Calculating the Jaccard Coefficient

Introduction

I used a small table of pathological test results to calculate the Jaccard distance and corresponding Jaccard similarity coefficient for three pairs of individuals. The three individuals are Jack, Mary and Jim, and the required comparisons are:

• Jack and Mary
• Jack and Jim
• Jim and Mary

What is the Jaccard coefficient?

The Jaccard similarity coefficient is a similarity measure for binary attributes. It compares two records by computing the ratio of matches where both records have a value of 1 to the number of attributes where at least one record has a value of 1.

The formula is:

J(A, B) = f₁₁ / (f₀₁ + f₁₀ + f₁₁)

The corresponding Jaccard distance, or dissimilarity, is the proportion of mismatches:

dⱼ(A, B) = (f₀₁ + f₁₀) / (f₀₁ + f₁₀ + f₁₁) = 1 - J(A, B)

Where:

• f₁₁ = number of attributes where both records have value 1
• f₀₁ = number of attributes where record A has 0 and record B has 1
• f₁₀ = number of attributes where record A has 1 and record B has 0
• Matches where both have 0 are ignored in the denominator.

For this activity, each person is treated as a set of attribute-value pairs, excluding the Name field. For example, Jack’s record is represented as:

• Gender = M
• Fever = Y
• Cough = N
• Test-1 = P
• Test-2 = N
• Test-3 = N
• Test-4 = A

Binary Conversion

Asymmetric variables are converted to binary:

• Y and P → 1
• N and A → 0

Gender is NOT converted because it is a symmetric variable (male and female have equal weight). It is retained only to check that the records are comparable.

After binary conversion, each person has 6 binary attributes (excluding Name and the unconverted Gender). For example, Jack's record becomes:

• Gender = M (not converted)
• Fever = Y → 1
• Cough = N → 0
• Test-1 = P → 1
• Test-2 = N → 0
• Test-3 = N → 0
• Test-4 = A → 0

Source Data

Name	Gender	Fever	Cough	Test-1	Test-2	Test-3	Test-4
Jack	M	Y	N	P	N	N	A
Mary	F	Y	N	P	A	P	N
Jim	M	Y	P	N	N	N	A

Calculation Method

For each pair of individuals, I converted asymmetric variables to binary (Y & P = 1; N & A = 0), excluded Gender from binary conversion, and compared the 6 binary attributes.

For each pair, I calculated:

1. f₁₁: Count of attributes where both individuals have value 1
2. f₀₁ + f₁₀: Count of attributes where one has 1 and the other has 0
3. The Jaccard distance using: dⱼ = (f₀₁ + f₁₀) / (f₀₁ + f₁₀ + f₁₁)
4. The Jaccard similarity coefficient using: J = f₁₁ / (f₀₁ + f₁₀ + f₁₁)

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Pair 1: Jack and Mary

Original Data

Jack: Gender=M, Fever=Y, Cough=N, Test-1=P, Test-2=N, Test-3=N, Test-4=A
Mary: Gender=F, Fever=Y, Cough=N, Test-1=P, Test-2=A, Test-3=P, Test-4=N

Binary Conversion

Jack (binary): Gender=M (not converted), 1, 0, 1, 0, 0, 0
Mary (binary): Gender=F (not converted), 1, 0, 1, 0, 1, 0

Attribute Comparison

Comparing the 6 binary attributes (skipping Gender):

Position	Attribute	Jack	Mary	Match Type
1	Fever	1	1	f₁₁ (both 1)
2	Cough	0	0	Ignored
3	Test-1	1	1	f₁₁ (both 1)
4	Test-2	0	0	Ignored
5	Test-3	0	1	f₀₁ (mismatch)
6	Test-4	0	0	Ignored

Calculation

• f₁₁ = 2 (Fever, Test-1)
• f₀₁ + f₁₀ = 1 (Test-3)

Jaccard distance:

dⱼ(Jack, Mary) = (1) / (1 + 2) = 1/3 = 0.333

Result

The Jaccard distance for Jack and Mary is:

