Project506-Caconical Correlation Analysis
Menghan Lin, Chunhui Lin, Xinyu Wu
November 26, 2018
Introduction
Canonical correlation is a method to assess correlations between two sets of variables. Canonical correlation is appropriate in the same situations where multiple regression would be, but where are there are many intercorrelated outcome variables.
The following example is an replication of the paper “Examination of the relationships between environmental exposures to volatile organic compounds(VOCs) and biochemical liver tests: Application of canonical correlation analysis”(Liu 2009).
Instead of repeating the examination, we further investigate the correlation in the subgroup controlling on people who drink more than 12 times per year.
The typical purposes of CCA are: 1. Data reduction: explain covariation between two sets of variables using a small number of linear combinations. 2. Data interpretation: find features (canonical variates) that are important for explaining covariation between sets of variables.
Note Canonical correlation terminology makes an important distinction between the words variables and variates. The term variables is reserved for referring to the original variables being analyzed. The term variates is used to refer to variables that are constructed as weighted averages of the original variables. Thus, a set of Y variates is constructed from the original Y variables.
Derivation
If we have two vectors \(X = (X_1 ,X_2,.X_n )^T\) and \(Y = (Y_1,Y_2,.Y_m )^T\) of random variables, and there are correlations among the variables, then canonical-correlation analysis seeks the vectors \(a(a\in R_n)\) and \(b(b\in R_m)\) such that the linear combinations \(a^TX\) and \(b^TY\) maximize the correlation \(\rho= corr(a^TX,b^TY)\). In short, it can be expressed as:
\[(a,b)=arg\max_{a,b}corr(a^TX,b^TY)\]
Let \(\Sigma_{XX}=cov(X,X)\) and \(\Sigma_{YY}=cov(Y,Y)\). The parameter to maximize is\(\rho={{a^T\Sigma_{XY}b}\over{\sqrt{a^T \Sigma_{XX}a}\sqrt{b^T \Sigma_{YY}b}}}\).
To define a change of basis, we set:
\[c=\Sigma_{XX}^{-\frac{1}{2}}a\]
\[d=\Sigma_{YY}^{-\frac{1}{2}}b\]
thus, we have:
\[\rho={{c^T \Sigma_{XX}^{-\frac{1}{2}}\Sigma_{XY}\Sigma_{YY}^{-\frac{1}{2}}d}\over {\sqrt{c^Tc}\sqrt{d^Td}}}\]
By the Cauchy-Schwarz inequality, we have:
\[(c^T \Sigma_{XX} ^{-\frac{1}{2}} \Sigma_{XY} \Sigma_{YY} ^{-\frac{1}{2}}) d \leq (c^T \Sigma_{XX} ^{-\frac{1}{2}} \Sigma_{XY} \Sigma_{YY} ^{-1} \Sigma_{YX} \Sigma_{XX} ^{-\frac{1}{2}} c)^{-\frac{1}{2}} (d^T d)^{-\frac{1}{2}}\]
After canceling the term \((d^T d)^{-\frac{1}{2}}\), we have:
\[\rho \leq {{(c^T\Sigma_{XX}^{-\frac{1}{2}}\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{XX}^{-\frac{1}{2}}c)^{-\frac{1}{2}}}\over{\sqrt{c^Tc}}}\]
There is equality if the vectors d and \((c^T \Sigma_{XX} ^{-\frac{1}{2}} \Sigma_{XY} \Sigma_{YY} ^{-\frac{1}{2}})^T\) are colinear. In addition, the maximum of correlation is attained if c is the eigenvector with the maximum eigenvalue for the matrix \(\Sigma_{XX} ^{-\frac{1}{2}} \Sigma_{XY} \Sigma_{YY} ^{-1} \Sigma_{YX} \Sigma_{XX} ^{-\frac{1}{2}}\).
The subsequent pairs are found by using eigenvalues of decreasing magnitudes. Orthogonality is guaranteed by the symmetry of the correlation matrices.
Data description
The VOC Project of personal exposures to air toxics was conducted among a subsample of NHANES 1999-2000 participants between the ages of 20 and 59 years. This tutorial include 565 observations with 10VOCs: benzene, chloroform, ethylbenzene, tetrachloroethene, toluene, trichloroethene, o-xylene, m-,p-xylene, 1,4-dichloro- benzene, andmethyl tert-butyl ether(MTBE). Besides, liver condition serves as the outcome variables, which was measured by albumin, totalbilirubin(TB), alanineaminotransfer- ase (ALT), aspartateaminotransferase(AST), lactatedehydrogenase(LDH), alkaline phosphatase (ALP) and g-glutamyltransferase(GGT). Controlling on people who drink more than 12 times per year, the subgroup descriptive statistics are also listed below.
Descriptive statistics of personal exposure to 10 VOCs and biochemical liver tests
| n | min | mean | median | 90th percentile | 95th percentile | |
|---|---|---|---|---|---|---|
| 1,4-Dichlorobenzene | 556 | 0.62 | 46.96 | 2.08 | 104.93 | 257.94 |
| Benzene | 557 | 1.25 | 5.36 | 2.88 | 11.16 | 17.22 |
| o-Xylene | 556 | 0.31 | 6.17 | 2.11 | 12.13 | 21.59 |
| m,p-Xylene | 556 | 0.34 | 17.76 | 5.97 | 33.99 | 61.17 |
| Ethylbenzene | 552 | 0.20 | 7.69 | 2.33 | 11.53 | 21.94 |
| MTBE | 555 | 0.60 | 6.13 | 0.60 | 13.69 | 23.47 |
| Toluene | 548 | 2.69 | 34.54 | 16.02 | 54.64 | 92.24 |
| Chloroform | 561 | 0.32 | 2.69 | 1.14 | 6.16 | 10.36 |
| Tetrachloroethene | 553 | 0.18 | 4.59 | 0.78 | 6.24 | 16.15 |
| Trichloroethene | 556 | 0.27 | 4.09 | 0.27 | 1.19 | 6.75 |
| n | min | mean | median | 90th percentile | 95th percentile | |
|---|---|---|---|---|---|---|
| Albumin | 555 | 33.0 | 44.37 | 45.0 | 49.0 | 50.0 |
| ALT | 555 | 7.0 | 26.46 | 20.0 | 45.6 | 57.0 |
| ALP | 555 | 28.0 | 81.47 | 78.0 | 112.6 | 129.0 |
| AST | 555 | 9.0 | 23.99 | 21.0 | 34.0 | 40.3 |
| GGT | 555 | 5.0 | 30.13 | 20.0 | 51.0 | 74.0 |
| LDH | 555 | 63.0 | 147.38 | 144.0 | 182.0 | 195.3 |
| TB | 555 | 1.7 | 9.21 | 8.6 | 13.7 | 17.1 |
Descriptive statistics of personal exposure to 10 VOCs and biochemical liver tests(subgroup)
| n | min | mean | median | 90th percentile | 95th percentile | |
|---|---|---|---|---|---|---|
| 1,4-Dichlorobenzene | 356 | 0.62 | 45.29 | 1.83 | 62.80 | 223.88 |
| Benzene | 359 | 1.25 | 5.48 | 2.88 | 10.95 | 17.66 |
| o-Xylene | 358 | 0.31 | 6.91 | 2.23 | 12.48 | 24.55 |
| m,p-Xylene | 358 | 0.34 | 20.78 | 6.34 | 37.13 | 80.74 |
| Ethylbenzene | 356 | 0.20 | 9.57 | 2.46 | 12.96 | 27.76 |
| MTBE | 356 | 0.60 | 5.96 | 0.60 | 13.23 | 22.58 |
| Toluene | 351 | 2.69 | 38.15 | 16.13 | 54.61 | 92.12 |
| Chloroform | 361 | 0.32 | 2.44 | 1.11 | 5.10 | 8.77 |
| Tetrachloroethene | 356 | 0.18 | 4.98 | 0.82 | 5.38 | 14.76 |
| Trichloroethene | 360 | 0.27 | 2.93 | 0.27 | 1.17 | 7.53 |
| n | min | mean | median | 90th percentile | 95th percentile | |
|---|---|---|---|---|---|---|
| Albumin | 357 | 33.0 | 44.79 | 45.0 | 49.0 | 50.00 |
| ALT | 357 | 8.0 | 28.67 | 22.0 | 51.4 | 65.00 |
| ALP | 357 | 28.0 | 79.68 | 76.0 | 109.0 | 122.40 |
| AST | 357 | 9.0 | 25.21 | 22.0 | 35.0 | 45.20 |
| GGT | 357 | 5.0 | 34.41 | 21.0 | 62.4 | 93.00 |
| LDH | 357 | 63.0 | 147.42 | 143.0 | 187.0 | 199.00 |
| TB | 357 | 3.4 | 9.55 | 8.6 | 15.4 | 17.44 |
Note: To avoid potential confounding, we excluded the patients who had liver conditions, heart disease, stroke, cancer or diabetes. Those tested serum positive to hepatitis C virus (HCV) were also excluded. One observation with extreme values in ALT(1163U/L) and AST(827U/L) is also excluded.
