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I read a couple post on interpreting polynomial coefficients here in cross validate however none of them touch on how to interpret multiple polynomial regression coefficients. Perhaps its the same but I wanted to ask the question for my own edification as well as others who may be wondering.

Here is a regression I just ran where there are four terms each with a corresponding polynomial term. How would one go about interpreting this output?

Call:
lm(formula = a ~ t + d + r + p + I(t^2) + 
    I(d^2) + I(r^2) + I(p^2), data = df)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.8466 -1.4200 -0.2556  1.8784  6.9382 

Coefficients:
                       Estimate         Std. Error t value Pr(>|t|)    
(Intercept)   1.071213896096506  0.897660289412562   1.193 0.244901    
t            -0.000016729186474  0.000012896669665  -1.297 0.207434    
d             0.000240787673662  0.000136472581690   1.764 0.090949 .  
r             0.000936217403829  0.000238344538301   3.928 0.000673 ***
p            -0.000410104711084  0.000260680628526  -1.573 0.129327    
I(t^2)        0.000000000005504  0.000000000024388   0.226 0.823423    
I(d^2)       -0.000000000948744  0.000000002529495  -0.375 0.711043    
I(r^2)       -0.000000006440508  0.000000002136199  -3.015 0.006170 ** 
I(p^2)        0.000000007091433  0.000000007243474   0.979 0.337761    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.893 on 23 degrees of freedom
Multiple R-squared:  0.8754,    Adjusted R-squared:  0.832 
F-statistic:  20.2 on 8 and 23 DF,  p-value: 0.00000001125
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The example that comes to mind is wind resistance. Wind resistance increases with the square of speed. If you double your speed, you must quadruple your force to maintain that speed. This analogy can be used to help understand a strong quadratic effect, like your I(r^2) term, except your coefficient is negative. I also notice in your model that the I(r^2) term is 10^6 smaller than your r coefficient, so only at large r values will the quadratic term play in.

The other coefficients are not really significant.

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  • $\begingroup$ d is significant but does it only mater if d and I(d^2) are both significant? $\endgroup$ – moku Jun 18 '15 at 18:23

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