How to make predictions with time-dependent covariates with Cox regression I learned about time-dependent covariates in Cox regression in R using the function survSplit of the package survival.
I use this as an interaction term for covariates which do not follow the Cox proportionality assumption which works fine.
The problem is now, that I want to make predictions for new datapoints, but how can I do this as I do not know how long they will survive? How do I know which time-value to use in my calculations? If it helps, I wish to calculate the probability of one-year survival. Also, how can I calculate the calibration/discrimination since certain patients will be duplicated in the dataset?
Code:
rm(list = ls(all=T))
library("rms")
library("pec")
data(veteran)

vet2 <- survSplit(Surv(time, status) ~ ., data= veteran, cut=c(90, 180),
                  episode= "tgroup", id="id")
ddist <- datadist(vet2); options(datadist='ddist')
vfit2 <- cph(Surv(tstart, time, status) ~ trt + prior + karno*strat(tgroup), data=vet2,surv = T,X=T,Y=T)
predictSurvProb(vfit2,newdata=vet2[vet2$id==2,],times = c(121,190))

 A: Calculating predicted probabilities using a Cox model
There is a way of obtaining prediction out of a Cox model, as survival probability at time $t$ ($S(t)$) depends on your cox model like so:
$S(t) = e^{-H_0(t) * exp(LP)}$
in this formula $H_0(t)$ is called the baseline hazard at time $t$; and $LP$ is the linear predictor.
If $X_i$ are the predictor variables $1,2,...,i$ in the Cox model, and $β_i$ are the corresponding coefficients from the Cox model, then the linear predictor is calculated like so:
$LP = X_1*β_1 + X_2*β_2 + ... + X_i*β_i$
You might already be familiar with what a linear predictor is, but I added this for clarity sake.
The baseline hazard is a little harder to obtain. Basically it is a function of time, which shows the hazard for an event of an individual who has a $LP$ of 0. Due to the way cox regression works this value is not estimated (look at this CV question or others, which sheds more light on baseline hazard). In R however (which I can see the OP uses), there is a function available called 'basehaz()' in the survival package, which will let you extract the baseline hazard at a specific timepoint based on your model's fit.
If you extract this baseline hazard you can complete the formula above for any individual in your data, and for unseen data, as long as you are able to calculate the linear predictor for these individuals.
Remember this formula results in the probability of survival or not having had the event at time $t$. If you want to know the probability of an event at time $t$, simply subtract the probability from 1:
$P(event|LP) = 1-S(t) = 1 - e^{-H_0(t) * exp(LP)}$
Final remark: Note that the basehaz function gives you a baseline hazard at time $t$ based on your specific data. So, as is the case for the coefficients, extrapolating this to new cases might not result in a good fit/prediction, due to overfitting, bias, etc.
Calibration & Discrimination
With the predicted probabilities and the event status at one year you can calculate various calibration statistics.
For the discrimination statistic (the c-index) there is a specific kind of rank correlation for a censored response variable. In R, the rcorr.cens function from the Hmisc package can provide you with it.
Whether these statistics are the most suitable for your research I can not say, as that depends mostly on the specifics (e.g. are you building a new and first model for this setting, or are you comparing to previously developed ones?)
A: How do I know which time-value to use in my calculations? 

This isn't a question that your model can answer you.  How did you choose the cutoff points?  
The result of a Cox-PH model is a survival distribution over time for each patient.  Without time dependence, one can look at the overall hazard of each patient to compare relative risks between patients.  With time dependence, one could use the cumulative hazard. 
If you are interested in time dependence in your data, I would suggest other approaches (such as Random Forest survival) rather than modified Cox-PH, which is ad hoc.  