Comparison of Bolm normal transformation
part <- c('o-Xylene','m,p-Xylene')
scatterplotMatrix(combine1[,..part])
scatterplotMatrix(combine2[,..part])
To satisfy the assumption of CCA, we also transformed the VOC and liver function test variables to Blom normal scores (\(s = \Phi(\frac{r-3/8} {n +1/4})\)) from their ranks to assure the multivariate normality is not violated. Here we only use o-Xylene and m,p-Xylene as an example, and it shows that after transformation, their distributions become normal.
Software
CCA output can be fairly complex. Quantities of interest include raw coefficients, structural correlations (or loadings) and standardized coefficients. The first two are of primary focuses in this project. If you want to know more, please read the discussion session and related references below.
R
This tutorial uses the following packages.
library(yacca)
library(data.table)
library(dplyr)Canonical structures of the first pair of canonical variate and F-test (full group)
#Display raw canonical coefficients
cca.fit$xcoef
cca.fit$ycoef| CV 1 | CV 2 | CV 3 | CV 4 | CV 5 | CV 6 | CV 7 | |
|---|---|---|---|---|---|---|---|
| 1,4-Dichlorobenzene | 0.08 | 0.83 | 0.08 | 0.37 | 0.11 | 0.09 | -0.25 |
| Benzene | 0.81 | 0.08 | -0.59 | -0.30 | 0.45 | 0.55 | 0.18 |
| o-Xylene | 0.62 | 0.69 | 0.95 | 0.34 | -0.64 | -0.88 | -1.04 |
| m,p-Xylene | -0.24 | -0.63 | -0.56 | -0.20 | -0.13 | 1.46 | 0.30 |
| Ethylbenzene | -0.03 | -0.29 | 0.08 | 0.82 | 0.60 | -0.39 | 0.42 |
| MTBE | 0.32 | -0.02 | 0.05 | -0.66 | -0.11 | -0.54 | -0.64 |
| Toluene | -0.18 | -0.22 | 0.16 | -0.21 | 0.01 | -0.98 | 0.55 |
| Chloroform | -0.35 | -0.29 | -0.04 | 0.07 | 0.81 | -0.07 | -0.46 |
| Tetrachloroethene | -0.07 | 0.21 | 0.82 | -0.39 | 0.21 | 0.49 | 0.47 |
| Trichloroethene | -0.03 | 0.51 | -0.74 | -0.06 | 0.08 | -0.49 | 0.39 |
| CV 1 | CV 2 | CV 3 | CV 4 | CV 5 | CV 6 | CV 7 | |
|---|---|---|---|---|---|---|---|
| Albumin | 0.97 | -0.24 | 0.64 | 0.20 | -0.09 | 0.06 | -0.07 |
| ALT | 0.09 | -0.63 | -0.69 | 0.27 | 0.17 | -0.90 | 1.16 |
| ALP | 0.52 | 0.61 | -0.51 | 0.15 | -0.39 | -0.22 | -0.28 |
| AST | -0.14 | 0.02 | 0.44 | -1.19 | -0.71 | -0.03 | -0.88 |
| GGT | 0.04 | 0.11 | -0.22 | -0.32 | 0.60 | 1.18 | 0.05 |
| LDH | -0.09 | 0.65 | 0.56 | 0.13 | 0.65 | -0.27 | 0.37 |
| TB | -0.63 | 0.40 | -0.01 | 0.34 | -0.66 | 0.37 | 0.51 |
The raw canonical coefficients are interpreted in a same manner analogous to interpreting regression coefficients. For example, a one unit increase in the concentration of Benzene is associated to a 0.81 increase in the first canonical variate in the set liver tests when all the other variables are held constant.
Next, we are going to compute the loadings of variables on the canonical dimensions(variates). Canonical loadings are known as the correlations between observed variables and canonical variables. These canonical variates are actually a type of latent variable.
