Logistic regression model with splines covariates

Question

I am trying to build a logistic regression model in R and I'm examining whether some covariates may follow a non linear distribution by using the splines function. The dataset I am using is the Wisconsin Breast Cancer: For example, I checked the linearity for the column fractal_dimension_mean by doing

fit.splines.fractal <-  lrm(diagnosis ~ 
    rcs(fractal_dimension_mean, 4), data=data)
print(fit.splines.fractal) # non linear

and I obtained this:

Coef                         S.E.   Wald     Z Pr(>|Z|)
Intercept                 -2.4383 0.4609 -5.29  <0.0001
fractal_dimension_mean    -2.1338 0.4876 -4.38  <0.0001
fractal_dimension_mean'    9.3886 2.3722  3.96  <0.0001    
fractal_dimension_mean'' -23.0268 6.2039 -3.71  0.0002

Now from what i can understand it seems that the distribution of this variable is non linear and therefore I should consider adding the complex form of this column into the model. So what I did was creating a function to add the complex forms of those variable and try using the forward selection method for the model variables:

spline_vars <- c("texture_se", "fractal_dimension_mean")

# formula:
formula <- as.formula(paste(
  "diagnosis ~", 
  paste(
    lapply(names(data), function(var) {
      if (var %in% spline_vars) {
        paste0("rcs(", var, ", 4)")  # rcs() for splines
      } else if (var != "diagnosis") {
        var  # keep the variables without transformation
      }
    }), 
    collapse = " + "
  )
))

model_null <- glm(diagnosis ~ 1, data = data, family = binomial) # null model
model_full <- glm(formula, data = data, family = binomial) # full model

# Forward selection
model_forward <- stepAIC(model_null, scope = list(lower = model_null, 
    upper = model_full), direction = "forward")
summary(model_forward)

Now because of the significance of the variables with the transformation I thought that the model would have had those as covariates, but instead I obtained a bigger AIC and log-likelihood than using a forward selection method without transformation. Am I doing something wrong?

Answer 1

Here are some options for automatically determining linear or non-linear effect and doing variable selection, while appropriately accounting for uncertainty in the entire process when doing statistical inference:

1. Bayesian logistic generalized additive model (GAM) with variable selection

2. The frequentist counterpart to (1) is here with instructions for variable selection here. For numeric variables, you can add them as + s(x) in the model formula, and this will automatically estimate the effect shape (linear/non-linear).

• I'm not 100% sure if the approach in (2) will result in valid pvalues/confidence intervals, but the approach in (1) will yield valid inferences via Bayes Theorem.
• An alternative frequentist approach for GAM with variable selection is here. I don't think they provide pvalues/CIs though.

EDIT: I should add, if you don't have that many predictors, why not just fit the full model (using your current approach with rcs() for numeric features OR fit a GAM with mgcv R pacakge), and do inference from that model? The full model will have pvalues/CIs that appropriately account for uncertainty in all effects. Getting correct inference after variable selection is hard, but the above Bayesian method is promising.