Raw data analysisRaw data analysis

S. Purcell & M. C. Neale

Twin Workshop,

IBG Colorado, March 2002

Raw data vs. summary Raw data vs. summary statisticsstatistics

Zyg T1 T2

1 1.2 0.8

1 -1.3 -2.2

2 0.7 1.9

2 0.2 -0.8

.. ... ...

MZ

1.03

0.87 0.98

DZ

0.95

0.57 1.08

Zyg T1 T2

1 1.2 0.8

1 -1.3 -2.2

2 0.7 1.9

2 0.2 -0.8

.. ... ...

Modelling raw data in Modelling raw data in MxMx

• Pros• Missing data

• Measures of individual fit

• Finite mixture distributions

• Continuous moderator variables

• Cons• Computationally more intensive

• Sensitivity to starting values

Likelihood analysis of raw Likelihood analysis of raw datadata

• What is the probability of observing a given twin pair, assuming a certain trait model?

• 1. e.g. genetic influences very important dissimilar MZ pairs less likely

• 2. e.g. no familial influences dissimilar pairs as likely as similar pairs

• How do we relate, statistically :• Sample-based observed statistics

• Model-based expectations : parameters ?

The Probability ModelThe Probability Model

Data

Mean

Variance

P(X)

X

Observed dataObserved data

P(X)

X

Probability of the data Probability of the data given the modelgiven the model

P(X)

X

Maximum LikelihoodMaximum Likelihood

P(X)

X

-estimate the 2 parametersmean variance

Twin modelTwin model

• Means vector

M1 M2

• Variance-covariance matrix

V1 C21

C12 V2

Bivariate normalBivariate normal

),(~ ΣμX N

Bivariate normal : MZ pairsBivariate normal : MZ pairs

),(~ ΣμX N

High positive correlationHigh positive correlation

-3 -2 -1 0 1 2 3 -3-2

-10

12

3

0.30.40.5

Bivariate normal : DZ pairsBivariate normal : DZ pairs

),(~ ΣμX N

Low correlationLow correlation

-3 -2 -1 0 1 2 3 -3-2

-10

12

3

0.30.40.5

LikelihoodLikelihood

MZ pair

DZ pair

LikelihoodLikelihood

MZ pair

DZ pair

ACE/AE model

LikelihoodLikelihood

MZ pair

DZ pair

CE model

LikelihoodLikelihood

MZ pair

DZ pair

E model

Summary statisticsSummary statistics

• Originally, model-fitting only on summary statistics• variances, covariances, means

• Maximum likelihood covariance matrix fit function

expected covariance matrix

• S observed covariance matrix

• p dimension of S and

ptrML ))((lnln 1SΣSΣ

Raw dataRaw data

• Individual likelihood • probability of the observation conditional on some model.

• x vector of scores (e.g. a twin pair) expected covariance matrix expected mean vector

• Sample log-likelihood = individual log-likelihoods• sum of log-likelihoods product of likelihoods

• assumes independence of observations

)()(log)2log( 1'iiii xx ΣΣ

Option MX%P=<file.name>Option MX%P=<file.name>

• Output individual fit statistics to a file• identify outliers, possible heterogeneity

• For each observation 8 values, including• -2 log likelihood

• Mahalanobis distance

• estimated z-score

• good for detection of outliers with missing data

• half-normal plot

Missing dataMissing data

Zyg A1 B1 C1 A2 B2 C2

MZ 12 9 23 13 7 29

MZ 6 5 22 7 9 19

MZ 10 11 26 10 10 30

MZ 9 8 29 11 9 24

DZ 5 10 21 12 9 28

DZ 10 7 24 7 8 29

DZ 9 6 23 5 12 25

DZ 12 8 25 10 7 21

Missing dataMissing data

Zyg A1 B1 C1 A2 B2 C2

MZ 12 9 23 13 7 29

MZ 6 . 22 7 9 19

MZ 10 11 26 . 10 30

MZ 9 8 . 11 9 24

DZ 5 10 21 12 9 28

DZ 10 7 24 . 8 29

DZ 9 6 23 5 12 25

DZ 12 8 25 . 7 21

Missing dataMissing data

Zyg A1 B1 C1 A2 B2 C2

MZ 12 9 23 13 7 29

MZ 6 -9 22 7 9 19

MZ 10 11 26 -9 10 30

MZ 9 8 -9 11 9 24

DZ 5 10 21 12 9 28

DZ 10 7 24 -9 8 29

DZ 9 6 23 5 12 25

DZ 12 8 25 -9 7 21

Mx implementationMx implementation

• Rectangular datatype• RE file=data.raw

• Means model• as well as a Covariance model

• missing keyword• Missing=-999

• treated as a string

• -999 does not equal -999.00

Example datasetExample dataset

1 1 0.361769 -0.35641

2 1 0.888986 1.46342

3 1 0.535161 0.636073

4 1 1.46187 0.663174

5 1 1.01716 0.346681

… … … … … … … … … … …

Example datasetExample dataset

• MZ covariance matrix

0.55

0.28 0.51

• DZ covariance matrix

0.56

0.15 0.54

• Correlations • MZ 0.53 (= 0.28 / ( 0.55 * 0.51 ) )

• DZ 0.27 (= 0.15 / ( 0.56 * 0.54 ) )

Example datasetExample dataset

• ACE• -2LL 2547.71

• df 1197

• a2 = 0.29

• c2 = 0.00

• e2 = 0.25

• CE• -2LL 2566.33

• df 1198

• c2 = 0.21

• e2 = 0.32

• Model comparison

• A test that the A component is

significantly nonzero is the

deterioration of fit from the ACE

to the CE model

• -2LL 2566.33 - 2547.71

= 18.62

• df 1198 - 1197

= 1

• p-value < 0.0001

Testing differences in meansTesting differences in means

• Do MZ and DZ twins have similar mean values?

• Equating MZ and DZ means• Joint zygosity mean -0.0014

• Model -2LL 2547.707

• df 1196

• Separate MZ and DZ means• MZ mean 0.0161

• DZ mean -0.0159

• Model -2LL 2547.304

• df 1195

Saturated modelSaturated model

• Expected covariance matrix = observed exactly• “Perfect fit”

• No constraints at all on the model

• e.g. variance separately estimated for each twin

• -2LL 2545.425

• df 1190

• (10 parameters : 4 variances, 2 covariances, 4 means)

• ACE model -2LL 2547.71

df 1197