cca.fit$corr
cca.fit$xstructcorr #Loadings
cca.fit$ystructcorr
F.test.cca(cca.fit)## CV 1 CV 2 CV 3 CV 4 CV 5 CV 6
## 0.31838580 0.23136020 0.19211314 0.11436996 0.10021086 0.06559313
## CV 7
## 0.02455458| CV 1 | CV 2 | CV 3 | CV 4 | CV 5 | CV 6 | CV 7 | |
|---|---|---|---|---|---|---|---|
| 1,4-Dichlorobenzene | 0.09 | 0.73 | 0.11 | 0.35 | 0.26 | -0.03 | -0.19 |
| Benzene | 0.87 | -0.10 | -0.12 | -0.01 | 0.39 | 0.08 | 0.13 |
| o-Xylene | 0.70 | -0.12 | 0.38 | 0.38 | 0.10 | -0.19 | 0.01 |
| m,p-Xylene | 0.67 | -0.21 | 0.29 | 0.37 | 0.16 | -0.08 | 0.12 |
| Ethylbenzene | 0.58 | -0.22 | 0.26 | 0.49 | 0.26 | -0.21 | 0.24 |
| MTBE | 0.43 | 0.01 | 0.14 | -0.55 | 0.11 | -0.35 | -0.32 |
| Toluene | 0.45 | -0.18 | 0.29 | 0.16 | 0.25 | -0.51 | 0.33 |
| Chloroform | -0.21 | -0.11 | 0.08 | 0.00 | 0.84 | -0.09 | -0.38 |
| Tetrachloroethene | 0.07 | 0.27 | 0.63 | -0.39 | 0.37 | 0.13 | 0.39 |
| Trichloroethene | 0.02 | 0.48 | -0.29 | -0.13 | 0.22 | -0.31 | 0.38 |
| CV 1 | CV 2 | CV 3 | CV 4 | CV 5 | CV 6 | CV 7 | |
|---|---|---|---|---|---|---|---|
| Albumin | 0.68 | -0.13 | 0.48 | 0.00 | -0.32 | 0.22 | 0.38 |
| ALT | 0.28 | -0.06 | -0.19 | -0.56 | -0.11 | -0.16 | 0.73 |
| ALP | 0.51 | 0.66 | -0.49 | -0.11 | -0.19 | -0.10 | 0.00 |
| AST | 0.13 | 0.14 | 0.14 | -0.87 | -0.29 | -0.09 | 0.32 |
| GGT | 0.33 | 0.16 | -0.23 | -0.55 | 0.16 | 0.54 | 0.45 |
| LDH | 0.06 | 0.64 | 0.38 | -0.33 | 0.32 | -0.26 | 0.40 |
| TB | -0.11 | 0.24 | 0.20 | 0.03 | -0.65 | 0.33 | 0.60 |
##
## F Test for Canonical Correlations (Rao's F Approximation)
##
## Corr F Num df Den df Pr(>F)
## CV 1 0.318386 1.821520 70.000000 3202.2 4.284e-05 ***
## CV 2 0.231360 1.244772 54.000000 2804.0 0.1096
## CV 3 0.192113 0.912402 40.000000 2400.2 0.6283
## CV 4 0.114370 0.557007 28.000000 1988.1 0.9709
## CV 5 0.100211 0.461607 18.000000 1561.8 0.9733
## CV 6 0.065593 0.272125 10.000000 1106.0 0.9871
## CV 7 0.024555 0.083556 4.000000 554.0 0.9875
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1A cutoff value of 0.35 are chosen to select important loadings.
The first canonical correlation coefficient was 0.318. The first canonical correlation was statistically significant (F = 1.82, p < 0.05) indicating that two sets of variables were correlated. From the result, we found that personal exposure to Benzene, o-Xylene, m,p-Xylene, Ethylbenzene, MTBE and Toluene as a group might affect the serum levels of Albumin and ALP.
Canonical structures of the first pair of canonical variate and F-test (subgroup)
#Display raw canonical coefficients (subgroup)
cca.fit1$xcoef
cca.fit1$ycoef| CV 1 | CV 2 | CV 3 | CV 4 | CV 5 | CV 6 | CV 7 | |
|---|---|---|---|---|---|---|---|
| 1,4-Dichlorobenzene | -0.09 | 0.79 | -0.35 | 0.44 | 0.19 | 0.05 | 0.30 |
| Benzene | 0.95 | -0.13 | -0.51 | -0.20 | 0.31 | -0.16 | 0.28 |
| o-Xylene | 0.39 | 1.14 | 0.80 | -0.70 | 0.03 | -0.53 | 0.17 |
| m,p-Xylene | -0.08 | -0.62 | -0.23 | 1.10 | -0.26 | 1.21 | -1.89 |
| Ethylbenzene | -0.20 | -0.19 | 0.13 | 0.24 | 0.77 | -0.79 | 1.00 |
| MTBE | 0.39 | 0.00 | -0.24 | -0.32 | -0.19 | -0.13 | -0.02 |
| Toluene | -0.28 | -0.13 | 0.47 | -0.26 | -0.97 | -0.32 | 0.60 |
| Chloroform | -0.38 | -0.22 | 0.12 | -0.26 | 0.51 | -0.63 | -0.29 |
| Tetrachloroethene | -0.16 | 0.47 | 0.22 | -0.69 | 0.28 | 0.48 | 0.06 |
| Trichloroethene | -0.12 | 0.18 | -0.61 | 0.16 | -0.71 | -0.61 | -0.55 |
| CV 1 | CV 2 | CV 3 | CV 4 | CV 5 | CV 6 | CV 7 | |
|---|---|---|---|---|---|---|---|
| Albumin | 0.85 | 0.23 | 0.81 | 0.05 | -0.01 | 0.13 | -0.05 |
| ALT | 0.00 | -0.71 | -0.06 | 0.39 | -1.51 | -0.43 | 0.71 |
| ALP | 0.30 | 0.32 | -0.39 | 0.67 | -0.19 | -0.10 | -0.63 |
| AST | -0.01 | -0.35 | 0.12 | -0.84 | 0.60 | 0.27 | -1.37 |
| GGT | 0.42 | 0.01 | -0.68 | -0.28 | 1.02 | 0.23 | 0.62 |
| LDH | -0.07 | 0.90 | -0.09 | -0.47 | -0.41 | 0.06 | 0.43 |
| TB | -0.69 | -0.01 | -0.17 | 0.41 | -0.11 | 0.91 | 0.11 |
Compared to the full group, a one unit increase in the concentration of Benzene has a larger effect, and is associated to a 0.95 increase in the first canonical variate in the set liver tests when all the other variables are held constant.
