School, Grades and the Collider

Another example of Berkson’s paradox, a collider bias, is the observed relationship, in surveys, between attending classes and grades. Here we illustrate the various possibilities and the results. Here, we explain the math using the following example.

Attend	Attend	Do Not Attend	Do Not Attend
Good Grade	Poor Grade	Good Grade	Poor Grade
300	200	200	300

And this leads to the following conclusions:

1. Probability of getting good grades, given the person attends classes, P(G|A) = # good grades and attend / total attend = 300/(300 + 200)= 0.6 
2. Probability of getting poor grades, given the person attends, P(P|A) = 1 -P(G|A) = 0.4
3. Probability of getting good grades, given the person doesn’t attend, P(G|N) = 200/(200 + 300) = 0.4
4. Probability of getting good grades, given the person doesn’t attend, P(P|N) = 1 – P(G|N) = 0.6

Attending classes helps! But this information is never known outside. And it is where the survey gets interesting.

Imagine the survey captured the following proportions for each category.

Attend	Attend	Do Not Attend	Do Not Attend
Good Grade	Poor Grade	Good Grade	Poor Grade
0.9	0.5	0.5	0.1

Leading to the following Survey table.

Attend	Attend	Do Not Attend	Do Not Attend
Good Grade	Poor Grade	Good Grade	Poor Grade
270	100	100	30

Now, calculate the probability tables, and compare with the actual.

	Survey	Actual
P(G|A)	0.73 (270/370)	0.6
P(P|A)	0.27	0.4
P(G|N)	0.77 (100/130)	0.4
P(P|N)	0.23	0.6

The survey tends to conclude the advantages of not attending classes!