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Categorical Data. Frühling Rijsdijk 1 & Caroline van Baal 2 1 IoP, London 2 Vrije Universiteit, A’dam Twin Workshop, Boulder Tuesday March 2, 2004. Aims. Introduce Categorical Data Define liability and describe assumptions of the liability model

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categorical data

Categorical Data

Frühling Rijsdijk1 & Caroline van Baal2

1IoP, London 2Vrije Universiteit, A’dam

Twin Workshop, Boulder

Tuesday March 2, 2004

slide2
Aims
  • Introduce Categorical Data
  • Define liability and describe assumptions of the liability model
  • Show how heritability of liability can be estimated from categorical twin data
  • Practical exercises
categorical data1
Categorical data

Measuring instrument is able to only discriminate between two or a few ordered categories : e.g. absence or presence of a disease

Data therefore take the form of counts, i.e. the number of individuals within each category

univariate normal distribution of liability
Univariate Normal Distribution of Liability

Assumptions:

(1) Underlying normal distribution of liability

(2) The liability distribution has 1 or more thresholds (cut-offs)

the standard normal distribution

68%

-2

3

-

0

1

1

2

-3

The standard Normal distribution

Liability is a latentvariable, the scale is arbitrary,

distribution is, therefore, assumed to be a

Standard Normal Distribution (SND) or z-distribution:

  • mean() = 0 and SD () = 1
  • z-values are the number of SD away from the mean
  • area under curve translates directly to probabilities > Normal Probability Density function ()
slide6

Area=P(z  z0)

z0

3

-3

Standard Normal Cumulative Probability in right-hand tail

(For negative z values, areas are found by symmetry)

Z0 Area

0 .50 50%

.2 .42 42%

.4 .35 35%

.6 .27 27%

.8 .21 21%

1 .16 16%

1.2 .12 12%

1.4 .08 8%

1.6 .06 6%

1.8 .036 3.6%

2 .023 2.3%

2.2 .014 1.4%

2.4 .008 .8%

2.6 .005 .5%

2.8 .003 .3%

2.9 .002 .2%

slide7

3

-3

When we have one variable it is possible to find a z-value (threshold) on the SND, so that the proportion exactly matches the observed proportion of the sample

  • i.e if from a sample of 1000 individuals, 150 have met a criteria for a disorder (15%): the z-value is 1.04

1.04

two categorical traits

Trait2

Trait2

Trait1

0

1

Trait1

0

1

00

01

545

(.76)

75

(.11)

0

0

10

11

56

(.08)

40

(.05)

1

1

Two categorical traits

When we have two categorical traits, the data are

represented in a Contingency Table, containing

cell counts that can be translated into proportions

0 = absent

1 = present

categorical data for twins

Twin2

Twin1

0

1

a

b

00

01

0

c

d

10

11

1

Categorical Data for twins:

When the measured trait is dichotomous i.e. a disorder either present or absent in an unselected sample of twins:

cell a: number of pairs concordant for unaffected

cell d: number of pairs concordant for affected

cell b/c: number of pairs discordant for the disorder

0 = unaffected

1 = affected

joint liability model for twin pairs
Joint Liability Model for twin pairs
  • Assumed to follow a bivariate normal distribution
  • The shape of a bivariate normal distribution is determined by the correlation between the traits
  • Expected proportions under the distribution can be calculated by numerical integration with mathematical subroutines
slide13

Liab 2

Liab 1

0

1

.87

.05

0

.05

.03

1

Expected Proportions of the BN, for R=0.6, Th1=1.4, Th2=1.4

slide14

Twin2

Twin1

0

1

a

b

00

01

0

c

d

10

11

1

Correlated dimensions:

The correlation (shape) and the two thresholds determine the relative proportions of observations in the 4 cells of the CT.