cca.fit1$corr
cca.fit1$xstructcorr
cca.fit$xstructcorr
cca.fit1$ystructcorr
cca.fit$ystructcorr
F.test.cca(cca.fit1)## CV 1 CV 2 CV 3 CV 4 CV 5 CV 6
## 0.34584676 0.27173286 0.20181075 0.14155267 0.08342351 0.07952745
## CV 7
## 0.03187424| CV 1 | CV 2 | CV 3 | CV 4 | CV 5 | CV 6 | CV 7 | |
|---|---|---|---|---|---|---|---|
| 1,4-Dichlorobenzene | -0.08 | 0.74 | -0.20 | 0.33 | 0.14 | -0.15 | 0.27 |
| Benzene | 0.82 | 0.11 | 0.09 | -0.06 | 0.09 | -0.32 | 0.05 |
| o-Xylene | 0.47 | 0.38 | 0.65 | 0.19 | -0.01 | -0.29 | -0.27 |
| m,p-Xylene | 0.46 | 0.28 | 0.60 | 0.27 | -0.01 | -0.25 | -0.33 |
| Ethylbenzene | 0.32 | 0.21 | 0.51 | 0.30 | 0.12 | -0.44 | -0.02 |
| MTBE | 0.41 | 0.10 | -0.16 | -0.42 | -0.10 | -0.21 | -0.10 |
| Toluene | 0.23 | 0.21 | 0.58 | -0.01 | -0.42 | -0.39 | 0.22 |
| Chloroform | -0.27 | -0.02 | 0.05 | -0.32 | 0.42 | -0.59 | -0.26 |
| Tetrachloroethene | -0.08 | 0.51 | 0.08 | -0.65 | 0.07 | 0.11 | -0.13 |
| Trichloroethene | -0.15 | 0.31 | -0.43 | -0.10 | -0.38 | -0.44 | -0.39 |
| CV 1 | CV 2 | CV 3 | CV 4 | CV 5 | CV 6 | CV 7 | |
|---|---|---|---|---|---|---|---|
| Albumin | 0.69 | 0.00 | 0.44 | 0.05 | -0.14 | 0.56 | 0.06 |
| ALT | 0.43 | -0.44 | -0.36 | -0.27 | -0.58 | 0.30 | 0.05 |
| ALP | 0.43 | 0.26 | -0.60 | 0.36 | -0.24 | 0.08 | -0.43 |
| AST | 0.29 | -0.28 | -0.28 | -0.57 | -0.31 | 0.43 | -0.40 |
| GGT | 0.58 | -0.22 | -0.59 | -0.20 | 0.07 | 0.40 | 0.24 |
| LDH | 0.11 | 0.57 | -0.28 | -0.57 | -0.46 | 0.23 | 0.02 |
| TB | -0.09 | -0.08 | -0.05 | 0.17 | -0.18 | 0.96 | 0.03 |
##
## F Test for Canonical Correlations (Rao's F Approximation)
##
## Corr F Num df Den df Pr(>F)
## CV 1 0.345847 1.428028 70.000000 2030.16 0.0124 *
## CV 2 0.271733 1.002686 54.000000 1779.05 0.4701
## CV 3 0.201811 0.671618 40.000000 1524.05 0.9428
## CV 4 0.141553 0.434890 28.000000 1263.37 0.9957
## CV 5 0.083424 0.280575 18.000000 993.26 0.9987
## CV 6 0.079527 0.259591 10.000000 704.00 0.9892
## CV 7 0.031874 0.089750 4.000000 353.00 0.9856
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The first canonical correlation coefficient was 0.34. The pooled sum of squares of all canonical correlation coefficients was 0.269, which was contributed by 44.5% by the first canonical correlation. Compared to the full group, the subgroup analysis narrowed down the relationship between the VOC exposure and liver function to fewer numbers of VOCs but more liver function tests. The first canonical correlation indicated that Benzene, o-Xylene, m,p-Xylene and MTBE as a group might affect the serum levels of albumin, ALP, ALT and GGT.
Stata
We need to load the data into STATA using import delimited.
. import delimited data.csv
(18 vars, 565 obs)To view the first three rows of data, we use list command.
. list in 1/3
+-----------------------------------------------------------------------+
1. | v1 | v2 | benzene | oxylene | mpxylene | ethylbe~e |
| 1 | 1.037667 | -1.060735 | 2.163308 | 2.163308 | 2.007744 |
|-----------------------------------------------------------------------|
| mtbe | toluene | chlorof~m | tetrac~e | trichl~e | albumin |
| 1.0924 | 1.88951 | .8161429 | 1.250906 | .5039024 | .0732355 |
|-----------+-----------+-----------+-----------+-----------+-----------|
| alt | alp | ast | ggt | ldh | tb |
| -.6783221 | -.1066309 | -.4419047 | -.6262044 | 1.088382 | -.4615485 |
+-----------------------------------------------------------------------+
+-----------------------------------------------------------------------+
2. | v1 | v2 | benzene | oxylene | mpxylene | ethylbe~e |
| 2 | -.0643453 | .2031609 | -.5292404 | -.3295934 | -.493858 |
|-----------------------------------------------------------------------|
| mtbe | toluene | chlorof~m | tetrac~e | trichl~e | albumin |
| -.5626988 | -.5652983 | .7528 | -.160328 | -.542037 | 1.007743 |
|-----------+-----------+-----------+-----------+-----------+-----------|
| alt | alp | ast | ggt | ldh | tb |
| -1.033876 | -.599451 | -.6154503 | -.4838632 | -.0754589 | .4615485 |
+-----------------------------------------------------------------------+
+-----------------------------------------------------------------------+
3. | v1 | v2 | benzene | oxylene | mpxylene | ethylbe~e |
| 3 | -.2326791 | -.3672921 | -2.211818 | -1.199255 | -.4370206 |
|-----------------------------------------------------------------------|
| mtbe | toluene | chlorof~m | tetrac~e | trichl~e | albumin |
| .5241461 | -1.475034 | -.2993069 | .6506712 | 1.605612 | .3696651 |
|-----------+-----------+-----------+-----------+-----------+-----------|
| alt | alp | ast | ggt | ldh | tb |
| 2.042195 | .6506712 | 1.876456 | 1.792977 | -.013304 | 1.100489 |
+-----------------------------------------------------------------------+We then run the canonical correlation analysis using canon command, specifying the exposure variables (volatile organic compounds) as the first set of variables and the outcome variables (biochemical liver tests) as the second set. From the output, we can see the coefficients, also called canonical weights for the two variable sets and the canonical correlations.
The canonical weights can be used to generate canonical variates. The number of possible canonical variate pairs is equal to the number of variables in the smaller set. This leads to seven possible canonical variate pairs and seven canonical correlations in the output.
. canon (v2 benzene oxylene mpxylene ethylbenzene mtbe toluene chloroform tetra
> chloroethene trichloroethene) (albumin alt alp ast ggt ldh tb)
Canonical correlation analysis Number of obs = 565
Raw coefficients for the first variable set
| 1 2 3 4 5 6 7
-------------+----------------------------------------------------------------------
v2 | 0.0756 0.8255 0.0848 0.3657 0.1051 -0.0897 0.2513
benzene | 0.8105 0.0800 -0.5946 -0.2967 0.4546 -0.5500 -0.1799
oxylene | 0.6197 0.6911 0.9462 0.3385 -0.6435 0.8811 1.0375
mpxylene | -0.2399 -0.6321 -0.5617 -0.1971 -0.1289 -1.4618 -0.3028
ethylbenzene | -0.0325 -0.2859 0.0801 0.8217 0.6042 0.3922 -0.4197
mtbe | 0.3226 -0.0154 0.0538 -0.6598 -0.1084 0.5373 0.6359
toluene | -0.1840 -0.2222 0.1589 -0.2089 0.0055 0.9820 -0.5544
chloroform | -0.3459 -0.2872 -0.0361 0.0722 0.8131 0.0680 0.4626
tetrachlor~e | -0.0666 0.2111 0.8209 -0.3917 0.2137 -0.4856 -0.4713
trichloroe~e | -0.0323 0.5114 -0.7368 -0.0611 0.0812 0.4931 -0.3929
------------------------------------------------------------------------------------
Raw coefficients for the second variable set
| 1 2 3 4 5 6 7
-------------+----------------------------------------------------------------------
albumin | 0.9655 -0.2429 0.6400 0.2047 -0.0927 -0.0558 0.0704
alt | 0.0933 -0.6251 -0.6853 0.2720 0.1690 0.8961 -1.1633
alp | 0.5209 0.6121 -0.5120 0.1461 -0.3885 0.2201 0.2831
ast | -0.1377 0.0208 0.4377 -1.1944 -0.7119 0.0288 0.8795
ggt | 0.0386 0.1082 -0.2176 -0.3201 0.5975 -1.1838 -0.0459
ldh | -0.0874 0.6455 0.5559 0.1323 0.6481 0.2671 -0.3746
tb | -0.6301 0.3990 -0.0146 0.3359 -0.6610 -0.3703 -0.5104
------------------------------------------------------------------------------------
----------------------------------------------------------------------------
Canonical correlations:
0.3184 0.2314 0.1921 0.1144 0.1002 0.0656 0.0246
----------------------------------------------------------------------------
Tests of significance of all canonical correlations
Statistic df1 df2 F Prob>F
Wilks' lambda .796381 70 3202.18 1.8215 0.0000 a
Pillai's trace .219833 70 3878 1.7962 0.0001 a
Lawley-Hotelling trace .236003 70 3824 1.8418 0.0000 a
Roy's largest root .112804 10 554 6.2494 0.0000 u
----------------------------------------------------------------------------
e = exact, a = approximate, u = upper bound on FIn order to find out how many possible canonical correlations would be statistically significant, we can use the test option in canon command as shown below. From the output, we discover that the first two canonical correlations are statistically significant (F = 1.82, P < 0.0000 and F = 1.24, P = 0.1096), indicating that the two sets of variables are correlated. The first canonical correlation is 0.3184 and the second was 0.2214.
