Calculating Blau Index for Board Skill Diversity

[Tiffany Theresia]

Hi Everyone,

Right now I am working on a variable that measures the degree of diversity/heterogeneity of the board skill. From several discussion with my colleagues, they suggest me to use the Blau Index.
Can anyone assist me on how to calculate Blau index for board skill diversity on stata? Or anyone has other recommendation on how to best calculate such variables? Thank you!

Here example of my data:

* Example generated by -dataex-. To install: ssc install dataex
clear
input int Year str7 Ticker str49 DirectorName str58 RoleName float(academic_skill_dummy company_business_skill_dummy governance_skills_dummy legal_skills_dummy management_skills_dummy)
2011 "MMM" "Sir George Buckley"             "Chairman/President/CEO"    0 0 0 0 0
2011 "MMM" "Aulana Peters"                  "Independent Director"      0 0 0 1 0
2011 "MMM" "Bob Morrison"                   "Independent Director"      1 0 0 0 0
2011 "MMM" "Bob Ulrich"                     "Independent Director"      0 0 0 0 1
2011 "MMM" "Doctor Vance Coffman"           "Lead Independent Director" 1 0 0 0 0
2011 "MMM" "Ed Liddy"                       "Independent Director"      1 0 0 0 0
2011 "MMM" "Herb Henkel"                    "Independent Director"      1 0 0 0 0
2011 "MMM" "Jim Farrell"                    "Independent Director"      0 0 0 0 0
2011 "MMM" "Linda Alvarado"                 "Independent Director"      0 0 0 0 1
2011 "MMM" "Mike Eskew"                     "Independent Director"      0 0 0 0 0
2012 "MMM" "Inge Thulin"                    "Chairman/President/CEO"    0 0 0 0 0
2012 "MMM" "Aulana Peters"                  "Independent Director"      0 0 0 1 0
2012 "MMM" "Bob Morrison"                   "Independent Director"      1 0 0 0 0
2012 "MMM" "Bob Ulrich"                     "Independent Director"      0 0 0 0 1
2012 "MMM" "Doctor Vance Coffman"           "Independent Director"      1 0 0 0 0
2012 "MMM" "Ed Liddy"                       "Independent Director"      1 0 0 0 0
2012 "MMM" "Herb Henkel"                    "Independent Director"      1 0 0 0 1
2012 "MMM" "Jim Farrell"                    "Independent Director"      0 0 0 0 0
2012 "MMM" "Linda Alvarado"                 "Independent Director"      0 0 0 0 1
2012 "MMM" "Mike Eskew"                     "Lead Independent Director" 0 0 0 0 0
2013 "MMM" "Inge Thulin"                    "Chairman/President/CEO"    0 0 0 0 0
2013 "MMM" "Aulana Peters"                  "Independent Director"      0 0 0 1 0
2013 "MMM" "Bob Morrison"                   "Independent Director"      1 0 0 0 0
2013 "MMM" "Bob Ulrich"                     "Independent Director"      0 0 0 0 1
2013 "MMM" "Doctor Vance Coffman"           "Independent Director"      1 0 0 0 0
2013 "MMM" "Ed Liddy"                       "Independent Director"      1 0 0 0 0
2013 "MMM" "Herb Henkel"                    "Independent Director"      1 0 0 0 1
