poly fit

fitpoly.R

> x<-runif(100,0,2)
> y<-1+x+x^2+x^3
> summary(lm(y~x))

Call:
lm(formula = y ~ x)

Residuals:
   Min     1Q Median     3Q    Max 
-1.334 -1.018 -0.213  0.677  2.742 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   -1.071      0.213   -5.03  2.2e-06 ***
x              6.397      0.206   31.06  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.16 on 98 degrees of freedom
Multiple R-squared:  0.908,	Adjusted R-squared:  0.907 
F-statistic:  965 on 1 and 98 DF,  p-value: <2e-16

> summary(lm(y~poly(x,2)))

Call:
lm(formula = y ~ poly(x, 2))

Residuals:
    Min      1Q  Median      3Q     Max 
-0.3308 -0.1444  0.0489  0.1176  0.3394 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   4.4838     0.0153   292.4   <2e-16 ***
poly(x, 2)1  35.9596     0.1533   234.5   <2e-16 ***
poly(x, 2)2  11.3604     0.1533    74.1   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.153 on 97 degrees of freedom
Multiple R-squared:  0.998,	Adjusted R-squared:  0.998 
F-statistic: 3.02e+04 on 2 and 97 DF,  p-value: <2e-16

> summary(lm(y~poly(x,3)))

Call:
lm(formula = y ~ poly(x, 3))

Residuals:
      Min        1Q    Median        3Q       Max 
-4.09e-15 -4.61e-16 -8.00e-17  3.45e-16  1.07e-14 

Coefficients:
            Estimate Std. Error  t value Pr(>|t|)    
(Intercept) 4.48e+00   1.55e-16 2.90e+16   <2e-16 ***
poly(x, 3)1 3.60e+01   1.55e-15 2.32e+16   <2e-16 ***
poly(x, 3)2 1.14e+01   1.55e-15 7.34e+15   <2e-16 ***
poly(x, 3)3 1.51e+00   1.55e-15 9.75e+14   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.55e-15 on 96 degrees of freedom
Multiple R-squared:     1,	Adjusted R-squared:     1 
F-statistic: 1.98e+32 on 3 and 96 DF,  p-value: <2e-16