. canon, test(1 2 3 4 5 6 7)
Canonical correlation analysis Number of obs = 565
Raw coefficients for the first variable set
| 1 2 3 4 5 6 7
-------------+----------------------------------------------------------------------
v2 | 0.0756 0.8255 0.0848 0.3657 0.1051 -0.0897 0.2513
benzene | 0.8105 0.0800 -0.5946 -0.2967 0.4546 -0.5500 -0.1799
oxylene | 0.6197 0.6911 0.9462 0.3385 -0.6435 0.8811 1.0375
mpxylene | -0.2399 -0.6321 -0.5617 -0.1971 -0.1289 -1.4618 -0.3028
ethylbenzene | -0.0325 -0.2859 0.0801 0.8217 0.6042 0.3922 -0.4197
mtbe | 0.3226 -0.0154 0.0538 -0.6598 -0.1084 0.5373 0.6359
toluene | -0.1840 -0.2222 0.1589 -0.2089 0.0055 0.9820 -0.5544
chloroform | -0.3459 -0.2872 -0.0361 0.0722 0.8131 0.0680 0.4626
tetrachlor~e | -0.0666 0.2111 0.8209 -0.3917 0.2137 -0.4856 -0.4713
trichloroe~e | -0.0323 0.5114 -0.7368 -0.0611 0.0812 0.4931 -0.3929
------------------------------------------------------------------------------------
Raw coefficients for the second variable set
| 1 2 3 4 5 6 7
-------------+----------------------------------------------------------------------
albumin | 0.9655 -0.2429 0.6400 0.2047 -0.0927 -0.0558 0.0704
alt | 0.0933 -0.6251 -0.6853 0.2720 0.1690 0.8961 -1.1633
alp | 0.5209 0.6121 -0.5120 0.1461 -0.3885 0.2201 0.2831
ast | -0.1377 0.0208 0.4377 -1.1944 -0.7119 0.0288 0.8795
ggt | 0.0386 0.1082 -0.2176 -0.3201 0.5975 -1.1838 -0.0459
ldh | -0.0874 0.6455 0.5559 0.1323 0.6481 0.2671 -0.3746
tb | -0.6301 0.3990 -0.0146 0.3359 -0.6610 -0.3703 -0.5104
------------------------------------------------------------------------------------
----------------------------------------------------------------------------
Canonical correlations:
0.3184 0.2314 0.1921 0.1144 0.1002 0.0656 0.0246
----------------------------------------------------------------------------
Tests of significance of all canonical correlations
Statistic df1 df2 F Prob>F
Wilks' lambda .796381 70 3202.18 1.8215 0.0000 a
Pillai's trace .219833 70 3878 1.7962 0.0001 a
Lawley-Hotelling trace .236003 70 3824 1.8418 0.0000 a
Roy's largest root .112804 10 554 6.2494 0.0000 u
----------------------------------------------------------------------------
Test of significance of canonical correlations 1-7
Statistic df1 df2 F Prob>F
Wilks' lambda .796381 70 3202.18 1.8215 0.0000 a
----------------------------------------------------------------------------
Test of significance of canonical correlations 2-7
Statistic df1 df2 F Prob>F
Wilks' lambda .886217 54 2803.96 1.2448 0.1096 a
----------------------------------------------------------------------------
Test of significance of canonical correlations 3-7
Statistic df1 df2 F Prob>F
Wilks' lambda .936336 40 2400.19 0.9124 0.6283 a
----------------------------------------------------------------------------
Test of significance of canonical correlations 4-7
Statistic df1 df2 F Prob>F
Wilks' lambda .972219 28 1988.08 0.5570 0.9709 a
----------------------------------------------------------------------------
Test of significance of canonical correlations 5-7
Statistic df1 df2 F Prob>F
Wilks' lambda .985104 18 1561.78 0.4616 0.9733 a
----------------------------------------------------------------------------
Test of significance of canonical correlations 6-7
Statistic df1 df2 F Prob>F
Wilks' lambda .995097 10 1106 0.2721 0.9871 e
----------------------------------------------------------------------------
Test of significance of canonical correlation 7
Statistic df1 df2 F Prob>F
Wilks' lambda .999397 4 554 0.0836 0.9875 e
----------------------------------------------------------------------------
e = exact, a = approximate, u = upper bound on FNow we focus on the first two sets of canonical weights and we might be interested in which coefficients in each set are significant (P < 0.1). By stderr option, we can call out the standard errors and significance test.
The first set of canonical weights of exposure variables mainly represent Benzene, Chloroform, and MTBE and the first set of canonical weights of outcome variables mainly represent Albumin, ALP, and TB.
The second set of canonical weights of exposure variables mainly represent 1,4-Dichlorobenzene, and Trichloroethene and the second set of canonical weights of outcome variables mainly represent ALT, ALP, LDH and TB.
These results help narrow down the relationship VOCs exposure and liver function tests outcome to fewer numbers of VOCs and liver function tests. This implies that exposure to a cluster of certain VOCs might be associated with certain biochemical liver tests as a group.
