With the new version of JMP 17, the Multivariate Correlations platform now supports Partial correlation diagram can now be drawn. In this article, I will explain what the partial correlation coefficient is by comparing it with the correlation coefficient, and explain the merits of drawing a partial correlation diagram for multivariate data.
What is partial correlation
As the name "partial correlation diagram" suggests, it shows the partial correlation coefficient. Partial correlation is the correlation of pairs after removing the influence of other variables. When there is a variable that confounds the pair of variables you want to examine (confounding variable), it is used to remove the influence of the confounding variable and examine the true relationship between the pairs.
JMP 17 and earlier can also calculate the partial correlation coefficient matrix as follows, but it is difficult to comprehensively see the relationships between all pairs for the six variables just by looking at these numbers. is. It becomes even more difficult with a larger number of variables. In such a case, a partial correlation diagram that visualizes the relationship of partial correlations is useful.
Partial Correlation Coefficient and Correlation Coefficient
The examples of the partial correlation diagram and partial correlation matrix shown above use the following data.
2019 Average values for test items by prefecture in specific medical examinations (male)
(Data source : Ministry of Health, Labor and Welfare NDB open data)
This is the average test value (abdominal girth, fasting blood sugar, blood pressure, etc.) for people who have undergone specific medical examinations (for those aged 40 and over) by prefecture. There are various test values in actual data, but here we are targeting six test values used to judge metabolic syndrome.
For these test values (continuous scale), use "multivariate correlation" to find the correlation coefficient for each pair.
What is required here is It is the correlation coefficient of two pairs and is completely unaffected by variables other than the pair of interest.
Let's add a partial correlation matrix report and compare it with the correlation coefficient matrix.
Comparing the correlation coefficient matrix and the partial correlation coefficient matrix, we can see that some pairs of coefficients are nearly identical, while others have very different coefficients.
For example, let's take a look at the relationship between "neutral fat" enclosed in a red frame and other test items (abdominal circumference, diastolic blood pressure) (Table 1).
| |
correlation coefficient
|
partial correlation coefficient
|
Neutral fat and waist circumference
|
0.4811
|
0.4465
|
Triglycerides and diastolic blood pressure
|
0.2041
|
-0.1616
|
table 1. Correlation coefficient and partial correlation coefficient
"Neutral fat" and "waist circumference" are correlation coefficients and partial correlation coefficients. There is no value difference. It shows little influence of other variables (HDL, systolic blood pressure, diastolic blood pressure, fasting blood glucose).
On the other hand, in the relationship between "neutral fat" and "diastolic blood pressure," the correlation coefficient shows a slightly positive correlation of 0.2041, but the partial correlation coefficient shows a slightly negative correlation of -0.1616. showing correlation. The correlation coefficient (absolute value) is low, so it does not indicate a strong relationship, but if one is told that one is positively correlated and the other is negatively correlated, the conclusion will differ depending on the statistic used. I don't know what to do."
Correlation and regression analysis
The relationship between correlation coefficients and partial correlation coefficients can also be thought of as the relationship between simple regression analysis and multiple regression analysis.
In simple regression analysis, one explanatory variable, X, creates a model that explains the variation in the target variable, Y.
Multiple regression analysis, on the other hand, creates a model that explains the variation in Y with multiple explanatory variables (e.g., X1 and X2). When there is a correlation between X1 and X2, the estimated coefficients of X1 have the effect of X2 removed, and similarly the estimated coefficients of X2 have the effect of X1 removed. For this reason, the coefficients of each term estimated by multiple regression analysis are called "partial regression coefficients." (In JMP, this is included in the Estimated Values report under Parameter Estimates.)
For this data, we will perform the following simple regression analysis and multiple regression analysis in the "fit model" platform.
Lower left (single regression analysis) Y: triglycerides, X: diastolic blood pressure
Lower right (multiple regression analysis) Y: neutral fat, X: diastolic blood pressure, waist circumference, HDL, systolic blood pressure, fasting blood sugar
Note the estimate of 'Diastolic Blood Pressure' (partial regression coefficient) in the report 'Parameter Estimates'. The single regression analysis has a positive coefficient of 1.009, while the multiple regression analysis has a negative coefficient of -0.8255.
If you also refer to the "prediction profile" below, the results of simple regression analysis show that triglycerides also increase as diastolic blood pressure increases, but the results of multiple regression analysis show that triglycerides increase as diastolic blood pressure increases. It's going to be a relationship that will be low in fat. You may feel that the conclusions are different between the two, but this result shows that the correlation coefficient between triglycerides and diastolic blood pressure shown in Table 1 is positive (=0.2041), but the partial correlation coefficient is negative. (= -0.1616).
Drawing a partial correlation diagram
Now that you understand the partial correlation, click the red triangle button on the upper left of the report. [Partial correlation diagram] Choose.
Set the options and properties for the displayed partial correlation chart as follows.
- Select Show Partial Correlation from the red triangle to the left of the partial correlation plot
- Make the following settings in "Edge properties" on the right side of the partial correlation diagram
- Set absolute value of partial correlation to 0.25
- Move the width slider to the right
The vertices of each variable in the partial correlation diagram are determined by the eigenvectors of the partial correlation coefficients, so pairs with large coefficients are close together. here Since the absolute value of the partial correlation is specified as 0.25, only correlations above 0.25 or below -0.25 are connected by lines.
When you look at the relationship of each variable in terms of partial correlation, you can see the following from the partial correlation diagram.
- High (positive) correlation between diastolic blood pressure (so-called lower blood pressure) and systolic blood pressure (so-called upper blood pressure)
- Triglycerides are (positively) correlated with waist circumference, systolic blood pressure, and fasting blood glucose
- HDL is negatively correlated with waist circumference and systolic blood pressure
When showing multivariable relationships, it is common to use partial correlations to examine the relationship between pairs excluding the influence of other variables. At that time, if you visualize it using the partial correlation diagram introduced this time, you can easily consider the overall relationship of variables.
In addition, you can check the operation method until drawing a partial correlation diagram in the following video. (about 3 minutes)
https://community.jmp.com/t5/video/gallerypage/video-id/6316240369112
