First off, your objective here is very backwards. It shouldn't be which method is the most interesting, it should be which method is applicable to my situation. In your situation, a GAM is complete overkill when you don't have to deal with nonlinearity, and is going to be more data hungry then typical techniques. A GLM with a specific residual distribution should really only be applied if you expect the conditional distribution to exhibit a specific pattern (e.g. Bernoulli distribution of a binary response). Linear regression is for modeling a Gaussian conditional distribution and would be applicable in a situation where you don't need to go beyond that assumption.
However, with such a finite amount of data, I can't expect you to get anything of value with $n = 11$ data points. What's worse, you are going to dangerously overfit the model by having almost as many predictors as observations. In fact, such a model should in principle be impossible to fit anyway given you have more predictors than observations, so I think some information may be missing from your question. A very simple demonstration in R with some simulated data shows the results are nonsensical:
#### Setup Parameters ####
set.seed(123) # random seed
n <- 11 # observations
num_x <- 14 # predictors
#### Simulate X Variables ####
x <- matrix(
rnorm(n * num_x),
nrow = n,
ncol = num_x
)
#### Merge X Data ####
x_df <- as.data.frame(x)
colnames(x_df) <- paste0("x", 1:num_x)
#### Simulate Y and Merge Data Again ####
y <- rnorm(n)
data <- cbind(x_df, y)
#### Fit Regression ###
fit <- lm(
formula = y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 + x14,
data = data
)
#### Print Model ####
summary(fit)
As shown below (notice it completely drops coefficients at $X_{11}$ and warns us that the matrix has become singular):
Call:
lm(formula = y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 +
x10 + x11 + x12 + x13 + x14, data = data)
Residuals:
ALL 11 residuals are 0: no residual degrees of freedom!
Coefficients: (4 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.028574 NaN NaN NaN
x1 1.148516 NaN NaN NaN
x2 0.421618 NaN NaN NaN
x3 -0.008065 NaN NaN NaN
x4 -0.278369 NaN NaN NaN
x5 2.076605 NaN NaN NaN
x6 -2.059897 NaN NaN NaN
x7 0.661934 NaN NaN NaN
x8 -1.367416 NaN NaN NaN
x9 -0.074724 NaN NaN NaN
x10 0.774342 NaN NaN NaN
x11 NA NA NA NA
x12 NA NA NA NA
x13 NA NA NA NA
x14 NA NA NA NA
No matter what technique you use, your results are going to be highly unreliable with such little data anyway. Since you already peaked at this data and threw different combinations of techniques at it, the results are not going to be very trustworthy. I would find a way to either get a new sample as well as find out how to get more data in the future. This will be problematic in your future research if you can't manage get more than eleven observations.