Quick Answer: Kendall’s Tau-b Result

The worked and strongest pair in this analysis is G2 vs G3. G2 is the second-period grade and G3 is the final grade. Kendall’s tau-b was selected because the grade variables have repeated values, meaning ties are common.

The result is Kendall’s tau-b = 0.869595, with N = 649 valid paired observations and p = 2.113285e-199. This is a very strong positive monotonic association. The null hypothesis of no monotonic association is rejected.

What Is Kendall’s Tau-b?

Kendall’s Tau-b is a rank-based correlation coefficient used to measure monotonic association between two variables while correcting for ties. A monotonic association means that as one variable increases, the other variable tends to increase or decrease consistently, even if the pattern is not perfectly linear.

Tau-b compares every pair of observations. If the order of two observations is the same on both variables, the pair is concordant. If the order is opposite, the pair is discordant. If one or both variables have equal values for the pair, ties are counted and corrected in the denominator.

Kendall’s Tau-b is closely related to Correlation in Python, Correlation in R, Correlation in SPSS, Correlation in Excel, Correlation Matrix, Correlation Assumptions, p-value, effect size and parametric vs nonparametric tests.

Kendall’s Tau-a vs Tau-b vs Spearman

Students often confuse Kendall’s tau-a, Kendall’s tau-b and Spearman correlation. They are all rank-based ideas, but they do not handle ties in the same way.

Kendall’s Tau-b Formula

The tie-corrected Kendall’s tau-b formula is:

In this formula, C is the number of concordant pairs, D is the number of discordant pairs, T_x is the number of pairs tied only on X, and T_y is the number of pairs tied only on Y.

The result is strongly positive because concordant pairs greatly outnumber discordant pairs. The tie correction is necessary because grade values repeat often.

Null and Alternative Hypotheses

Kendall’s tau-b tests whether the population monotonic rank association is zero.

Dataset and Variables Used

The analysis used 16 numeric or ordinal variables from the student performance dataset. The strongest worked pair was G2 vs G3. The matrix also included age, Medu, Fedu, traveltime, studytime, failures, famrel, freetime, goout, Dalc, Walc, health, absences, G1, G2 and G3.

The diagnostic table shows that all 16 variables had N = 649 valid values and no missing values. Several ordinal-style variables had large tie groups, such as failures, Dalc, traveltime and studytime. This supports the use of Kendall’s tau-b instead of a tie-free rank correlation.

Verified Kendall’s Tau-b Results

The strongest association is G2 vs G3, followed by G1 vs G2 and G1 vs G3. The grade variables form the strongest positive rank-correlation block in the matrix.

The practical conclusion is that grade variables have the strongest positive monotonic relationships, while failures has the strongest negative relationship with grades.

Python Chart-by-Chart Interpretation

The Python report includes six charts: tau-b heatmap, p-value heatmap, top tau-b pairs, worked pair jittered scatterplot, tie profile and concordant-discordant-tie counts.

Python Chart 1: Kendall’s Tau-b Heatmap

The tau-b heatmap shows the direction and strength of monotonic association across the full variable set. Positive cells indicate that higher ranks on one variable tend to match higher ranks on the other variable. Negative cells indicate that higher ranks on one variable tend to match lower ranks on the other variable.

The strongest positive block appears among the grade variables G1, G2 and G3. The strongest negative pattern appears between failures and the grade variables. This visual summary supports the ranked top-pair table.

Python Chart 2: Kendall’s Tau-b p-value Heatmap

The p-value heatmap shows where the statistical evidence is strongest. The grade relationships have extremely small p-values, especially G2 vs G3, G1 vs G2 and G1 vs G3.

This chart should be read alongside the tau-b heatmap. The tau-b heatmap gives strength and direction, while the p-value heatmap gives statistical evidence. A small p-value does not automatically mean a strong effect, so both charts are necessary.

Python Chart 3: Top Kendall’s Tau-b Pairs

The top-pairs chart clearly ranks the strongest monotonic relationships. G2 vs G3 is the strongest with tau-b about 0.8696. G1 vs G2 and G1 vs G3 follow closely, showing that the grade measurements are highly ordered together.

