Looking at the definition of adjusted r-squared may be helpful ( https://miro.medium.com/max/1100/1*kW2xSxbIb7laGRoEcgx8vg.jpeg )

Dear Caty Gonçalves,

When one begins with the linear model with no regressor and you add the regressors one after another, R² never decreases when a regressor is added to the model, regardless of the value of the contribution of that variable. Therefore, it is difficult to judge whether the increase in R2 is really telling us anything important.

Some regression model builders prefer to use an adjusted R² statistic. The

adjusted R² will only increase upon adding a variable to the model, if the addition of the variable reduces the residual mean square, otherwise it will reduce.

So when we add a regressor to the model that is not useful in predicting Y, R² increases for sure but the adjusted R² will decrease.

So in multiple regression, when you have multiple predictors, always use Adjusted R².

Reading the following post, and the list of posts that appeared to the right of the opened page, might be useful:

https://stats.stackexchange.com/questions/52517/why-is-adjusted-r-squared-less-than-r-squared-if-adjusted-r-squared-predicts-the

Good luck,

Hamid

In the equation Sal Mangiafico directed you to, the (N-1)/(N-p-1) is a weight that is applied to (1 - R2), i.e., to the portion of the total variance that is not accounted for by the model, or the error variance.

A good way to understand the equation is to fix the values of N and R2, and vary the value of p, the number of predictors. You could try that in a spreadsheet, for example. If you do, you'll see that the more variables you include in the model, the greater the weight that is applied to (1 - R2), and therefore, the smaller smaller the adjusted R2 becomes. HTH.

The usual adjustment means that you can get negative adjusted R^2 values, which just means that the associated F statistic is less than 1. For example:

Hamid Ghorbani in my model the R-squared is 0.78 and the adjusted r-squared 0.77. The closer they are in a Multiple linear regression model what that means?

Dear Caty Gonçalves,

About your question, I do not know what it means.

I would try to make inferences about the model not just by investigating either R^2 values (have a look at p-values for testing the significant contribution of individual regressors when other regressors are assumed to be included).

About your question, I would check whether the regressors in the full model (the at-hand model) are significantly contributed or not.

beginning from the null model y=beta0, toward the full model have a look at the conditional residual sum of squares to test the significant contribution of new regressors in the model.

It could happen that you find regressors that after their removal R^2 decreases slightly and the adjusted R^2 doest not adjust too much.

Hamid

Hello Caty,

This answer is late in the thread, but what I would tell you is:

The larger the N of cases, the less of a difference there will be between the standard R-squared and the "adjusted" R-squared, regardless of the number of predictors (independent variables) in the model. (This is implied by several of the answers already given, so I'm not trying to take credit for something novel here.)

Really, the "adjusted" R-squared is intended to penalize one for over-fitting a model by virtue of having too many predictors relative to the sample size (N). In fact, if you were to build a regression model using k random variates as predictors when the sample size is k + 1 cases, you'll get perfect prediction! Both the R-squared, and the "adjusted" R-squared will be 1.0, though there's virtually no chance that the resultant model would generalize to a new sample.

So, be happy that your sample size was good enough that there was little need to penalize the obtained R-squared. As mentioned by others, focus on what your model tells you about the respective variables you've incorporated.

Good luck with your work.

Earlier, I suggested fixing N and R2 and observing how Adj R2 changes as a function of p. Here is an example of that with N=100, Rsq=0.25, and p ranging from 1 to 10:

p N Rsq AdjRsq

1 100 .25 0.2423

2 100 .25 0.2345

3 100 .25 0.2266

4 100 .25 0.2184

5 100 .25 0.2101

6 100 .25 0.2016

7 100 .25 0.1929

8 100 .25 0.1841

9 100 .25 0.1750

10 100 .25 0.1657

What David Morse suggests is fixing p and Rsq and letting N vary. Here is an example with p and Rsq fixed at 5 and .25, and with N varying from 50 to 500 in steps of 50.

N p Rsq AdjRsq

50 5 .25 0.1648

100 5 .25 0.2101

150 5 .25 0.2240

200 5 .25 0.2307

250 5 .25 0.2346

300 5 .25 0.2372

350 5 .25 0.2391

400 5 .25 0.2405

450 5 .25 0.2416

500 5 .25 0.2424

Notice that as N increases, Adj Rsq gets closer to Rsq (when p and Rsq are fixed).

Here is my Stata code, in case you find it helpful.

// Illustrate how p (# of predictors) affects Adj Rsq

// when N and R^2 are fixed at 100 and .25

clear

set obs 10

generate p = _n // p = row number

generate N = 100

generate Rsq = .25

generate AdjRsq = 1 - ((1-Rsq)*(N-1)/(N-p-1))

format AdjRsq %8.4f

list, clean noobs

// Illustrate how N affects Adj Rsq when p and Rsq

// are fixed at 5 and .25

clear

set obs 10

generate N = (_n/2)*100

generate byte p = 5

generate Rsq = .25

generate AdjRsq = 1 - ((1-Rsq)*(N-1)/(N-p-1))

format AdjRsq %8.4f

list, clean noobs

This thread has been going on for awhile. Let me tweak the problem for David Morse , Bruce Weaver , Sal Mangiafico and others. Suppose I do a multiple regression and have used some (non-shrinkage, i.e., not lasso-like, that might be a different tweak) model selection procedure like forward selection (which I know all of you don't like, but ...). Are there off-the-shelf algorithms for less-biased estimates (i.e., adjusted) of the population R? People often report the R^2 and adj R^2 as if this was the only model considered.

Daniel Wright , I have no idea. It sounds like a good simulation project. If the results are interesting, maybe a little publication.

Sal Mangiafico , it could be fun. I wonder how closely related the bias of the adj R2 would be just to the number of steps. Hmmm. dm me if you want to pursue this!

@Daniel if you are to suggest anything at all suggest Mallow's Cp which for OLS is equivalent to AIC. NOW IF you want a good chance of getting the wrong answer simply follow the procedure you suggested. There's no point in suggesting procedures that don't work. For anyone that doesn't believe me see the attachments below David Booth

Here's another paper that you might want to see besides the references in the first paper.Also see the references in both papers. David Booth

At those who believe adjR** isn't ever negative see Daniel's example. David Booth

Caty Gonçalves , if you are still following this thread:

I think the upshot is that --- for the same model --- all things being equal --- the r-squared and the adjusted r-squared will be similar if there are relatively few terms on the right side of the model. And will be more different if there are relatively few terms on the right side of the model.

I don't think there's much to interpret here.

If possible, it may be a good idea to present both the r-squared and the adjusted r-squared.