Kendall Tau-b Rank Correlation Calculator

Description

Calculate tie-adjusted Kendall tau-b and disjoint concordant, discordant, and tied pair counts, without a p-value approximation.

Kendall tau compares whether pairs of observations have the same relative ordering in two series. Tau-b adjusts for ties in either series; pairs tied in both are counted separately rather than treated as agreement.^[1]

Calculate tie-adjusted Kendall tau-b and disjoint concordant, discordant, and tied pair counts, without a p-value approximation.

When to use this tool

Use this tool to calculate kendall tau under the units, normalization, and boundary conventions below.

How to use the tool and how it works

Supply first, second in the field shapes below.
Run the calculation. Correct format, shape, range, or undefined-statistic errors before interpreting a result.
Read concordant, discordant, tauB, tiedInBoth, tiedInFirstOnly, tiedInSecondOnly. Use the conventions below.

Inspect every unordered paired observation. Count concordant, discordant, tied-in-first-only, tied-in-second-only and tied-in-both cases separately, then calculate tau-b=(C−D)/sqrt((C+D+Tfirst)(C+D+Tsecond)).^[2]

Input field	Meaning and units	Accepted input
first	First paired series	Required. Array of Finite number.; minimum items 2; maximum items 512.
second	Second paired series	Required. Array of Finite number.; minimum items 2; maximum items 512.

Output field	Interpretation	Returned shape
concordant	Concordant pairs	Required. Safe integer; minimum 0.
discordant	Discordant pairs	Required. Safe integer; minimum 0.
tauB	Kendall tau-b	Required. Finite number; minimum -1; maximum 1.
tiedInBoth	Pairs tied in both series	Required. Safe integer; minimum 0.
tiedInFirstOnly	Pairs tied only in the first series	Required. Safe integer; minimum 0.
tiedInSecondOnly	Pairs tied only in the second series	Required. Safe integer; minimum 0.

json

{
  "input": {
    "first": [
      1,
      2,
      3,
      4,
      5
    ],
    "second": [
      1,
      3,
      2,
      4,
      5
    ]
  },
  "output": {
    "concordant": 9,
    "discordant": 1,
    "tauB": 0.8,
    "tiedInBoth": 0,
    "tiedInFirstOnly": 0,
    "tiedInSecondOnly": 0
  }
}

Limitations and assumptions

Both lists must have equal length. A zero denominator is an undefined-correlation error. Joint ties are excluded from the single-series tie terms, and the tool returns no p-value or significance approximation.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

Choose the statistic according to the question and data conventions. Inspect the supplied series or matrix and the stated assumptions before comparing outputs from implementations with different defaults.

References

SciPy: Kendall tau-b and tie conventions — SciPy
Tool behavior and example: Kendall Tau-b Rank Correlation Calculator — tool.io

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