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According to this output, the independent variables are orthogonal.

Please tell me, when doing the backward selection, why it can be directly known that it should be reduced to 5th order model?

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  • $\begingroup$ Paste the code, rather than an image. The image seems to have messed things up. $\endgroup$ – Jeremy Miles Dec 27 '19 at 18:12
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With orthogonal polynomial terms as predictors in linear regression (orthogonality is enforced by the poly() function in the formula), the lack of correlations between predictors means that when you choose to remove a high-order term from the model you don't face the problem with omitted-variable bias that occurs when you remove a predictor associated both with outcome and with the included predictors. So in your particular case you could start from the highest-order term and remove sequentially all those that don't pass your criterion of statistical significance without biasing the maintained coefficients. Once you get down to a "significant" term you stop: at the 5th-order term in this case.

Note, however, that stepwise selection is generally not a good idea even for linear regression. That approach doesn't directly correct p-values or confidence intervals for the fact that you used the data to choose the model, and the selected model often doesn't generalize well to new samples from the same population. And in other types of regressions (e.g., logistic regression, Cox survival models) lack of correlation of a removed predictor with the included predictors doesn't even protect against omitted-variable bias.

Finally, for curve fitting you almost always should use a spline fit instead of a polynomial.

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In addition to @EdM 's excellent answer, I'd like to address your question:

why it can be directly known that it should be reduced to 5th order model?

If you go strictly by p values, then all the terms of higher order are nonsignificant. But that's not really a good test. If you had similar effect sizes with a very large N, then the 6th or 7th or whatever order term might well be significant.

Instead, you can look, graphically, at how well each model fits the data. You can do this by getting the predicted values from various models and graphing them against the actual values using scatter plots, quantile plots, Tukey mean difference plots and whatever else you find useful.

But why are you fitting a 10th degree polynomial in the first place? As Ed says, a spline is almost always better. The main advantages of polynomials over splines are:

  1. They are easier to interpret.
  2. They may represent substantive concerns (e.g. especially in the "hard" sciences we often have hypotheses about specific terms).

But both these tend to be irrelevant when the exponent is over 3. If you use the 10th (or 5th) order model, it's not going to be easy to interpret. And, since you are willing to remove the higher order terms purely on statistical grounds, they clearly don't have important connections to theory (and it's hard to imagine a case where they would have such connections, although maybe it's possible).

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