How to calculate a "Predicted line"? [duplicate]

Question

Here is a regression realized with R

> degree=2    
> frml = formula(A~factor(B)*poly(C,degree=degree))
> m = lm(frml,data=data)
> summary(m)$coeff

                                               Estimate Std. Error t value Pr(>|t|)    
(Intercept)                                    0.050882   0.001591  31.980  < 2e-16 ***
factor(B)2                                     0.090513   0.002250  40.227  < 2e-16 ***
factor(B)3                                     0.245776   0.002250 109.231  < 2e-16 ***
factor(B)4                                     0.483829   0.002250 215.030  < 2e-16 ***
factor(B)5                                     0.741211   0.002250 329.419  < 2e-16 ***
factor(B)6                                     0.907053   0.002250 403.125  < 2e-16 ***
poly(C, degree)1                               0.008182   0.016988   0.482  0.63114    
poly(C, degree)2                              -0.004527   0.016988  -0.266  0.79044    
factor(B)2:poly(C, degree)1                   -0.014671   0.024024  -0.611  0.54284    
factor(B)3:poly(C, degree)1                    0.010721   0.024024   0.446  0.65640    
factor(B)4:poly(C, degree)1                    0.037756   0.024024   1.572  0.11933    
factor(B)5:poly(C, degree)1                    0.031446   0.024024   1.309  0.19368    
factor(B)6:poly(C, degree)1                    0.011876   0.024024   0.494  0.62220    
factor(B)2:poly(C, degree)2                    0.006151   0.024024   0.256  0.79846    
factor(B)3:poly(C, degree)2                   -0.033512   0.024024  -1.395  0.16625    
factor(B)4:poly(C, degree)2                   -0.064164   0.024024  -2.671  0.00889 ** 
factor(B)5:poly(C, degree)2                   -0.008938   0.024024  -0.372  0.71068    
factor(B)6:poly(C, degree)2                   -0.024527   0.024024  -1.021  0.30985 

What do we call a "predicted line"? How can we calculate this function from these data?

I think it should be something like y=0.050882+0.090513+0.008182x-0.004527x^2+...Is it correct? What would be the following. Are there other name for this "predicted line"?

What is the difference between adding this predicting line on a plot than adding a simple lm (of the first or second degree)?

What would be the predicted line if I computed B as a covariate (measuring a surface) instead of a factor?

Thanks a lot for your help!

Answer 1

In my opinion showing six polynomial lines (each corresponds to certain level of factor B) is better than an interpolated surface. This can be done as

par(mfrow = c(2,3))
sapply(levels(factor(data$B)), function(x){
    C_val=seq(min(data$C),max(data$C),by=(max(data$C)-min(data$C))/100)
    plot(C_val,
         predict(m, newdata = data.frame(C=C_val,B=rep(x,101))),
         type="l",
         ylab = x) 
  }
)

Answer 2

Well, you could call it the "line of best fit" or the "trend line".

The simple lm simply fits a straight line relationship, the polynomial line posits a more complex, non-linear relationship between the variables. Can you think what processes might drive such a relationship?

EDIT: As you say if you treat B as another variable you'll get a 2D surface for predicting the value of A, with interpolated points. Does this make sense or does B only take discrete values?