. canon (v2 benzene oxylene mpxylene ethylbenzene mtbe toluene chloroform tetra
> chloroethene trichloroethene) (albumin alt alp ast ggt ldh tb), first(2) stde
> rr
Linear combinations for canonical correlations Number of obs = 565
------------------------------------------------------------------------------
| Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
u1 |
v2 | .0756267 .1390955 0.54 0.587 -.1975818 .3488351
benzene | .8105217 .1831084 4.43 0.000 .4508641 1.170179
oxylene | .6196655 .3964418 1.56 0.119 -.1590173 1.398348
mpxylene | -.2399039 .4662774 -0.51 0.607 -1.155756 .6759485
ethylbenzene | -.0325145 .2653383 -0.12 0.903 -.5536864 .4886574
mtbe | .3225807 .1650388 1.95 0.051 -.001585 .6467463
toluene | -.1839661 .1851906 -0.99 0.321 -.5477136 .1797815
chloroform | -.3458607 .1372016 -2.52 0.012 -.6153493 -.0763722
tetrachlor~e | -.0666163 .1499327 -0.44 0.657 -.361111 .2278783
trichloroe~e | -.0323482 .1683552 -0.19 0.848 -.3630279 .2983315
-------------+----------------------------------------------------------------
v1 |
albumin | .9655298 .1524076 6.34 0.000 .666174 1.264886
alt | .0932795 .2231235 0.42 0.676 -.344975 .5315341
alp | .5209347 .1387179 3.76 0.000 .2484679 .7934014
ast | -.1377092 .2155219 -0.64 0.523 -.5610329 .2856144
ggt | .0385658 .1749642 0.22 0.826 -.3050952 .3822267
ldh | -.0874316 .1483353 -0.59 0.556 -.3787886 .2039255
tb | -.6300689 .1546978 -4.07 0.000 -.933923 -.3262148
-------------+----------------------------------------------------------------
u2 |
v2 | .8255232 .1964452 4.20 0.000 .4396696 1.211377
benzene | .0800258 .2586049 0.31 0.757 -.4279204 .5879721
oxylene | .6910712 .5598968 1.23 0.218 -.4086663 1.790809
mpxylene | -.6320947 .6585259 -0.96 0.338 -1.925557 .6613681
ethylbenzene | -.2859168 .3747385 -0.76 0.446 -1.02197 .4501368
mtbe | -.0153545 .2330851 -0.07 0.948 -.4731753 .4424663
toluene | -.2222011 .2615456 -0.85 0.396 -.7359235 .2915213
chloroform | -.2872412 .1937705 -1.48 0.139 -.6678412 .0933588
tetrachlor~e | .2110777 .2117507 1.00 0.319 -.2048386 .6269939
trichloroe~e | .5113532 .2377688 2.15 0.032 .0443327 .9783738
-------------+----------------------------------------------------------------
v2 |
albumin | -.2429123 .215246 -1.13 0.260 -.665694 .1798693
alt | -.6251191 .3151185 -1.98 0.048 -1.244068 -.00617
alp | .6120708 .195912 3.12 0.002 .2272646 .996877
ast | .0208182 .3043827 0.07 0.945 -.5770439 .6186803
ggt | .1082178 .2471027 0.44 0.662 -.3771362 .5935719
ldh | .6454798 .2094947 3.08 0.002 .2339948 1.056965
tb | .3989909 .2184804 1.83 0.068 -.0301437 .8281256
------------------------------------------------------------------------------
(Standard errors estimated conditionally)
Canonical correlations:
0.3184 0.2314 0.1921 0.1144 0.1002 0.0656 0.0246
----------------------------------------------------------------------------
Tests of significance of all canonical correlations
Statistic df1 df2 F Prob>F
Wilks' lambda .796381 70 3202.18 1.8215 0.0000 a
Pillai's trace .219833 70 3878 1.7962 0.0001 a
Lawley-Hotelling trace .236003 70 3824 1.8418 0.0000 a
Roy's largest root .112804 10 554 6.2494 0.0000 u
----------------------------------------------------------------------------
e = exact, a = approximate, u = upper bound on FFinally, we use the estat loadings command to display the structure correlation coefficients, also called canonical loadings. These loadings are correlations between variables and the canonical variates, used to interpret the importance of each original variable in the canonical variates.
. estat loadings
Canonical loadings for variable list 1
| 1 2
-------------+--------------------
v2 | 0.0873 0.7347
benzene | 0.8667 -0.1024
oxylene | 0.6973 -0.1190
mpxylene | 0.6694 -0.2100
ethylbenzene | 0.5779 -0.2223
mtbe | 0.4336 0.0067
toluene | 0.4493 -0.1767
chloroform | -0.2142 -0.1082
tetrachlor~e | 0.0706 0.2696
trichloroe~e | 0.0185 0.4834
----------------------------------
Canonical loadings for variable list 2
| 1 2
-------------+--------------------
albumin | 0.6800 -0.1265
alt | 0.2808 -0.0569
alp | 0.5064 0.6637
ast | 0.1298 0.1422
ggt | 0.3257 0.1612
ldh | 0.0558 0.6438
tb | -0.1148 0.2439
----------------------------------
Correlation between variable list 1 and canonical variates from list 2
| 1 2
-------------+--------------------
v2 | 0.0278 0.1700
benzene | 0.2759 -0.0237
oxylene | 0.2220 -0.0275
mpxylene | 0.2131 -0.0486
ethylbenzene | 0.1840 -0.0514
mtbe | 0.1381 0.0015
toluene | 0.1430 -0.0409
chloroform | -0.0682 -0.0250
tetrachlor~e | 0.0225 0.0624
trichloroe~e | 0.0059 0.1118
----------------------------------
Correlation between variable list 2 and canonical variates from list 1
| 1 2
-------------+--------------------
albumin | 0.2165 -0.0293
alt | 0.0894 -0.0132
alp | 0.1612 0.1536
ast | 0.0413 0.0329
ggt | 0.1037 0.0373
ldh | 0.0178 0.1489
tb | -0.0366 0.0564
----------------------------------Repeat the above steps using subgroup data. This time we only report the output of related to CCA.
. canon (v2 benzene oxylene mpxylene ethylbenzene mtbe toluene chloroform tetra
> chloroethene trichloroethene) (albumin alt alp ast ggt ldh tb)
Canonical correlation analysis Number of obs = 364
Raw coefficients for the first variable set
| 1 2 3 4 5 6 7
-------------+----------------------------------------------------------------------
v2 | -0.0864 0.7927 -0.3509 -0.4380 0.1915 -0.0461 -0.2965
benzene | 0.9536 -0.1311 -0.5125 0.2035 0.3055 0.1649 -0.2804
oxylene | 0.3931 1.1373 0.7959 0.6987 0.0267 0.5340 -0.1730
mpxylene | -0.0786 -0.6241 -0.2293 -1.0983 -0.2600 -1.2125 1.8878
ethylbenzene | -0.1973 -0.1926 0.1325 -0.2430 0.7675 0.7911 -1.0040
mtbe | 0.3857 0.0035 -0.2383 0.3222 -0.1886 0.1266 0.0232
toluene | -0.2762 -0.1281 0.4657 0.2602 -0.9704 0.3166 -0.6035
chloroform | -0.3778 -0.2199 0.1192 0.2600 0.5099 0.6297 0.2853
tetrachlor~e | -0.1556 0.4707 0.2210 0.6919 0.2814 -0.4807 -0.0608
trichloroe~e | -0.1163 0.1813 -0.6071 -0.1560 -0.7052 0.6053 0.5468
------------------------------------------------------------------------------------
Raw coefficients for the second variable set
| 1 2 3 4 5 6 7
-------------+----------------------------------------------------------------------
albumin | 0.8539 0.2318 0.8054 -0.0546 -0.0056 -0.1301 0.0476
alt | -0.0039 -0.7102 -0.0580 -0.3882 -1.5081 0.4344 -0.7116
alp | 0.2985 0.3195 -0.3945 -0.6661 -0.1914 0.0953 0.6314
ast | -0.0061 -0.3476 0.1249 0.8405 0.5994 -0.2740 1.3703
ggt | 0.4186 0.0072 -0.6775 0.2759 1.0163 -0.2260 -0.6180
ldh | -0.0719 0.8987 -0.0876 0.4705 -0.4063 -0.0612 -0.4309
tb | -0.6927 -0.0054 -0.1744 -0.4061 -0.1110 -0.9081 -0.1127
------------------------------------------------------------------------------------
----------------------------------------------------------------------------
Canonical correlations:
0.3458 0.2717 0.2018 0.1416 0.0834 0.0795 0.0319
----------------------------------------------------------------------------
Tests of significance of all canonical correlations
Statistic df1 df2 F Prob>F
Wilks' lambda .755585 70 2030.16 1.4280 0.0124 a
Pillai's trace .268514 70 2471 1.4081 0.0153 a
Lawley-Hotelling trace .29288 70 2417 1.4447 0.0099 a
Roy's largest root .13586 10 353 4.7959 0.0000 u
----------------------------------------------------------------------------
e = exact, a = approximate, u = upper bound on FIn order to find out how many possible canonical correlations would be statistically significant, we can use the test option in canon command as shown below. From the output, we discover that the first canonical correlations are statistically significant (F = 1.43, P < 0.0124), indicating that this set of variables is correlated. The first canonical correlation is 0.3458.
