`I`

*isolates* or *insulates* the contents of `I( ... )`

from the gaze of R’s formula parsing code. It allows the standard R operators to work as they would if you used them outside of a formula, rather than being treated as special formula operators.

For example:

```
y ~ x + x^2
```

would, to R, mean “give me:

`x`

= the main effect of`x`

, and`x^2`

= the main effect and the second order interaction of`x`

“,

not the intended `x`

plus `x`

-squared:

```
> model.frame( y ~ x + x^2, data = data.frame(x = rnorm(5), y = rnorm(5)))
y x
1 -1.4355144 -1.85374045
2 0.3620872 -0.07794607
3 -1.7590868 0.96856634
4 -0.3245440 0.18492596
5 -0.6515630 -1.37994358
```

This is because `^`

is a special operator in a formula, as described in `?formula`

. You end up only including `x`

in the model frame because the main effect of `x`

is already included from the `x`

term in the formula, and there is nothing to cross `x`

with to get the second-order interactions in the `x^2`

term.

To get the usual operator, you need to use `I()`

to isolate the call from the formula code:

```
> model.frame( y ~ x + I(x^2), data = data.frame(x = rnorm(5), y = rnorm(5)))
y x I(x^2)
1 -0.02881534 1.0865514 1.180593....
2 0.23252515 -0.7625449 0.581474....
3 -0.30120868 -0.8286625 0.686681....
4 -0.67761458 0.8344739 0.696346....
5 0.65522764 -0.9676520 0.936350....
```

(that last column is correct, it just looks odd because it is of class `AsIs`

.)

In your example, `-`

when used in a formula would indicate *removal* of a term from the model, where you wanted `-`

to have it’s usual binary operator meaning of *subtraction*:

```
> model.frame( y ~ x - mean(x), data = data.frame(x = rnorm(5), y = rnorm(5)))
Error in model.frame.default(y ~ x - mean(x), data = data.frame(x = rnorm(5), :
variable lengths differ (found for 'mean(x)')
```

This fails for reason that `mean(x)`

is a length 1 vector and `model.frame()`

quite rightly tells you this doesn’t match the length of the other variables. A way round this is `I()`

:

```
> model.frame( y ~ I(x - mean(x)), data = data.frame(x = rnorm(5), y = rnorm(5)))
y I(x - mean(x))
1 1.1727063 1.142200....
2 -1.4798270 -0.66914....
3 -0.4303878 -0.28716....
4 -1.0516386 0.542774....
5 1.5225863 -0.72865....
```

Hence, where you want to use an operator that has special meaning in a formula, but you need its *non-formula* meaning, you need to wrap the elements of the operation in `I( )`

.

Read `?formula`

for more on the special operators, and `?I`

for more details on the function itself *and* its other main use-case within data frames (which is where the `AsIs`

bit originates from, if you are interested).