Conversely, the sample proportions in the 4 cells can be used to estimate the correlation and the thresholds.

c

c

d

d

a

b

b

a

twin models
Twin Models
  • A variance decomposition (A, C, E) can be applied to liability, where the correlations in liability are determined by path model
  • This leads to an estimate of the heritability of the liability
ace liability model

¯

¯

Aff

Aff

ACE Liability Model

1

1/.5

E

C

A

A

C

E

L

L

1

1

Unaf

Unaf

Twin 1

Twin 2

summary
Summary

It is possible to estimate a correlation

between categorical traits from simple

counts (CT), because of the assumptions

we make about their joint distributions

how can we fit ordinal data in mx
How can we fit ordinal data in Mx?
  • Summary statistics: CT
  • Mx has a built-in fit function for the maximum-likelihood analysis of 2-way Contingency Tables
  • >analyses limited to only two variables
  • Raw data analyses
  • - multivariate
  • - handles missing data
  • - moderator variables
model fitting to ct
Model-fitting to CT
  • Mx has a built in fit function for the maximum-likelihood
  • analysis of 2-way Contingency Tables
  • The Fit Function is twice the log-likelihood
  • of the observed frequency data calculated as:

nij is the observed frequency in cell ij

pij is the expected proportion in cell ij

expected proportions
Expected proportions

Are calculated by numerical integration of the

bivariate normal over two dimensions:

the liabilities for twin1 and twin2

e.g. the probability that both twins are affected :

Φ is the bivariate normal probability density function, L1and L2 are the liabilities of twin1 and twin2, with means 0, and  is the correlation matrix of the two liabilities.

slide21

d

B

a

L2

B

L1

For example: for a correlation of .9 and thresholds (z-values)

of 1, the probability that both twins are above threshold

(proportion d) is around .12

The probability that both twins are are below threshold

(proportion a) is given by another integral function with

reversed boundaries :

and is around .80 in this example

slide22

χ² statistic:

log-likelihood of the data under the model subtracted fromlog-likelihood of the observed frequencies themselves:
slide23

The model’s failure to predict the observed

data i.e. a bad fitting model,is reflected

in a significant χ²

model fitting to raw ordinal data
Model-fitting to Raw Ordinal Data
  • ordinal ordinal
  • Zyg respons1 respons2
  • 1 0 0
  • 1 0 0
  • 1 0 1
  • 2 1 0
  • 2 0 0
  • 1 1 1
  • 2 . 1
  • 2 0 .
  • 2 0 1
model fitting to raw ordinal data1
Model-fitting to Raw Ordinal Data
  • The likelihood of a vector of ordinal responses is
  • computed by the Expected Proportion in
  • the corresponding cell of the MN

Expected proportion are calculated by

numerical integration of the MN normal

over n dimensions. In this example it will

be two, the liabilities for twin1 and twin2

slide26

(0 0)

(1 1)

(0 1)

(1 0)

slide27

 is the MN pdf, which is a function of

, the correlation matrix of the variables

By maximizing the likelihood of the data

under a MN distribution, the ML estimate of the

correlation matrix and the thresholds are obtained

practical exercise 1
Practical Exercise 1

Simulated data for 625 MZ and 625 DZ pairs

(h2 = .40 c2 = .20 e2 = .40 > rmz=.60 rdz=.40)

Dichotomized 0 = bottom 88%, 1 = top 12%

This corresponds to threshold (z-value) of 1.18

Observed counts:

MZDZ

0 1 0 1

0 508 48 0 497 59

1 35 34 1 49 20

Raw ORD File: bin.dat

Scripts: tetracor.mx and ACEbin.mx

practical exercise 2
Practical Exercise 2

Same simulated data

Categorized 0 = bottom 22%, 1 = mid 66%,

2 = top 12%

This corresponds to thresholds (z-values) of

-0.75 1.18

Observed counts:

