Interpret the key results for Correlation

Complete the following steps to interpret a correlation analysis. Key output includes the Pearson correlation coefficient, the Spearman correlation coefficient, and the p-value.

Step 1: Examine the linear relationship between variables (Pearson)

Use the Pearson correlation coefficient to examine the strength and direction of the linear relationship between two continuous variables.

Strength

The correlation coefficient can range in value from −1 to +1. The larger the absolute value of the coefficient, the stronger the relationship between the variables.

For the Pearson correlation, an absolute value of 1 indicates a perfect linear relationship. A correlation close to 0 indicates no linear relationship between the variables.
Direction

The sign of the coefficient indicates the direction of the relationship. If both variables tend to increase or decrease together, the coefficient is positive, and the line that represents the correlation slopes upward. If one variable tends to increase as the other decreases, the coefficient is negative, and the line that represents the correlation slopes downward.

The following plots show data with specific correlation values to illustrate different patterns in the strength and direction of the relationships between variables.

No relationship: Pearson r = 0

The points fall randomly on the plot, which indicates that there is no linear relationship between the variables.

Moderate positive relationship: Pearson r = 0.476

Some points are close to the line but other points are far from it, which indicates only a moderate linear relationship between the variables.

Large positive relationship: Pearson r = 0.93

The points fall close to the line, which indicates that there is a strong linear relationship between the variables. The relationship is positive because as one variable increases, the other variable also increases.

Large negative relationship: Pearson r = −0.968

The points fall close to the line, which indicates that there is a strong negative relationship between the variables. The relationship is negative because, as one variable increases, the other variable decreases.

Consider the following points when you interpret the correlation coefficient:
  • It is never appropriate to conclude that changes in one variable cause changes in another based on correlation alone. Only properly controlled experiments enable you to determine whether a relationship is causal.
  • The Pearson correlation coefficient is very sensitive to extreme data values. A single value that is very different from the other values in a data set can greatly change the value of the coefficient. You should try to identify the cause of any extreme value. Correct any data entry or measurement errors. Consider removing data values that are associated with abnormal, one-time events (special causes). Then, repeat the analysis.
  • A low Pearson correlation coefficient does not mean that no relationship exists between the variables. The variables may have a nonlinear relationship. To check for nonlinear relationships graphically, create a Scatterplot or use Fitted Line Plot.

Correlation: Hydrogen, Porosity, Strength

Correlations Hydrogen Porosity Porosity 0.625 0.017 Strength -0.790 -0.527 0.001 0.053

Cell Contents: Pearson correlation P-Value

Key Result: Pearson correlation

In these results, the Pearson correlation between porosity and hydrogen is 0.625, which indicates that there is a moderate positive relationship between the variables. The Pearson correlation between strength and hydrogen is −0.790, and between strength and porosity is −0.527. The relationship between these variables is negative, which indicates that, as hydrogen and porosity increase, strength decreases.

Step 2: Determine whether the correlation coefficient is significant

To determine whether the correlation between variables is significant, compare the p-value to your significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. An α of 0.05 indicates that the risk of concluding that a correlation exists—when, actually, no correlation exists—is 5%. The p-value tells you whether the correlation coefficient is significantly different from 0. (A coefficient of 0 indicates that there is no linear relationship.)
P-value ≤ α: The correlation is statistically significant
If the p-value is less than or equal to the significance level, then you can conclude that the correlation is different from 0.
P-value > α: The correlation is not statistically significant
If the p-value is greater than the significance level, then you cannot conclude that the correlation is different from 0.

Correlation: Hydrogen, Porosity, Strength

Correlations Hydrogen Porosity Porosity 0.625 0.017 Strength -0.790 -0.527 0.001 0.053

Cell Contents: Pearson correlation P-Value

Key Result: P-Value

In these results, the p-values for the correlation between porosity and hydrogen and between strength and hydrogen are both less than the significance level of 0.05, which indicates that the correlation coefficients are significant. The p-value between strength and porosity is 0.053. Because the p-value is greater than the significance level of 0.05, there is inconclusive evidence about the significance of the association between the variables.

Step 3: Examine the monotonic relationship between variables (Spearman)

Use the Spearman correlation coefficient to examine the strength and direction of the monotonic relationship between two continuous or ordinal variables. In a monotonic relationship, the variables tend to move in the same relative direction, but not necessarily at a constant rate. To calculate the Spearman correlation, Minitab ranks the raw data. Then, Minitab calculates the correlation coefficient on the ranked data.

Strength

The correlation coefficient can range in value from −1 to +1. The larger the absolute value of the coefficient, the stronger the relationship between the variables.

For the Spearman correlation, an absolute value of 1 indicates that the rank-ordered data are perfectly linear. For example, a Spearman correlation of −1 means that the highest value for Variable A is associated with the lowest value for Variable B, the second highest value for Variable A is associated with the second lowest value for Variable B, and so on.

Direction

The sign of the coefficient indicates the direction of the relationship. If both variables tend to increase or decrease together, the coefficient is positive, and the line that represents the correlation slopes upward. If one variable tends to increase as the other decreases, the coefficient is negative, and the line that represents the correlation slopes downward.

The following plots show data with specific Spearman correlation coefficient values to illustrate different patterns in the strength and direction of the relationships between variables.

No relationship: Spearman rho = 0

The points fall randomly on the plot, which indicates that there is no relationship between the variables.

Strong positive relationship: Spearman rho = 0.948

The points fall close to the line, which indicates that there is a strong relationship between the variables. The relationship is positive because the variables increase concurrently.

Strong negative relationship: Spearman rho = 1.0

The points fall close to the line, which indicates that there is a strong relationship between the variables. The relationship is negative because as one variable increases, the other variable decreases.

It is never appropriate to conclude that changes in one variable cause changes in another based on correlation alone. Only properly controlled experiments enable you to determine whether a relationship is causal.

Spearman Rho: Hydrogen, Porosity, Strength

Correlations Hydrogen Porosity Porosity 0.590 0.026 Strength -0.859 -0.675 0.000 0.008

Cell Contents: Spearman rho P-Value

Key Result: Spearman Rho

In these results, the Spearman correlation between porosity and hydrogen is 0.590, which indicates that there is a positive relationship between the variables. The Spearman correlation between strength and hydrogen is -0.859 and between strength and porosity is -0.675. The relationship between these variables is negative, which indicates that as hydrogen and porosity increase, strength decreases.

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