KENDALL’S TAU

Introduction to Kendall’s Tau

Kendall’s Tau, often simply denoted as Tau ($tau$), is a fundamental non-parametric statistic employed extensively within the fields of psychology, statistics, and social sciences. It serves as a robust measure of the association, or dependence, between two measured variables, specifically designed for use when those variables are measured on an ordinal scale. Unlike parametric measures, such as Pearson’s product-moment correlation coefficient ($r$), Kendall’s Tau makes no assumptions about the underlying distribution of the data, making it particularly valuable when data is non-normally distributed or when sample sizes are small. Its core purpose is to quantify the strength and direction of the monotonic relationship between two sets of rankings, determining the extent to which the observed data reflects a consistent order across both variables. A strong positive Tau indicates that high ranks on one variable correspond to high ranks on the second variable, while a strong negative Tau suggests an inverse relationship, where high ranks on one variable correspond to low ranks on the other.

The necessity for a non-parametric measure arises frequently in psychological research, especially when dealing with variables such as attitude ratings, subjective severity scales, or educational achievement levels, which naturally yield ranked data rather than true interval or ratio data. When researchers are unable or unwilling to assume that the distance between ranks is equal (a core assumption violated by ordinal data), relying on Tau provides a statistically sound method for correlation analysis. This coefficient is conceptually distinct from other rank-based measures, such as Spearman’s Rho, because its interpretation is fundamentally rooted in probability theory: it measures the difference between the probability that the observed data agree in their ordering (concordance) versus the probability that they disagree (discordance). This probabilistic interpretation often grants Tau a greater degree of intuitive meaning for researchers interpreting complex relationships.

The coefficient was first introduced by the British statistician Maurice G. Kendall in 1938 and has since become a cornerstone tool for analyzing bivariate relationships where ranking information is paramount. Its mathematical elegance and interpretability have ensured its continued relevance, particularly in environments where traditional regression assumptions are difficult to meet. The statistic is inherently grounded in the comparison of all possible pairs of observations within the dataset, forming the basis for determining the overall direction and magnitude of the association. This foundational approach—the counting of agreements and disagreements among pairs—is what distinguishes Tau and anchors its application primarily within datasets characterized by discrete or ranked categories.

The Conceptual Basis: Concordance and Discordance

The calculation of Kendall’s Tau is built upon the examination of all possible distinct pairs of observations drawn from the sample. For any given dataset containing $N$ observations, there are $N(N-1)/2$ possible pairs that can be formed. Each of these pairs is then categorized based on whether the rankings for the two variables ($X$ and $Y$) are consistent or inconsistent with each other. This categorization results in three distinct types of pairs: concordant pairs, discordant pairs, and tied pairs. Understanding these concepts is essential for grasping the underlying mechanism of the Tau coefficient, as the resulting statistic is simply a standardized difference between the count of the agreeing pairs and the count of the disagreeing pairs.

A pair of observations is classified as concordant (or in agreement) if the ranks for both variables follow the same order. Specifically, if observation $i$ has a higher rank on variable $X$ than observation $j$, and observation $i$ also has a higher rank on variable $Y$ than observation $j$, the pair is concordant. Conversely, if observation $i$ has a lower rank on $X$ and a lower rank on $Y$ than observation $j$, the pair is still concordant because the relative ordering is maintained across both dimensions. These concordant pairs contribute positively to the resulting Tau statistic, indicating evidence of a positive association between the variables. The higher the number of concordant pairs ($C$) relative to the total number of pairs, the closer Tau will be to $+1$.

In contrast, a pair of observations is categorized as discordant (or in disagreement) if the rankings are inverted between the two variables. If observation $i$ has a higher rank on variable $X$ than observation $j$, but observation $i$ has a lower rank on variable $Y$ than observation $j$, the pair is discordant. This indicates a disruption in the monotonic relationship; as one variable increases, the other tends to decrease. These discordant pairs contribute negatively to the Tau statistic. The count of discordant pairs ($D$) determines the magnitude of the negative correlation. When $D$ is significantly greater than $C$, the Tau coefficient will be negative, approaching $-1$, signaling a strong inverse relationship.

