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![Page 1: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/1.jpg)
Summarizing Variation Matrix Algebra
Benjamin NealeAnalytic and Translational Genetics Unit,
Massachusetts General HospitalProgram in Medical and Population Genetics,
Broad Institute of Harvard and MIT
![Page 2: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/2.jpg)
Goals of the session
• Introduce summary statistics
• Learn basics of matrix algebra
• Reinforce basic OpenMx syntax
• Introduction to likelihood
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Computing Mean
• Formula Σ(xi)/N
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Computing Mean
• Formula Σ(xi)/N
Sum
![Page 5: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/5.jpg)
Computing Mean
• Formula Σ(xi)/N
Sum
Each individual score (the i refers to the ith individual)
![Page 6: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/6.jpg)
Computing Mean
• Formula Σ(xi)/N
Sum
Each individual score (the i refers to the ith individual)
Sample size
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Computing Mean
• Formula Σ(xi)/N• Can compute with
![Page 8: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/8.jpg)
Computing Mean
• Formula Σ(xi)/N• Can compute with
• Pencil• Calculator• SAS• SPSS• Mx• R, others….
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Means and matrices
• For all continuous traits we will have a mean• Let’s say we have 100 traits (a lot!)• If we had to specify each mean
separately we would have something like this: • M1 M2 M3 … M99 M100 (without “…”!)
• This is where matrices are useful: they are boxes to hold numbers
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Means and matrices cont’d
• So rather than specify all 100 means in OpenMx we can define a matrix to hold all of our means:
![Page 11: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/11.jpg)
Means and matrices cont’d
• So rather than specify all 100 means in OpenMx we can define a matrix to hold all of our means:
mxMatrix(type=“Full”, nrow=1, ncol=100,
free=TRUE, values=5, name=“M”)
![Page 12: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/12.jpg)
Means and matrices cont’d
• So rather than specify all 100 means in OpenMx we can define a matrix to hold all of our means:
mxMatrix(type=“Full”, nrow=1, ncol=100,
free=TRUE, values=5, name=“M”)
Quick note on specification of matrices in text:
Matrix M has dimension 1,100 meaning 1 row
And 100 columns – two mnemonics – like a procession
rows first then columns, or RC = Roman Catholic
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To the Variance!
We’ll start with coin tossing…
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One Coin toss2 outcomes
Heads TailsOutcome
0
0.1
0.2
0.3
0.4
0.5
0.6Probability
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Two Coin toss3 outcomes
HH HT/TH TT
Outcome
0
0.1
0.2
0.3
0.4
0.5
0.6Probability
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Four Coin toss5 outcomes
HHHH HHHT HHTT HTTT TTTT
Outcome
0
0.1
0.2
0.3
0.4Probability
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Ten Coin toss11 outcomes
Outcome
0
0.05
0.1
0.15
0.2
0.25
0.3Probability
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Fort Knox Toss
0 1 2 3 4-1
-2
-3
-4 Heads-Tails
0
0.1
0.2
0.3
0.4
0.5
Gauss 1827 de Moivre 1738Series 1
Infinite outcomes
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Variance
• Measure of Spread
• Easily calculated
• Individual differences
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Average squared deviationNormal distribution
0 1 2 3 4-1
-2
-3
-4
(Heads-Tails)
0
0.1
0.2
0.3
0.4
0.5
Gauss 1827 de Moivre 1738
xi
Φ(xi)
μ
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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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Variance calculation
1. Calculate mean
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Variance calculation
1. Calculate mean
2. Calculate each squared deviation:
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Variance calculation
1. Calculate mean
2. Calculate each squared deviation:1. Subtract the mean from each observations
and square individually: (xi-μ)2
![Page 26: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/26.jpg)
Variance calculation
1. Calculate mean
2. Calculate each squared deviation:1. Subtract the mean from each observations
and square individually: (xi-μ)2
2. Sum up all the squared deviations
![Page 27: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/27.jpg)
Variance calculation
1. Calculate mean
2. Calculate each squared deviation:1. Subtract the mean from each observations
and square individually: (xi-μ)2
2. Sum up all the squared deviations
3. Divide sum of squared deviations by (N-1)
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Your turn to calculate
Data
Subject Weight in kg
1 80
2 70
3 75
4 85
5 90
1. Calculate mean
2. Calculate each deviation from the mean
3. Square each deviation individually
4. Sum the squared deviations
5. Divide by number of subjects - 1
