Is Big Worse than Bad?

So far on StatisticalBullshit.com, I’ve written about general Statistical Bullshit, Statistical Bullshit that I’ve come across in consulting, or Statistical Bullshit that readers have sent into the website.  I don’t *think* that I’ve written about Statistical Bullshit that was pointed out by an academic article.  For this reason, today’s post is about the concept of how being big could be worse than being bad in regards to Equal Employment Opportunity Enforcement Policies (EEOP).  Most of the material for this post comes from Jacobs, Murphy, and Silva’s (2012) article, entitled “Unintended Consequences of EEO Enforcement Policies: Being Big is Worse than Being Bad,” which was published in the Journal of Business and Psychology.  Rick Jacobs was my advisor at Penn State, so I am happy to have his article discussed on StatisticalBullshit.com.  For more information about this concept, please email me at MHoward@SouthAlabama.edu or check out Jacobs et al. (2012).

As stated by Jacobs et al. (2012), “The Equal Employment Opportunity Commission (EEOC) is the chief Federal agency charged with enforcing the Civil Rights Acts of 1964 and 1991 and other federal laws that forbid discrimination against a job applicant or an employee because of the person’s race, color, religion, sex (including pregnancy), national origin, age (40 or older), disability, or genetic information” (p. 467).  In other words, the EEOC ensures that businesses do not discriminate against protected classes, and this includes in business employment practices.

When a disproportionately low number of peoples from a protected class are hired, most often relative to the majority class, this is called disparate impact.  But how do we know whether a “disproportionately low number” has occurred?  In EEO cases, there are many methods, but the 80% rule and statistical significance testing are among the most popular.

The 80% rule specifies that disparate impact occurs when members of a protected class are hired at a rate that is less than 80% the rate of the majority class.  Let’s take a look at the following example to figure out what this means:

In this example, 40 Caucasian people and 20 African Americans applied for the same job.  The organization selected 20 Caucasians and 5 African Americans for the job.  This results in 50% of the Caucasians being hired, but only 25% of the African Americans being hired.  To determine whether disparate impact occurred, we take .50 (ratio for Caucasians) and multiply it by .80 (80% rule).  This results in .40.  We then compare this number to the ratio of African Americans hired, .25.  Since .40 is greater than .25, we can determine that disparate impact occurred on the basis of the 80% rule.

Although it may seem relatively simple, the 80% rule works quite well across most situations.  But what about the other method – statistical significance testing?

Many different tests could be used to identify disparate impact, but the chi-square test may be the simplest.  The chi-square test can be used to determine whether the association between two categorical variables is significant, such as whether the association between race and hiring decisions is significant.  So, we can use this to test whether disparate impact may have occurred.

To do so, you can use the following calculator: https://www.graphpad.com/quickcalcs/contingency2/ .  Let’s enter the data above, which would look like the following in a chi-square calculator:

The resultant p-value is .06, which is greater than .05.  Not statistically significant!  Although this is the exact same data as the 80% rule example, the chi-square test determined that it was not a case of disparate impact.  Interesting!

But what happens when we double the size of each group?  The 80% rule table would look like this:

Again, the resultant ratio for Caucasians is .50, which is .40 when multiplied by .80 (80% rule).  The resultant ratio for African Americans is .25, which is smaller than .40.  This suggests that disparate impact occurred on the basis of the 80% rule.

On the other hand, let’s enter this data into the chi-square calculator, which would look like this:

The resultant p-value is .009, which is much less than .05.  Statistically significant!  While the ratios are identical for the two examples, the latter chi-square test determines that disparate impact occurred.

This is the idea behind “Being Big is Worse than Being Bad.”  Although both examples had the same ratio, and thereby were just as bad, the chi-square test indicated that disparate impact only occurred in the latter example, which was bigger.  Thus, significance testing has concerns when identifying disparate impact, because the sample size strongly influences whether a result is significant or not.

So, do we just apply the 80% rule?  Not necessarily.  Jacobs et al. (2012) call for “a more dynamic definition of adverse impact, one that considers sample size in light of other important factors in the specific selection situation” (p. 470), and they also call for a less-simplified view of disparate and adverse impact.  While the 80% rule can certainly address certain problems that significance testing encounters, it cannot satisfy all the needs within this call.

Issues like these is why we need more statistics-savvy people in the world.  Disparate and adverse impact are huge issues that impact millions of people.  And many of these decisions are not made by statisticians.  Instead, they are made by courts and companies.  Even if you aren’t interested in becoming a statistician, the world still needs people that understand statistics – and know how to watch out for Statistical Bullshit in significance testing!

That’s all for today.  If you have any questions, please email me at MHoward@SouthAlabama.edu.  Until next time, watch out for Statistical Bullshit!