The third category, tied pairs, introduces a critical complexity, especially common in real-world ordinal data where multiple respondents might select the same category (e.g., “Neutral” on a Likert scale). A pair is tied if the two observations have the same rank on either variable $X$, variable $Y$, or both. In the strictest mathematical sense of Tau-a, tied pairs are problematic, but the practical variants of Tau (Tau-b and Tau-c) are specifically designed to handle these ties by making adjustments to the denominator of the correlation formula. Tied pairs neither contribute to concordance nor discordance; they are typically removed from the total pair count used in the denominator, reflecting the fact that they provide ambiguous information about the direction of the relationship.

Variations of the Tau Coefficient

While the fundamental principle of comparing concordant and discordant pairs remains consistent, Kendall’s Tau exists in several important variations, developed to address specific data structures and the pervasive issue of tied observations. The three most common forms are Tau-a, Tau-b, and Tau-c, each serving a distinct purpose and having specific constraints regarding data arrangement and sample characteristics. Choosing the appropriate variation is crucial for accurate statistical inference, particularly when analyzing contingency tables or dealing with variables that have numerous identical scores.

Kendall’s Tau-a is the most basic and theoretically pure form of the coefficient. It is calculated simply as the difference between concordant and discordant pairs divided by the total number of possible pairs: $tau_a = (C – D) / [N(N-1)/2]$. This formulation assumes that there are no ties in the data for either variable. Because ties are extremely common in behavioral and social science research utilizing ordinal scales, Tau-a is rarely used in practical applications. It functions primarily as a theoretical baseline for measuring association, often utilized in contexts where variables are inherently continuous but have been rank-ordered without score duplication. If ties are present, the use of Tau-a results in an attenuation of the correlation magnitude, meaning the calculated coefficient will underestimate the true strength of the relationship.

Kendall’s Tau-b is by far the most widely reported and used version of the coefficient. It explicitly addresses the presence of ties by adjusting the denominator. The adjustment involves subtracting the number of pairs tied on $X$ only and the number of pairs tied on $Y$ only from the total possible pairs. This correction ensures that the magnitude of the coefficient can still reach the theoretical limits of $+1$ and $-1$ when there is a perfect monotonic relationship, even in the presence of ties. Tau-b is the preferred measure when analyzing square contingency tables (where the number of categories for variable $X$ equals the number of categories for variable $Y$). Its robustness to tied data makes it the standard choice for most research involving Likert scales or other fixed-category ordinal variables.

The third major variation is Stuart’s Tau-c, which is specifically designed for use with rectangular contingency tables, where the number of categories for variable $X$ is unequal to the number of categories for variable $Y$ (i.e., the table is $R times C$ where $R ne C$). When the table is rectangular, Tau-b might not be able to achieve the maximum value of $|1|$, even if the association is perfect, due to the structural limitations imposed by the unequal number of ranks. Tau-c addresses this by standardizing the denominator using a multiplier that accounts for the minimum number of rows or columns, ensuring that the coefficient can still reach $+1$ or $-1$ under conditions of perfect association, regardless of the unequal marginal totals. This ensures an accurate representation of association strength when working with asymmetrical ordinal data structures.

Calculation Methodology and Formulation

The calculation of Kendall’s Tau, particularly the practical Tau-b, involves a systematic, multi-step process centered on the identification and enumeration of the three types of pairs. Assuming a dataset of $N$ paired observations $(x_i, y_i)$, the first step is to list all possible $N(N-1)/2$ pairs. Once listed, each pair is assessed to determine its classification as concordant ($C$), discordant ($D$), or tied ($T$). The total count of concordant pairs, $C$, and the total count of discordant pairs, $D$, form the numerator of the Tau equation, representing the net agreement: $(C – D)$.

