interpreting polynomial regression output when the regressors are orthogonal (vs. raw)

Question

I want to show an inverted U-shape relationship between two variables: "minutes spent in a room A" and "trustworthiness in others". The hypothesis is that those who have low and high trustworthiness are the ones who spend the least amount of time in room A, whereas those with medium level-trustworthiness spend the most time in that room. The inverted U shape relationship is visible when I plot the two variables (x=trustworthiness and y=minutes spent in a room A).

To statisitcally test this inverted U-shape, I calculated a polynomial regression in R using the poly function. I have been reading about the different arguements that the function can have, depending on whether the linear and quadratic regressor should be considered as orthogonal or raw regressors.

The output for the "raw" polynomial regression is as follows:

The output for the "orthogonal" polynomial regression is as follows:

Now, reading through questions (and answers) of others, in my model, the linear and quadratic regressors seem to be highly correlated as the raw and orthogonal output is vastly different considering their own p-values and beta-weights. I would interpret that in the "raw" model, both predictors are significant but not on their own, their significance is dependent on both regressors being in the model (because they are strongly correlated, (actually r = 0.94)). In the "orthogonal" model, the regressors are considered independently. Therefore one could conclude that the orthogonal model is the right choice and the result shows that the relationship can be significantly described by a quadratic fit, rather than a linear fit, so a U-shape relationship seems fair. (note: calculating the linear fit along (lm(y~x)) results in an insignificant fit with an Rsuared under 1%, whereas in the polynomial fits, it goes up to 18% - which is still not fantastic, but still better).

However, when interpreting the beta-weights, I would still need to use the coefficients from the raw model in order to make sensible predictions. So if one cannot interpret the beta-weights in the orthogonal models because in fact the two regressors are inherently correlated, can the above conclusion about better fit really be made?

ps. the QQ plots for the residuals as well as the normality checks using Shapiro-Wilk indicate that the prerequisits for the regression model are met

Answer 1

A polynomial term in a regression represents the product of a continuous predictor with itself. It's just one form of an interaction term in a regression, which in general is the product of two predictors.

You should be very cautious about interpreting individual coefficients when predictors are involved in interactions. The value of an individual coefficient for a predictor involved in an interaction depends on the coding of the predictor(s) with which it interacts. Thus, for a predictor involved in an interaction, the apparent "significance" of its individual coefficient (whether it differs from 0, which is what the p-value reports) is a function of the way the interacting predictors are coded. In this case, the coding differs between the choice for raw. To evaluate the significance of such a predictor, you need to include all of the terms that involve the predictor, for example in a likelihood-ratio test or a multi-coefficient Wald test.

However you code the predictor, the model will still provide the same predictions about outcome. The intercept and the linear terms in your two models differ due to the reasons explained above, but the models are identical in terms of fitting the data and any predictions you would make from them. Don't waste time fretting about the "significance" of its individual coefficient.

One further suggestion: a polynomial fit imposes a strong structure over the shape of the predictor-outcome relationship. You might consider a more flexible fit, for example with a regression spline.