0.33

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Pair 2: Jack and Jim

Original Data

Jack: Gender=M, Fever=Y, Cough=N, Test-1=P, Test-2=N, Test-3=N, Test-4=A
Jim: Gender=M, Fever=Y, Cough=P, Test-1=N, Test-2=N, Test-3=N, Test-4=A

Binary Conversion

Jack (binary): Gender=M (not converted), 1, 0, 1, 0, 0, 0
Jim (binary): Gender=M (not converted), 1, 1, 0, 0, 0, 0

Attribute Comparison

Comparing the 6 binary attributes (skipping Gender):

Position	Attribute	Jack	Jim	Match Type
1	Fever	1	1	f₁₁ (both 1)
2	Cough	0	1	f₀₁ (mismatch)
3	Test-1	1	0	f₁₀ (mismatch)
4	Test-2	0	0	Ignored
5	Test-3	0	0	Ignored
6	Test-4	0	0	Ignored

Calculation

• f₁₁ = 1 (Fever)
• f₀₁ + f₁₀ = 2 (Cough, Test-1)

Jaccard distance:

dⱼ(Jack, Jim) = (2) / (2 + 1) = 2/3 = 0.667

Result

The Jaccard distance for Jack and Jim is:

0.67

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Pair 3: Jim and Mary

Original Data

Jim: Gender=M, Fever=Y, Cough=P, Test-1=N, Test-2=N, Test-3=N, Test-4=A
Mary: Gender=F, Fever=Y, Cough=N, Test-1=P, Test-2=A, Test-3=P, Test-4=N

Binary Conversion

Jim (binary): Gender=M (not converted), 1, 1, 0, 0, 0, 0
Mary (binary): Gender=F (not converted), 1, 0, 1, 0, 1, 0

Attribute Comparison

Comparing the 6 binary attributes (skipping Gender):

Position	Attribute	Jim	Mary	Match Type
1	Fever	1	1	f₁₁ (both 1)
2	Cough	1	0	f₁₀ (mismatch)
3	Test-1	0	1	f₀₁ (mismatch)
4	Test-2	0	0	Ignored
5	Test-3	0	1	f₀₁ (mismatch)
6	Test-4	0	0	Ignored

Calculation

• f₁₁ = 1 (Fever)
• f₀₁ + f₁₀ = 3 (Cough, Test-1, Test-3)

Jaccard distance:

dⱼ(Jim, Mary) = (3) / (3 + 1) = 3/4 = 0.75

Result

The Jaccard distance for Jim and Mary is:

0.75

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Summary of Results

Jaccard Distance

Pair	f₁₁	f₀₁ + f₁₀	Jaccard Distance
Jack and Mary	2	1	0.33
Jack and Jim	1	2	0.67
Jim and Mary	1	3	0.75

Jaccard Similarity Coefficient

The Jaccard similarity coefficient is the complement of the Jaccard distance: J = 1 - dⱼ.

Pair	Calculation	Jaccard Similarity Coefficient
Jack and Mary	2 / (2 + 1)	0.67
Jack and Jim	1 / (1 + 2)	0.33
Jim and Mary	1 / (1 + 3)	0.25

Interpretation

Jack and Mary have the highest similarity, with a Jaccard similarity coefficient of 0.67 and a Jaccard distance of 0.33. They have the fewest mismatches among attributes where at least one individual has a value of 1.

Jack and Jim have moderate similarity, with a Jaccard similarity coefficient of 0.33 and a Jaccard distance of 0.67. They differ on the Cough and Test-1 attributes while sharing the Fever characteristic.

Jim and Mary have the lowest similarity, with a Jaccard similarity coefficient of 0.25 and a Jaccard distance of 0.75. They have the most mismatches among 1-valued attributes: Cough, Test-1, and Test-3.

Higher Jaccard similarity coefficients indicate greater similarity, while lower Jaccard distances indicate greater similarity.