2013 "MMM" "Jim Farrell"                    "Independent Director"      0 0 0 0 0
2013 "MMM" "Linda Alvarado"                 "Independent Director"      0 0 0 0 1
2013 "MMM" "Mike Eskew"                     "Lead Independent Director" 0 0 0 0 0
2013 "MMM" "Muhtar Kent"                    "Independent Director"      0 0 0 0 0
2013 "MMM" "Tony Brown"                     "Independent Director"      0 0 0 0 0
2014 "MMM" "Inge Thulin"                    "Chairman/President/CEO"    0 0 0 0 0
2014 "MMM" "Bob Ulrich"                     "Independent Director"      0 0 0 0 1
2014 "MMM" "Doctor Vance Coffman"           "Independent Director"      1 0 0 0 0
2014 "MMM" "Ed Liddy"                       "Independent Director"      1 0 0 0 0
2014 "MMM" "Herb Henkel"                    "Independent Director"      1 0 0 0 1
2014 "MMM" "Linda Alvarado"                 "Independent Director"      0 0 0 0 1
2014 "MMM" "Mike Eskew"                     "Lead Independent Director" 0 0 0 0 0
2014 "MMM" "Muhtar Kent"                    "Independent Director"      0 0 0 0 0
2014 "MMM" "Sondra Barbour"                 "Independent Director"      0 0 0 0 0
2014 "MMM" "Tony Brown"                     "Independent Director"      0 0 0 0 0
2015 "MMM" "Inge Thulin"                    "Chairman/President/CEO"    0 0 0 0 0
2015 "MMM" "Bob Ulrich"                     "Independent Director"      0 0 0 0 1
2015 "MMM" "Dave Dillon"                    "Independent Director"      0 0 0 0 0
2015 "MMM" "Doctor Vance Coffman"           "Independent Director"      1 0 0 0 0
2015 "MMM" "Ed Liddy"                       "Independent Director"      1 0 0 0 0
2015 "MMM" "Herb Henkel"                    "Independent Director"      1 0 0 0 1
2015 "MMM" "Linda Alvarado"                 "Independent Director"      0 0 0 0 1
2015 "MMM" "Mike Eskew"                     "Lead Independent Director" 0 0 0 0 0
2015 "MMM" "Muhtar Kent"                    "Independent Director"      0 0 0 0 0
2015 "MMM" "Sondra Barbour"                 "Independent Director"      0 0 0 0 0
2015 "MMM" "Tony Brown"                     "Independent Director"      0 0 0 0 1
2016 "MMM" "Inge Thulin"                    "Chairman/President/CEO"    0 0 0 0 0
2016 "MMM" "Bob Ulrich"                     "Independent Director"      0 0 1 0 1
2016 "MMM" "Dave Dillon"                    "Independent Director"      0 0 0 0 0
2016 "MMM" "Doctor Vance Coffman"           "Independent Director"      1 0 0 0 0
2016 "MMM" "Ed Liddy"                       "Independent Director"      1 0 1 0 1
2016 "MMM" "Greg Page"                      "Independent Director"      0 0 0 0 0
2016 "MMM" "Herb Henkel"                    "Independent Director"      1 0 0 0 1
2016 "MMM" "Mike Eskew"                     "Lead Independent Director" 0 0 1 0 0
2016 "MMM" "Muhtar Kent"                    "Independent Director"      0 0 0 0 0
2016 "MMM" "Pat Woertz"                     "Independent Director"      0 0 0 0 0
2016 "MMM" "Sondra Barbour"                 "Independent Director"      0 0 0 0 1