The complexity lies primarily in accurately calculating the denominator, especially when applying the necessary tie adjustments for Tau-b. The calculation of tied pairs requires two separate components: $T_x$, the number of pairs tied only on the $X$ variable, and $T_y$, the number of pairs tied only on the $Y$ variable. These tie counts are derived by summing up tie correction factors for each rank category that contains ties. Specifically, for any group of $t_i$ observations tied at a particular rank on variable $X$, the number of tied pairs generated by that group is $t_i(t_i – 1)/2$. This calculation is performed for every set of tied ranks on $X$ and summed to get $T_x$, and the same process is repeated for $Y$ to get $T_y$.

The final formulation for Kendall’s Tau-b integrates these components into a standardized measure: $tau_b = (C – D) / sqrt{(C + D + T_x)(C + D + T_y)}$. This denominator, which uses the geometric mean of the total non-tied pairs plus the ties on $X$ and the total non-tied pairs plus the ties on $Y$, is designed to ensure that the coefficient is properly scaled. The denominator effectively represents the maximum possible value of $(C – D)$ that could occur given the observed marginal frequency distributions (the tie structure). By dividing the net agreement by this maximum potential agreement, the resulting $tau_b$ coefficient is guaranteed to fall within the range of $-1$ to $+1$, providing a clear, normalized metric of the strength of the monotonic association.

Interpretation of the Tau Coefficient

Like other correlation coefficients, Kendall’s Tau ranges from $-1.0$ to $+1.0$, providing both a measure of the strength and the direction of the relationship between the two ordinal variables. The sign of the coefficient immediately indicates the direction of the association. A positive Tau value signifies a positive monotonic relationship, meaning that as the rank of one variable increases, the rank of the second variable also tends to increase. Conversely, a negative Tau value indicates a negative or inverse monotonic relationship, where an increase in the rank of one variable is generally accompanied by a decrease in the rank of the other.

The magnitude of the coefficient indicates the strength of this relationship. A Tau value close to $+1.0$ suggests a perfect or near-perfect positive association, indicating that the relative rankings of all observed pairs are nearly identical across both variables. A value close to $-1.0$ denotes a near-perfect negative association, meaning the rankings are almost perfectly inverted. A Tau value close to zero, typically between $-0.1$ and $+0.1$, indicates a very weak or negligible monotonic association between the variables; the rankings of one variable provide little predictive information about the rankings of the other.

Crucially, the interpretation of Tau extends beyond mere magnitude and direction; it carries a distinct probabilistic meaning. Specifically, $tau$ can be interpreted as the probability of observing a concordant pair minus the probability of observing a discordant pair, given a randomly selected pair of observations from the population. For instance, if $tau = 0.60$, it means that if we randomly select two individuals, the probability that their rankings are in the same order is 0.60 greater than the probability that their rankings are in the inverse order. This probabilistic interpretation is often seen as a significant advantage over the interpretation of Spearman’s Rho, which is based on the variance of rank differences. Furthermore, researchers must typically perform a significance test (often using a Z-statistic approximation for larger samples) to determine whether the observed Tau value is sufficiently far from zero to conclude that a genuine association exists in the underlying population.

Advantages and Key Characteristics

Kendall’s Tau holds several distinct advantages that make it a superior choice in various statistical contexts, particularly those involving ordinal data or situations where robustness to distributional assumptions is critical. One of the most significant advantages is its basis in the comparison of pairs, which yields the highly interpretable probabilistic meaning discussed previously. This interpretation—the excess probability of concordance over discordance—is often more straightforward for researchers and practitioners to communicate than the variance-based interpretation of other correlation methods.

A second major advantage is Tau’s robustness against outliers. Since Tau depends only on the relative ranks of the data points and not the magnitude of the differences between them (unlike Pearson’s $r$), an extreme outlier in raw score space does not exert the disproportionate influence on the coefficient that it might in parametric analyses. This resilience makes Tau highly reliable for messy, real-world data where extreme scores might be measurement errors or natural variations that do not reflect the central tendency of the relationship.

