The Residual Fallacy

Douglas P. McManaman
March 27, 2019
Reproduced with Permission

That women are still paid less than men for doing the same work is a claim that has been tested and has been found wanting. The root of this claim is the residual fallacy, a pervasive statistical fallacy that involves assuming that all unexamined factors that might contribute to an explanation of a particular phenomenon are equal, such that all remaining differences in outcome can be attributed to discrimination.

Allow me to explain using the conditional reasoning. The logic of the scientific method requires testing a conclusion, because the method is fundamentally inductive. This means we begin with the evidence and proceed to account for that evidence through conjecture. Corroborating evidence does not prove a hypothesis; rather, confirming a hypothesis leaves a conclusion underdetermined (a matter of affirming the consequent), and so rigorous testing is required in order to determine which alternative possibilities are the most plausible. You are familiar with the following argument:

If p, then q
Therefore, p

In the context of scientific experimentation, we speak of independent and dependent variables. If we were to put this in the context of the conditional syllogism, it would look like the following:

If (independent variable), then (dependent variable)
dependent variable
Therefore, independent variable.

For example,

If I eat cookies that are high in fiber (p), then I will experience stomach pains (q)
I am experiencing stomach pains (q).
Therefore, it is the result of eating cookies that are high in fiber (p)

The argument requires testing because it is deductively invalid. The independent variable in this case is the high fiber cookies; the dependent variable is the symptoms that I am experiencing, that is, the stomach pains. The problem is there are other factors that can account for the dependent variable, so to test this particular hypothesis, we must control for these other variables, that is, we must try to reduce the effect of confounding variables, which are variables that influence both the dependent variable and the independent variable; this will enable us to determine, with greater plausibility, whether high fiber cookies are in fact the reason for the stomach pains. For example, I like to drink diet soda when snacking on a high fiber cookie; hence, we will need to try to reduce the effect of soda, which is a possible factor in having a stomach ache as well as factor in my reaching for a cookie. What this means is that the effects of all other variable predictors need to be taken into account as much as possible. Other variable predictors might include soda pop, grains, sugar, certain dairy products, fish, processed foods, turkey, etc.

What we want to do now is assess the effect of changing or manipulating one or more of the independent variables on the dependent variable. What is the effect of manipulating the independent variable "cookies high in fiber" on stomach pains? I am instructed by my doctor to cut out cookies high in fiber from my diet; in time we discover that it has no effect on my stomach pains. We conclude that the high fiber cookies are an irrelevant variable.

If p, then q

Hence, p is an irrelevant variable. However, if the effect of manipulating the independent variable brought about a change in the dependent variable, that would still not prove the hypothesis:

If p, then q

This is a matter of denying the antecedent, which is also deductively invalid. It certainly corroborates the hypothesis that it was the fiber, but it does not prove it, so further testing is required.

Control variables must be held constant in order to discern the relationship between independent and dependent variables. If we are comparing the flights of two badminton birdies made by two different companies, we must hold constant certain control variables if our conclusion is to have plausibility. For example, if we test one birdie on a bright and sunny day, without the slightest breeze in the air and the other birdie on a cloudy and windy day, our conclusion about their quality would be unwarranted; for the differences in their flight could very well be due to factors other than the quality of the birdie, such as the weather, or possibly the person testing the birdie, the kind of racquet he is using, etc. So, the weather, the player himself, his equipment, etc., are control variables that must be held constant in this case. Both birdies must be tested in the same conditions. If the flights of the two birdies are significantly different within the same conditions (with the control variables held constant), we know the flight difference was not due to the differences in weather, player, equipment, etc., for these factors were held constant.

Now, the claim that income disparity between men and women is the result of unjust discrimination must undergo the same kind of testing. Hence,

If there is income discrimination against women in the workforce, there will be income disparity between men and women
There is income disparity between men and women.
Hence, women are victims of income discrimination in the workforce

Once again, we know this is a deductively invalid argument, so it must be tested. Other variables might account for the outcome (income disparity), such as quantity of education, quality of education, type of degree one has, years of experience (often interrupted by maternity leave), skill level, number of hours worked on a weekly basis, etc. All these must be held constant. If income disparity disappears when we control for certain confounding variables, then discrimination becomes an irrelevant variable.

The two groups under consideration are "men with university degrees" and "women with university degrees". There are, however, other variable predictors that come to the fore as a result of a more precise analysis of the general category of university or college degree. To hold years of education (quantity) constant is not enough, for there are qualitative differences as well that have to be take into consideration. Qualitative differences are a variable predictor, and quality can be measured by academic performance, the ranking of the university, or the difficulty and remuneration factor in the particular field of study, etc. In terms of remuneration capacity, it is unreasonable of me to expect a master's degree in philosophy to equal a master's degree in biochemistry or electrical engineering degree; most industries have no use for a person with a graduate degree in philosophy but do have a place for a person with a master's in biochemistry or engineering - philosophy majors are more likely to be waiting on tables after they earn their degree; engineers are usually working as engineers.

The category of "university educated" women and men is problematic from another angle. University graduates include people who go on to postgraduate study, and this too influences income. The ratio of women and men with postgraduate degrees differs from the ratios of those with university degrees. At the bachelor's degree level, women outnumber men, but men outnumber women by more than a two to one ratio at the master's level and by 59% at the PhD level. So when we compare university educated men and women, which includes those who have gone on to pursue postgraduate work, we are really comparing apples and oranges.