2016 "MMM" "Tony Brown"                     "Independent Director"      0 0 0 0 1
2017 "MMM" "Inge Thulin"                    "Chairman/President/CEO"    0 0 0 0 0
2017 "MMM" "Amy Hood"                       "Independent Director"      0 0 0 0 0
2017 "MMM" "Dave Dillon"                    "Independent Director"      0 0 1 1 1
2017 "MMM" "Doctor Vance Coffman"           "Independent Director"      1 0 0 0 0
2017 "MMM" "Ed Liddy"                       "Independent Director"      1 0 1 0 1
2017 "MMM" "Greg Page"                      "Independent Director"      1 0 1 0 0
2017 "MMM" "Herb Henkel"                    "Independent Director"      1 0 0 0 1
2017 "MMM" "Mike Eskew"                     "Lead Independent Director" 0 0 1 0 0
2017 "MMM" "Muhtar Kent"                    "Independent Director"      0 0 0 0 0
2017 "MMM" "Pat Woertz"                     "Independent Director"      1 0 1 0 0
2017 "MMM" "Sondra Barbour"                 "Independent Director"      0 0 0 0 1
2017 "MMM" "Tony Brown"                     "Independent Director"      0 0 0 0 1
2018 "MMM" "Inge Thulin"                    "Chairman (Executive)"      0 0 0 0 0
2018 "MMM" "Mike Roman"                     "CEO"                       0 0 0 0 0
2018 "MMM" "Amy Hood"                       "Independent Director"      0 0 0 0 0
2018 "MMM" "Dave Dillon"                    "Independent Director"      0 0 1 1 1
2018 "MMM" "Doctor Dambisa Moyo"            "Independent Director"      0 0 0 0 0
2018 "MMM" "Ed Liddy"                       "Independent Director"      1 0 1 0 1
2018 "MMM" "Greg Page"                      "Independent Director"      1 0 1 0 0
2018 "MMM" "Herb Henkel"                    "Independent Director"      1 0 0 0 1
2018 "MMM" "Mike Eskew"                     "Lead Independent Director" 0 0 1 0 0
2018 "MMM" "Muhtar Kent"                    "Independent Director"      0 0 0 0 0
2018 "MMM" "Pat Woertz"                     "Independent Director"      1 0 1 0 0
2018 "MMM" "Sondra Barbour"                 "Independent Director"      0 0 0 0 1
2018 "MMM" "Tony Brown"                     "Independent Director"      0 0 0 0 1
2011 "ABT" "Miles White"                    "Chairman/CEO"              1 0 0 0 0
2011 "ABT" "Bill Osborn"                    "Independent Director"      0 0 0 0 0
2011 "ABT" "Ed Liddy"                       "Independent Director"      0 0 0 0 0
2011 "ABT" "Glenn Tilton"                   "Independent Director"      0 0 0 1 0
2011 "ABT" "Harry Fuller"                   "Lead Independent Director" 0 0 0 0 0
2011 "ABT" "Jim Farrell"                    "Independent Director"      0 0 0 0 0
2011 "ABT" "Nancy McKinstry"                "Independent Director"      0 0 0 0 0
2011 "ABT" "Phebe Novakovic"                "Independent Director"      0 0 0 0 0
2011 "ABT" "Professor Doctor Robert Alpern" "Independent Director"      1 0 0 0 0
2011 "ABT" "Roxanne Austin"                 "Independent Director"      0 0 0 0 0
end

