# Model selection on a system of ordinary differential equations

FAMoS was originally designed to perform model selection on systems of ODEs, which is why this tool is especially suited to tackle these kind of problems. Assume we have three different cell populations, A, B and C. All of these cells types can divide and therefore increase in their numbers. Furthermore, it is possible that all of these cells can turn into each other. The underlying dynamics of these cell populations can be described by a system of ordinary differential equations:

``````#define the global model dynamics
global.dynamics <- function(t,x,parms){
with(as.list(c(x, parms)), {

dA <- rho_A*A + mu_BA*B + mu_CA*C - (mu_AB + mu_AC)*C
dB <- rho_B*B + mu_AB*A + mu_CB*C - (mu_BA + mu_BC)*B
dC <- rho_C*C + mu_AC*A + mu_BC*B - (mu_CA + mu_CB)*C

return(list(c(dA,dB,dC)))
}
)
}``````

We will use this function to simulate some data using the deSolve library:

``````#define simulation parameter set
pars <- c(rho_A = 0, rho_B = 0, rho_C = 0.1,
mu_AB = 0.2, mu_AC = 0, mu_BA = 0,
mu_CA = 0, mu_BC = 0.05, mu_CB = 0)

#set initial values for cells
init.vals <- c(A = 100, B = 0, C = 0)

#simulate data
library(deSolve)

sim.data <- lsoda(y = init.vals,
times = 0:10,
func = global.dynamics,
parms = pars)``````

Now, we will define the cost function for FAMoS:

``````#cost function for famos
cost.function <- function(parms, binary, data, inits){
#simulate the data with the current parameter set
fit.data <- lsoda(y = inits,
times = 0:10,
func = global.dynamics,
parms = parms)
#calculate the aic
ls2 <- sum((sim.data[,-1] - fit.data[,-1])^2)
out.aic <- ls2 + 2*sum(binary == 1)

return(out.aic)
}``````

To call FAMoS, we need to specify a vector containing the names and the values of the start parameters. All that’s left is calling the model selection routine then. The fitting might take a couple of minutes. To see more of the model selection process, set verbose = TRUE.

``````
library(FAMoS)

start.vals <- c(rho_A = 0.1, rho_B = 0.1, rho_C = 0.1,
mu_AB = 0.1, mu_AC = 0.1, mu_BA = 0.1,
mu_CA = 0.1, mu_BC = 0.1, mu_CB = 0.1)

famos.fit <- famos(init.par = start.vals,
fit.fn = cost.function,
homedir = tempdir(),
init.model.type = c("rho_A"),
data = sim.data,
inits = init.vals)

print(famos.fit)``````

# Model selection based on logistic regression models

There are many R packages available that tackle the problem of model selection based on regression models and FAMoS is not meant to replace them. However, it is also possible (and not even that difficult) to apply FAMoS to these kind of problems. For our example, we will use the birthwt data set from the MASS package, a standard test set for model selection. We will follow the modifications by Venables and Ripley in their book Modern Applied Statistics with S-PLUS and transform the data set in the following way before using it:

``````library(MASS)

attach(birthwt)
race <- factor(race, labels = c("white","black","other"))
ptd <- factor(ptl > 0)
levels(ftv)[-c(1:2)] <- "2+"
bwt <- data.frame(low = factor(low), age, lwt, race,
smoke = (smoke>0), ptd, ht = (ht>0), ui =(ui>0), ftv)
detach();rm(race,ptd,ftv)``````

Now, we want to perform a model selection based on logistic regression models with interactions. To this end, we will use the fitting routine glm instead of the standard optimiser optim, which is the default option in FAMoS. To tell FAMos, that we’re using a different optimiser, we will set the option use.optim = FALSE and since we don’t need multiple repeats with random starting values, we also set optim.runs = 1. Now, we need to supply and fitting function to FAMoS that - since we are using another fitting routine - needs to return a list, that contains the value of the selection criterion as well as the vector containing the names and the values of the fitted variables. As we will only work with the names in our example, having to return the parameter values might seem a bit clunky. However, as FAMoS is intended to offer a lot of flexibility, this feature is useful in many other cases (see below). Our fitting function looks like this:

``````fit_func <- function(parms, data, binary){
#First transform the parameter names into a formula.
#The to-be-fitted parameters are identified using the binary vector
fitted.pars <- names(parms[which(binary == 1)])
glm_formula <- as.formula(
paste0("low ~ ", paste0(fitted.pars, collapse = "+"))
)
#fit the logistic model using glm
out <- summary(
glm(glm_formula,
data = data
)
)

#prepare output parameters, in this case only the names of the fitted
#parameters are relevant. As FAMoS also expects values with these, we will just
#return the value 1 for all parameters
out.par <- rep(1, length(fitted.pars))
names(out.par) <- fitted.pars

#return a list containing the first the selection criterion value and second the
#vector of the fitted parameters
return(list(SC = out\$aic, params = out.par))
}``````

The only thing left is to specify the initial parameter vector. As we also want to have interactions, we need to create entries for all parameter combinations. Since glm expects them to be in the form of par1:par2, we will name the entries accordingly.

``````#first we define the available parameters. As the values are not important, we
#will just set them equal to 1.
inits <- c(age = 1, lwt = 1, race = 1, smoke = 1, ptd = 1, ht = 1, ui = 1, ftv = 1)
#Now, we calculate all possible interactions and name the corresponding vector
#entries accordingly
combinations <- combn(names(inits),2)
for(i in 1:ncol(combinations)){
inits <- c(inits,1)
names(inits)[i + 8] <- paste0(combinations[1,i],":",combinations[2,i])
}``````

All that’s left is calling FAMoS now:

``````library(FAMoS)
famos.glm <- famos(init.par = inits,
fit.fn = fit_func,
init.model.type = names(inits[1:8]),
data = bwt,
use.optim = FALSE,
optim.runs = 1)``````