Adjusted R²

Should we learn this formula by heart? I’m overdosing on formulas, if i can let go of one, I will. As per LOS: “distinguish between and interpret the R2 and adjusted R2 in multiple regression;”

I’m pretty sure that I wouldn’t.

(Note: I have only limited space left in my brain; your mileage may vary.)

Adj R^2=1-((n-1/n-k-1)*(1-R^2))

Term 1 =1

Term2 ((n-1)/(n-k-1))*(1-R^2)

Even though it is difficult to remember, few points can help to select the right answers:

  1. Adj will be always be less than R^2 (one of the choices will be greater than R^2, we can rule out that one…;)).

  2. Term 2 should be large enough so that ouput (1-term2) will be smaller than R^2.

  3. To make Term 2 larger, Numerator (n-1) should be larger than denominator (n-k-1).

  4. Numerator(n-1) is the “degree of freedom” of total variation.

  5. Denominator(n-k-1).is the “degree of freedom” of unexplained error.

I have (I think) an easier way to remember the “formula” for Adjusted R-squared.

MSE (Mean Squared Error) = SSE/(n-k-1) and VAR(Y) = SST/(n-1)… where SSE is the sum of squared errors and SST(Y) is the total sum of squares for Y (total variation in Y)… These are just variances (MSE and VAR(Y)), so this should seem less daunting.

Adjusted R-squared = 1 - [MSE/VAR(Y)] … try it out with a couple of practice problems— after accounting for the degrees of freedom (which is partly the point of adjusting R-squared ), the explained variance is total variance of Y minus the unexplained variance of Y (SER squared, MSE). To calculate adjusted R-squared, simply put this difference over the total variance of Y— this simplifies to 1 - [MSE/VAR(Y)].

Regular R-squared is 1 – [SSE/SST(Y)].

Realistically, you’re not dealing with anything new, exactly. If you remember it in terms of the variances, I don’t think it will be as difficult to remember or work with going forward.

For the LOS, I think it’s safe to assume they want you familiar with some of the distinctions between the two forms and to know when one is appropriate versus the other.

Hope this helps!

I ain’t remembering it, screw it. If they wanted us to calculate it, they would’ve put “calculate” in the LOS, as they did for all other formulas.

Well, I tried haha… Honestly, I can’t see them wasting their time asking you to calculate adjusted R-squared, but, then again, I could see someone saying “a calculation would imply a basic understanding” or something like that. Either way, we’re not going to know everything. I think this LOS is pretty low-yield-- aside from knowing why you might make an adjustment to R-squared (just my opinion, though).

Don’t memorize a formula included in a stat package software…

I’m not sure which formulas to memorize and which ones I shouldn’t.

Here’s another one when calculating prediction intervals:

sf^2 = s^2 [1+(1/n)+(x - xbar)/(n-1)sx^2]

(don’t have the book with me right now so I’m not even sure that’s correct).

EDIT: corrected formula

^ Yeah that one is a pain. And there are multiple CFAI EOC questions that require it to be calculated…

I remember the adjusted ra2 formula by keeping it symmetrical. 1-ra2= ((n-1)/(n-k-1)) (1-r2)

What is this formula? I’ve never seen it anywhere.

It’s from LOS 9i- calculate and interpret a confidence interval for the predicted value of the dependent variable. I believe we are suppose to know it.


In the real world, I agree, but being familiar with the calculation is certainly helpful when you run into issues or are evaluating your model. However, this doesn’t seem to be the best rule of thumb. If they ask you to calculate and interpret the global f-test for total model utility, I would say you don’t understand the basic idea of the f-statistic for the test if you’re unable to calculate it (same goes for r-squared, t-tests, etc.). If you truly have intuition about these ideas, the calculation should be easier to remember.

Remarkable statement.

Softwares should be tools only, not a replacement of our minds.