12.2.2 - Correlation Matrix

12.2.2 - Correlation Matrix

When examining correlations for more than two variables (i.e., more than one pair), correlation matrices are commonly used. In Minitab, if you request the correlations between three or more variables at once, your output will contain a correlation matrix with all of the possible pairwise correlations. For each pair of variables, Pearson's r will be given along with the p value. The following pages include examples of interpreting correlation matrices. 

12.2.2.1 - Example: Student Survey

12.2.2.1 - Example: Student Survey

Construct a correlation matrix to examine the relationship between how many hours per week students exercise, their heights, their weights, and their resting pulse rates.

This example uses the 'StudentSurvey' dataset from the Lock5 textbook. The data was collected from a sample of 362 college students.

  To construct a correlation matrix in Minitab...

  1. Open the Minitab file: StudentSurvey.mpx
  2. Select Stat > Basic Statistics > Correlation
  3. Enter the variables Exercise, Height, Weight and Pulse into the Variables box
  4. Select the Graphs... button and select Correlations and p-values from the dropdown
  5. Select the Results... button and verify that the Correlation matrix and the Pairwise correlation table boxes are checked
  6. Click OK and OK

This should result in the following output:

Correlation
  Exercise Height Weight
Height 0.118    
Weight 0.118 0.619  
Pulse -0.172 -0.082 -0.120
Pairwise Pearson Correlations
Sample 1 Sample 2 N Correlation 95% CI for ρ P-Value
Height Exercise 354 0.118 (0.014, 0.220) 0.026
Weight Exercise 356 0.118 (0.015, 0.220) 0.026
Pulse Exercise 361 -0.172 (-0.271, -0.071) 0.001
Weight Height 352 0.619 (0.551, 0.680) 0.000
Pulse Height 355 -0.082 (-0.184, 0.023) 0.124
Pulse Weight 357 -0.120 (-0.221, -0.016) 0.023

Interpretation

When we look at the matrix graph or the pairwise Pearson correlations table we see that we have six possible pairwise combinations (every possible pairing of the four variables). Let's say we wanted to examine the relationship between exercise and height. We would find the row in the pairwise Pearson correlations table where these two variables are listed for sample 1 and sample 2. In this case, that is the first row. The correlation between exercise and height is 0.118 and the p-value is 0.026.

If we were conducting a hypothesis test for this relationship, these would be step 2 and 3 in the 5 step process.

12.2.2.2 - Example: Body Correlation Matrix

12.2.2.2 - Example: Body Correlation Matrix

Construct a correlation matrix using the variables age (years), weight (Kg), height (cm), hip girth, navel (or abdominal girth), and wrist girth.

This example is using the body dataset. These data are from the Journal of Statistics Education data archive.

For this example, you can use the following Minitab file: body.dat.mpx

  To construct a correlation matrix in Minitab...

  1. Open the Minitab file: StudentSurvey.mpx
  2. Select Stat > Basic Statistics > Correlation
  3. Enter the variables Age(years), Weight (Kg), Height (cm), Hip girth at level of bitrochan, Navel (or "Abdominal") girth, and Wrist minimum girth into the Variables box
  4. Select the Graphs... button and select Correlations and p-values from the dropdown
  5. Select the Results... button and verify that the Correlation matrix and Pairwise correlation table boxes are checked
  6. Click OK and OK

This should result in the following partial output:

Pairwise Pearson Correlations
Sample 1 Sample 2 N Correlation 95% CI for ρ P-Value
Weight (Kg) Age (years) 507 0.207 (0.122, 0.289) 0.000
Height (cm) Age (years) 507 0.068 (-0.019, 0.154) 0.127
Hip girth at level of bitrochan Age (years) 507 0.227 (0.143, 0.308) 0.000
Navel (or "Abdominal") girth at Age (years) 507 0.422 (0.348, 0.491) 0.000
Wrist minimum girth Age (years) 507 0.192 (0.107, 0.275) 0.000
Height (cm) Weight (Kg) 507 0.717 (0.672, 0.757) 0.000
Hip girth at level of bitrochan Weight (Kg) 507 0.763 (0.724, 0.797) 0.000
Navel (or "Abdominal") girth at Weight (Kg) 507 0.712 (0.666, 0.752) 0.000
Wrist minimum girth Weight (Kg) 507 0.816 (0.785, 0.844) 0.000
Hip girth at level of bitrochan Height (cm) 507 0.339 (0.259, 0.413) 0.000
Navel (or "Abdominal") girth at Height (cm) 507 0.313 (0.232, 0.390) 0.000
Wrist minimum girth Height (cm) 507 0.691 (0.642, 0.734) 0.000
Navel (or "Abdominal") girth at Hip girth at level of bitrochan 507 0.826 (0.796, 0.852) 0.000
Wrist minimum girth Hip girth at level of bitrochan 507 0.459 (0.387, 0.525) 0.000
Wrist minimum girth Navel (or "Abdominal") girth at 507 0.435 (0.362, 0.503) 0.000

Cell contents grouped by Age, Weight, Height, Hip Girth, and Abdominal Girth; First row: Pearson correlation, Following row: P-Value

Cell contents grouped by Age, Weight, Height, Hip Girth, and Abdominal Girth; First row: Pearson correlation, Following row: P-Value

This correlation matrix presents 15 different correlations. For each of the 15 pairs of variables, the 'Correlation' column contains the Pearson's r correlation coefficient and the last column contains the p value. 

The correlation between age and weight is \(r=0.207\). This correlation is statistically significant (\(p=0.000\)). That is, there is evidence of a relationship between age and weight in the population.

The correlation between age and height is \(r=0.068\). This correlation is not statistically significant (\(p=0.127\)). There is not enough evidence of a relationship between age and height in the population.

The correlation between weight and height is \(r=0.717\). This correlation is statistically significant (\(p<0.000\)). That is, there is evidence of a relationship between weight and height in the population.

And so on.

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