Approach for comparing linear, non-linear and different parameterization non-linear models

Question

I search for one approach for comparing linear, non-linear and different parameterization non-linear models. For this:

#Packages
library(nls2)
library(minpack.lm)

# Data set - Diameter in function of Feature and Age
Feature<-sort(rep(c("A","B"),22))
Age<-c(60,72,88,96,27,
36,48,60,72,88,96,27,36,48,60,72,
88,96,27,36,48,60,27,27,36,48,60,
72,88,96,27,36,48,60,72,88,96,27,
36,48,60,72,88,96)
Diameter<-c(13.9,16.2,
19.1,19.3,4.7,6.7,9.6,11.2,13.1,15.3,
15.4,5.4,7,9.9,11.7,13.4,16.1,16.2,
5.9,8.3,12.3,14.5,2.3,5.2,6.2,8.6,9.3,
11.3,15.1,15.5,5,7,7.9,8.4,10.5,14,14,
4.1,4.9,6,6.7,7.7,8,8.2)
d<-dados <- data.frame(Feature,Age,Diameter)
str(d)

I will create three different models, two non-linear models with specific parametization and one linear model. In my example a suppose that all the coefficients of each mode were significant (and not considering real results).

# Model 1 non-linear
e1<- Diameter ~ a1 * Age^a2 
#Algoritm Levenberg-Marquardt
m1 <-  nlsLM(e1, data = d,
     start = list(a1 = 0.1, a2 = 10),
     control = nls.control(maxiter = 1000))

# Model 2 linear
m2<-lm(Diameter ~ Age, data=d)

# Model 3 another non-linear
e2<- Diameter ~ a1^(-Age/a2)
m3 <-  nls2(e2, data = d, alg = "brute-force",
     start = data.frame(a1 = c(-1, 1), a2 = c(-1, 1)),
     control = nls.control(maxiter = 1000))

Now, my idea is comparing the "better" model despite the different nature of each model, than I try a proportional measure and for this I use each mean square error of each model comparing of total square error in data set, when a make this I have if a comparing model 1 and 2:

## MSE approach (like pseudo R2 approach)

#Model 1
SQEm1<-summary(m1)$sigma^2*summary(m1)$df[2]# mean square error of model 
SQTm1<-var(d$Diameter)*(length(d$Diameter)-1)#total square error in data se
R1<-1-SQEm1/SQTm1
R1

#Model 2
SQEm2<-summary(m2)$sigma^2*summary(m2)$df[2]# mean square error of model 
R2<-1-SQEm2/SQTm1
R2

In my weak opinion model 1 is "better" that model 2. My question is, does this approach sounds correct? Is there any way to compare these models types?

Thanks in advance!

Answer 1

#First cross-validation approach ------------------------------------------

#Cross-validation model 1
set.seed(123) # for reproducibility

n <- nrow(d)
frac <- 0.8
ix <- sample(n, frac * n) # indexes of in sample rows

e1<- Diameter ~ a1 * Age^a2 
#Algoritm Levenberg-Marquardt
m1 <-  nlsLM(e1, data = d,
     start = list(a1 = 0.1, a2 = 10),
     control = nls.control(maxiter = 1000), subset = ix)# in sample model

BOD.out <- d[-ix, ] # out of sample data
pred <- predict(m1, new = BOD.out)
act <- BOD.out$Diameter
RSS1 <- sum( (pred - act)^2 )
RSS1
#[1] 56435894734

#Cross-validation model 2
m2<-lm(Diameter ~ Age, data=d,, subset = ix)# in sample model
BOD.out2 <- d[-ix, ] # out of sample data
pred <- predict(m2, new = BOD.out2)
act <- BOD.out2$Diameter
RSS2 <- sum( (pred - act)^2 )
RSS2
#[1] 19.11031

# Sum of squares approach -----------------------------------------------
deviance(m1)
#[1] 238314429037

deviance(m2)
#[1] 257.8223

Based in gfgm and G. Grothendieck comments, RSS2 has lower error that RSS1 and comparing deviance(m2) and deviance(m2) too, than model 2 is better than model 1.