E , z)fE(z)dz = P(D|E)

π•1 =

∫z

P(D|E , z)fE(z)dz = Pm(D|E)

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 8 / 19

Theoretical Results McNemar’s Test

Outline

1 Related BackgroundMatchingMatched Pairs Designs

2 Theoretical ResultsTheory of Matched StudyMarginal Homogeneity TestMcNemar’s TestLogit Model

3 ImplementationX-rays and Leukemia Example

4 Limitation

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 8 / 19

Theoretical Results McNemar’s Test

McNemar’s Large Sample Test

To test the null, the z-statistic is used:

ZM =π12 − π21√

(π12 + π21)/N=

f − g√f + g

,

which is asymptotically normally distributed under H0. For a two-sidedtest, the equivalent chi-square statistic can be used:

χ2M =

(f − g)2

f + g,

which is asymptotically distributed as chi-square on 1 df. This isMcNemar’s test (McNemar, 1947).

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 9 / 19

Theoretical Results Logit Model

Outline

1 Related BackgroundMatchingMatched Pairs Designs

2 Theoretical ResultsTheory of Matched StudyMarginal Homogeneity TestMcNemar’s TestLogit Model

3 ImplementationX-rays and Leukemia Example

4 Limitation

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 9 / 19

Theoretical Results Logit Model

Logit Model for Matched Pairs

The model is now:

logπij

1− πij= αi + βxij , i = 1, . . . ,N; j = 1,2,

where

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 10 / 19

Theoretical Results Logit Model

Likelihoods of Logit

Li(αi ,β) ∝(eαi+β

1 + eαi+β

)yi1 ( 11 + eαi+β

)1−yi1(

eαi

1 + eαi

)yi2(

11 + eαi

)1−yi2

The corresponding log likelihood is

` =∑

i

αi(yi1 + yi2) +∑

i

yi1β −∑

i

log (1 + eαi )−∑

i

log (1 + eαi+β)

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 11 / 19

Theoretical Results Logit Model

1:M Design Logit Model

Suppose we have n matched sets and that we take i = 0 to representthe case and i = 1, . . . ,M to represent the controls. The logisticregression model is the following:

logit(pj(xij)) = αj + βT xij ,

where αj models the effect of the confounding variables in the j-thmatched set. The conditional probability of the observed outcome, orthat subject i = 0 is the case and the rest are controls is

Lj(β) =exp(βT x0j)∑Mi=0 exp(βT xij)

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 12 / 19

Theoretical Results Logit Model

Likelihood for 1:M Matching

The likelihood for the model is

L(β) =n∏

j=1

Lj(β) =n∏

j=1

{1 +M∑

i=1

exp[βT (xij − x0j)]}−1

Based on the conditional likelihood, we can make inference for themodel using the information matrix.

Note that the log Odds Ratio is β:

logπi1

1− πi1− log

πi2

1− πi2= αi + βxi1 − (αi + βxi2) = β

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 13 / 19

Implementation X-rays and Leukemia Example

Outline

1 Related BackgroundMatchingMatched Pairs Designs

2 Theoretical ResultsTheory of Matched StudyMarginal Homogeneity TestMcNemar’s TestLogit Model

3 ImplementationX-rays and Leukemia Example

4 Limitation

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 13 / 19

Implementation X-rays and Leukemia Example

X-rays and Leukemia Data

Look at the data:

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 14 / 19

Implementation X-rays and Leukemia Example

Data Manipulation

For variable Down syndrome, there are only seven subjects which areall cases without control. This coefficient is infinite if the variable isincluded. So we remove these cases for simplicity here.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 15 / 19

Implementation X-rays and Leukemia Example

Initial Model

Mray, MupRay and MlowRay can be correlated so only Mray is pickedup. The matched sets must be designated by the strata function.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 16 / 19

Implementation X-rays and Leukemia Example

Final Model

We pick out the significant factors and fit the model again with onlylinear CnRay. This time we move out Frayyes.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 17 / 19

Implementation X-rays and Leukemia Example

Ignoring the Matching Structure

If we ignore the matching structure, we may get an incorrect analysis,which is biased.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 18 / 19

Limitation

Limitation

The more confounding variables one specifies, the more difficult tomatch.

It loosens the matching requirements.

It loses the possibility of discovering the effects of the variablesused to determine the matches.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 19 / 19

Limitation

Limitation

The more confounding variables one specifies, the more difficult tomatch.

It loosens the matching requirements.

It loses the possibility of discovering the effects of the variablesused to determine the matches.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 19 / 19

Limitation

Limitation

The more confounding variables one specifies, the more difficult tomatch.

It loosens the matching requirements.

It loses the possibility of discovering the effects of the variablesused to determine the matches.

Wang, W. and Liu, X. An Introduction to Matched Pairs Designs February 25, 2013 19 / 19

Appendix References

References I

[1] Faraway, J. Extending the Linear Model with R. Chapman & Hall/CRC, 2006.

[2] Lachin, J. Biostatistical Methods: The Assessment of Relative Risks SecondEdition. Wiley, 2011.

[3] Casella, G. Statistical Design. Springer, 2008.

Appendix Thank You

Thank you!