If we wish to compare men and women at the PhD level, we discover once again disparities between men and women, and changing ratios. Women receive 37% of all PhDs; moreover, the areas of study differ significantly from those of the 63% of males who receive a PhD. The PhDs which men receive tend to be more heavily concentrated in math and science and other fields of greater remunerative capacity. Women received almost half of the PhDs in the social sciences, and more than half in the area of education. Men received more than 80% of the PhDs in the sciences and more than 90% in engineering. Not even the social sciences are equally remunerative; two people with a social science degree may show a difference in income if the one degree is in sociology while the other is in econometrics; a degree in econometrics has greater remunerative capacity - and more men enter econometrics than do women.

It is simply not the case that the disparity of income between men and women is due to discrimination any more than the disparity of income between me and Oprah Winfrey is due to a pervasive discrimination against philosophy teachers.

The fallacy at the heart of the claim that women make less than men for doing the same work is the fallacy of assuming that all variables left unexamined must be equal so that all residual differences in outcome (in this case, income) can be attributed to discrimination. Such a conclusion is always underdetermined, and it is very often unwarranted.

The same can be said for the contention that "white privilege" is the reason that a majority are successful and white, while a minority are less successful and "not white". To illustrate, I will simplify society into two groups: East Indian minority that makes up 30% of the population, leaving the white majority at 70%. Let it also be that 20% of the Indian population are very successful (2 out of 10). Since the Indian population constitutes 30% of the society that is a white majority, it follows that 6% of the entire population is East Indian and very successful. We will also assume 50% of the Indian population are middle class (5 out of 10). That's 15% of the entire population is East Indian and middle class. Finally, we'll say that 30% of the East Indian population are poor (3 out of 10). That's 9% of the entire society are poor East Indians.

In the white population, which is the remaining 70%, 10% are white and very poor (1 out of 10), or 7% of the entire population, while 60% of the white population are middle class, or 6 out of 10, which means 42% of the entire population are white middle class (since 70% of the population is white). And finally, we'll say that 30% of the white population are very rich, or 3 out of 10. That means 21% of the entire population is white and very rich.



As we can see when considering their respective populations, a greater percentage of East Indians are poor (⅔ more), and a greater percentage of whites are rich (⅓ more) and middle class (16.7% more).

The residual fallacy consists in assuming that all unexamined factors that might contribute to an explanation of this particular phenomenon are equal, such that all remaining differences in outcome can be attributed to discrimination, or white privilege.

But such an assumption needs to be tested. Other possible variables include: educational backgrounds, educational standards (i.e., medical schools in India are not the same as medical schools here, different degree requirements, which requires extra schooling here in Canada, etc.), different educational opportunities in India, so those who emigrate here have a different distribution of low skilled versus skilled labor; initial language barriers, lack of connections (family or business connections), the time it takes to acquire credit, or differing initial economic conditions (i.e., my parents arrived from India with nothing, so they did not inherit anything from anyone, while your parents inherited money when their parents died, which permitted you to open your own business, because your parents put that inheritance in savings, etc.).

Consider this last condition: parents arrived from India with nothing, did not inherit anything from anyone, while some white students inherited money when their parents died, which permitted them to open their own business, etc. This privilege, without question, but it is not essentially white. Whiteness is incidental. It is simply the result of circumstances belonging to one country (Canada) that are different from those of another country (India).

We can compare your parents with the parents of that white student, but we can also compare your parents with the parents of another white student whose parents did not inherit anything. Both are white, but both do not enjoy the same level of privilege. Thus, privilege, like income, is circumstantial, while "white privilege" is categorical. The residual fallacy in this case involves assuming that the distribution of "privilege" is due to the color of one's skin, or the "category" to which one belongs by virtue of one's color. However, there are a multitude of factors that prevent a large percentage of white people from "privilege", namely, a poor work ethic, poor choices, a sense of entitlement, lack of talent, drug use that has had adverse effects on brain development, etc. Such factors also keep some members of a minority group from achieving a level of "privilege" available to them in this country. But it has been factors opposite the aforementioned, namely, a good work ethic, hard work, character, talent, persistence and good decisions, etc., that account for the level of privilege that many people, white or brown, currently enjoy. Indeed, some who enjoy privilege may not fit that description, but may have inherited all they have. However, such privilege cannot be sustained without those factors.

Once a generation has passed and the language barrier is no longer a factor - because the children of East Indian parents went to public school and now speak fluently and they have the same educational opportunities as white children, and are achieving comparably with white children and in many cases better than the white students, since the Indian work ethic is significantly better - , we should see a change in the differences in percentages in the latter table above; they should eventually equal out at the very least. If white privilege is nothing more than a fallacy rooted in prejudice, then not only will they equal out, they might even tip in favor of some minorities. And this is just what has happened among the Asian population. In 2001, 31% of Chinese in Canada - both those born in Canada and foreign born - had a university education, while the national average was 18%. Furthermore, Chinese who immigrated to Canada in the 90s and who were of prime working age had an employment rate of 61%, 19% lower than the national average only eleven years later in 2001. The fundamental issue here, however, was the recognition of foreign qualifications. But, the employment rate for Canadian-born Chinese men who were of prime working age was slightly above the national average (86%), and the employment rate for Canadian born Chinese women of prime working age was 83%, which was higher than the national average, which was 76% (Chinese Canadians: Enriching the cultural mosaic," Canadian Social Trends, Spring 2005, no. 76)