Jaccard Coefficient Calculations

Activity: The table shows the pathological test results for three individuals.

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

Calculate Jaccard coefficient for the following pairs:

  • (Jack, Mary)
  • (Jack, Jim)
  • (Jim, Mary)

Solution: The Jaccard coefficient, also known as the Jaccard index or Jaccard similarity coefficient, is a statistic used in clustering and other forms of data analysis to measure the similarity and diversity of sample sets and it is given by:

Jaccard = (f01 + f10) / (f01 + f10 + f11)

  • To calculate the Jaccard coefficient, we first convert the asymmetric variables to binary values and re-write the table. Since Gender is a symmetric variable (that is, male, female have the same weight), it is not converted.

  • So, let Y & P = 1; N & A = 0 Let’s recalculate it using the binary values:

  • Now, we’ll rewrite the table with these binary values:

    Name Gender Fever Cough Test-1 Test-2 Test-3 Test-4
    Jack M 1 0 1 0 0 0
    Mary F 1 0 1 0 1 0
    Jim M 1 1 0 0 0 0

  • For (Jack, Mary)

    Jack: (1, 0, 1, 0, 0, 0)

    Mary: (1, 0, 1, 0, 1, 0)

    Jaccard = (1+0)/(1+0+2)= 0.33

  • For (Jack, Jim)

Jack: (1, 0, 1, 0, 0, 0)

Jim: (1, 1, 0, 0, 0, 0)

Jaccard = (1+1)/(1+1+1)= 0.67

  • For (Jim, Mary)

Jim: (1, 1, 0, 0, 0, 0)

Mary: (1, 0, 1, 0, 1, 0)

Jaccard = (2+1)/(2+1+1)= 0.75

Conclusions

  • Jack and Mary have a Jaccard coefficient of 0.33, which means that they have a relatively low degree of similarity in their test results. Only 33% of the binary attributes are the same between them.

  • Jack and Jim have a Jaccard coefficient of 0.67, indicating a higher degree of similarity compared to Jack and Mary. 67% of their binary attributes match, suggesting a moderate level of similarity.

  • Jim and Mary have the highest Jaccard coefficient among the pairs, with a value of 0.75. This suggests a relatively high degree of similarity in their test results, with 75% of their binary attributes being the same.

Note that, in practical applications, a Jaccard coefficient of 1 is often used to represent complete similarity or a perfect match between two sets.