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Your turn to calculate
Data
Subject Weight in kg
1 80
2 70
3 75
4 85
5 90
1. Calculate mean (80)
2. Calculate each deviation from the mean (0,-10,-5,5,10)
3. Square each deviation individually (0,100,25,25,100)
4. Sum the squared deviations (250)
5. Divide by number of subjects – 1 (250/(5-1))=(250/4)=62.5
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Covariance
• Measure of association between two variables
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Covariance
• Measure of association between two variables
• Closely related to variance
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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 dimensionsμx
μy
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Deviations in two dimensionsμx
μy
dx
dy
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Different covariance
Covariance .3 Covariance .5 Covariance .7
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Covariance calculation
1. Calculate mean for each variables: (μx,μy)
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Covariance calculation
1. Calculate mean for each variables: (μx,μy)
2. Calculate each deviation:
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Covariance calculation
1. Calculate mean for each variables: (μx,μy)
2. Calculate each deviation:1.Subtract the mean for
variable 1 from each observations of variable 1 (xi - μx )
![Page 39: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/39.jpg)
Covariance calculation
1. Calculate mean for each variables: (μx,μy)
2. Calculate each deviation:1. Subtract the mean for
variable 1 from each observations of variable 1 (xi - μx )
2. Likewise for variable 2, making certain to keep the variables paired within subject (yi - μy )
![Page 40: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/40.jpg)
Covariance calculation
1. Calculate mean for each variables: (μx,μy)
2. Calculate each deviation:
1. Subtract the mean for variable 1 from each observations of variable 1 (xi - μx )
2. Likewise for variable 2, making certain to keep the variables paired within subject (yi - μy )
3. Multiply the deviation for variable 1 and variable 2:
(xi - μx ) * (yi - μy )
![Page 41: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/41.jpg)
Covariance calculation
1. Calculate mean for each variables: (μx,μy)
2. Calculate each deviation:
1.Subtract the mean for variable 1 from each observations of variable 1 (xi - μx )
2.Likewise for variable 2, making certain to keep the variables paired within subject (yi - μy )
3. Multiply the deviation for variable 1 and variable 2: (xi - μx ) * (yi - μy )
4. Sum up all the multiplied deviations: Σ(xi - μx ) * (yi - μy )
![Page 42: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/42.jpg)
Covariance calculation
1. Calculate mean for each variables: (μx,μy)
2. Calculate each deviation:
1.Subtract the mean for variable 1 from each observations of variable 1 (xi - μx )
2.Likewise for variable 2, making certain to keep the variables paired within subject (yi - μy )
3. Multiply the deviation for variable 1 and variable 2: (xi
- μx ) * (yi - μy )
4. Sum up all the multiplied deviations: Σ(xi - μx ) * (yi - μy )
5. Divide sum of the multiplied deviations by (N-1):
Σ((xi - μx ) * (yi - μy ))/(N-1)
![Page 43: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/43.jpg)
Your turn to calculate
1. Calculate means
2. Calculate each deviation:1. Subtract the mean X from
each observations of X
2. Likewise for Y
3. Multiply the deviation for X and Y pair-wise
4. Sum up all the multiplied deviation
5. Divide sum of the multiplied deviations by (N-1)
Subject X Y
1 80 180
2 70 175
3 75 170
4 85 190
5 90 185
Data:
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Your turn to calculate
Data: 1. Calculate means: (80,180)
2. Calculate each deviation:1.Subtract the mean for
variable 1 from each observations of variable 1 (0,-10,-5,5,10)
2.Likewise for variable 2, making certain to keep the variables paired within subject (0,-5,-10,10,5)
Subject X Y
1 80 180
2 70 175
3 75 170
4 85 190
5 90 185
![Page 45: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/45.jpg)
Your turn to calculate
Data: 3. Multiply the deviation for variable 1 and variable 2: (0,-10,-5,5,10)*(0,-5,-10,10,5) (0,50,50,50,50)
4. Sum up all the multiplied deviation 0+50+50+50+50=200
5. Divide sum of the multiplied deviations by (N-1): 200/4=50
Subject X Y
1 80 180
2 70 175
3 75 170
4 85 190
5 90 185
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Measuring CovariationCovariance Formula
σxy = Σ(xi - μx) * (yi - μy)
(N-1)
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Measuring CovariationCovariance Formula
σxy = Σ(xi - μx) * (yi - μy)
(N-1)Sum
![Page 48: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/48.jpg)
Measuring CovariationCovariance Formula
σxy = Σ(xi - μx) * (yi - μy)
(N-1)
Each individual score on trait x (the i refers to the ith pair)
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Measuring CovariationCovariance Formula
σxy = Σ(xi - μx) * (yi - μy)
(N-1)Mean of x
