Supplementary material:
A systems analysis of age-related changes
in some cardiac aging traits
Govindaraju et. al.
Sample Framingham Heart Study.
Approach: The null hypothesis of equal covariance among cardiovascular disease risk factors across broad age groups (less than 65 vs. 65 years and older) is tested with a multiple group exploratory structural equation models (ESEM) and tests of mea-surement invariance (MI/ESEM).
Table 1. Source Variables
Mean (SD) Observed
Characteristic or n (%) range
Total number of individuals in the 6th exam / 5124 / (100)Left ventricular mass [M (SD)] / 0.0 / (36.3) / [-94.1-202.5]
Aortic root size [M (SD)] / 0.0 / (0.3) / [-1.3-1.9]
Left aortic diameter [M (SD)] / 0.0 / (0.5) / [-1.4-2.4]
Systolic blood pressure [M (SD)] / 0.0 / (17.6) / [-52.2-113.3]
Diastolic blood pressure [M (SD)] / 0.0 / (9.2) / [-33.8-43.1]
Low density lipoprotein [M (SD)] / 0.0 / (33.1) / [-99.3-223.8]
High density lipoprotein [M (SD)] / 0.0 / (13.5) / [-61.1-80.0]
Triglycerides [M (SD)] / 0.0 / (130.8) / [-212.9-5109.0]
C reactive protein [M (SD)] / 0.0 / (9.1) / [-8.3-248.9]
Fibrinogen [M (SD)] / 0.0 / (65.6) / [-214.0-469.1]
Weight [M (SD)] / 0.0 / (32.7) / [-120.5-172.2]
Height [M (SD)] / 0.0 / (2.6) / [-20.4-8.5]
Age [M (SD)] / 0.0 / (9.1) / [-28.0-28.9]
Hip girth [M (SD)] / 0.0 / (3.8) / [-17.4-18.3]
Individuals use in this study / 3412 / (66.6)
Individuals eliminated due to many missing or incomplete data / 1712 / (33.4)
1712 observations dropped from subsequent analyses due to missing data on the outcomes or standardization variables (HDL or height).
First, anthropometric measures were residualized (weight, waist circumference, and heart anatomy measures) with respect to height (and height squared).
Moreover, inclulding weight and waist circumference in the same model is problematic, because they are dependent upon one another. I will be using weight residualized with respect to height only.
Second, a variable MAP (mean arterial pressure) was created that combines diastolic and systolic blood pressure:
MAP = 2/3 DP + 1/3 SP
Calculation of LDL:
LDL was estimated as TC – HDL – TG/5. (Friedewald et al. 1972)
Third, LDL and Triglyclerides were residualized with respect to HDL and did not include HDL in the model.
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C-Reactive Protein
C-reactive protein was included; when it was included in multivariate (factor analytic) models, Mplus claimed it had a negative residual variance. This can happen when it is co-linear with another variable. CRP was not unduly correlated with other variables, and none could be identified that suggested a non-linear relationship existed.
:
Aortic Root Size Omitted
Aortic Root Size was weakly correlated with other heart anatomy measures and did not load in either of the factors in our two factor model.
Blom Transformation
The Blom transformation is useful to address skew in the distribution variables. The Blom transformation is the normalized percentile rank approximation of the normal distribution (Blom, 1958; Statistical estimates and transformed beta variables. New York: John Wiley Sons, Inc.) The Blom score for observation ı, zi , with corresponding observed value yi and rank ri , is :
−1 ri −3/8 n+1/4
where Φ−1 is the inverse of the cumulative function, n is the sample size of non-missing observa- tions, and the scaling for the rank of observation i, ri ,is 1 ri n. Variables transformed using the Blom have the property of having a mean 0 and unit standard deviation, and the normal(z) scores have the property of being interpretable as approximate percentile ranks using a normal back-transformation.
Table 2. Blom-standardized Versions of Source Variables
Mean (SD) Observed
Characteristic or n (%) range
Total [n (%)] / 3412 / (100)Left ventricular mass (rwrt hgt) [Blom] [M (SD)] / 0.0 / (1.0) / [-3.5-3.5]
Aortic root size (rwrt hgt) [Blom] [M (SD)] / 0.0 / (1.0) / [-3.6-3.6]
Left aortic diameter (rwrt hgt) [Blom] [M (SD)] / 0.0 / (1.0) / [-3.6-3.6]
Mean arterial pressure [Blom] [M (SD)] / 0.0 / (1.0) / [-3.6-3.6]
Low density lipoprotein (rwrt HDL) [Blom] [M (SD)] / 0.0 / (1.0) / [-3.6-3.6]
Triglycerides (rwrt HDL) [Blom] [M (SD)] / 0.0 / (1.0) / [-3.6-3.6]
C reactive protein [Blom] [M (SD)] / 0.0 / (1.0) / [-3.6-3.6]
Fibrinogen [Blom] [M (SD)] / 0.0 / (1.0) / [-3.6-3.6]
Weight (rwrt hgt) [Blom] [M (SD)] / 0.0 / (1.0) / [-3.6-3.6]
