One r2 for each beta column/predictor

Hi,

The 'stats' output from regress returns a 1x4 vector, first value of which is r2. If you do regress(Y,X) where X is not one column vector, but a matrix of predictors (columns), then you would get as many beta columns as predictors, am I right?

Would you also get as many r2 as beta columns (or predictors)? Because I am only getting one r2 for X and Y, even though X is not one predictor, but many. Is this correct? Or am I indexing wrongly the stats output and missing data?

Thank you all

Answer 1

[Greg Heath]

With n points and p predictors you get p+1 betas (b0,b1,...bp) and a R^2 quantifying prformance

For any subset of predictors the corresponding R^2 will be less.

Although there is no universally accepted way to divide R^2 p+1 ways and attribute each part to a single predictor, I am satisfied to use the function stepwisefit in the backward mode to obtain such a result.

help stepwisefit

doc stepwisefit

Hope this helps.

Greg

[Nuchto]

Thanks, this is what I was looking for, except... I can't find 'r2' from any of the outputs of stepwisefit!

Answer 2

[Mark Whirdy]

Hi Nuchto

No, you're correct - its the R^2 of the overall model that is output as stats(1).

Kind Rgds, Mark

[Nuchto]

Thanks. Any chance to get the individual r2 for each beta value? Same way that betas are the individual contribution of each variable to explaining the data, would be nice to have the respective r2 of that contribution...

[Mark Whirdy]

This is tantamount to individual regressions so would have to actually do that with regress(). Alternatively, forget the regress() function and write your own code around "\" operator calculating the SSE for each regression etc etc

[Nuchto]

I see. But then why we get the beta values for each predictor and not just a beta value for the whole set of predictors? If we wanted just one beta we would just run regress with one predictor at a time, right?

[Mark Whirdy]

Hi Nuchto

y = b1*x1 + b2*x2 + b3*x3

Beta's are coefficients of the predictor X variables, so by definition there must be as many coefficients as variables (plus an optional intercept). How would you calculate a "single model beta" number?

The R^2 on the other hand refers more to the predicted Y variable than to the predictor X variables (at least its helpful at the start maybe to think of it like this), describing how much of Y's variance is explained by your model. Lets say its 69%, then 69% of its variance is explained. What would an R^2 like [45% 36% 54%] mean - how much of Y is your model explaining then? ... you don't know. (i.e. for the concept of model explanatory power to have meaning it must be a single number). 3 individual R^2 will be the explanatory power of 3 individual univariate models respectively then - its useful/interesting information, but doesn't describe the overall 3-variable model as such.

This isn't a pecularity of matlab really but more concepts around linear regression itself.

Does this make sense at all?

Kind Rgds, Mark

[Nuchto]

Thanks for your explanation. When I said "if we wanted just one beta value, we would run regress with one predictor at a time", I meant that only with one predictor you get one beta, obviously. Indeed, the r2 is the proportion of variance that the model accounts for. But can't this be broken down to the specific contributions of each predictor? That is what I was asking. I know it makes sense to get the overall percentage of the whole model's contribution to Y, but also what is the contribution of each predictor in terms of r2?