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In a linear regression, we can use R-Squared to check if a model fits

But what if I have a polynomial regression with to variable $var_1$ and $var_2$ and a model that goes like $$y=x_0+ x_1\cdot var_1 + x_2\cdot var_1^2 + x_3\cdot var_2 + x_4\cdot var_2^2$$

can we use r squared to check if fits well (or adjusted r squared)

i looked up and find both answers, yes and no...

also can we say that $$y=x_0+ x_1\cdot var_1 + x_2\cdot var_2$$ is a simple linear regression and use r squared?

if no, how can we check if a polynomial regression fits well?

callculus42
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pengj
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  • Without seeing your data it is hazardous to suggest anything, but I might consider whether a term $x_5\mathrm{var}_1 \mathrm{var}_2$ should be included. $\qquad$ – Michael Hardy Mar 09 '16 at 22:55
  • One thing to consider in judging goodness of fit or its lack is whether any pattern is visible in the residuals. $\qquad$ – Michael Hardy Mar 09 '16 at 22:56

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It is an error to think that polynomial regression is not linear regression:

Why is polynomial regression considered a kind of linear regression?

It would also be a mistake to think that just by looking at $R^2$ you can tell whether a model fits. The value of $R^2$ may be used in a significance test if you also know the sample size, but what is significant depends on the sample size.

The value of $R^2$ does tell you how much of the variability in the response variable is explained by the model.

Community
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Michael Hardy
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  • if my sample size is 20 000 observations? is it enough to use r²? also if i have a polynomial regression like x0+ x1var1 + x2var1² + x3var2 + x4var2² and a r² adjusted of 65%. Can i say that the model fits? and what is a way to say if the model fits if i can't use r squared? – pengj Mar 09 '16 at 19:16
  • also the fact that there is two variable var1 and var2 still makes x0+ x1var1 + x2var1² + x3var2 + x4var2² a linear regression? I couldn't tell from that link – pengj Mar 09 '16 at 19:32
  • @pengj : One can validly use $R^2$ in some kinds of hypothesis tests even with small samples, but the value of $R^2$ that would be significant evidence against the null hypothesis depends on the sample size. – Michael Hardy Mar 09 '16 at 22:50
  • but is a 20000 sample size enough? – pengj Mar 10 '16 at 06:38