. canon, test(1 2 3 4 5 6 7)
Canonical correlation analysis Number of obs = 364
Raw coefficients for the first variable set
| 1 2 3 4 5 6 7
-------------+----------------------------------------------------------------------
v2 | -0.0864 0.7927 -0.3509 -0.4380 0.1915 -0.0461 -0.2965
benzene | 0.9536 -0.1311 -0.5125 0.2035 0.3055 0.1649 -0.2804
oxylene | 0.3931 1.1373 0.7959 0.6987 0.0267 0.5340 -0.1730
mpxylene | -0.0786 -0.6241 -0.2293 -1.0983 -0.2600 -1.2125 1.8878
ethylbenzene | -0.1973 -0.1926 0.1325 -0.2430 0.7675 0.7911 -1.0040
mtbe | 0.3857 0.0035 -0.2383 0.3222 -0.1886 0.1266 0.0232
toluene | -0.2762 -0.1281 0.4657 0.2602 -0.9704 0.3166 -0.6035
chloroform | -0.3778 -0.2199 0.1192 0.2600 0.5099 0.6297 0.2853
tetrachlor~e | -0.1556 0.4707 0.2210 0.6919 0.2814 -0.4807 -0.0608
trichloroe~e | -0.1163 0.1813 -0.6071 -0.1560 -0.7052 0.6053 0.5468
------------------------------------------------------------------------------------
Raw coefficients for the second variable set
| 1 2 3 4 5 6 7
-------------+----------------------------------------------------------------------
albumin | 0.8539 0.2318 0.8054 -0.0546 -0.0056 -0.1301 0.0476
alt | -0.0039 -0.7102 -0.0580 -0.3882 -1.5081 0.4344 -0.7116
alp | 0.2985 0.3195 -0.3945 -0.6661 -0.1914 0.0953 0.6314
ast | -0.0061 -0.3476 0.1249 0.8405 0.5994 -0.2740 1.3703
ggt | 0.4186 0.0072 -0.6775 0.2759 1.0163 -0.2260 -0.6180
ldh | -0.0719 0.8987 -0.0876 0.4705 -0.4063 -0.0612 -0.4309
tb | -0.6927 -0.0054 -0.1744 -0.4061 -0.1110 -0.9081 -0.1127
------------------------------------------------------------------------------------
----------------------------------------------------------------------------
Canonical correlations:
0.3458 0.2717 0.2018 0.1416 0.0834 0.0795 0.0319
----------------------------------------------------------------------------
Tests of significance of all canonical correlations
Statistic df1 df2 F Prob>F
Wilks' lambda .755585 70 2030.16 1.4280 0.0124 a
Pillai's trace .268514 70 2471 1.4081 0.0153 a
Lawley-Hotelling trace .29288 70 2417 1.4447 0.0099 a
Roy's largest root .13586 10 353 4.7959 0.0000 u
----------------------------------------------------------------------------
Test of significance of canonical correlations 1-7
Statistic df1 df2 F Prob>F
Wilks' lambda .755585 70 2030.16 1.4280 0.0124 a
----------------------------------------------------------------------------
Test of significance of canonical correlations 2-7
Statistic df1 df2 F Prob>F
Wilks' lambda .858239 54 1779.05 1.0027 0.4701 a
----------------------------------------------------------------------------
Test of significance of canonical correlations 3-7
Statistic df1 df2 F Prob>F
Wilks' lambda .926663 40 1524.05 0.6716 0.9428 a
----------------------------------------------------------------------------
Test of significance of canonical correlations 4-7
Statistic df1 df2 F Prob>F
Wilks' lambda .966006 28 1263.37 0.4349 0.9957 a
----------------------------------------------------------------------------
Test of significance of canonical correlations 5-7
Statistic df1 df2 F Prob>F
Wilks' lambda .985757 18 993.263 0.2806 0.9987 a
----------------------------------------------------------------------------
Test of significance of canonical correlations 6-7
Statistic df1 df2 F Prob>F
Wilks' lambda .992666 10 704 0.2596 0.9892 e
----------------------------------------------------------------------------
Test of significance of canonical correlation 7
Statistic df1 df2 F Prob>F
Wilks' lambda .998984 4 353 0.0898 0.9856 e
----------------------------------------------------------------------------
e = exact, a = approximate, u = upper bound on F
Now we focus on the first set of canonical weights and we might be interested in which coefficients in each set are significant (P < 0.1). By stderr option, we can call out the standard errors and significance test.
This set of canonical weights of exposure variables mainly represent Benzene, MTBE, and Chloroform and the first set of canonical weights of outcome variables mainly represent Albumin, ALP, GGT, and TB.
. canon (v2 benzene oxylene mpxylene ethylbenzene mtbe toluene chloroform tetra
> chloroethene trichloroethene) (albumin alt alp ast ggt ldh tb), first(1) stde
> rr
Linear combinations for canonical correlations Number of obs = 364
------------------------------------------------------------------------------
| Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
u1 |
v2 | -.0863798 .161699 -0.53 0.594 -.4043642 .2316047
benzene | .9535735 .2029701 4.70 0.000 .5544285 1.352718
oxylene | .393107 .4568681 0.86 0.390 -.5053334 1.291547
mpxylene | -.0786449 .5300306 -0.15 0.882 -1.120961 .9636712
ethylbenzene | -.1973275 .2819226 -0.70 0.484 -.7517342 .3570792
mtbe | .3857021 .1862609 2.07 0.039 .0194162 .7519879
toluene | -.2761614 .2033818 -1.36 0.175 -.6761159 .1237931
chloroform | -.3778441 .1561604 -2.42 0.016 -.6849368 -.0707515
tetrachlor~e | -.1556496 .1689163 -0.92 0.357 -.4878271 .1765279
trichloroe~e | -.116344 .1957082 -0.59 0.553 -.5012083 .2685202
-------------+----------------------------------------------------------------
v1 |
albumin | .8538873 .1733685 4.93 0.000 .5129546 1.19482
alt | -.0038843 .2738774 -0.01 0.989 -.5424699 .5347012
alp | .2984847 .1597807 1.87 0.063 -.0157273 .6126968
ast | -.0060705 .2554019 -0.02 0.981 -.5083237 .4961826
ggt | .4185558 .2121205 1.97 0.049 .0014165 .8356951
ldh | -.0719023 .1699071 -0.42 0.672 -.4060281 .2622235
tb | -.6927423 .1775667 -3.90 0.000 -1.041931 -.3435538
------------------------------------------------------------------------------
(Standard errors estimated conditionally)
Canonical correlations:
0.3458 0.2717 0.2018 0.1416 0.0834 0.0795 0.0319
----------------------------------------------------------------------------
Tests of significance of all canonical correlations
Statistic df1 df2 F Prob>F
Wilks' lambda .755585 70 2030.16 1.4280 0.0124 a
Pillai's trace .268514 70 2471 1.4081 0.0153 a
Lawley-Hotelling trace .29288 70 2417 1.4447 0.0099 a
Roy's largest root .13586 10 353 4.7959 0.0000 u
----------------------------------------------------------------------------
e = exact, a = approximate, u = upper bound on FFinally, we use the estat loadings command to display the structure correlation coefficients, also called canonical loadings.