The failures pairs are negative, meaning students with more past failures tend to rank lower on grade outcomes. This should be reported as association, not causation.

Python Chart 4: Worked Pair Jittered Scatterplot

The jittered scatterplot shows the relationship between G2 and G3. Because grades are repeated integer values, many points would overlap without jittering. Jitter makes the dense tied values easier to see while preserving the overall pattern.

The upward pattern is very clear: higher G2 values are strongly associated with higher G3 values. This matches the very high tau-b value of 0.8696.

Python Chart 5: Tie Profile

The tie profile explains why Kendall’s tau-b is needed. Some variables have very large tie groups. For example, failures has many observations at zero, Dalc has many observations at one, and several ordinal variables have repeated category values.

When ties are common, tau-b is preferred because it adjusts the denominator for ties in each variable. Without tie correction, the coefficient can be misleading.

Python Chart 6: Concordant, Discordant and Tie Counts

This chart directly explains the formula. For G2 vs G3, there are 169,536 concordant pairs and only 5,460 discordant pairs. This large excess of concordant pairs produces the strong positive tau-b value.

The chart also shows X-only ties, Y-only ties and both-variable ties. These tied values are part of the reason tau-b is used instead of tau-a.

R Chart-by-Chart Interpretation

The R report validates the Python results using colorful charts. The same six ideas are repeated: tau-b heatmap, p-value heatmap, top pairs, worked scatterplot, tie profile and pair classification counts.

R Chart 1: Colorful Kendall’s Tau-b Heatmap

The R heatmap confirms the same pattern: grade variables form the strongest positive cluster, while failures has negative associations with G1, G2 and G3. This confirms that the Python heatmap is not a software-specific result.

R Chart 2: Colorful Kendall’s Tau-b p-value Heatmap

The R p-value heatmap confirms the extremely small p-values for the strongest grade relationships. G2 vs G3, G1 vs G2 and G1 vs G3 show very strong statistical evidence of monotonic association.

R Chart 3: Colorful Top Kendall’s Tau-b Pairs

The R top-pairs chart confirms that G2 vs G3 is the strongest pair, followed by G1 vs G2 and G1 vs G3. The same ordering appears in the Python report and Excel workbook.

R Chart 4: Colorful Worked Pair Jittered Scatterplot

The R scatterplot again shows a strong upward monotonic pattern between G2 and G3. Jittering helps reveal repeated grade combinations where many points overlap.

R Chart 5: Colorful Tie Profile

The R tie profile reinforces the methodological choice. Variables with repeated categories or repeated grade values need tie correction. Kendall’s tau-b handles this directly.

R Chart 6: Colorful Concordant, Discordant and Tie Counts

The R pair-classification chart confirms the formula logic: concordant pairs dominate discordant pairs. That is why the final coefficient is strongly positive.

SPSS Output Interpretation

The SPSS output provides a formal Kendall’s tau-b correlation matrix. SPSS confirms the strongest grade relationships, including G2 with G3, G1 with G2 and G1 with G3.

Open the SPSS Kendall’s Tau-b Output PDF

The SPSS result should be reported with rounded values. For example, G2 and G3 showed a very strong positive monotonic relationship, τ_b = .870, p < .001, N = 649.

Excel Worked File Explanation

The Excel workbook is a fully worked Kendall’s tau-b analysis. It includes a report sheet, raw data, numeric data, variable diagnostics, pairwise tau-b table, tau-b matrix, p-value matrix, top-pairs table and worked formula example.

Download the Kendall’s Tau-b Fully Worked Excel File

The workbook’s worked example uses G2 vs G3. It calculates 169,536 concordant pairs, 5,460 discordant pairs, 13,777 G2-only ties, 13,593 G3-only ties and 7,910 ties on both variables. These values produce Kendall’s tau-b = 0.8695948161.