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Measuring CovariationCovariance Formula
σxy = Σ(xi - μx) * (yi - μy)
(N-1)
Number of pairs
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Measuring CovariationCovariance Formula
σxy = Σ(xi - μx) * (yi - μy)
(N-1)
Each individual score on trait y (the i refers to the ith pair)
![Page 52: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/52.jpg)
Measuring CovariationCovariance Formula
σxy = Σ(xi - μx) * (yi - μy)
(N-1)Mean of y
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Measuring CovariationCovariance Formula
σxy = Σ(xi - μx) * (yi - μy)
(N-1)Sum
Each individual score on trait x (the i refers to the ith pair)
Mean of x
Number of pairs
Mean of y
Each individual score on trait x (the i refers to the ith pair)
![Page 54: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/54.jpg)
Variances, covariances, and matrices
For multiple traits we will have:
![Page 55: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/55.jpg)
Variances, covariances, and matrices
For multiple traits we will have:a variance for each trait a covariance for each pair
![Page 56: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/56.jpg)
Variances, covariances, and matrices
For multiple traits we will have:a variance for each trait a covariance for each pair
If we have 5 traits we have:
![Page 57: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/57.jpg)
Variances, covariances, and matrices
For multiple traits we will have:a variance for each trait a covariance for each pair
If we have 5 traits we have:5 variances (V1, V2, … V5)10 covariances (CV1-2, CV1-3,…CV4-5
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Variances, covariances, and matrices cont’d
• Just like means, we can put variances and covariances in a box• In fact this is rather convenient,
because:• Cov(X,X) = Var(x) so the organization
is natural
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Variances, covariances, and matrices cont’d
Trait 1 Trait 2 Trait 3 Trait 4 Var(T1)Trait 1
Trait 2
Trait 3
Trait 4
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Variances, covariances, and matrices cont’d
Trait 1 Trait 2 Trait 3 Trait 4 Var(T1)
Var(T2)
Var(T3)
Var(T4)
Trait 1
Trait 2
Trait 3
Trait 4
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Variances, covariances, and matrices cont’d
Trait 1 Trait 2 Trait 3 Trait 4 Cov(T1,T2)Trait 1
Trait 2
Trait 3
Trait 4
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Variances, covariances, and matrices cont’d
Trait 1 Trait 2 Trait 3 Trait 4 Cov(T1,T2) Cov(T1,T3) Cov(T1,T4)
Cov(T2,T1) Cov(T2,T3) Cov(T2,T4)
Cov(T3,T1) Cov(T3,T2) Cov(T3,T4)
Cov(T4,T1) Cov(T4,T2) Cov(T4,T3)
Trait 1
Trait 2
Trait 3
Trait 4
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Variances, covariances, and matrices cont’d
Trait 1 Trait 2 Trait 3 Trait 4 Cov(T1,T2) Cov(T1,T3) Cov(T1,T4)
Cov(T2,T1) Cov(T2,T3) Cov(T2,T4)
Cov(T3,T1) Cov(T3,T2) Cov(T3,T4)
Cov(T4,T1) Cov(T4,T2) Cov(T4,T3)
Trait 1
Trait 2
Trait 3
Trait 4
Note Cov(T3,T1)=Cov(T1,T3)
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Variances, covariances, and matrices cont’d
Trait 1 Trait 2 Trait 3 Trait 4Var(T1) Cov(T1,T2) Cov(T1,T3) Cov(T1,T4)
Cov(T2,T1) Var(T2) Cov(T2,T3) Cov(T2,T4)
Cov(T3,T1) Cov(T3,T2) Var(T3) Cov(T3,T4)
Cov(T4,T1) Cov(T4,T2) Cov(T4,T3) Var(T4)
Trait 1
Trait 2
Trait 3
Trait 4
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Variances, covariances, and matrices in Mx
A variance covariance matrix is symmetric, which means the elements above and below the diagonal are identical
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Variances, covariances, and matrices in Mx
A variance covariance matrix is symmetric, which means the elements above and below the diagonal are identical
A B C D
B E F G
C F H I
D G I J
![Page 67: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/67.jpg)
Variances, covariances, and matrices in Mx
A variance covariance matrix is symmetric, which means the elements above and below the diagonal are identical
A B C D
B E F G
C F H I
D G I J
The Diagonal
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Variances, covariances, and matrices in Mx
A variance covariance matrix is symmetric, which means the elements above and below the diagonal are identical
A B C D
B E F G
C F H I
D G I J
Identical
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Variances, covariances, and matrices in Mx
A variance covariance matrix is symmetric, which means the elements above and below the diagonal are identical
A B C D
B E F G
C F H I
D G I J
Identical
Element 1,2 = Element 2,1
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Variances, covariances, and matrices in Mx
A variance covariance matrix is symmetric, which means the elements above and below the diagonal are identical
A B C D
B E F G
C F H I
D G I J
Identical
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Variances, covariances, and matrices in Mx
A variance covariance matrix is symmetric, which means the elements above and below the diagonal are identical:
Considered another way, a symmetric matrix’s transpose is equal to itself.