. estat loadings
Canonical loadings for variable list 1
| 1
-------------+----------
v2 | -0.0787
benzene | 0.8187
oxylene | 0.4725
mpxylene | 0.4565
ethylbenzene | 0.3241
mtbe | 0.4090
toluene | 0.2334
chloroform | -0.2706
tetrachlor~e | -0.0831
trichloroe~e | -0.1468
------------------------
Canonical loadings for variable list 2
| 1
-------------+----------
albumin | 0.6854
alt | 0.4265
alp | 0.4337
ast | 0.2890
ggt | 0.5831
ldh | 0.1124
tb | -0.0906
------------------------
Correlation between variable list 1 and canonical variates from list 2
| 1
-------------+----------
v2 | -0.0272
benzene | 0.2831
oxylene | 0.1634
mpxylene | 0.1579
ethylbenzene | 0.1121
mtbe | 0.1415
toluene | 0.0807
chloroform | -0.0936
tetrachlor~e | -0.0287
trichloroe~e | -0.0508
------------------------
Correlation between variable list 2 and canonical variates from list 1
| 1
-------------+----------
albumin | 0.2371
alt | 0.1475
alp | 0.1500
ast | 0.1000
ggt | 0.2017
ldh | 0.0389
tb | -0.0313
------------------------SAS
Canonical structures of the first pair of canonical variate and F-test (full group)
proc cancorr data=project.Combine2;
var Benzene o_Xylene m_p_Xylene Ethylbenzene MTBE _1_4_Dichlorobenzene Toluene Chloroform Tetrachloroethene Trichloroethene;
with Albumin ALT ALP AST GGT LDH TB;
run;The output below gives the canonical correlations and the multivariate tests of the dimensions, and also includes the multivariate criteria and the F approximations.
Note The F statistics vary depending on the criteria.
Next, the raw canonical coefficients are shown below.
The raw canonical coefficients are interpreted in a manner analogous to interpreting regression coefficients. For the variable o-Xylene, a one unit increase in o-Xylene leads to a 0.6197 increase in the first canonical variate of set 1 when all of the other variables are held constant.
The raw coefficients are followed by the standardized canonical coefficients shown below. After standardizing, the coefficients are easier to compare because their values don’t depend on the their units. However, the raw coefficients are more interpretable.
Below are correlations between observed variables and canonical variables which are known as the canonical loadings, which SAS labels as the canonical structure.
Through the graphes above, we can narrow down the relationship between the VOC exposure and liver function to fewer numbers of VOCs and liver function tests in the full group. in addition, it can imply that exposure to a cluster of certain VOCs mignt be associated with certain biochemical liver tests as a group.
Canonical structures of the first pair of canonical variate and F-test (subgroup)
proc cancorr data=project.moredrink;
var Benzene o_Xylene m_p_Xylene Ethylbenzene MTBE _1_4_Dichlorobenzene Toluene Chloroform Tetrachloroethene Trichloroethene;
with Albumin ALT ALP AST GGT LDH TB;
run;The output below gives the canonical correlations and the multivariate tests of the dimensions, and also includes the multivariate criteria and the F approximations.
Next, the raw canonical coefficients are shown below.
The raw canonical coefficients are interpreted in a manner analogous to interpreting regression coefficients. For the variable o-Xylene, a one unit increase in o-Xylene leads to a 0.3931 increase in the first canonical variate of set 1 when all of the other variables are held constant.
The raw coefficients are followed by the standardized canonical coefficients shown below.
Below are correlations between observed variables and canonical variables which are known as the canonical loadings, which SAS labels as the canonical structure.
We can compare these graphes with the graphes of full group and see whether the change of group has influence on the correlation between the VOC exposure and liver function tests.
Discussion
The advantages of CCA in this case:
The liver damage caused by isolated VOCs maybe be even worse when facing a cluster of VOCs.
The liver injuries would be better captured by the combination of liver function tests.
Compared to the full model, subgroup implies that liver injuries may be caused by a narrower cluster of VOCs.
Take-home notes
Canonical correlation terminology makes an important distinction between the words variables and variates. The term variables is reserved for referring to the original variables being analyzed. The term variates is used to refer to variables that are constructed as weighted averages of the original variables.
CCA output can be fairly complex. Quantities of particular interest include the correlations between the original variables in each set and their respective canonical variates (structural correlations or loadings). The canonical correlations provide the concordance between the transformed variables, while the loadings reveal the extent to which each canonical variate is associated with particular variables in each set.
In general, the number of canonical dimensions is equal to the number of variables in the smaller set; however, the number of significant dimensions may be even smaller. Canonical dimensions, also known as canonical variates, are latent variables that are analogous to factors in factor analysis.
Alternative Methods
- Multivariate multiple regression
- Separate OLS Regressions
Citation
*Burch, J. B., Everson, T. M., Seth, R. K., Wirth, M. D., & Chatterjee, S. (2015). Trihalomethane exposure and biomonitoring for the liver injury indicator, alanine aminotransferase, in the United States population (NHANES 1999-2006). Science of The Total Environment, 521-522, 226-234. doi:10.1016/j.scitotenv.2015.03.050*
*Liu, J., Drane, W., Liu, X., & Wu, T. (2009). Examination of the relationships between environmental exposures to volatile organic compounds and biochemical liver tests: Application of canonical correlation analysis. Environmental Research, 109(2), 193-199. doi:10.1016/j.envres.2008.11.002*
*Jang, E. S., Jeong, S., Hwang, S. H., Kim, H. Y., Ahn, S. Y., Lee, J.,… Lee, D. H. (2012). Effects of coffee, smoking, and alcohol on liver function tests: A comprehensive cross-sectional study. BMC Gastroenterology, 12(1). doi:10.1186/1471-230x-12-145*
Caconical Correlation by Wikipedia
The Algorithm of CCA(Chinese version)
NCSS Statistical Software Chapter 400 Canonical Correlation