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When using the forward stepwise approach to select variables, is the end model guaranteed to have the highest possible $R^2$? Said another way, does the stepwise approach guarantee a global optimum or only a local optimum?

As an example, if I have 10 variables to select from and want to build a 5-variable model, will the end result 5-variable model built by the stepwise approach have the highest $R^2$ of all possible 5-variable models that could have been built?

Note that this question is purely theoretical, i.e. we are not debating whether a high $R^2$ value is optimal, whether it leads to overfit, etc.

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    $\begingroup$ I think stepwise selection is going to give you the highest possible $R^2$ in the sense that it will be biased to be much higher than the true model (that is, it will not result in the optimal model). You may want to read this. $\endgroup$ – gung - Reinstate Monica Jun 5 '12 at 21:21
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    $\begingroup$ A maximum $R^2$ is attained when all variables are included. This is clearly the case because including a new variable cannot decrease $R^2$. Indeed, in what sense do you mean "local" and "global"? Variable selection is a discrete problem--choose one out of $2^k$ subsets of $k$ variables--so what would a local neighborhood of a subset be? $\endgroup$ – whuber Jun 5 '12 at 21:30
  • $\begingroup$ Re the edit: Could you please describe the "stepwise approach" you have in mind? (The ones I am familiar with do not arrive at a specified number of variables: part of their purpose is to help you decide how many variables to use.) $\endgroup$ – whuber Jun 5 '12 at 21:50
  • $\begingroup$ Do you think that a higher (raw) $R^2$ is a good thing? That's why they have adjusted $R^2$, AIC, etc. $\endgroup$ – Wayne Jun 6 '12 at 17:38
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    $\begingroup$ For maximum R2, include all 2-way and 3-way interactions, various transformations (log, inverse, square, etc.), phases of the moon, etc. $\endgroup$ – Zach Jun 7 '12 at 19:52
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You will not necessarily get the highest R$^2$ because you only compare a subset of possible models and may miss the one with the highest R$^2$ which would include all the variables.. To get that model you would need to look at all subsets. But the best model may not be the one with the highest R$^2$ because it may be that you over fit because it includes all the variables.

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    $\begingroup$ I believe this too, but to be convincing--because you have not provided a rigorous argument--it would be very nice to see an actual example. It would be even nicer to understand why a stepwise procedure converging to $k$ variables (say) might fail to converge to the highest-$R^2$ combination of $k$ variables (which would not require searching all subsets). $\endgroup$ – whuber Jun 6 '12 at 13:20
  • $\begingroup$ Stepwise procedrues depend on where you start. If you start with two different initial sets of variables it could lead you to different solutions. The point is that at each step there is a criterion on the F statistic for a variable to enter and als for a variable to leave. The F statistic depends on the variables that are currently in the model. The procedure stops when neither the F to enter nor the F to exit are statistically significant at the specified threshold. So that easily could happen before you add all the variables to the model. $\endgroup$ – Michael R. Chernick Jun 6 '12 at 15:59
  • $\begingroup$ This could easily be demonstrated with an example say in SAS with the output pasted into the answer. $\endgroup$ – Michael R. Chernick Jun 6 '12 at 15:59
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    $\begingroup$ I agree--but finding the counterexample is the hard part, @Michael, not using the software! $\endgroup$ – whuber Jun 6 '12 at 16:48
  • $\begingroup$ Either way it is a lot of work! $\endgroup$ – Michael R. Chernick Jun 6 '12 at 16:52
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Here is a counter example using randomly generated data and R:

library(MASS)
library(leaps)

v <- matrix(0.9,11,11)
diag(v) <- 1

set.seed(15)
mydat <- mvrnorm(100, rep(0,11), v)
mydf <- as.data.frame( mydat )

fit1 <- lm( V1 ~ 1, data=mydf )
fit2 <- lm( V1 ~ ., data=mydf )

fit <- step( fit1, formula(fit2), direction='forward' )
summary(fit)$r.squared

all <- leaps(mydat[,-1], mydat[,1], method='r2')
max(all$r2[ all$size==length(coef(fit)) ])

plot( all$size, all$r2 )
points( length(coef(fit)), summary(fit)$r.squared, col='red' )

