Interpreting multiple polynomial regression coefficients

Question

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

Answer 1

The first thing I would do is rescale the independent variables so there are fewer leading zeroes after the decimal. Maybe multiply each by 1000.

Next, center your variables.

Then, I note that you seem to have an N of 31 which means your model is very overfit. So, I'd either gather a lot more data or make a much simpler model.

Finally, to your question: The interpretation of a polynomial regression is the same, whether there is one or more; what I like to do in cases like this is make lots of graphs of the predicted value at different combinations of the input variables.

Answer 2

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.