# Comparing regression models with different coding of the same linear predictor

How can I determine whether one coding of a linear predictor leads to a better fit of the corresponding regression model than the other? In the following example, the restricted cubic spline coding of albumin leads to a higher chi-square value of the resulting model compared to the linear coding. However, it has also more degrees of freedom. As I understand it, I cannot use the log likelihood test in this case, since both models are not nested.

What should I do?

> library(rms)
>
> data(pbc)
> d <- pbc
> rm(pbc, pbcseq)
> d$status <- ifelse(d$status != 0, 1, 0)
>
>
> # linear model
> m1 <- cph(Surv(time, status) ~  albumin, data=d)
> anova(m1)
Wald Statistics          Response: Surv(time, status)

Factor     Chi-Square d.f. P
albumin    73.51      1    <.0001
TOTAL      73.51      1    <.0001
>
> # rcs model
> m2 <- cph(Surv(time, status) ~  rcs(albumin, 4), data=d)
> anova(m2)
Wald Statistics          Response: Surv(time, status)

Factor     Chi-Square d.f. P
albumin    82.80      3    <.0001
Nonlinear  4.73      2    0.094
TOTAL      82.80      3    <.0001


## UPDATE #1

I thought plotting both models would be a good way to decide whether a linear coding or a restricted cubic spline coding would be best. In this case (see below), I would think that the more complex coding is not better. However, the core of my question aimed to reinforce the eyeballing by a statistical test. But as I understand you correct, this is prone to over-fitting?