MZDZ

0 1 2 0 1 2

0 80 58 1 0 63 74 2

1 68 302 47 1 71 289 57

2 1 34 34 2 4 45 20

Raw ORD File: cat.dat

Adjust the correlation and ACE script

threshold specification in mx
Threshold Specification in Mx

2 Categories

Threshold Matrix : 1 x 2

T(1,1) T(1,2) threshold twin1 & twin2

3 Categories

Threshold Matrix : 2 x 2

T(1,1) T(1,2) threshold 1 for twin1 & twin2

T(2,1) T(2,2) threshold 2 for twin1 & twin2

threshold specification in mx1
Threshold Specification in Mx

3 Categories

nthresh=2 nvar=2

Matrix T: nthresh nvar (2 x 2)

T(1,1) T(1,2) threshold 1 for twin1 & twin2

T(2,1) T(2,2) increment

L LOW nthresh nthresh

Value 1 T 1 1 to T nthresh nthresh

Threshold Model L*T /

1 0

1 1

t11 t12

t21 t22

t11 t12

t11 + t21 t12 + t22

=

*

using frequency weights
Using Frequency Weights
  • ordinal ordinal
  • Zyg respons1 respons2 FREQ
  • 1 0 0 508
  • 1 0 1 48
  • 1 1 0 35
  • 1 1 1 34
  • 2 0 0 497
  • 2 0 1 59
  • 2 1 0 49
  • 2 1 1 20

The 1250 lines data file (bin.dat) can be summarized like this

using frequency weights1
Using Frequency Weights

G1: Data and model for MZ correlation

DAta NGroups=2 NInput_vars=4

Missing=.

Ordinal File=binF.dat

Labels

zyg bin1 bin2 freq

SELECT IF zyg = 1

SELECT bin1 bin2 freq /

DEFINITION freq /

Begin Matrices;

R STAN 2 2 FREE !Correlation matrix

T FULL nthresh nvar FREE !thresh tw1, thresh tw2

L LOW nthresh nthresh

F FULL 1 1

End matrices;

Value 1 L 1 1 to L nthresh nthresh ! initialize L

COV R / !Predicted Correlation matrix for MZ pairs

Thresholds L*T / !to ensure t1>t2>t3 etc.......

FREQ F /

slide34

Example: 2 Variables measured in twins:

  • x has 2 cat > 0 below Tx1 , 1 above Tx1
  • y has 3 cat > 0 below Ty1 , 1 Ty1 - Ty2 , 2 above Ty2
  • Ordinal respons vector (x1, y1, x2, y2)
  • For example (1 2 0 1)
  • The likelihood of this vector of observations is
  • the Expected Proportion in
  • the corresponding cell of the MN :
proband ascertained samples
Proband-Ascertained Samples

For rare disorders (e.g. Schizophrenia), selecting a random sample of twins will lead to the vast majority of pairs being unaffected.

A more efficient design is to ascertain twin pairs through a register of affected individuals.

When an affected twin (the proband) is identified, the cotwin is followed up to see if he or she is also affected.

There are several types of ascertainment

types of ascertainment
Types of ascertainment

Single

Ascertainment

Complete

Ascertainment

ascertainment correction
Ascertainment Correction

Omission of certain classes from observation leads to an increase

of the likelihood of observing the remaining individuals

Mx corrects for incomplete ascertainment by dividing the likelihood

by the proportion of the population remaining after ascertainment

CT from ascertained data can be analysed in Mx

by simply substituting a –1 for the missing cells

CTable 2 2

-1 11

-1 13

summary1
Summary

For a 2 x 2 CT:

3 observed statistics, 3 parameters

(1 correlation, 2 threshold) df=0 

any pattern of observed frequencies

can be accounted for, no goodness of fit

of the normal distribution assumption.

This problem is solved when we have a CT

which is at least 3 x 2: df>0

A significant 2 reflects departure from

normality.

slide39

Summary

Power to detect certain effects increases

with increasing number of categories >

continuous data most powerful

For raw ordinal data analyses, the first category

must be coded 0!

Threshold specification when analyzing CT are

different