One r2 for each beta column/predictor

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Nuchto
Nuchto 2012-7-10
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

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Greg Heath
Greg Heath 2012-7-12
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
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Nuchto
Nuchto 2012-7-14
Thanks, this is what I was looking for, except... I can't find 'r2' from any of the outputs of stepwisefit!

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Mark Whirdy
Mark Whirdy 2012-7-10
Hi Nuchto
No, you're correct - its the R^2 of the overall model that is output as stats(1).
Kind Rgds, Mark
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Mark Whirdy
Mark Whirdy 2012-7-11
编辑：Mark Whirdy 2012-7-11
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
Nuchto 2012-7-11
编辑：Nuchto 2012-7-11
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?

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