Roger D Metcalf DDS, JD
PO Box 137442
Fort Worth, TX 761361442
ph: +18173713312
roger
In statistics we often have to deal with the “degrees of freedom” in a calculation. Where do those come from? What does that mean? I pondered over that for a long time before “the light bulb went on."
Consider:









Suppose I asked you to fill out the above line of boxes (a vector, actually) with the numbers from 1 to 9, using each number only once. How many ways are there to do that? Well…a lot. We could do the math, if we were so inclined, but I'm not, and right now I really don’t care about that, anyway. It’s just that there’s a whole lot of ways for you to do it.
Ok, now suppose we had this one:
1  2  3  4  5  6  7  8 

...and, as before, I ask you to finish filling out the vector with numbers from 1 to 9 and we have the same caveat that each number can be used only once…how many ways are there to complete this vector, Victor? One. That’s it…one. And that’s your basic concept of degrees of freedom.
Under the conditions I just gave, I had choices about which numbers I could decide to put in boxes #1 thru #8. But when we got to box #9, you were no longer free to make your choice—the value was constrained by the conditions—you had to put a 9 there.
Now an example that's a little more complicated, but a little more realistic—here’s some totally made up data about blood type and eye color:
 A  B  O  AB  Total 
green eyes 



 400 
blue eyes 



 100 
Total  210  50  220  20  500 
…if I told you the above totals for blood type and eye color had been found in a particular sample of folks, could you tell me how to fill out the blank spots in the table to break down the frequencies of each blood type/eye color combination? No, there's too much data missing from the table.
Ok, what if I helped you out just a little bit?
 A  B  O  AB  Total 
green eyes  150 


 400 
blue eyes 



 100 
Total  210  50  220  20  500 
Well, this isn’t much better than the original, but at least you could deduce from this incomplete table that there must be 60 people with blue eyes and type A blood in this group (210 total A  150 green A = 60 blue A).
So I’ll help out with some more numbers…
 A  B  O  AB  Total 
green eyes  150  35 

 400 
blue eyes 

 20 
 100 
Total  210  50  220  20  500 
When you look closely at the table now, you’ll find I’ve actually provided you with all the information you really need in order to fill out the rest of the blank spots...sort of like doing a puzzle.
 A  B  O  AB  Total 
green eyes  150  35  200  15  400 
blue eyes  60  15  20  5  100 
Total  210  50  220  20  500 
Thus in the above table, the degrees of freedom are 3—that is, once I fill in values for three boxesany three of the empty boxesthen, given the constraints of the totals in the margins, the rest are automatically determined—we no longer have freedom to "choose" their valuesthe die has been cast, the moving finger has writ and moved on.
Anyway, this works out to be: degrees of freedom = (#rows – 1) x (#columns – 1) = (2 – 1) x (4 1) = 1 x 3 = 3.
* * *
If you like to do puzzles, does this remind you of...something? I suspect it does.
Hmmm. But there aren’t (9 1) x (9 – 1) = 64 degrees of freedom in Sudoku because of the additional conditions that the 3 x 3 squares, the rows, and the columns are also constrained. The computations are way beyond my ken, but Prof. Gary McGuire of Dublin College says that one must have at least 17 “clues” to solve a 9 x 9 Sudoku, and that 16 just won’t do it.
See: http://www.math.ie/McGuire_V1.pdf
I'm sure he will be glad to know that I have no reason to doubt him.
OK, now on to the Bayesian part.
Here is the table from above repeated...
 A  B  O  AB  Total 
green eyes  150  35  200  15  400 
blue eyes  60  15  20  5  100 
Total  210  50  220  20  500 
The numbers in the cells are counts, so let's change those in to proportions....
 A  B  O  AB  Total 
green eyes  0.30  0.07  0.40  0.03  0.80 
blue eyes  0.12  0.03  0.04  0.01  0.20 
Total  0.42  0.10  0.44  0.04  1.00 
The numbers in red are the marginal probabilities. The numbers in blue are the joint probabilitiesfor example, the probability of having blue eyes and type O blood here is 0.088...that is, the marginal probability of blue eyes (0.20) times the marginal probability of O blood type (0.44).
So, 0.80 x 0.42 = 0.3 (Ok, ok, it's really 0.34, but we are victims of rounding off here.) This is marginal x marginal = joint.
Now we may be interested in knowing the probability of someone from this population who we know has green eyes also has type A blood  that is the conditional probability of A blood type given that the person has green eyes ( or in math noptation: P(A  green) ). So the conditional probabilty is the (joint probability of green and A) / (the marginal probability of green) > 0.30 / 0.80 = 0.0375
So the cool thing about this comes to us from the degrees of freedom discussion above: if I know enough of the marginal probabilities here (the red numbers), then I can easily calculate all the joint probabilities (the blue numbers). Ok, well, that's pretty easy. And once I have the joint probabilities, I can calculate the conditional probabilities, also pretty easy.
Now, you may ask "why the interest in these statistical things?"
For one thing, I just really like doing statistics. In my PhD program I actually have a declared minor in Statisticsnot many PhD programs have real "minors," but mine does.
And, not to be too coy, it also may have something to do with dental age assessment. Stay tuned.....
© Copyright 2013, 2019 Roger D Metcalf DDS, JD. All worldwide rights reserved. No reproduction without permission. Neither the Tarrant County Medical Examiner's District, Tarrant County, the American Board of Forensic Odontolgy, the American Society of Forensic Odontology, the Royal College of Physicians, Oklahoma State University, nor any other organizaion mentioned here necessarily supports or endorses any information on this website. Any opinions, errors, or omissions are my responsibility, and mine alone. This site DOES NOT REPRESENT the official views of any of theseor any other organizations. Similarly, those other organizations may not fully represent my views, either.
Roger D Metcalf DDS, JD
PO Box 137442
Fort Worth, TX 761361442
ph: +18173713312
roger