![Page 72: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/72.jpg)
Variances, covariances, and matrices in Mx
A variance covariance matrix is symmetric, which means the elements above and below the diagonal are identical:
Considered another way, a symmetric matrix’s transpose is equal to itself.
Transpose is exchanging rows and columns
![Page 73: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/73.jpg)
Variances, covariances, and matrices in Mx
A variance covariance matrix is symmetric, which means the elements above and below the diagonal are identical:
Considered another way, a symmetric matrix’s transpose is equal to itself.
Transpose is exchanging rows and columns e.g.:
A = 1 2 3
4 5 6
![Page 74: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/74.jpg)
Variances, covariances, and matrices in Mx
A variance covariance matrix is symmetric, which means the elements above and below the diagonal are identical:
Considered another way, a symmetric matrix’s transpose is equal to itself.
Transpose is exchanging rows and columns e.g.:
A = A’ =orAT
1 2 3
4 5 6
![Page 75: Summarizing Variation Matrix Algebra Benjamin Neale Analytic and Translational Genetics Unit, Massachusetts General Hospital Program in Medical and Population.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649e525503460f94b480dc/html5/thumbnails/75.jpg)
Variances, covariances, and matrices in Mx
A variance covariance matrix is symmetric, which means the elements above and below the diagonal are identical:
Considered another way, a symmetric matrix’s transpose is equal to itself.
Transpose is exchanging rows and columns e.g.:
A = A’ =orAT
1 2 3
4 5 6
1
2
3
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Variances, covariances, and matrices in Mx
A variance covariance matrix is symmetric, which means the elements above and below the diagonal are identical:
Considered another way, a symmetric matrix’s transpose is equal to itself.
Transpose is exchanging rows and columns e.g.:
A = A’ =orAT
1 2 3
4 5 6
1 4
2 5
3 6
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Variances, covariances, and matrices in Mx
OpenMx uses mxFIMLObjective to calculate the variance-covariance matrix for the data (let’s stick with matrix V)
mxMatrix(type=“Sym”, nrow=2, ncol=2, free=TRUE, values=.5, name=“V”)
type specifies what kind of matrix – symmetric
nrow specifies the number of rows
ncol specifies the number of columns
free means that OpenMx will estimate
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How else can we get a symmetric matrix?
We can use matrix algebra and another matrix type.
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How else can we get a symmetric matrix?
We can use matrix algebra and another matrix type.
We will be using matrix multiplication and lower matrices
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How else can we get a symmetric matrix?
We can use matrix algebra and another matrix type.
We will be using matrix multiplication and lower matrices
Let’s begin with matrix multiplication
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Matrix multiplication
Surprisingly, everyone in this room has completed matrix multiplication
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Matrix multiplication
Surprisingly, everyone in this room has completed matrix multiplication—it’s just been of two 1x1 matrices
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Matrix multiplication
Surprisingly, everyone in this room has completed matrix multiplication—it’s just been of two 1x1 matrices
Let’s extend this to a simple case: Two 2x2 matrices
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Matrix multiplication
Surprisingly, everyone in this room has completed matrix multiplication—it’s just been of two 1x1 matrices
Let’s extend this to a simple case: Two 2x2 matrices
A B
C D
E F
G H%*%
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Matrix multiplication example
A B
C D
E F
G H
%*%
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Matrix multiplication example
A B
C D
E F
G H
RULE: The number of columns in the 1st matrix must equal the number of rows in the 2nd matrix.
