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Let's say we have a model as given below:

$Y_1 = \beta_0 + \beta_1 X_1 + \beta_2 X_2+ \beta_3 \frac{X_2}{X_1} + \beta_4\frac{X_2^2}{X_1}$, $R^2= 0.98$

Here, $X_1$ & $X_2$ are positive integers.

The value of $\beta_3$ and $\beta_4$ are high compared to $\beta_1$ and $\beta_2$.

If we fit multiple regression model for $Y_1$ with only the linear terms, the R-squared is very high.

$Y_1 = \gamma_0 + \gamma_1 X_1 + \gamma_2 X_2$, $R^2= 0.9$

So, can we say the interaction terms or ratio terms here are not significant? Also how can we find the significance/ contribution of each term to the model prediction?

Following are my observations when fitting only the interaction terms

$Y_1 = \alpha_0 + \alpha_1 \frac{X_2}{X_1}$, $R^2= 0.4$

$Y_1 = \rho_0 + \rho_1 \frac{X_2^2}{X_1}$, $R^2= 0.7$

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    $\begingroup$ "The value of β3 and β4 are high compared to β1 and β2." their units are different, so their relative size is a case of apples and oranges $\endgroup$
    – Glen_b
    Commented Aug 26, 2022 at 1:40

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Technically, this is still a linear model as it is linear in the coefficients, the $\beta$ values. As @Glen_b notes in a comment, the magnitudes of the coefficients aren't to be compared when the associated predictors are on different scales. The sub-models on their own also aren't reliable guides when the outcome-associated predictors are correlated, due to omitted-variable bias.

You can evaluate the statistical significance of each coefficient with standard techniques; start with the full model and look at the model summary to see which coefficients have significant t-test values. You apply your knowledge of the subject matter to evaluate the practical significance.

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    $\begingroup$ +1. FWIW, these things have been pointed out (albeit more briefly) to the OP in comments to the four previous versions of this question they have deleted (!). $\endgroup$
    – whuber
    Commented Aug 26, 2022 at 15:51

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