The main goal of the package `bigstep`

is to allow you to
select a regression model using the stepwise procedure when data is very
big, potentially larger than available RAM in your computer. What is
more, the package gives you a lot of control over how this procedure
should look like. At this moment, you can use one of these functions:
`stepwise()`

, `forward()`

,
`backward()`

, `fast_forward()`

,
`multi_backward()`

and combinations of them. They can be
treated as blocks from which the whole procedure of finding the best
model is built.

```
# generate data
set.seed(1)
<- 200
n <- 20
p <- matrix(rnorm(n * p), ncol = p)
X colnames(X) <- paste0("X", 1:p)
<- 1 + 0.4 * rowSums(X[, c(5, 10, 15, 20)]) + rnorm(n) y
```

First, you have to convert your data to a proper format, an object of
class `big`

. It can be done using the function
`prepare_data()`

. In most cases you will only need to specify
a vector of responses, `y`

, and a matrix of predictors,
`X`

.

```
library(bigstep)
<- prepare_data(y, X) data
```

Then, you can use the stepwise procedure with, for example, Akaike
Information Criterion and `summary()`

to get more information
about the final model. The function `get_model()`

returns the
object of class `lm`

.

```
<- stepwise(data, crit = aic)
results $model
resultssummary(results)
get_model(results)
```

You can use only one forward step (for example if you want to choose the best predictor).

`forward(data, crit = aic)`

What is important, results are in the same format as input data
(class `big`

), so you can use `forward()`

again or
in combination with other functions (with different criteria if you
like). The pipe (`%>%`

) operator will be helpful. For
every step the actual number of variables in a model, mean squared error
(*MSE*) or accuracy (*ACC*; if you use the logistic
regression) and a value of the chosen criterion (*crit*) are
given.

```
%>%
data forward(aic) %>%
forward(aic) %>%
forward(aic) %>%
backward(bic)
```

It may seem unnecessary or even unjustified for small data, but can be useful if you have a lot of predictors (see the next paragraph).

```
# generate data
set.seed(1)
<- 1e3
n <- 1e4
p <- matrix(rnorm(p * n), ncol = p)
X colnames(X) <- paste0("X", 1:p)
<- matrix(rnorm(5 * n), n, 5) # additional variables
Xadd colnames(Xadd) <- paste0("Xadd", 1:5)
<- 0.2 * rowSums(X[, 1000 * (1:10)]) + Xadd[, 1] - 0.1 * Xadd[, 3] + rnorm(n) y
```

If you have a lot o predictors, it can be a good idea to remove those
that are not related with `y`

. You can do that using
`reduce_matrix()`

. This function calculates p-values for the
Pearson correlation test and every variable from `X`

(separately). Variables with p-values larger than `minpv`

will not be considered in the next steps (formally, they are removed
from `candidates`

, one of components of class
`big`

). Thanks to that, the whole stepwise procedure will be
much quicker. What is more, `reduce_matrix()`

changes the
order of predictors in such a way that at the beginning there will be
variables with the smallest p-values. It is important if you want to use
`fast_forward()`

.

Another problem is choosing an appropriate criterion to such data.
Classical ones like AIC or BIC are bad choice because they will almost
certainly select a model with too many variables^{1}. You can use
modifications of them like mBIC^{2}, mBIC2^{3}, mAIC or mAIC2. In brief, these criteria
have much heavier penalty for the number of parameters, so they prefer
smaller models than their classic versions.

Additionally, in the example below we add variables from other matrix
to a model (`Xadd`

). It can be a good idea if there are
predictors which are important for us and want them to remain at every
stage of building the model.

```
<- prepare_data(y, X, Xadd = Xadd)
data %>%
data reduce_matrix(minpv = 0.15) %>%
stepwise(mbic) ->
resultssummary(results)
%>%
data reduce_matrix(0.15) %>%
stepwise(bic) # bad idea...
```

Sometimes it will be reasonable to start with a model with some good
predictors and then use the stepwise procedure. It can be achieved if we
use `fast_forward()`

which adds every variable that reduces a
criterion (not necessarily the best one). It is important for that
function to search for variables in a reasonable order (first, the most
correlated with `y`

), so you should use
`fast_forward()`

after `reduce_matrix()`

(you can
set `minpv = 1`

if you do not want to remove any predictor,
just change the order). It is good idea to run
`fast_forward()`

with a criterion which does not have a heavy
penalty for the size of a model, so for example BIC is better than mBIC.
After adding a lot of variables, most of them will be useless, so it can
be a good idea to perform the backward elimination—as long as there are
variables reducing a criterion. Run `multi_bacward()`

to do
that.

```
%>%
data reduce_matrix() %>%
fast_forward() %>%
multi_backward() %>%
stepwise()
```

If you are lucky, you do not have to run `stepwise()`

after `fast_forward()`

and `multi_backward()`

because you will already have the best model. It is important because
`stepwise()`

consists of potentially many backward and
forward steps and forward takes most time in whole procedure of building
a model (we have to check every predictor to find the best one). So the
fewer such steps, the faster you will get the model. It can be crucial
if you have big data.

