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The issue with the approach you described is that the breakpoint is a parameter, which appears to have been chosen by hand to fit the data. But, it's treated as a known value when calculating the AIC. This fails to account for the uncertainty in estimating the breakpoint from the data. As a result, the estimated AIC will be overoptimistically biased, and ...
2
AIC looks at variance explained while penalizing complexity of model by number of features used.
If I'm understanding this correctly I think what you're seeing is when you merge the two the model is overall better, but I believe that's just because the model has more data to work with in approach two.
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Stepwise model selection, particularly forward stepwise, is not very reliable. This page provides much general discussion.
With Cox models the problem is even worse than for ordinary linear regression.
In ordinary linear regression there is no bias if you omit a predictor that is uncorrelated to the predictors you include. In other types of regressions ...
1
Consider scalar parameters $\theta_0$ and the corresponding scalar estimate $\hat \theta$ for simplicity.
I will answer Q1 and Q3 which are essentially asking why is the mean of the score function $\Bbb{E}_{\theta}(s(\theta)) =0 $. This is a widely known result.. To put it simply, Notice that score function $s(\theta)$ depends of the random observations $X$...
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