Regular readers know Uncertainty proposes we go back to the old way of examining and making conclusions about data, and eschew many innovations of the 20th Century. No p-values, no tests, no posteriors. Just plain probability statements about observables and a rigorous separation of probability from decision.
These criticisms you know (or ought to by now). So why not let’s do a case study or three, and take our time doing so. Case Study 1 uses the same data presented in Uncertainty. We’re interested in quantifying our uncertainty in a person’s end-of-first-year College GPA given we know their SAT score, high school GPA, and perhaps another measure we might have.
Now right off, we know we haven’t a chance to discover the cause—actually causes—of a person’s CGPA. These are myriad. A GPA is comprised of scores/grades per class, and the causes of the score in each class are multitudinous. How much one drank the evening before a quiz, how many hours put in on a term paper, whether a particular book was available at a certain time, and on and on.
It is equally obvious a person’s HGPA or SAT does not and cannot cause a person’s CGPA. Some of the same causes responsible for the HGPA, SAT might appear in the list of causes for CGPA, but it’s a stretch to say they’re identical. We could say “diligence” or “sloth” are contributory causes, but since these cannot be quantified (even though some might attempt such a maneuver), they cannot take their place in a numerical analysis.
Which brings up the excellent question: why do a numerical analysis at all?
Do no skip lightly over this. For in that query is the foundation of all we’ll do. We’re doing a numerical, as opposed to the far more common qualitative (which form most of our judgments), study because we have in mind a decision we will make. Everything we do must revolve around that decision. Since, of course, different people will make different decisions, the method of analysis would change in each case.
It should be clear the decision we cannot make is about what causes CGPA. Nor can we decide how much “influence” SAT or HGPA has on CGPA, because “influence” is a causal word. We cannot “control” for SAT or HGPA on CGPA because, again, “control” is a causal word, and anyway HGPA and SAT were in no way caused, i.e. controlled, by any experimenter.
All we can do, then, if a numerical analysis is our goal, is to say how much our uncertainty in CGPA changes given what we know about SAT or HGPA. Anything beyond that is beyond the data we have in hand. And since we can make up causal stories until the trump of doom, we can always come up with a causal explanation for what we see. But our explanation could be challenged by somebody else who has their own story. Presuming no logical contradiction (say a theory insists SAT scores that we observed are impossible), our “data” would support all causal explanations.
This point is emphasized to the point we’re sick of hearing it because the classic way of doing statistics is saturated in incorrect causal language. We’re trying to escape that baggage.
So just what decision do I want to make about CGPA?
I could be interested in my own or in another individual’s. Let’s start with that by thinking what CGPA is. Well, it’s a score. Every class, in the fictional college we’re imagining, awards a numerical grade, F (= 0) up to A+ (A = 4, A+ = 4.33, and so on). CGPA = score per class divided by number of classes. That’s several numbers we need to know.
How many classes will there be? In this data, I don’t know. That is to say, I do not know the precise number for any individual, but I do know it must be finite. Experience (which is not part of the data) says it’s probably around 10-12 for a year. But who knows? We also can infer that each person has at least one class—but it could be that some have only one class. Again, who knows?
So number of classes is equal to or greater than one and finite. So, given the scoring system for grades, that means CGPA must be of finite precision. Suppose a person has only one class, then the list of possible CGPAs is 0, 0.33, …, 4, 4.33 and none other. If a person has two classes, then the possibilities are 0, 0.165, 0.33, and so forth. However many classes there are, the final list will be a discrete, finite set of possible CGPAs, which will be known to us given the premises about the grading system.
Suppose a student had 12 classes, then his score (CGPA) might be (say) 2.334167. That’s 7 digits of precision! This number is one of lots of different possible grades (these begin with 0, 0.0275, 0.055, 0.0825, …). And there is more than one way to get some of these grades. A person with a CGPA of 2 might have had 12 classes with all C’s (= 2), or 12 with half A’s and half F’s; and there are other combinations that lead to CGPA = 2. And so now we have to ask ourselves just what about the CGPA we want to know.
We’ve reached our first branching point! And the end of today’s lesson. See if you can guess where this is going.
I’ll answer all pertinent questions, but please look elsewhere on the site (or in Uncertainty) for criticisms of classical methods. Non-pertinent objections will be ignored.