Given the cuadratic models for the odor dataset from the faraway package:
> lmod <- lm(odor ~ I(temp) + I(gas) + I(pack)+I(temp^2)+I(gas^2)+I(pack^2)+I(temp*gas)+I(gas*pack)+I(pack*temp),odor)
> lmod6 <- lm(odor ~ polym(temp,gas,pack,degree = 2),odor)
Both models have the same fitted values:
> lmod$fitted
1 2 3 4 5 6 7 8 9 10 11 12
86.62500 45.87500 36.12500 28.37500 42.50000 15.25000 -3.25000 -24.50000 59.87500 29.37500 20.62500 -16.87500
13 14 15
-30.66667 -30.66667 -30.66667
> lmod6$fitted
1 2 3 4 5 6 7 8 9 10 11 12
86.62500 45.87500 36.12500 28.37500 42.50000 15.25000 -3.25000 -24.50000 59.87500 29.37500 20.62500 -16.87500
13 14 15
-30.66667 -30.66667 -30.66667
However, when comparing these fitted values to each other they are not identical, why is that?
> table(lmod6$fitted==lmod$fitted)
FALSE TRUE
13 2
This is a classic example of the floating point trap as described in chapter 1 of R Inferno by Patric Burns or R FAQ
The differences are zero up to floating point accuracy:
options(digits=20)
lmod$fitted - lmod6$fitted
1 2
0.0000000000000000000e+00 0.0000000000000000000e+00
3 4
7.1054273576010018587e-15 1.0658141036401502788e-14
5 6
-7.1054273576010018587e-15 -3.5527136788005009294e-15
7 8
4.4408920985006261617e-16 -3.5527136788005009294e-15
9 10
-1.4210854715202003717e-14 -1.4210854715202003717e-14
11 12
7.1054273576010018587e-15 3.5527136788005009294e-15
13 14
7.1054273576010018587e-15 7.1054273576010018587e-15
15
7.1054273576010018587e-15
all.equal() function is designed for testing if the two vectors are "almost equal" (up to fp accuracy):
all.equal(lmod$fitted,lmod6$fitted)
[1] TRUE