What Does the Capital Letter "I" in R Linear Regression Formula Mean

What does the capital letter I in R linear regression formula mean?

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:

1. x = the main effect of x, and
2. 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).

Significance of 'I' keyword in lm model in R

I() prevents the formula-interface from interpreting the argument, so it gets passed along instead to the expression-parsing part.

In the formula interface -x means 'remove x from the predictors'. So I can do y~.-x to mean 'fit y against everything but x'.

You don't want it to do that - you actually want to make a variable that is the difference of two variables and regress on that, so you don't want the formula interface to parse that expression.

I() achieves that for you.

Terms with squaring in them (x^2) also need the same treatment. The formula interface does something special with powers, and if you actually want a variable squared you have to I() it.

I() has some other uses in other contexts as well. See ?I

In R formulas, why do I have to use the I() function on power terms, like y ~ I(x^3)

The tilde operator is actually a function that returns an unevaluated expression, a type of language object. The expression then gets interpreted by modeling functions in a manner that is different than the interpretation of operators operating on numeric objects.

The issue here is how formulas and specifically the "+, ":", and "^" operators in them are interpreted. (A side note: the correct statistical procedure would be to use the function poly when attempting to make higher order terms in a regression formula.) Within R formulas the infix operators "+", "*", ":" and "^" have entirely different meanings than when used in calculations with numeric vectors. In a formula the tilde (~) separates the left hand side from the right hand side. The ^ and : operators are used to construct interactions so x = x^2 = x^3 rather than becoming perhaps expected mathematical powers. (A variable interacting with itself is just the same variable.) If you had typed (x+y)^2 the R interpreter would have produced (for its own good internal use), not a mathematical: x^2 +2xy +y^2 , but rather a symbolic: x + y +x:y where x:y is an interaction term without its main effects. (The ^ gives you both main effects and interactions.)

?formula

The I() function acts to convert the argument to "as.is", i.e. what you expect. So I(x^2) would return a vector of values raised to the second power.

The ~ should be thought of as saying "is distributed as" or "is dependent on" when seen in regression functions. The ~ is an infix function in its own right. You can see that LHS ~ RHS is almost shorthand for formula(LHS, RHS) by typing this at the console:

`~`(LHS,RHS)
#LHS ~ RHS

class( `~`(LHS,RHS) )
#[1] "formula"

identical( `~`(LHS,RHS), as.formula("LHS~RHS") )
#[1] TRUE   # cannot use `formula` since it interprets its first argument

In regression functions the an error term in model descriptions will be in whatever form that regression function presumes or is specifically called for in the parameters for family. The mean for the base level will generally be labelled (Intercept). The function context and arguments may also further determine a link function such as log() or logit() from the family value, and it is also possible to have a non-canonical family/link combination.

The "+" symbol in a formula is not really adding two variables but is usually an implicit request to calculate a regression coefficient(s) for that variable in the context of the rest of the variables that are on the RHS of a formula. The regression functions use `model.matrix and that function will recognize the presence of factors or character vectors in the formula and build a matrix that expand the levels of the discrete components of the formula.

In plot()-ting functions it basically reverses the usual ( x, y ) order of arguments that the plot function usually takes. There was a plot.formula method written so that formulas could be used as a more "mathematical" mode of communicating with R. In the graphics::plot.formula, curve, and 'lattice' and 'ggplot' functions, it governs how multiple factors or numeric vectors are displayed and "facetted".

The overloading of the "+" operator is discussed in the comments below and is also done in the plotting packages: ggplot2 and gridExtra where is it separating functions that deliver object results. There it acting as a pass-through and layering operator. Some aggregation functions have a formula method which use "+" as an "arrangement" and grouping operator.

I() equivalent (used in R), what is the Python equivalent?

I found the answer, seems to be as simple as:

f = 'medv~lstat + I(lstat**2)'
fit3 = smf.ols(f, data=data).fit()
print(fit3.summary())

What does the capital letter I in R linear regression formula mean?

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:

1. x = the main effect of x, and
2. 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).

Time Series and Linear Regression

1. The reason you can't use $ is that the z object shown in the question is not a zoo object. It is a ts object. You can use class(z), str(z) and dput(z) to determine what you have. Also, $ works on zoo objects but not on ts objects. Convert it to zoo and then $ will work.

   library(zoo)
   zz <- zoo(z, as.yearqtr(time(z)))
   zz$GDPGrowth
   ##     2482 Q1     2482 Q2     2482 Q3     2482 Q4     2483 Q1     2483 Q2 
   ##  1.20000000 -0.20000000 -0.15000000 -0.10000000  0.40000000  0.33333333 
   ##     2483 Q3     2483 Q4     2484 Q1     2484 Q2     2484 Q3     2484 Q4 
   ##  0.26666667  0.20000000  0.50000000  0.80000000  1.10000000  1.40000000 
   ## # ... snip ...
   

   The times in your object are way into the future but unless we know how you created them we cannot know how that happened. You possibly were playing with Date objects and made some error in converting them to ts.

2. You have quarterly data and the 0, 0.25, 0.5 and 0.75 are how ts objects represent the 4 quarters internally. If this refers to the not wanting to apply na.approx to certain columns then if ix is a vector of column names or numbers to convert then zz[, ix] <- na.approx(zz[, ix]) applies na.approx only to those columns.

3. ts and zoo represent the index via tsp and index attributes respectively so they are still there. time(z) and time(zz) will retrieve the index.

4. If you want to do statistical tests, compute confidence intervals, etc. then you need to take the correlations into account; however, if you just want to get point estimates you don't need to concern yourself with that. The dyn package (also the dynlm package) can be used to facilitate running lm with zoo objects.

   library(dyn)
   fm <- dyn$lm(GDPGrowth ~ ApprovalGOV, zz)
   fm
   
   ## Call:
   ## lm(formula = dyn(GDPGrowth ~ ApprovalGOV), data = zz)
   ##
   ## Coefficients:
   ## (Intercept)  ApprovalGOV  
   ##     -1.9717       0.3575
   

   Either of these also work and make use of with.zoo and fortify.zoo.

   with(zz, lm(GDPGrowth ~ ApprovalGOV))
   
   lm(GDPGrowth ~ ApprovalGOV, fortify.zoo(zz))
   

   To plot the points and draw in a regression line:

   plot(formula(fm), zz)
   abline(fm)
   

Other points are:

• R is case sensitive so GDPGrowth is not the same as GDPGROWTH.

• do not use random code snippets that you have found on the net without first reading the help files for each function used so that you know whether it makes sense for your problem. Also read all the vignettes (pdf or html documents) for each package that you are using. In particular, the zoo package has 5 vignettes and a reference manual.

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