Furthermore, Kendall’s Tau is highly effective in dealing with data that inherently contain many ties, provided the appropriate variation (Tau-b or Tau-c) is used. The explicit adjustment for ties ensures that the coefficient remains a valid measure of association strength, preventing the systematic underestimation of the true relationship that would occur if the simpler Tau-a formula were applied to tied data. This feature is paramount when working with psychological assessment instruments that rely on discrete Likert scales or ranking systems with limited categories.

Finally, Tau is considered statistically efficient for testing the independence of two variables. When comparing the statistical power of tests based on Tau versus those based on Spearman’s Rho, Tau often demonstrates superior performance, especially when the underlying relationship is known to be monotonic. This efficiency makes it a preferred choice for hypothesis testing aimed at definitively establishing the presence or absence of an association between two ranked measures.

Comparison with Spearman’s Rho and Pearson’s r

When analyzing associations between variables, researchers typically choose among three primary correlation coefficients: Pearson’s $r$, Spearman’s Rho ($rho$), and Kendall’s Tau ($tau$). The choice depends entirely on the level of measurement of the variables and the distributional assumptions that can be reasonably made. Pearson’s $r$ is the standard parametric measure, requiring that both variables be measured on interval or ratio scales and that their relationship be linear, with data following a bivariate normal distribution. When these stringent conditions are met, Pearson’s $r$ is the most powerful measure.

However, when the data are ordinal or when the distributional assumptions for $r$ are violated, researchers turn to the non-parametric alternatives: Spearman’s Rho and Kendall’s Tau. Both Rho and Tau measure the strength and direction of the monotonic relationship between two ranked variables, meaning they assess whether the variables tend to move in the same direction, regardless of linearity. Despite their shared goal, they calculate association using fundamentally different mathematical approaches. Spearman’s Rho is based on the Pearson correlation formula applied to the numerical ranks of the data. It measures the linear relationship between the ranks themselves, utilizing the squared differences between the ranks of paired observations.

In contrast, Kendall’s Tau, as discussed, focuses on the counting of inversions—the difference between concordant and discordant pairs. Because of this difference in methodology, Tau and Rho calculated on the same dataset will almost always yield different numerical results. Generally, the magnitude of Spearman’s Rho tends to be larger than that of Kendall’s Tau for the same set of data. This does not mean Rho is “more accurate”; it simply reflects the different scaling used by the two statistics. Researchers often prefer Tau because its probabilistic interpretation is often deemed conceptually superior and less abstract than Rho’s interpretation, which is based on the variance of rank differences. Furthermore, Tau is typically preferred in statistical inference because its sampling distribution properties are often better behaved than those of Rho, particularly in the presence of ties.

Applications in Research and Practice

The utility of Kendall’s Tau spans numerous disciplines, owing to its ability to handle ranked data accurately and its independence from strict distributional assumptions. In psychological measurement and psychometrics, Tau is frequently utilized to assess inter-rater reliability, especially when judges or clinicians are ranking subjects or behaviors based on subjective criteria (e.g., severity of symptoms, quality of performance). If two raters provide ordinal scores for a group of clients, Tau quantifies the extent to which their rankings agree, providing a robust measure of consistency.

In social sciences and market research, Tau is indispensable for analyzing survey data, particularly those relying heavily on Likert scales (e.g., strongly disagree to strongly agree). When researchers investigate the relationship between two attitude variables—for example, the correlation between agreement on political policy $A$ and agreement on political policy $B$—Tau provides the appropriate measure of association, since Likert scores are inherently ordinal. Using Pearson’s $r$ in such contexts would violate key statistical assumptions and potentially lead to inaccurate conclusions.

Furthermore, in medical and epidemiological studies, Tau is employed when severity indices or disease stages are measured ordinally. For instance, correlating the stage of a disease (Stage I, II, III) with a specific risk factor rating (low, medium, high) relies on Tau to quantify the association accurately. Its non-parametric nature ensures that the analysis remains valid even if the underlying distribution of the disease stages or risk factors is highly skewed. The flexibility and interpretability of Kendall’s Tau thus solidify its position as an essential tool for statistical analysis across any domain where variables are naturally or necessarily measured through ranking or ordered categorization.