2 x 2 2 x 2
%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 1: Determine the size of the box (matrix) needed for the answer
%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 1: Determine the size of the box (matrix) needed for the answer
•The resulting matrix will have dimension: the number of rows of the 1st matrix by the number of columns of the 2nd matrix
%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 1: Determine the size of the box (matrix) needed for the answer
•The resulting matrix will have dimension: the number of rows of the 1st matrix by the number of columns of the 2nd matrix
2 x 2 2 x 2
=
2
%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 1: Determine the size of the box (matrix) needed for the answer
•The resulting matrix will have dimension: the number of rows of the 1st matrix by the number of columns of the 2nd matrix
2 x 2 2 x 2
=
2 x 2
%*%
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Matrix multiplication example
A B
C D
E F
G H
RULE: The number of columns in the 1st matrix must equal the number of rows in the 2nd matrix.
2 x 2 2 x 2
=
2 x 2
%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 2: Calculate each element of the resulting matrix.
2 x 2 2 x 2
=
2 x 2
%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 2: Calculate each element of the resulting matrix.
There are 4 elements in our new matrix: 1,1
2 x 2 2 x 2
=
2 x 2
%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 2: Calculate each element of the resulting matrix.
There are 4 elements in our new matrix: 1,1; 1,2;
2 x 2 2 x 2
=
2 x 2
%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 2: Calculate each element of the resulting matrix.
There are 4 elements in our new matrix: 1,1; 1,2; 2,1;
2 x 2 2 x 2
=
2 x 2
%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 2: Calculate each element of the resulting matrix.
There are 4 elements in our new matrix: 1,1; 1,2; 2,1; 2,2
2 x 2 2 x 2
=
2 x 2
%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 2: Calculate each element of the resulting matrix. Each element is the cross-product of the corresponding row of the 1st matrix and the corresponding column of the 2nd matrix (e.g. element 2,1 will be the cross product of the 2nd row of matrix 1 and the 1st column of matrix 2)
2 x 2 2 x 2
=
2 x 2
%*%
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What’s a cross-product?
•Used in the context of vector multiplication:
•We have 2 vectors of the same length (same number of elements)
•The cross-product ( x ) is the sum of the products of the elements, e.g.:
•Vector 1 = { a b c } Vector 2 = { d e f }
•V1 x V2 = a*d +b*e +c*f
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Matrix multiplication example
A B
C D
E F
G H
Step 2, element 1,1: cross-product row1 matrix1 with column1 matrix2
{ A B } x { E G } = ( A*E + B*G )
2 x 2 2 x 2
=
2 x 2
%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 2, element 1,2: cross-product row1 matrix1 with column2 matrix2
{ A B } x { F H } = ( A*F + B*H )
2 x 2 2 x 2
=
2 x 2
AE + BG%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 2, element 2,1: cross-product row2 matrix1 with column1 matrix2
{ C D } x { E G } = ( C*E + D*G )
2 x 2 2 x 2
=
2 x 2
AE + BG AF + BH%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 2, element 2,2: cross-product row2 matrix1 with column2 matrix2
{ C D } x { F H } = ( C*F + D*H )
2 x 2 2 x 2
=
2 x 2
AE + BG AF + BH
CE + DG
%*%
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Matrix multiplication example
A B
C D
E F
G H
Step 2, element 2,2: cross-product row2 matrix1 with column2 matrix2
{ C D } x { F H } = ( C*F + D*H )
2 x 2 2 x 2
=
2 x 2
AE + BG AF + BH
CE + DG CF + DH
%*%
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Matrix multiplication exercises
2 0
1 4
L
Calculate:
1. L%*% t(L) (L times L transpose)
2. B%*%A
3. A%*%B
4. C%*%A
1
2
5
4 7 -4
A B C -4 2
-3 2
2 -2
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Matrix multiplication answers
4 2
2 17
1
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Matrix multiplication answers
4 2
2 17
1-2
2
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Matrix multiplication answers
4 2
2 17
1-2
2 34 7 4
8 14 8
20 35 20
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Matrix multiplication answers
4 2
2 17
1-2
2 3 4Incomputable4 7 4
8 14 8
20 35 20
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What have we learned?
