How to Choose the Right Correlation for Your Data

The Fundamentals of Correlation Analysis

In the field of statistics, the concept of correlation encapsulates both the magnitude and the trajectory of the association between two or more variables. This relationship is quantified by a correlation coefficient, a standardized value that facilitates easy interpretation.

The range of a correlation coefficient is strictly defined, spanning from -1 to +1. A value of -1 signifies a perfect negative relationship, meaning that as one variable increases, the other decreases consistently. Conversely, a value of +1 represents a perfect positive relationship, where both variables increase or decrease together proportionally. A coefficient of 0 indicates no linear relationship whatsoever between the variables under observation. It is vital to remember that correlation measures association, but does not imply causation.

Selecting the Appropriate Correlation Measure

The decision regarding which correlation coefficient to employ is driven primarily by the type of data involved and the assumptions regarding its distribution. Researchers must first determine if their variables are continuous, ordinal (ranked), or if the relationship is linear or monotonic. While all correlation coefficients aim to describe bivariate relationships, their underlying mathematical foundations differ significantly.

We typically rely on three primary methodologies to measure correlation, each suited for specific data characteristics:

Pearson Correlation: This parametric test is used exclusively to measure the linear relationship between two continuous variables (e.g., relating a person’s height to their weight). It requires the data to be approximately normally distributed and free from extreme outliers.

Spearman Correlation: A non-parametric alternative, the Spearman coefficient is designed to measure the monotonic relationship between two ranked variables, or when the assumption of normality for continuous data is violated. It assesses how well the relationship between the variables can be described using a monotonic function. A common example is correlating the rank of a student’s math exam score versus the rank of their science exam score in a class.

Kendall’s Correlation: This non-parametric test is often preferred over Spearman’s when dealing with smaller sample sizes or when the data contains a large number of tied ranks. It measures the probability that two variables are in the same order (concordant) versus the probability that they are in different orders (discordant).

The subsequent sections of this tutorial detail the practical application of these three types of correlation analysis using the statistical software package, Stata.

Setting Up Your Analysis Environment in Stata

To demonstrate the calculation of these correlation coefficients, we will utilize a built-in dataset commonly used for introductory examples in Stata. This dataset, named auto, provides various characteristics of automobiles.

To load this dataset into your active Stata session, execute the following command in the Command window:

use http://www.stata-press.com/data/r13/auto

Once the data is successfully loaded, it is good practice to perform a quick summary check to familiarize ourselves with the dataset’s structure, variables, and data completeness. This is achieved by typing the following command:

summarize

The output confirms that the auto dataset contains 74 total observations across 12 different variables, ready for our subsequent correlation analyses.

Calculating Pearson Correlation in Stata

The Pearson correlation coefficient (r) measures the linear association between two continuous, normally distributed variables. In Stata, the primary command for calculating pairwise correlations is pwcorr.

To find the Pearson correlation between the variables weight and length, which are both continuous measures, we use the pwcorr command followed by the names of the variables:

pwcorr weight length

The initial output provides the correlation value itself. However, in rigorous statistical analysis, determining the statistical significance of the coefficient is paramount. We can easily obtain the corresponding p-value by adding the sig option to the command:

pwcorr weight length, sig

Upon reviewing the output, we observe that the p-value is reported as 0.000. Since this value is considerably less than the conventional significance level of 0.05 (alpha), we conclude that the correlation observed between weight and length is highly statistically significant. This suggests that the relationship is unlikely to have occurred by random chance.

If the goal is to examine the relationships among several variables simultaneously, the pwcorr command efficiently handles lists of variables. To obtain the Pearson Correlation Coefficients and associated p-values for weight, length, and displacement, simply list them after the command, retaining the sig option:

pwcorr weight length displacement, sig

The resulting matrix provides the pairwise correlation coefficients and their significance levels for all combinations of the selected variables. Here is a detailed interpretation of this comprehensive output:

• The Pearson Correlation between weight and length is 0.9460, indicating a very strong positive association. The corresponding p-value is 0.000.
• The Pearson Correlation between weight and displacement is 0.8949, also showing a strong positive relationship. The p-value is 0.000.
• The Pearson Correlation between displacement and length is 0.8351, representing a strong positive link. The p-value is 0.000.