Now, let consider data which is larger than RAM you have in your
computer. It is impossible to read it in a normal way, but in a process
of building regression model it is not necessary to have access to all
predictors at the same time. Instead, you can read only a part of the
matrix `X`

, check all variables from that part and then read
another one. To do that, you only need to read the matrix `X`

using `read.big.matrix()`

from `bigmemory`

package. The `prepare_data()`

has a parameter
`maxp`

which represents the maximum size (that is the number
of elements) of one part. If `X`

is bigger, it will be split.
It will be done even if your matrix is big but you have enough RAM to
read it in a normal way. It may seem unnecessary, but it is worth to do
because R is not very efficient in dealing with big matrices. Remember
that `maxp`

cannot be smaller than the number of observations
(rows in `X`

), by default it is 1e6.

In the code below we assume that you have a big matrix in a file
*X.txt*. Reading such matrix can be slow, but if you set
`backingfile`

and `descriptorfile`

you have to do
that once and next time you can use `attach.big.matrix()`

which is much faster.

```
<- read.big.matrix("X.txt", sep = " ", header = TRUE,
Xbig backingfile = "X.bin", descriptorfile = "X.desc")
# Xbig <- attach.big.matrix("X.desc") # much faster
<- read.table("y.txt")
y # data <- prepare_data(y, Xbig) # slow because of checking NA
<- prepare_data(y, Xbig, na = FALSE) # set if you know that you do not have NA
data %>%
data reduce_matrix(minpv = 0.001) %>%
fast_forward(crit = bic, maxf = 50) %>%
multi_backward(crit = mbic) %>%
stepwise(crit = mbic) -> m
summary(m)
```

We set `minpv`

to a low value to speed up the whole
procedure, but be careful because you can lose some important
predictors.

You can easily define your own criterion. It can depend on
log-likelihood (`loglik`

), the number of observations
(`n`

), the number of all variables (`p`

), the
number of variables currently in a model (`k`

), the matrix
with variables which are currently in a model (`Xm`

) and
constants. These parameters will be sent to your criterion, exactly with
such names (but you do not have to use all).

```
<- function(loglik, k, n, c1 = 0.5, c2 = 8) {
my_crit -c1*loglik + 10*sqrt(k*c2)
}<- reduce_matrix(data, minpv = 0.15) # data from the paragraph "Bigger data"
m stepwise(m, crit = my_crit)
stepwise(m, crit = function(loglik, k, n) -0.4*loglik + 10*sqrt(k*8))
```

Because your criterion has access to a model matrix
(`Xm`

), you can use untypical ones which depend, for example,
on the average correlation between variables which are in a model.

The package allows you to fit logistic and Poisson models. All you
need to do is setting the parameter `type`

when you prepare
data. Take note that the `reduce_matrix()`

always calculates
the Pearson correlation, even if you use general linear models. What’s
more, it is a known fact that even if one is interested in a logistic
regression model, the stepwise regression can be performed based on
linear regression — very often the same variables will be selected (and
the whole procedure is faster).

```
# Poisson model
set.seed(1)
<- 50
n <- 1000
p <- matrix(runif(n * p), ncol = p)
X colnames(X) <- paste0("X", 1:p)
<- rowSums(X[, 100 * (1:5)])
mu <- rpois(n, exp(mu))
y <- prepare_data(y, X, type = "linear")
data1 <- prepare_data(y, X, type = "poisson")
data2 %>%
data1 reduce_matrix() %>%
stepwise() # did not see any variables
%>%
data2 reduce_matrix() %>%
stepwise()
# logistic model
set.seed(2)
<- 100
n <- matrix(runif(n * p, -5, 5), ncol = p)
X colnames(X) <- paste0("X", 1:p)
<- 0.8 * rowSums(X[, 100 * (1:5)])
mu <- 1 /( 1 + exp(-mu))
prob <- rbinom(n, 1, prob)
y <- prepare_data(y, X, type = "linear")
data1 <- prepare_data(y, X, type = "logistic")
data2 %>%
data1 reduce_matrix() %>%
stepwise()
%>%
data2 reduce_matrix() %>%
stepwise()
```

M. Bogdan, J.K. Ghosh, M. Zak-Szatkowska.

*Selecting explanatory variables with the modified version of Bayesian Information Criterion*. Quality and Reliability Engineering International, 24:989–999, 2008.↩︎M. Bogdan, J.K. Ghosh, R.W. Doerge.

*Modifying the Schwarz Bayesian Information Criterion to locate multiple interacting quantitative trait loci*. Genetics, 167:989–999, 2004.↩︎F. Frommlet, A. Chakrabarti, M. Murawska, M. Bogdan.

*Asymptotic Bayes optimality under sparsity for general distributions under the alternative*, Technical report, arXiv:1005.4753v2, 2011.↩︎