### R: Calculating all possible linear regression models for a given set of predictors

Although the graphic at the left might not seem a 100% appropriate, it gives a hint to what I am about to do. I want to calculate all possible linear regression models with one dependent and several independent variables. I do not want to address bias and fitting issues or the question if this makes sense from a statistical point of view in this posting. Here I want to emphasize the technical issues only.

To solve the task, several approaches are possible. The first one is a step-by-step approach using a lot of code. Another one would be to make use of a specialized package. The packages `leaps` and `meifly` would be appropriate for the task but have some slight drawbacks in terms of flexibility. I will not address solutions using these packages here, but I would like to point out that in contrast to the below only a few lines of code would do the job.

## The step-by-step approach

Let’s suppose we have the following set of four possible regressors.

`regressors <- c("y1", "y2", "y3", "y4")`

Now we want to construct a formula that contains the first and third regressor.

```
vec <- c(T, F, T, F)
paste(regressors[vec])
> [1] "y2" "y3"
```

So the paste commmand works vectorwise which helps a lot in this case. Now we add a plus sign between the regressors…

`paste(regressors[vec], collapse=" + ")`

… and add the left side of the equation. The 1 in the formula models the intercept , 0 would be a model without intercept.

`paste(c("y ~ 1", regressors[vec]), collapse=" + ")`

Now let’s make a formula out of it.

`as.formula(paste(c("y ~ 1", regressors[vec]), collapse=" + "))`

So we can construct a formula from each row of a TRUE /FALSE matrix which determines if a regressor is used or not. Now we need a TRUE / FALSE matrix of all the possible regressor combinations. The `expand.grid()` function produces one (see `?expand.grid`).

regMat <- expand.grid(c(TRUE,FALSE), c(TRUE,FALSE), c(TRUE,FALSE), c(TRUE,FALSE)) # > regMat # Var1 Var2 Var3 Var4 # 1 TRUE TRUE TRUE TRUE # 2 FALSE TRUE TRUE TRUE # 3 TRUE FALSE TRUE TRUE # 4 FALSE FALSE TRUE TRUE # 5 TRUE TRUE FALSE TRUE

The last line describes a trivial model as it does not contain any regressors (as it contains only FALSE values), thus it is removed.

regMat <- regMat[-(dim(regMat)[1]),] # let's name the columns names(regMat) <- paste("y", 1:4, sep="")

Now we can apply the above way of formula construction to each row of the matrix so we get a list with all the possible models.

# x1 will be dependent regressors <- c("y1", "y2", "y3", "y4") allModelsList <- apply(regMat, 1, function(x) as.formula( paste(c("x1 ~ 1", regressors[x]), collapse=" + ")) ) # > allModelsList # [[1]] # x1 ~ 1 + y1 + y2 + y3 + y4 # # [[2]] # x1 ~ 1 + y2 + y3 + y4 # # [[3]] # x1 ~ 1 + y1 + y3 + y4

The last step is to use each list element for the calculation.

```
data=anscombe
allModelsResults <- lapply(allModelsList,
function(x) lm(x, data=data))
```

So basically, here our computation work is done, but as in most cases a lot of work follows to prepare the data in a nice way. So now let’s get all the important information into one dataframe. Let’s say we want a data frame like the following.

```
+-------+-----------------------------------------------------+
| model | no. of | coefficients | se coef. | t-Val | etc. |
| | regressors | x1 | x2 ... | | | |
| | | | | | |
```

So we need to extract all the following information (coefficients, SE etc.) and cast them into one data frame.

```
x <- allModelsResults[[1]]
coef(x)
coef(summary(x))[, "Std. Error"]
### ... and so on
```

