Summarizing Variation Matrix Algebra

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Summarizing VariationMatrix Algebra Mx
Michael C Neale PhDVirginia Institute for
Psychiatric and Behavioral GeneticsVirginia
Commonwealth University19th International
workshop on Methodology Twin and Family Studies
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Overview

• Mean/Variance/Covariance
• Calculating
• Estimating by ML
• Matrix Algebra
• Normal Likelihood Theory
• Mx script language

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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
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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
De Moivre 1733 Gauss 1827
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Dinosaur (of a) Joke

• Elk
• The Theory by A. Elk brackets Miss brackets.
  My theory is along the following lines.
• Host
• Oh God.
• Elk
• All brontosauruses are thin at one end, much
  MUCH thicker in the middle, and then thin again
  at the far end.

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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 friend Chevalier de Mere 1654 Huygens
1657 Cardan 1501-1576
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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?

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Measuring Variation
Ways Means

• Squared differences

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

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

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Deviations in two dimensions
x














y



















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Deviations in two dimensions
x
dx

dy
y
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Measuring Covariation
Concept Area of a rectangle

• A square, perimeter 4
• Area 1

1
1
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Measuring Covariation
Concept Area of a rectangle

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

.25
1.75
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Measuring Covariation
Concept Area of a rectangle

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

1.75
-.25
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Measuring Covariation
Covariance Formula Average cross-product of
deviations from mean
F E(xi - x)(yi - y)
xy
N
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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 for sample statistics i1N observations
(Exi)/N
Fx E (xi - x ) / (N)
2
2
Fxy E(xi-x )(yi-y ) / (N)
r F
xy
xy
2
2
F F
x
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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Variance covariance matrix
Univariate Twin Data
Var(Twin1) Cov(Twin1,Twin2)
Cov(Twin2,Twin1) Var(Twin2) Only
suitable for complete data Good conceptual
perspective
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Conclusion

• Means and covariances
• Basic input statistics for Traditional SEM
• Easy to compute
• Can use raw data instead

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Likelihood computation
Calculate height of curve
-1
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Height of normal curve
Probability density function
x
N(xi)
0
1
2
3
-1
-2
-3
xi
N(xi) is the likelihood of data point xi for
particular mean variance estimates
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Height of bivariate normal curve
An unlikely pair of (x,y) values
y
yi
x
xi
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Exercises Compute Normal PDF

• Get used to Mx script language
• Use matrix algebra
• Taste of likelihood theory