Height [M (SD)] / 0.0 / (2.6) / [-20.4-8.5]
Age [M (SD)] / 0.0 / (9.1) / [-28.0-28.9]
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Table 3. Blom-standardized Versions of Source Variables, by Age and Sex Group
Total [n (%)] / 1268 / (100) / 1115 / (100) / 543 / (100) / 486 / (100)Left ventricular mass (rwrt hgt) [Blom] [M (SD)] / -0.1 / (0.8) / -0.1 / (1.1) / 0.3 / (0.9) / 0.1 / (1.2) / F=14.7; P 0.001
Aortic root size (rwrt hgt) [Blom] [M (SD)] / -0.1 / (0.9) / -0.1 / (1.1) / 0.3 / (0.9) / 0.2 / (1.1) / F=30.2; P 0.001
Left aortic diameter (rwrt hgt) [Blom] [M (SD)] / -0.1 / (0.9) / -0.1 / (1.0) / 0.2 / (1.0) / 0.2 / (1.2) / F=16.4; P 0.001
Mean arterial pressure [Blom] [M (SD)] / -0.1 / (1.0) / 0.0 / (1.0) / 0.2 / (1.0) / 0.0 / (1.0) / F=11.4; P 0.001
Low density lipoprotein (rwrt HDL) [Blom] [M (SD)] / -0.1 / (1.0) / 0.0 / (1.0) / 0.2 / (1.0) / 0.0 / (0.9) / F=6.0; P 0.001
Triglycerides (rwrt HDL) [Blom] [M (SD)] / 0.0 / (1.0) / 0.0 / (1.0) / 0.1 / (1.0) / 0.0 / (1.0) / F=0.8; P = 0.472
C reactive protein [Blom] [M (SD)] / -0.1 / (1.0) / 0.0 / (0.9) / 0.0 / (1.2) / 0.2 / (0.9) / F=12.5; P 0.001
Fibrinogen [Blom] [M (SD)] / -0.1 / (1.0) / -0.1 / (1.0) / 0.3 / (1.0) / 0.3 / (1.0) / F=34.9; P 0.001
Weight (rwrt hgt) [Blom] [M (SD)] / 0.0 / (1.0) / 0.1 / (1.0) / -0.1 / (1.0) / -0.2 / (0.9) / F=13.2; P 0.001
Age [M (SD)] / -4.4 / (6.5) / -4.3 / (6.7) / 10.3 / (5.3) / 9.9 / (5.5) / F=1285.5; P 0.001
Group
Younger women Younger men Older women Older Figure 1. Scatterplot M
Matrix of Source Outcome Variables
igure 2. Scatterplot Matrix of Blom-standardized Outcome Variables
Table 4. Exploratory SEM - Configural Invariance (no parameter constraints)
Young Young Older Older
Women Men Women Men
Indicator F1 F2 F1 F2 F1 F2 F1 F2
Unstandardized Factor LoadingsLeft ventricular mass (rwrt hgt) [Blom] / 0.52 / -0.01 / 0.64 / 0.01 / 0.68 / 0.14 / 0.78 / 0.00
Left aortic diameter (rwrt hgt) [Blom] / 0.53 / -0.04 / 0.65 / -0.13 / 0.54 / -0.12 / 0.66 / -0.10
Mean arterial pressure [Blom] / 0.16 / 0.34 / 0.04 / 0.45 / 0.12 / 0.29 / 0.07 / 0.28
Low density lipoprotein (rwrt HDL) [Blom] / -0.01 / 0.50 / -0.11 / 0.23 / 0.00 / 0.40 / -0.11 / 0.26
Triglycerides (rwrt HDL) [Blom] / -0.03 / 0.19 / -0.09 / 0.40 / -0.06 / 0.32 / -0.12 / 0.42
Fibrinogen [Blom] / 0.24 / 0.30 / 0.12 / 0.12 / 0.18 / -0.03 / 0.05 / 0.05
Weight (rwrt hgt) [Blom] / 0.74 / 0.25 / 0.53 / 0.24 / 0.61 / -0.01 / 0.51 / 0.40
Unstandardized Factor Mean / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00
Unstandardized Factor Variance / 1.00 / 1.00 / 1.00 / 1.00 / 1.00 / 1.00 / 1.00 / 1.00
Unstandardized Factor Covariance 0.13 0.21 0.02 0.04
Standardized Factor LoadingsLeft ventricular mass (rwrt hgt) [Blom] / 0.62 / -0.01 / 0.57 / 0.01 / 0.78 / 0.16 / 0.63 / 0.00
Left aortic diameter (rwrt hgt) [Blom] / 0.58 / -0.04 / 0.66 / -0.13 / 0.55 / -0.13 / 0.55 / -0.09
Mean arterial pressure [Blom] / 0.16 / 0.34 / 0.04 / 0.47 / 0.12 / 0.28 / 0.07 / 0.28
Low density lipoprotein (rwrt HDL) [Blom] / -0.01 / 0.49 / -0.11 / 0.24 / 0.00 / 0.40 / -0.12 / 0.27
Triglycerides (rwrt HDL) [Blom] / -0.03 / 0.20 / -0.08 / 0.38 / -0.06 / 0.34 / -0.12 / 0.42
Fibrinogen [Blom] / 0.24 / 0.31 / 0.12 / 0.12 / 0.18 / -0.03 / 0.05 / 0.05
Weight (rwrt hgt) [Blom] / 0.72 / 0.25 / 0.53 / 0.24 / 0.62 / -0.01 / 0.57 / 0.45
Standardized Factor Covariance 0.13 0.21 0.02 0.04
RMSEA / 0.067CFI / 0.928
χ2 (df ) / 152.852 / (32)
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Table 5. Summary of Measurement Noninvariance ESEM Models.
Model / Description / CFI / RMSEA / χ2(df ) / ∆χ2 (df ) / P1 / Configural Invariance / 0.928 / 0.067 / 152.852 (32)
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4 Strict Factorial Invariance (Metric and Scalar In- variance with Equality Constraints on Residual Variances)
no convergence
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5 Weak Structural Invariance (Strict Factorial In- variance with Equality Constraints on Com- mon Factor Variances, Covariances [But Not Means])
not estimated
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Notes: CFI, Confirmatory Fit Index; RMSEA, Root Mean Squared Error of Approximation. Five planned models are listed. The first three converged. The model constraints were too restrictive for the fourth model, and convergence was not achieved. The fourth and fifth models are therefore not estimated.
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