Executing Spearman Rank Correlation in Stata

When dealing with ordinal data, non-normally distributed continuous data, or when seeking a relationship that is merely monotonic (not strictly linear), the Spearman’s rank correlation coefficient (rho) is the appropriate non-parametric test. This method works by ranking the data for each variable and then applying the Pearson formula to the ranks.

In Stata, the dedicated command for this procedure is spearman. Let’s calculate the Spearman coefficient between trunk size and the rep78 (repair record 1978) variable:

spearman trunk rep78

The output generated by the spearman command provides key metrics necessary for interpretation:

• Number of obs: This crucial statistic indicates the number of paired observations used in the calculation. Since the variable rep78 had missing values, Stata employed only 69 pairwise observations out of the total 74 records, utilizing pairwise deletion.
• Spearman’s rho: This is the Spearman correlation coefficient itself. In this instance, rho is -0.2235, which suggests a weak negative monotonic relationship. As the rank of one variable increases, the rank of the other tends to decrease slightly.
• Prob > |t|: This represents the p-value associated with the test of the hypothesis that the true correlation is zero. With a p-value of 0.0649, which is greater than the standard significance threshold (α = 0.05), we conclude that there is not a statistically significant correlation between these two variables.

To calculate Spearman Correlation Coefficients for multiple variables, list them following the command. Furthermore, to structure the output neatly into a matrix that includes both the correlation coefficient (rho) and the p-value, we employ the stats(rho p) option:

spearman trunk rep78 gear_ratio, stats(rho p)

Interpretation of the multiple variable output:

• Spearman Correlation between trunk and rep78 = -0.2235, with a p-value of 0.0649 (not significant).
• Spearman Correlation between trunk and gear_ratio = -0.5187, indicating a moderate negative relationship. The p-value of 0.0000 confirms high statistical significance.
• Spearman Correlation between gear_ratio and rep78 = 0.4275, suggesting a moderate positive relationship. The p-value of 0.0002 confirms statistical significance.

Using Kendall’s Tau for Small Samples and Ties

The third major method for measuring rank correlation is Kendall’s Tau correlation coefficient (τ). This non-parametric statistic is especially useful when the sample size is small, or when the data features a significant number of identical values, known as “tied ranks,” which can potentially bias the Spearman coefficient. Kendall’s Tau often provides a more robust estimate of the population correlation.

In Stata, we use the ktau command to perform this analysis. We will calculate the Kendall’s Tau coefficient between trunk and rep78 again for comparison:

ktau trunk rep78

Interpreting the ktau command output:

• Number of obs: Consistent with the Spearman calculation, 69 pairwise observations were used due to missing data in the rep78 variable.
• Kendall’s tau-b: This value, τ_b, is the standard measure reported for Kendall’s rank correlation when ties are present in the data (as opposed to tau-a, which assumes no ties). In this case, tau-b = -0.1752, indicating a weak negative correlation between the two variables, similar in direction to the Spearman result, but typically lower in magnitude.
• Prob > |z|: This is the p-value derived from the hypothesis test. With a p-value of 0.0662, which is marginally greater than the alpha level of 0.05, we fail to reject the null hypothesis, concluding that the correlation is not statistically significant.

To calculate Kendall’s Correlation Coefficient for multiple variable pairs, list them after the ktau command. To display the coefficient and the corresponding p-value in a clean matrix format, use the stats(taub p) option, ensuring the use of tau-b for handling ties:

ktau trunk rep78 gear_ratio, stats(taub p)

The pairwise results for the multiple variable Kendall’s Tau test are as follows:

• Kendall’s Correlation between trunk and rep78 = -0.1752, with a p-value of 0.0662 (not significant).
• Kendall’s Correlation between trunk and gear_ratio = -0.3753, indicating a moderate negative rank correlation. The p-value of 0.0000 confirms strong statistical significance.
• Kendall’s Correlation between gear_ratio and rep78 = 0.3206, showing a moderate positive rank correlation. The p-value of 0.0006 confirms statistical significance.

Summary of Correlation Choice

Selecting the correct correlation method is essential for obtaining meaningful results. If your data consists of two continuous variables that meet parametric assumptions (linearity and normality), Pearson’s r is preferred. If your data is ordinal, or if the continuous data violates normality assumptions, the non-parametric measures—Spearman’s rho or Kendall’s Tau—are appropriate. Kendall’s Tau offers added robustness when ties are frequent or the sample size is limited, making it a powerful tool for reliable rank correlation estimation.