# What is the difference between a polynomial regression and a generalized linear model?

I have seen that a polynomial linear regression can have this form:

$y = c_0 + c_1 x_1 + c_2 x_2 + \dots + c_k x_k$

but I have read that the general lineal model which is a form of the multiple linear regression, can have this form:

$Y_i=b_0+b_1X_{i1}+\dots+b_pX_{ip}$

I see that almost both of them are the same. When I program in R the second case, I can use the function abline, that draws a line of "best fit" as it is called. But when I want to use it in the first case (the polynomial one) it is not possible. Why is that? If both returns a set of scattered points over the plane

Examples of both cases:

First case it could be the index of houses bought by clients over years and bimonthly. Second case, a classical one, it would be the weight prediction of a person considering the weight of their hips, arms, etc.

Why I can use abline in the second case, but not in the first one?

Thanks

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Shouldn't the polynomial regression involve some exponents? Also the use of abline seems to indicate that you're only looking at the relationsship between two variables. – Stefan Hansen Jan 22 '13 at 15:08
exactly, but I saw a case in which there were no exponents. I think that it has to be that the predictor variables have no relationship within each other, but in the second case they have. For example first case could be about prediction of consumer interest yearly and each two months; while the second one could be about weight prediction for a person taking the measures of other parts of their body – Manolo Jan 22 '13 at 15:12
@StefanHansen, I have included two examples, maybe you can give me a hint with that – Manolo Jan 22 '13 at 15:16
So what is $y$, $x_1,\ldots,x_k$ in the first case? – Stefan Hansen Jan 22 '13 at 15:18
@StefanHansen, it could be something like this client_pref = c0 + c1 * year + c2 * bimonthly. For the second case it would be bodyfat=age+breast+arms. I dont know why I cannot use abline in the first case. For what some people told me is because in the first case I will have a hyperplane, but in the second one not. In the second case I got three variables so why I cannot get an hyperplane also – Manolo Jan 22 '13 at 15:22

You seem to be confusing "general" with "generalized". General linear models are linear models. You have $$Y = X\beta+\varepsilon$$ where

• $X\in\mathbb R^{n\times k}$ is fixed and observable,
• $\beta\in\mathbb R^{k\times 1}$ is fixed and unobservable,
• $\varepsilon\in\mathbb R^{n\times 1}$ is random and unobservable, but you may have assumptions about its distribution, such as homoscedasticity and uncorrelatedness, or sometimes (often) normality and independence,
• $Y\in\mathbb R^{n\times 1}$ is observable, and of course random.

Among such general linear models is polynomial regression. In that case, the entries $j$th column of $X$ are the $(j-1)$th powers of those in second column.

In these models, the least-squares estimate of $\beta$ is $\hat\beta= (X^TX)^{-1}X^T Y$.

A generalized linear model is not a linear model. You have a link function. For example, in logistic regression you have $$\operatorname{logit} E(Y) = X\beta,$$ where $X$ and $\beta$ may be as above, and every element of $Y$ is $0$ or $1$, and the logit is of course $\operatorname{logit} p = \log\dfrac{p}{1-p}$. The logit function is the link function. This is not a linear model.

In these models, one usually uses maximum-likelihood estimates found by iterative numerical methods. These are generally not least-squares estimates.

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