Python, R, SPSS and Excel Workflows

Code Blocks and Excel Formulas

Python Code for Kendall’s Tau-b

import pandas as pd
from scipy.stats import kendalltau

df = pd.read_csv("dataset.csv")

x_var = "G2"
y_var = "G3"

work = df[[x_var, y_var]].dropna()

tau_b, p_value = kendalltau(work[x_var], work[y_var])

print("Complete pairs:", len(work))
print("Kendall's tau-b:", tau_b)
print("Two-tailed p-value:", p_value)

if p_value < 0.05:
    print("Reject H0: significant monotonic association")
else:
    print("Fail to reject H0")

Python Code for Concordant, Discordant and Tie Counts

import numpy as np

x = work[x_var].to_numpy()
y = work[y_var].to_numpy()

C = 0
D = 0
Tx = 0
Ty = 0
Tboth = 0

for i in range(len(x)):
    for j in range(i + 1, len(x)):
        dx = np.sign(x[j] - x[i])
        dy = np.sign(y[j] - y[i])

        if dx == 0 and dy == 0:
            Tboth += 1
        elif dx == 0:
            Tx += 1
        elif dy == 0:
            Ty += 1
        elif dx == dy:
            C += 1
        else:
            D += 1

tau_b_manual = (C - D) / np.sqrt((C + D + Tx) * (C + D + Ty))

print("Concordant pairs:", C)
print("Discordant pairs:", D)
print("X-only ties:", Tx)
print("Y-only ties:", Ty)
print("Both ties:", Tboth)
print("Manual Kendall's tau-b:", tau_b_manual)

R Code for Kendall's Tau-b

df <- read.csv("dataset.csv", stringsAsFactors = FALSE)

x_var <- "G2"
y_var <- "G3"

work <- na.omit(df[, c(x_var, y_var)])

test <- cor.test(
  work[[x_var]],
  work[[y_var]],
  method = "kendall",
  exact = FALSE
)

cat("Complete pairs:", nrow(work), "\n")
cat("Kendall's tau-b:", unname(test$estimate), "\n")
cat("p-value:", test$p.value, "\n")

print(test)

SPSS Syntax for Kendall's Tau-b

* Kendall's Tau-b in SPSS.
OUTPUT CLOSE ALL.
OUTPUT NEW NAME=Kendalls_Tau_b_Output.

DESCRIPTIVES VARIABLES=G1 G2 G3
  /STATISTICS=MEAN STDDEV MIN MAX.

NONPAR CORR
  /VARIABLES=G2 G3
  /PRINT=KENDALL TWOTAIL
  /MISSING=PAIRWISE.

NONPAR CORR
  /VARIABLES=age Medu Fedu traveltime studytime failures famrel freetime goout Dalc Walc health absences G1 G2 G3
  /PRINT=KENDALL TWOTAIL
  /MISSING=PAIRWISE.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE='Kendalls-Tau-b-SPSS-Output.pdf'.

Excel Formula Patterns for Kendall's Tau-b

For each observation pair i < j:

Difference in X:
=X_j-X_i

Difference in Y:
=Y_j-Y_i

Sign of X difference:
=SIGN(X_j-X_i)

Sign of Y difference:
=SIGN(Y_j-Y_i)

Pair class:
=IF(AND(sign_dx=0,sign_dy=0),"Both tied",
 IF(sign_dx=0,"Tie X only",
 IF(sign_dy=0,"Tie Y only",
 IF(sign_dx=sign_dy,"Concordant","Discordant"))))

Concordant pairs:
=COUNTIF(pair_class_range,"Concordant")

Discordant pairs:
=COUNTIF(pair_class_range,"Discordant")

X-only ties:
=COUNTIF(pair_class_range,"Tie X only")

Y-only ties:
=COUNTIF(pair_class_range,"Tie Y only")

Kendall's tau-b:
=(C-D)/SQRT((C+D+Tx)*(C+D+Ty))

Decision:
=IF(p_value<0.05,"Significant monotonic association","Not significant")

How to Report Kendall's Tau-b

A complete Kendall's tau-b report should include the variables, sample size, coefficient, p-value, direction, strength and tie handling.

Common Mistakes in Kendall's Tau-b Interpretation

Downloads and Resources

External References

For software documentation, see SciPy documentation for kendalltau, R documentation for cor.test, IBM SPSS documentation for nonparametric correlations and Microsoft Excel documentation for COUNTIF and SQRT formulas.

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