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Summarizing Variation Matrix Algebra & Mx Michael C Neale PhD Virginia Institute for Psychiatric and Behavioral Genetics Virginia Commonwealth University 19 th 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 (x i )/N Can compute with Pencil Calculator SAS SPSS Mx

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One Coin toss 2 outcomes HeadsTails Outcome 0 0.1 0.2 0.3 0.4 0.5 0.6 Probability

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Two Coin toss 3 outcomes HHHT/THTT Outcome 0 0.1 0.2 0.3 0.4 0.5 0.6 Probability

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Four Coin toss 5 outcomes HHHHHHHTHHTTHTTTTTTT Outcome 0 0.1 0.2 0.3 0.4 Probability

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Ten Coin toss 9 outcomes Outcome 0 0.05 0.1 0.15 0.2 0.25 0.3 Probability

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Fort Knox Toss 01234-2-3-4 Heads-Tails 0 0.1 0.2 0.3 0.4 0.5 De Moivre 1733 Gauss 1827 Infinite outcomes

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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 Pascal's friend Chevalier de Mere 1654; Huygens 1657; Cardan 1501-1576 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 Frequency Probability

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Variance Measure of Spread Easily calculated Individual differences

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Average squared deviation Normal distribution 0123-2-3 : xixi didi Variance = G d i 2 /N

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Measuring Variation Absolute differences? Squared differences? Absolute cubed? Squared squared? Weighs & Means

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Measuring Variation Squared differences Ways & Means 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:x :y:y

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:x:x :y:y dx dy

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Measuring Covariation A square, perimeter 4 Area 1 Concept: Area of a rectangle 1 1

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Measuring Covariation A skinny rectangle, perimeter 4 Area.25*1.75 =.4385 Concept: Area of a rectangle 1.75.25

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Measuring Covariation Points can contribute negatively Area -.25*1.75 = -.4385 Concept: Area of a rectangle 1.75 -.25

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Measuring Covariation Covariance Formula: Average cross-product of deviations from mean F = E (x i - : x )(y i - : y ) xy N

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Correlation Standardized covariance Lies between -1 and 1 r = F xy 2 2 y x F * F

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Summary Formulae for sample statistics; i=1…N observations : = ( E x i )/N F x = E (x i - : x ) / (N) 2 2 r = F xy 2 2 x x F F xy = E (x i -: x )(y i -: y ) / (N)

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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

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Height of normal curve Probability density function 0123-2-3 :x:x xixi N (x i ) N (x i ) is the likelihood of data point x i for particular mean & variance estimates

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Height of bivariate normal curve An unlikely pair of (x,y) values :x:x xixi yiyi :y:y

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Exercises: Compute Normal PDF Get used to Mx script language Use matrix algebra Taste of likelihood theory

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