[Nick Cox]

The Blau index is, if I recall correctly, either a sum of squared probabilities or its complement. I forget which, but this shows some technique. On Statalist the name often used for the same measure (again, or its complement) is Herfindahl although Gini has a stronger claim than either to inventing it -- and that in turn might confuse because the name Gini is attached at least three different measures.

More generally, assuming that Statalist readers work in the same sub-sub-field as you and hence know the same jargon is sometimes optimistic.

There are canned commands for this kind of thing and I am guilty of writing some, but it is always a good idea to know how to do it from first principles, which means calculating numerator and denominator and making sure that you don't include repeated values more than once.

. bysort Year Ticker RoleName : gen freq = _N if _n == 1 
(72 missing values generated)

. by Year Ticker : gen total = _N 

. 
. by Year Ticker : egen Blau = total((freq/total)^2)

. 
. list Year Ticker RoleName freq total Blau, sepby(Year Ticker)

     +---------------------------------------------------------------------+
     | Year   Ticker                    RoleName   freq   total       Blau |
     |---------------------------------------------------------------------|
  1. | 2011      ABT                Chairman/CEO      1      10        .66 |
  2. | 2011      ABT        Independent Director      8      10        .66 |
  3. | 2011      ABT        Independent Director      .      10        .66 |
  4. | 2011      ABT        Independent Director      .      10        .66 |
  5. | 2011      ABT        Independent Director      .      10        .66 |
  6. | 2011      ABT        Independent Director      .      10        .66 |
  7. | 2011      ABT        Independent Director      .      10        .66 |
  8. | 2011      ABT        Independent Director      .      10        .66 |
  9. | 2011      ABT        Independent Director      .      10        .66 |
 10. | 2011      ABT   Lead Independent Director      1      10        .66 |
     |---------------------------------------------------------------------|
 11. | 2011      MMM      Chairman/President/CEO      1      10        .66 |
 12. | 2011      MMM        Independent Director      8      10        .66 |
 13. | 2011      MMM        Independent Director      .      10        .66 |
 14. | 2011      MMM        Independent Director      .      10        .66 |
 15. | 2011      MMM        Independent Director      .      10        .66 |
 16. | 2011      MMM        Independent Director      .      10        .66 |
 17. | 2011      MMM        Independent Director      .      10        .66 |
 18. | 2011      MMM        Independent Director      .      10        .66 |
 19. | 2011      MMM        Independent Director      .      10        .66 |
 20. | 2011      MMM   Lead Independent Director      1      10        .66 |
     |---------------------------------------------------------------------|
 21. | 2012      MMM      Chairman/President/CEO      1      10        .66 |
 22. | 2012      MMM        Independent Director      8      10        .66 |
 23. | 2012      MMM        Independent Director      .      10        .66 |
 24. | 2012      MMM        Independent Director      .      10        .66 |
 25. | 2012      MMM        Independent Director      .      10        .66 |
 26. | 2012      MMM        Independent Director      .      10        .66 |
 27. | 2012      MMM        Independent Director      .      10        .66 |
 28. | 2012      MMM        Independent Director      .      10        .66 |
 29. | 2012      MMM        Independent Director      .      10        .66 |
 30. | 2012      MMM   Lead Independent Director      1      10        .66 |
     |---------------------------------------------------------------------|
 31. | 2013      MMM      Chairman/President/CEO      1      12   .7083333 |
 32. | 2013      MMM        Independent Director     10      12   .7083333 |
 33. | 2013      MMM        Independent Director      .      12   .7083333 |
 34. | 2013      MMM        Independent Director      .      12   .7083333 |
 35. | 2013      MMM        Independent Director      .      12   .7083333 |
 36. | 2013      MMM        Independent Director      .      12   .7083333 |
 37. | 2013      MMM        Independent Director      .      12   .7083333 |
 38. | 2013      MMM        Independent Director      .      12   .7083333 |
 39. | 2013      MMM        Independent Director      .      12   .7083333 |
 40. | 2013      MMM        Independent Director      .      12   .7083333 |
 41. | 2013      MMM        Independent Director      .      12   .7083333 |
 42. | 2013      MMM   Lead Independent Director      1      12   .7083333 |
     |---------------------------------------------------------------------|
 43. | 2014      MMM      Chairman/President/CEO      1      10        .66 |
 44. | 2014      MMM        Independent Director      8      10        .66 |
 45. | 2014      MMM        Independent Director      .      10        .66 |
 46. | 2014      MMM        Independent Director      .      10        .66 |
 47. | 2014      MMM        Independent Director      .      10        .66 |
 48. | 2014      MMM        Independent Director      .      10        .66 |
 49. | 2014      MMM        Independent Director      .      10        .66 |
 50. | 2014      MMM        Independent Director      .      10        .66 |
 51. | 2014      MMM        Independent Director      .      10        .66 |
