Adjusted R^2

CFA Level II
#1

Do we have to know this formula? If so, can someone break it down for me to show the meaning behind it

#2

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

Adj R^2 is less than =R^2.

I don’t know how to make it easy for you.

#3

As far as I know, you don’t need to know the formula. (Though I’m in Philly right now, away from my Level II books, so I cannot confirm that.) You do need to know that if the adjusted R² decreases, you need to remove one or more independent variables from your model.

#4

why do you have to remove the independent variables from the models? Can you replace them? Doesn’t R^2 go up regardless if they are correlated or not?

#5

If you replace them, you’re still removing them; whether you add something else is another matter.

#6

Adjusted R-square=1- [(n-1)/(n-k-1)]*(1-R^2)

if you are using this formula, I hope this explanation provides some intuition about what is going on.

1 represents the total proportion of variation in your DV and you are subtracting out the proportion of unexplained variation in your DV to get an idea of approximately how much sampling variation in the dependent variable can be explained by the model.

SO (1-Rsquared)= proportion of variation in DV Not explained by the model, call this U.

When you actually calculate R-square adjusted, you can use a formula that allows you to see that you divide the numerator by n-k-1 and the denominator by n-1, which is the same as multiplying the formula by (n-1)/(n-k-1), call this A for adjustment for degrees of freedom in the model.

Now we are left with: 1 - A*U ; recall A will usually be greater than 1. So we are likely increasing the proportion of unexplained variance to account for our degrees of freedom in the regression (it is unlikely that R-square and the adjusted version will be identical). Now, 1- A*U is no bigger than the original R-square ( 1- U ). This means that after we accounted for the sample size and number of terms in the model, we are explaining less of the variation in the DV (our extra X variables are not as useful as we hoped).

#7

bascially adjusted R squared adjust down if you add additional variables that have very little predictive power.

#8

Helped a lot… Thanks so much…

#9

Good to hear.

#10

I just find a way to remember this, fyi, in case you really need it.

R^2 = SSR/ SST= 1 - SSE/SST

Adjusted R^2 = 1 - [SSE/(n-k-1)]/[SST/(n-1)]