Compare the two results: Check that the fitted values and residuals from the two calculations are the same, and that the t-statistic and p-value are the same for the coefficient labeled poly(rate, 2)2...

Compare the two results:

Check that the fitted values and residuals from the two calculations are the same, and that the t-statistic and p-value are the same for the coefficient labeled poly(rate, 2)2 in the polynomial regression as for the coefficient labeled I(rateˆ2) in the regression on rate and rateˆ2. Check that the coefficients remain the same if, in the calculation of seedrates.lm, rate + I(rateˆ2) is replaced by poly(rate, 2, raw=TRUE). Regress the second column of model. Matrix(seedrates.pol) on rate and I(rateˆ2), and similarly for the third column of model. Matrix(seedrates.pol). Hence, express the first and second orthogonal polynomial terms as functions of rate and rateˆ2.