This used to be one of the nasty tasks in R. Here Hadley Wickhams `plyr` package really helps a lot. ldply takes a list, applies a function and casts the results into ONE data frame (see `?ldply`). As function return value it expects a data frame or a vector. The advantage to return data frames is that the `ldply()` function uses rbind.fill for combining the results when they are data frames. `rbind.fill()` allows a different number of columns in each data frame. Here this is the case as a different number of regressors are used each time. So we have to make sure that the function returns a data frame. Thus we use as.data.frame paying attention to the orientation of the data frame, using `t()` in case it is outputted as one column.

```
library(plyr)
dfCoefNum <- ldply(allModelsResults, function(x) as.data.frame(
t(coef(x))))
dfStdErrors <- ldply(allModelsResults, function(x) as.data.frame(
t(coef(summary(x))[, "Std. Error"])))
dftValues <- ldply(allModelsResults, function(x) as.data.frame(
t(coef(summary(x))[, "t value"])))
dfpValues <- ldply(allModelsResults, function(x) as.data.frame(
t(coef(summary(x))[, "Pr(>|t|)"])))
# rename DFs so we know what the column contains
names(dfStdErrors) <- paste("se", names(dfStdErrors), sep=".")
names(dftValues) <- paste("t", names(dftValues), sep=".")
names(dfpValues) <- paste("p", names(dfpValues), sep=".")
# p-value for overall model fit
calcPval <- function(x){
fstat <- summary(x)$fstatistic
pVal <- pf(fstat[1], fstat[2], fstat[3], lower.tail = FALSE)
return(pVal)
}
# Before creating ONE data frame with all important entries,
# we need to compute some more indices
NoOfCoef <- unlist(apply(regMat, 1, sum))
R2 <- unlist(lapply(allModelsResults, function(x)
summary(x)$r.squared))
adjR2 <- unlist(lapply(allModelsResults, function(x)
summary(x)$adj.r.squared))
RMSE <- unlist(lapply(allModelsResults, function(x)
summary(x)$sigma))
fstats <- unlist(lapply(allModelsResults, calcPval))
# now we can combine all the data into one data frame
results <- data.frame( model = as.character(allModelsList),
NoOfCoef = NoOfCoef,
dfCoefNum,
dfStdErrors,
dftValues,
dfpValues,
R2 = R2,
adjR2 = adjR2,
RMSE = RMSE,
pF = fstats )
# round the results
results[,-c(1,2)] <- round(results[,-c(1,2)], 3)
results
```

This was really a lot of code. But now we have assembled all the information and indices that were important for my task. To choose what is needed is simple now. And we have the flexibility to add any indices. Next time I will try to extend this example doing a k-fold estimation for each set of regressors.

Cheers,

Mark Heckmann

Filed under: R / R-Code | 10 Comments

Tags: regression, plyr, permutation

You might also want to have a look at the code in the meifly package (http://github.com/hadley/meifly/) for some similar ideas (code is pre-plyr though).

Now you just need Furnival and Wilson’s leaps and bounds (Technometrics 1974, iirc) algorithm so that when the number of models gets too big you can avoid running the ones you can already do better than.

Hi efrique,

sure, this would be a great enhancement. Code is welcome ;)

Mark

Mark, great post. I like your coding style and the way you are using ldply. I should use it more in what I am doing.

It always feels like I am giving up control when using functions like apply / ldply / ddply, but your post outlines how to keep control over the math and use these awesome features of R to streamline the hard things that otherwise chew up a lot of time and effort.

If you don’t mind, I’d like to add the your post URL to my blog.

Thanks again,

-Alex

Hi Alex,

thanks! I am happy you liked it!

Of course you are welcome to link the blog!

-Mark

Thanks! Just what I was looking for to run my lm models. Now, can you tell me why glm can’t use my ‘formula’ object?

Hi Mark, I am new to R. Thanks for the code. Is there a way to calculate and attach the vif measures along with the models? I am struggling with it.

Hi Lina,

you might add the following lines to get a dataframe containing the variance inflation factors.

The VIF is of course only defined for models with more than one regressor variable. Hence for our case the code needs a bit of tweaking to take care of the one regressor models. Thus I use vif_2 that will return y1=NA when the vif function fails. This is of course a quick-and-dirty hack and no general solution.

library(plyr)

library(car)

vif_2 <- function(x) {

tryCatch(vif(x), error=function(e) c(y1=NA))

}

dfVif <- ldply(allModelsResults, function(x)

as.data.frame(t(vif_2(x))))

-Mark

Hi,

Thanks for this wonderful tutorial. I am quite new and need to use a very difficult package(GAM, mgcv)

I have several pollutants (about 7 over several lags) and want to regress each pollutant at a time and extract important estimates like coefficient, Gcv score, AIC for ozone and save it as file. I also want to check the model of each fit to be saves for later assessment. How can I achieve this?

fit1<- gam(mortality~s(ozone)+s(temperature)+s(humidity),family=poisson,data=dust)

Thanks in advance

This was exactly the article I was looking for! Very, very helpful – although with 17 variables my laptop struggled a wee bit on generation the results for the 130,000 different possible models! (in the end I reduced my variables to 13 which it coped a bit better with!)

Once I have selected the best linear model for my data do you have any suggestion as to how to incorporate interactions without manually going through and including them?