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 ...

Optimistic is an adjective that describes a hopeful and positive outlook or attitude. Optimistic individuals expect the best possible outcome in every situation, believe that good will prevail over bad, and see challenges or difficulties as opportunities for growth or improvement.

If a situation or event happens to someone or something, it has an effect on that person or thing:

as it happens/happened used when you say something that is surprising, or something connected with what somebody else has just said I agree with you, as it happens. As it happens, I have a spare set of keys in my office.

When something happens, it occurs or is done without being planned. Tell me what happened.

1. to take place; come to pass; occur. 2. to come to pass by chance; occur without apparent reason or design. 3. to have the fortune or lot (to do or be as specified); chance: I happened to see him on the street. 4. to befall, as to a person or thing: Don't worry; nothing happened to her.