•Matrix multiplication is not commutative:
•E.g. A*B ≠ B*A
•The product of a lower matrix and its transpose is symmetric
•Not all matrices were made to multiply with one another
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Correlation
• Standardized covariance• Calculate by taking the covariance and dividing
by the square root of the product of the variances• Lies between -1 and 1
r = σxyxy
σx*σ y
2 2
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Calculating Correlation from data
Data:
Subject X Y
1 80 180
2 70 175
3 75 170
4 85 190
5 90 185
1. Calculate mean for X and Y
2. Calculate variance for X and Y
3. Calculate covariance for X and Y
4. Correlation formula is covxy/(√(varx*vary))
5. Variance of X is 62.5, and Variance of Y is 62.5, covariance(X,Y) is 50
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Calculating Correlation from data
Data:
Subject X Y
1 80 180
2 70 175
3 75 170
4 85 190
5 90 185
1. Correlation formula is covxy/(√(varx*vary))
2. Variance of X is 62.5, and Variance of Y is 62.5, covariance(X,Y) is 50
3. Answer: 50/(√(62.5*62.5))=.8
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Standardization
• We can standardize an entire variance covariance matrix and turn it into a correlation matrix
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Standardization
• We can standardize an entire variance covariance matrix and turn it into a correlation matrix
• What will we need?
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Standardization
• We can standardize an entire variance covariance matrix and turn it into a correlation matrix
• What will we need?
r = σxyxy
σx*σ y
2 2
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Standardization
• We can standardize an entire variance covariance matrix and turn it into a correlation matrix
• What will we need?
r = σxyxy
σx*σ y
2 2
Covariance(X,Y)
Variance(Y)Variance(X)
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Standardization back to boxes
• We’ll start with our Variance Covariance matrix:
V1 CV12 CV13
CV21 V2 CV23
CV31 CV32 V3
V =
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Standardization back to boxes
• We’ll start with our Variance Covariance matrix:
•We’ll use 4 operators:
•Dot product (*)
•Inverse (solve)
•Square root (\sqrt)
•Pre-multiply and post-multiply by the transpose (%&%)
V =
V1 CV12 CV13
CV21 V2 CV23
CV31 CV32 V3
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Standardization formula
Formula to standardize V:solve (sqrt (I*V) ) %&% V
Extracts the variances
The dot product does element by element multiplication, so the dimensions of the two matrices must be equal, resulting in a matrix of the same size
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Standardization formula
Formula to standardize V:solve (sqrt (I*V) ) %&% V
Extracts the variances In this case I is an identity matrix of
same size as V (3,3):1 0 0
0 1 0
0 0 1
Identity matrices have the property of returning the same matrix as the original in standard multiplication
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Standardization formula
Formula to standardize V:solve (sqrt (I*V) ) %&% V
Extracts the variances So I*V =
1 0 0
0 1 0
0 0 1
V1 CV12 CV13
CV21 V2 CV23
CV31 CV32 V3
•=
•V1 0 0
•0 V2 0
•0 0 V3
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Standardization formula
Formula to standardize V:solve (sqrt (I*V) ) %&% V
Square root
Take the square root of each element
V1 0 0
0 V2 0
0 0 V3
√V1 0 0
0 √V2 0
0 0 √V3
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Standardization formula
Formula to standardize V:solve (sqrt (I*V) ) %&% V
Inverts the matrix
The inverse of matrix B is the matrix that when multiplied by B yields the identity matrix:
B %*% B-1 = B-1 %*%B = I
Warning: not all matrices have an inverse. A matrix with no inverse is a singular matrix. A zero matrix is a good example.
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Standardization formula
Formula to standardize V:solve (sqrt (I*V) ) %&% V
Inverts the matrix
√V1 0 0
0 √V2 0
0 0 √ V3
-1
•=
1/√V1 0 0
0 1/√V2 0
0 0 1/√V3
Let’s call this K:
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Operator roundup
Operator Function Rule%*% Matrix multiplication C1=R2
* Dot product R1=R2 & C1=C2
/ Element-by- R1=R2 & C1=C2 element division
solve Inverse non-singular+ Element-by- R1=R2 & C1=C2
element addition- Element-by- R1=R2 & C1=C2
element subtraction%&% Pre- and post- C1=R2=C2
multiplicationFull list can be found:
http://openmx.psyc.virginia.edu/wiki/matrix-operators-and-functions