Companies should be wary of any pay equity analysis that offers to look at wage disparities after controlling for legitimate business factors. The term “after” is open to interpretation, but a defensible, reliable pay equity assessment must measure wage disparities in a full model — one that also measures the effects of legitimate business factors at the same time.

The alternative to such a full model of wages is one in which the effects of legitimate business factors on employee wages are measured first. Under this alternative, the net effects of such legitimate business factors are evaluated for each employee to arrive at the employee’s predicted wage. The differences between each employee’s actual and the predicted wage for that employee are then averaged by protected class, for example by gender. This average difference in disparities is treated as the “pay gap.”

This post provides a simple, cautionary example as to why this two-step approach is likely misleading. If there is a class wage disparity, it is under-represented by this approach, which also mis-measures the role of legitimate business factors in determining wages.

In this example there are 100 employees, 50 are men (M) and 50 are women (F). There is just one legitimate factor affecting wages, which we could call “Experience.” Those with this factor are paid 20% more than those without. Also, suppose that there is a 10% gender wage gap in the data. Women are paid 10% less than men. These wage differences can be summarized as:

Percent Wage Differences:

|    |   M |    F |
| E  | 20% |  10% |
| !E | 0%  | -10% |

Where “E” indicates “Experience” and “!E” indicates No Experience. To be concrete, let’s assume we have only four wage rates across all employees. To be consistent with the percentage differences above, these could be as follows:

Percent Wage Differences:

|    | M      | F      |
| E  | $24/hr | $22/hr |
| !E | $20/hr | $18/hr |

In this example, 80% of those with Experience (E) are women and 20% are men. The numbers are reversed for No Experience (!E). For simplicity, half the employees (50) have Experience (E) while half do not (!E). Then counts of employees in the 4 categories above are:

Employee Counts

|    |  M |  F |
| E  | 10 | 40 |
| !E | 40 | 10 |

Ignoring gender, the weighted average wage difference between Experienced workers (E) and non-Experienced workers (!E) is the difference in the weighted average wages of E and !E. The weighted average wage premium for (E) is:

(20% x 10 + 10% x 40)/(10 + 40) = 12%.

while the weighted average wage for (!E) is:

(-10% x 10 + 0% x 40)/(10 + 40)= -2%.

In dollar terms, the weighted average E wage is $22.40 while the weighted average !E wage is $19.60.

The average wage difference between workers with Experience and without appears to be 12% – -2% = 14%, which is less than the true Experience wage premium of 20%. By ignoring the effect of gender in the first stage, the role of the legitimate business factor Experience in determining wages is underestimated.

If we look at wages net of this E vs !E effect for each of the four employee wage categories, we have:

Apparent Wage Differences Net of E Effects

|    | M              | F                |
| E  | 20% - 12% = 8% | 10% - 12% = -2%  |
| !E | 0% - -2% = 2%  | -10% - -2% = -8% |

Next, if we use these first-stage results to calculate the gender-specific wage, after controlling for apparent E effects we get:

(8% x 10 + 2% x 40)/(10 + 40) = 3.2% for M (Males) and
(-8% x 10 + -2% x 40)/(10 + 40) = -3.2% for F (Females).

That is, after controlling for E (i.e., the legitimate business factor), the gender wage gap appears to be 3.2% – -3.2% = 6.4%, which underestimates the true gender disparity of 10%.

Note that both the effect of E and the effect of gender are mismeasured. The E wage premium as reflected in the weighted average difference in wages between E and !E is 12% – -2% or 14%. The actual E effect is larger, namely 20%. Similarly, in the second stage, since the role of E on wages was mismeasured in the first step, the gender disparity is also underestimated. This two-step approach implies a gender disparity of 6.4% rather than the actual disparity of 10%.

This example used differences in weighted average wages, but these are precisely the results that are obtained when one applies a regression model first only to measuring legitimate business factor effects (like the E effect) and then uses wages net of those effects to measure a gender (or other protected class) disparity. The underlying statistical theory discrediting such two-step analyses is well known to regulators and plaintiff experts. Don’t risk being misled by consultants who first offer to measure how your legitimate business factors explain wages without also measuring the role of protected class wage effects.

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