 52. | 2014      MMM   Lead Independent Director      1      10        .66 |
     |---------------------------------------------------------------------|
 53. | 2015      MMM      Chairman/President/CEO      1      11   .6859504 |
 54. | 2015      MMM        Independent Director      9      11   .6859504 |
 55. | 2015      MMM        Independent Director      .      11   .6859504 |
 56. | 2015      MMM        Independent Director      .      11   .6859504 |
 57. | 2015      MMM        Independent Director      .      11   .6859504 |
 58. | 2015      MMM        Independent Director      .      11   .6859504 |
 59. | 2015      MMM        Independent Director      .      11   .6859504 |
 60. | 2015      MMM        Independent Director      .      11   .6859504 |
 61. | 2015      MMM        Independent Director      .      11   .6859504 |
 62. | 2015      MMM        Independent Director      .      11   .6859504 |
 63. | 2015      MMM   Lead Independent Director      1      11   .6859504 |
     |---------------------------------------------------------------------|
 64. | 2016      MMM      Chairman/President/CEO      1      12   .7083333 |
 65. | 2016      MMM        Independent Director     10      12   .7083333 |
 66. | 2016      MMM        Independent Director      .      12   .7083333 |
 67. | 2016      MMM        Independent Director      .      12   .7083333 |
 68. | 2016      MMM        Independent Director      .      12   .7083333 |
 69. | 2016      MMM        Independent Director      .      12   .7083333 |
 70. | 2016      MMM        Independent Director      .      12   .7083333 |
 71. | 2016      MMM        Independent Director      .      12   .7083333 |
 72. | 2016      MMM        Independent Director      .      12   .7083333 |
 73. | 2016      MMM        Independent Director      .      12   .7083333 |
 74. | 2016      MMM        Independent Director      .      12   .7083333 |
 75. | 2016      MMM   Lead Independent Director      1      12   .7083333 |
     |---------------------------------------------------------------------|
 76. | 2017      MMM      Chairman/President/CEO      1      12   .7083333 |
 77. | 2017      MMM        Independent Director     10      12   .7083333 |
 78. | 2017      MMM        Independent Director      .      12   .7083333 |
 79. | 2017      MMM        Independent Director      .      12   .7083333 |
 80. | 2017      MMM        Independent Director      .      12   .7083333 |
 81. | 2017      MMM        Independent Director      .      12   .7083333 |
 82. | 2017      MMM        Independent Director      .      12   .7083333 |
 83. | 2017      MMM        Independent Director      .      12   .7083333 |
 84. | 2017      MMM        Independent Director      .      12   .7083333 |
 85. | 2017      MMM        Independent Director      .      12   .7083333 |
 86. | 2017      MMM        Independent Director      .      12   .7083333 |
 87. | 2017      MMM   Lead Independent Director      1      12   .7083333 |
     |---------------------------------------------------------------------|
 88. | 2018      MMM                         CEO      1      13   .6094674 |
 89. | 2018      MMM        Chairman (Executive)      1      13   .6094674 |
 90. | 2018      MMM        Independent Director     10      13   .6094674 |
 91. | 2018      MMM        Independent Director      .      13   .6094674 |
 92. | 2018      MMM        Independent Director      .      13   .6094674 |
 93. | 2018      MMM        Independent Director      .      13   .6094674 |
 94. | 2018      MMM        Independent Director      .      13   .6094674 |
 95. | 2018      MMM        Independent Director      .      13   .6094674 |
 96. | 2018      MMM        Independent Director      .      13   .6094674 |
 97. | 2018      MMM        Independent Director      .      13   .6094674 |
 98. | 2018      MMM        Independent Director      .      13   .6094674 |
 99. | 2018      MMM        Independent Director      .      13   .6094674 |
100. | 2018      MMM   Lead Independent Director      1      13   .6094674 |
     +---------------------------------------------------------------------+

[Mike Lacy]

The Blau Index has more aliases than a high grade con-artist. Even among Blau's fellow sociologists, at least 2 other names are in use for what are essentially similar measures, the Gibbs-Martin Index and the Index of Qualitative Variation, the latter being probably most common. Add to this terms that reference Gini, Herfindahl, Simpson, and others I have missed ... . I think we need an interdisciplinary treaty of sorts among economists, sociologists, biologists, statisticians, geographers, and whatever other disciplinary countries claim this territory about what will be its rightful name. <grin> My impression is that discussions of the measure within discipline tend to ignore predecessors in other places. I'm violating our local norm of "complete references" to keep this short, and just say that interested persons can search with these terms in Wikipedia, and find more than one relevant entry.

[Nick Cox]

Stigler's Law of Eponymy runs that nothing is named after the people who really invented or discovered it. (Yes, it's self-referential.)

Lacy's Law of Eponymy is, evidently, that it has to be reinvented by people in your literature to be noticed.

Other names that don't mention people are repeat rate, match probability and quadratic entropy.

As before, assume any measure, its complement and the reciprocal of either are equivalent between friends.