Summarizing Variation

1
Summarizing Variation
Michael C Neale PhDVirginia Institute for
Psychiatric and Behavioral GeneticsVirginia
Commonwealth University
2
Overview

• Mean
• Variance
• Covariance
• Not always necessary/desirable

3
Computing Mean

• Formula E(xi)/N
• Can compute with
• Pencil
• Calculator
• SAS
• SPSS
• Mx

4
One Coin toss
2 outcomes
Probability
0.6
0.5
0.4
0.3
0.2
0.1
0
Heads
Tails
Outcome
5
Two Coin toss
3 outcomes
Probability
0.6
0.5
0.4
0.3
0.2
0.1
0
HH
HT/TH
TT
Outcome
6
Four Coin toss
5 outcomes
Probability
0.4
0.3
0.2
0.1
0
HHHH
HHHT
HHTT
HTTT
TTTT
Outcome
7
Ten Coin toss
9 outcomes
Probability
0.3
0.25
0.2
0.15
0.1
0.05
0





Outcome
8
Pascal's Triangle
Probability
Frequency
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6
15 20 15 6 1 1 7 21 35 35 21 7 1
1/1 1/2 1/4 1/8 1/16 1/32 1/64 1/128
Pascal's friendChevalier de Mere 1654 Huygens
1657 Cardan 1501-1576
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Fort Knox Toss

Infinite outcomes
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
-1
-2
-3
-4
Heads-Tails
Series 1
Gauss 1827
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Variance

• Measure of Spread
• Easily calculated
• Individual differences

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Average squared deviation
Normal distribution

xi
di
0
1
2
3
-1
-2
-3
Variance G di2/N
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Measuring Variation
Weighs Means

• Absolute differences?
• Squared differences?
• Absolute cubed?
• Squared squared?

13
Measuring Variation
Ways Means

• Squared differences

Fisher (1922) Squared has minimum variance under
normal distribution
14
Covariance

• Measure of association between two variables
• Closely related to variance
• Useful to partition variance

15
Deviations in two dimensions
x














y



















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Deviations in two dimensions
x
dx

dy
y
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Measuring Covariation
Area of a rectangle

• A square, perimeter 4
• Area 1

1
1
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Measuring Covariation
Area of a rectangle

• A skinny rectangle, perimeter 4
• Area .251.75 .4385

.25
1.75
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Measuring Covariation
Area of a rectangle

• Points can contribute negatively
• Area -.251.75 -.4385

1.75
-.25
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Measuring Covariation
Covariance Formula
F E(xi - x)(yi - y)
xy
(N-1)
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Correlation

• Standardized covariance
• Lies between -1 and 1

r F
xy
xy
2
2
F F
y
x
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Summary
Formulae
(Exi)/N
Fx E(xi - )/(N-1)
2
2
Fxy E(xi-x)(yi-y)/(N-1)
r F
xy
xy
2
2
F F
y
x
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Variance covariance matrix
Several variables
Var(X) Cov(X,Y) Cov(X,Z) Cov(X,Y)
Var(Y) Cov(Y,Z) Cov(X,Z) Cov(Y,Z) Var(Z)
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Conclusion

• Means and covariances
• Conceptual underpinning
• Easy to compute
• Can use raw data instead

25
Biometrical Model of QTL
m
d
a
-
a
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Biometrical model for QTL
Diallelic locus A/a with p as frequency of a
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Classical Twin Studies
Information and analysis

• Summary rmz rdz
• Basic model A C E
• rmz A C
• rdz .5A C
• var A C E
• Solve equations

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Contributions to Variance
Single genetic locus

• Additive QTL variance
• VA 2p(1-p) a - d(2p-1) 2
• Dominance QTL variance
• VD 4p2 (1-p)2 d2
• Total Genetic Variance due to locus
• VQ VA VD

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Origin of Expectations
Regression model

• P aA cC eE
• Standardize A C E
• VP a2 c2 e2
• Assumes A C E independent

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Path analysis
Elements of a path diagram

• Two sorts of variable
• Observed, in boxes
• Latent, in circles
• Two sorts of path
• Causal (regression), one-headed
• Correlational, two-headed

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Rules of path analysis

• Trace path chains between variables
• Chains are traced backwards, then forwards, with
  one change of direction at a double headed arrow
• Predicted covariance due to a chain is the
  product of its paths
• Predicted total covariance is sum of covariance
  due to all possible chains

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ACE model
MZ twins reared together
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ACE model
DZ twins reared together
34
ACE model
DZ twins reared apart
35
Model fitting

• Takes care of replicate statistics
• Maximum likelihood estimates
• Confidence intervals on parameters
• Overall fit of model
• Comparison of nested models

36
Fitting models to covariance matrices

• MZ covariances
• 3 statistics V1 CMZ V2
• DZ covariances
• 3 statistics V1 CDZ V2
• Parameters a c e
• Df nstat - npar 6 - 3 3

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Model fitting to covariance matrices

• Inherently compares fit to saturated model
• Difference in fit between A C E model and A E
  model gives likelihood ratio test with df
  difference in number of parameters

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Confidence intervals

• Two basic forms
• covariance matrix of parameters
• likelihood curve
• Likelihood-based has some nice properties
  squares of CIs on a give CI's on a2 Meeker
  Escobar 1995 Neale Miller, Behav Genet 1997

39
Multivariate analysis

• Comorbidity
• Partition into relevant components
• Explicit models
• One disorder or two or three
• Longitudinal data analysis
• Partition into new/old
• Explicit models
• Markov
• Growth curves

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Cholesky Decomposition
Not a model

• Provides a way to model covariance matrices
• Always fits perfectly
• Doesn't predict much else

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Perverse Universe
A
E
.7
.7
P
NOT!
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Perverse Universe
A
E
.7
.7
.7
-.7
X
Y
r(X,Y)0 Problem for almost any multivariate
method
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Analysis of raw data

• Awesome treatment of missing values
• More flexible modeling
• Moderator variables
• Correction for ascertainment
• Modeling of means
• QTL analysis

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Technicolor Likelihood Function
For raw data in Mx
m
ln Li fi 3 ln wj g(xi,ij,Gij)
j1
xi - vector of observed scores on n
subjects ij - vector of predicted means Gij -
matrix of predicted covariances - functions
of parameters
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Pihat Linkage Model for Siblings
Each sib pair i has different COVARIANCE
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Mixture distribution model
Each sib pair i has different set of WEIGHTS
rQ.0
rQ1
rQ.5
weightj x Likelihood under model j
p(IBD2) x P(LDL1 LDL2 rQ 1 )
p(IBD1) x P(LDL1 LDL2 rQ .5 )
p(IBD0) x P(LDL1 LDL2 rQ 0 ) Total
likelihood is product of weighted likelihoods
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Conclusion

• Model fitting has a number of advantages
• Raw data can be analysed with greater flexibility
  
• Not limited to continuous normally distributed
  variables

48
Conclusion II

• Data analysis requires creative application of
  methods
• Canned analyses are of limited use
• Try to answer the question!