R代码运算符：〜0 + a表示什么？

Question

I have seen how to use ~ operator in formula. For example y~x means: y is distributed as x.

However I am really confused of what does ~0+a means in this code:

require(limma)
a = factor(1:3)
model.matrix(~0+a)

Why just model.matrix(a) does not work? Why the result of model.matrix(~a) is different from model.matrix(~0+a) ? And finally what is the meaning of ~ operator here?

Answer 1

~ creates a formula - it separates the righthand and lefthand sides of a formula

From ?`~`

Tilde is used to separate the left- and right-hand sides in model formula

Quoting from the help for formula

The models fit by, eg, the lm and glm functions are specified in a compact symbolic form. The ~ operator is basic in the formation of such models. An expression of the form y ~ model is interpreted as a specification that the response y is modelled by a linear predictor specified symbolically by model. Such a model consists of a series of terms separated by + operators. The terms themselves consist of variable and factor names separated by : operators. Such a term is interpreted as the interaction of all the variables and factors appearing in the term.

In addition to + and :, a number of other operators are useful in model formulae. The * operator denotes factor crossing: a*b interpreted as a+b+a:b. The ^ operator indicates crossing to the specified degree. For example (a+b+c)^2 is identical to (a+b+c)*(a+b+c) which in turn expands to a formula containing the main effects for a, b and c together with their second-order interactions. The %in% operator indicates that the terms on its left are nested within those on the right. For example a + b %in% a expands to the formula a + a:b. The - operator removes the specified terms, so that (a+b+c)^2 - a:b is identical to a + b + c + b:c + a:c. It can also used to remove the intercept term: when fitting a linear model y ~ x - 1 specifies a line through the origin. A model with no intercept can be also specified as y ~ x + 0 or y ~ 0 + x.

So regarding specific issue with ~a+0

• You creating a model matrix without an intercept. As a is a factor, model.matrix(~a) will return an intercept column which is a1 (You need n-1 indicators to fully specify n classes)

The help files for each function are well written, detailed and easy to find!

why doesn't model.matrix(a) work

model.matrix(a) doesn't work because a is a factor variable, not a formula or terms object

From the help for model.matrix

object an object of an appropriate class. For the default method, a model formula or a terms object.

R is looking for a particular class of object, by passing a formula ~a you are passing an object that is of class formula . model.matrix(terms(~a)) would also work, (passing the terms object corresponding to the formula ~a

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general note

@BenBolker helpfully notes in his comment, This is a modified version of Wilkinson-Rogers notation.

There is a good description in the Introduction to R .

Answer 2

After reading several manuals, I was confused by the meaning of model.matrix(~0+x) ountil recently that I found this excellent book chapter .

In mathematics 0+a is equal to a and writing a term like 0+a is very strange. However we are here dealing with linear models: A simple high-school equation such as y=ax+b that uncovers the relationship between the predictor variable (x) and the observation (y).

So we can think of ~0+x or equally ~x+0 as an equation of the form: y=ax+b . By adding 0 we are forcing b to be zero, that means that we are looking for a line passing the origin (no intercept). If we indicated a model like ~x+1 or just ~x , there fitted equation could possibily contain a non-zero term b . Equally we may restrict b by a formula ~x-1 or ~-1+x that both mean: no intercept (the same way we exclude a row or column in R by negative index). However something like ~x-2 or ~x+3 is meaningless.

Thanking @mnel for the useful comment, finally what's the reason to use ~ and not = ? In standard mathematical terminology / symbology y~x denotes that y is equivalent to x, it is somewhat weaker that y=x . When you are fitting a linear model, you aren't really saying y=x , but more that you can model y as a linear function of x ( y = ax+b for example)

Answer 3

To answer part of your question, tilde is used to separate the left- and right-hand sides in model formula. See ?"~" for more help.