R2

whuber wanted the thought process: it is mostly a contrast between curiosity and laziness. The original post talked about having 10 predictor variables, so that is what I used. The 0.9 correlation was a nice round number with a fairly high correlation, but not too high (if it is too high then stepwise would most likely only pick up 1 or 2 predictors), I figured the best chance of finding a counter example would include a fair amount of collinearity. A more realistic example would have had various different correlations (but still a fair amount of collinearity) and a defined relationship between the predictors (or a subset of them) and the response variable. The sample size of 100 was also the 1st I tried as a nice round number (and the rule of thumb says you should have at least 10 observations per predictor). I tried the code above with seeds 1 and 2, then wrapped the whole thing in a loop and had it try different seeds sequentially. Actually it stopped at seed 3, but the difference in $R^2$ was in the 15th decimal point, so I figured that was more likely round-off error and restarted it with the comparison first rounding to 5 digits. I was pleasantly surprised that it found a difference as soon as 15. If it had not found a counter example in a reasonable amount of time I would have started tweaking things (the correlation, the sample size, etc.).

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    $\begingroup$ +1 Well done! Examples are much more convincing then invoking hypothetical possibilities. But if you ever get a chance, please consider sharing the thought process you went through in constructing this counterexample. $\endgroup$ – whuber Jun 7 '12 at 19:54
  • $\begingroup$ Your account of the procedure is invaluable: this is the sort of revealing, practical stuff that shows up only in the most lucid papers, if at all, and otherwise has to be learned directly from others or re-invented. (I wish I could add another upvote.) $\endgroup$ – whuber Jun 8 '12 at 19:49
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If you really want to get the highest $R^2$ you have to look (as @Michael said) at all subsets. With a lot of variables, that's sometimes not feasible, and there are methods for getting close without testing every subset. One method is called (IIRC) "leaps and bounds" and is in the R package leaps.

However, this will yield very biased results. p-values will be too low, coefficients biased away from 0, standard errors too small; and all by amounts that are impossible to estimate properly.

Stepwise selection also has this problem.

I strongly recommend against any automated variable selection method, because the worst thing about them is they stop you from thinking; or, to put it another way, a data analyst who uses automated methods is telling his/her boss to pay him/her less.

If you must use an automated method, then you should separate your data into training and test sets, or possibly training, validating, and final sets.

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    $\begingroup$ stepwise selection is not as bad as you make out if the purpose is for prediction, or for using the sequence of models produced. in fact many rj mcmc algorithms for model selection are basically "random stepwise" as the proposals usually consist of adding or removing one variable. $\endgroup$ – probabilityislogic Jun 6 '12 at 7:44
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    $\begingroup$ Stepwise has been shown to be horrid. For details, see Frank Harrell's book Regression Modeling Strategies. What is RJ? It's true that the sequence of models may say something useful, but then what? I also have lots of problems with p-values, but that's another issue (or see The Cult of Significance Testing) $\endgroup$ – Peter Flom - Reinstate Monica Jun 6 '12 at 10:59
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    $\begingroup$ @PeterFlom - RJ is reversible jump. stepwise is simply a fast way to search the model space, making the problem less prone to combinatorical explosion. But it needs "multiple restarts" to help avoid it getting stuck in a local mode. I'll have to get this book I think though. $\endgroup$ – probabilityislogic Jun 6 '12 at 20:15
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    $\begingroup$ @Macro, Even in the orthogonal case, you see why the (naive) $p$-values would be off, correct? You also see why $|\hat\beta|$ of the "selected" model would tend to be (i.e., stochastically) larger than in the unselected case, correct? Say you had just two orthogonal variables, both $\beta_i = 0$, and your model selection was to choose the one with the lower $p$-value to keep (same as choosing the one of largest magnitude). $\endgroup$ – cardinal Jun 8 '12 at 19:04
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    $\begingroup$ @cardinal, I see. So this basically just a result of the fact that when you have an iid sample $X_1, ..., X_n$, then $$E(\min\{X_1, ..., X_n \}) < E(X_1)$$ if I'm understanding you correctly. That makes sense. $\endgroup$ – Macro Jun 8 '12 at 19:59

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