Thread by @kareem_carr

Statistics can never be completely objective.

This is not just my opinion. It's a *mathematical* fact.

Read on if you want to learn a deep fundamental truth about data and its relationship to the universe we live in.

[At the end of this thread, you should also understand why robust social science research is fundamental to the correct interpretation of data related to racial disparities.]

SIMPSON'S PARADOX

"Every statistical relationship between two variables X and Y has the potential to be reversed when we include a third variable Z into the analysis."

This is called Simpson's paradox.

A famous example of this is data at the heart of the 1973 UC Berkeley Gender-Bias case.

There seemed to be a clear bias in the admissions rates of men vs women. About 45% of men who applied were admitted compared to only 30% of the women.

But once choice of major was added to the analysis, it was revealed that women were being accepted at higher rates in almost every field.

This seems like a contradiction at first but the solution is rather simple.

A disproportionate percentage of men were applying to easier majors to get into, like Departments A and B.

Approximately 1000 more men than women had applied to those departments alone and this difference shows up in the aggregated data as a higher admissions rate for men.

To summarize, being male seemed to be associated with an increase in the admissions rate but adding a third factor, choice of major, *reversed* the association.

Being male was actually associated with a decrease in admissions rate.

We call this a "reversal".

REVERSALS

The phenomenon of "reversal" should disturb you. It means the conclusion of any data analysis could be completely flipped just by adding an extra variable to the statistical model.

The situation is actually even worse than it first seems.

It is indeed possible to have chains of reversals where *every* new variable added to a data analysis *reverses* the direction of the relationship established in the previous analysis.

"Did being black help Obama become president?

For most of his life, almost certainly no, but perhaps during the primary debates it helped him stand out, but once he was running it probably lost him voters that he would have otherwise won over had he been white.

"Do masks reduce covid infections?"

Maybe in general they do, but if you use them badly they don't, but if you always socially distance, perhaps it doesn't change much either way.

PREDICTION

It might seem at first that this means that statistics is useless as a source of knowledge, but each additional variable actually tends to make our statistical model better.

This might seem like a contradiction but it's not.

Generally speaking, every new variable added to the model will improve the accuracy of its predictions. It is only the interpretation of the relationships between the outcome and input variables that might change.

In cases where interpretation doesn't matter, this can be enough.

CAUSATION

The correct relationship between X and Y can only be found by performing statistical analyses which respect the causal relationship between all the variables in the analysis.

Therefore, causal information is fundamental to the accuracy of statistical analyses.

Let me explain what I mean by "casual information".

Imagine we're trying to quantify the effect of our summer math camp program on student's math grades during the year, and we have information on math camp attendance, final grades, GRE scores and GPAs.

Adding causal information to our analysis means we need to produce a diagram like this one.

For each variable in the analysis, we need to identify all the relevant variables it is influenced by and all the relevant variables it influences.

Three important sources of causal information are:

1. Commonsense Assumptions. The analyst can make reasonable assumptions and try their best to justify them in their report.

2. Expertise/Scientific knowledge. Having the infrastructure of an entire science behind you can be extremely useful in making the tough decisions about which causal relationships are real.

3. Experimental Design. We can collect data in a way that makes all the causal paths that we don't care about physically impossible, and allows us to isolate the causal path that we do care about.

OBJECTIVITY

None of these ways of getting causal information is infallible. Therefore, no statistical analysis is infallible. All rely on some prior knowledge.

Many statistical analyses do not state what prior information was used which makes them seem "objective".

This does not mean that none were used.

It only means the audience was robbed of the ability to fully assess the quality of the causal information used and of the analysis.

A statistical analysis can never be completely objective because it lives or dies on the quality of the analyst's prior knowledge.

This is a mathematical constraint on humans, AI or anything else that needs to analyze data.

It is a fundamental property of our universe.

For more information, check out:
- "The Book of Why" by Judea Pearl [Accessible to a General Audience]
- "Understanding Simpson's Paradox" also by Pearl [Short. Technical. Good discussion of Reversals]
- "What If" by Hernán and Robins [Free Online Textbook]

I enjoy explaining math and statistics ideas in a way that regular people can understand.

Follow me for more content like this, and don't forget to click the little notification bell so you don't miss out on future threads.