Categorical Data

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Categorical Data

• Frühling Rijsdijk1 Caroline van Baal2
• 1IoP, London 2Vrije Universiteit, Adam
• Twin Workshop, Boulder
• Tuesday March 2, 2004

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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

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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
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Univariate Normal Distribution of Liability

• Assumptions
• (1) Underlying normal distribution of liability
  
• (2) The liability distribution has 1 or more
  thresholds (cut-offs)

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The standard Normal distribution

• Liability is a latent variable, 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 gt Normal Probability Density
  function (?)

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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
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• 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
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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
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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
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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

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Bivariate Normal
R.90
R.00
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Bivariate Normal (R0.6) partitioned at threshold
1.4 (z-value) on both liabilities
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Expected Proportions of the BN, for R0.6,
Th11.4, Th21.4
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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
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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

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ACE Liability Model
1
1/.5
E
C
A
A
C
E
L
L
1
1
Unaf
Unaf
Twin 1
Twin 2
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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
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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 gtanalyses limited to
only two variables Raw data analyses -
multivariate - handles missing data - moderator
variables
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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
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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
F is the bivariate normal probability density
function, L1 and L2 are the liabilities of twin1
and twin2, with means 0, and ? is the correlation
matrix of the two liabilities.
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B
L2
a
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
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log-likelihood of the data under the model
subtracted fromlog-likelihood of the observed
frequencies themselves
?² statistic
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The models failure to predict the observed data
i.e. a bad fitting model,is reflected in a
significant ?²
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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
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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
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(0 0)
(1 1)
(0 1)
(1 0)
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? 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
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Practical Exercise 1
Simulated data for 625 MZ and 625 DZ pairs (h2
.40 c2 .20 e2 .40 gt rmz.60 rdz.40)
Dichotomized 0 bottom 88, 1 top 12 This
corresponds to threshold (z-value) of
1.18 Observed counts MZ DZ 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
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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 MZ DZ 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
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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
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Threshold Specification in Mx
3 Categories nthresh2 nvar2 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 LT /
1 0 1 1
t11 t12 t21 t22
t11 t12 t11 t21 t12 t22


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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
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Using Frequency Weights
G1 Data and model for MZ correlation DAta
NGroups2 NInput_vars4 Missing. Ordinal
FilebinF.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 LT / !to ensure t1gtt2gtt3
etc....... FREQ F /
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Example 2 Variables measured in twins x has 2
cat gt 0 below Tx1 , 1 above Tx1 y has 3 cat gt 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
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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
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Types of ascertainment
Single Ascertainment
Complete Ascertainment
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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
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Summary
For a 2 x 2 CT 3 observed statistics, 3
parameters (1 correlation, 2 threshold) df0
? 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 dfgt0 A
significant ?2 reflects departure from normality.
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Summary
Power to detect certain effects increases with
increasing number of categories gt continuous data
most powerful For raw ordinal data analyses,
the first category must be coded 0! Threshold
specification when analyzing CT are different