Linear Regression Models (3)

Mean Joe

TS Contributor
Say you have E(Y) = b_0 + b_1 * W^2. As a model of W^2, it is linear. Just because you have measured weight, doesn't mean the model can't be based on weight^2 (I don't think we have an instrument for measuring such a thing...).

There is a truth you are trying to get at with your model. You do some measurements and fit the model. But sometimes, your measurements are in "non-optimal" units. But for your model, you can use the "optimal units".

I'd like to know how to answer this too :)

You could be modelling SAT score (Y) as a function of age (X_1), *** (X_2), GPA (X_3), etc.

You can fit other kinds of curves besides lines. But a linear model refers specifically to a model that fits a line/plane. As an example of a non-linear model, there's polynomial-fit.
2) What I think is that the definition of "simple linear model" is not very well-defined. It's ambiguous. I looked at the definitions in 3 different textbooks, but still can't really figure out whether e.g. Y = β0 + β1*X + β2*(X^2) + ε is a simple linear model or mutliple linear model. There seems to be only ONE independent variable X (X^2 is also determined by X, I think it's not a DIFFERENT variable, once we've measured X, we can determine the values of both X and X2), but it has a β2 in there. Is there a nicer definition of a "simple linear model"?

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3) "A linear regression model is of the form:
Y = β0 + β1*X1 + β2*X2 + ... + βk*Xk + ε "

(i) Y = β0 + β1*X + β2*exp(X) + ε
(ii) Y = β0 + β1*X1 + β2*X2 + β3*(X1*X2) + β4*(X1)^2 + β5*(X2)^2 + ε

For (i), X1=X, X2=exp(X)
For (ii), X3=X1*X2, X4=X1^2, X5=X2^2
The latter X's depends on the previous X's. In particular, X3 depends on TWO of the previous X's: X1 AND X2, are those allowed? Somehow I am having a lot of troubles understanding this...I understand the general form of a multiple linear regression model, but I don't seem to understand the specific examples of it like (i) and (ii).