How do I interpret a Cox hazard model survival curve? How do you interpret a survival curve from cox proportional hazard model?
In this toy example, suppose we have a cox proportional hazard model on age variable in kidney data, and generate the survival curve.
library(survival)
fit <- coxph(Surv(time, status)~age, data=kidney)
plot(conf.int="none", survfit(fit))
grid()


For example, at time $200$, which statement is true? or both are wrong?


*

*Statement 1: we will have 20% subjects left (e.g., if we have $1000$ people, by day $200$, we should have approximately $200$ left),  

*Statement 2: For one given person, he/she has a $20\%$ chance to survive at day $200$.

My attempt: I do not think the two statements are the same (correct me if I am wrong), since we do not have the i.i.d. assumption (survival time for all people is NOT drawing from one distribution independently). It is similar to logistic regression in my question here, each person's hazard rate depends on $\beta^Tx$ for that person. 
 A: Since the hazard depends on the covariates, so does the survival function.  The model assumes that the hazard function
of an individual with covariate vector $x$ is
$$
h(t;x) = h_0(t) e^{\beta'x}.
$$
Hence, the cumulative hazard of this individual is
$$
H(t;x) = \int_0^t h(u;x) du=\int_0^t h_0(u) e^{\beta'x} du = H_0(t)e^{\beta'x},
$$
where we may define $H_0(t)=\int_0^t h_0(u) du$ as the baseline cumulative hazard.  The survival function for an individual with covariate vector $x$ is in turn
$$
S(t;x) = e^{-H(t;x)}=e^{-H_0 e^{\beta'x}}=S_0(t)^{e^{\beta'x}}
$$
where we define $S_0(t) = e^{-H_0(t)}$ as the baseline survival function.
Given estimates $\hat\beta$ and $\hat S_0(t)$ of the regression coefficients and the baseline survival function, an estimate the survival function for an individual with covariate vector $x$ is given by $\hat S(t;x)=\hat S_0(t)^{e^{\hat\beta'x}}$.
To compute this in R you specify the value of your covariates in the newdata argument.  For example if you want the survival function for individuals of age=70, do
plot(survfit(fit, newdata=data.frame(age=70)))

If you, as you do, omit the newdata argument, its default value equals the average values of the covariates in the sample (see ?survfit.coxph).  So what is shown in your graph is an estimate of $S_0(t)^{e^{\beta'\bar x}}$.
A: 
We will have 20% subjects left (e.g., if we have 1000 people, by
  day 200, we should have 200 left)? or For a given person, it has
  20% chance to survive at day 200?

In its most pure form, the Kaplan-Meier curve in your example doesn't make any of the above statements.
The first statement makes a forward looking projection will have. Basic survival curve only describes the past, your sample. Yes, 20% of your sample survived by day 200. Will 20% survive in next 200 days? Not necessarily. 
In order to make that statement you have to add more assumptions, build a model etc. The model doesn't even have to be statistical in a sense like logistic regression. For instance, it could PDE in epidemiology etc.
Your second statement is probably based on some kind of homogeneity assumption: all people are the same. 
A: Thanks for Jarle Tufto's answer. I think I should be able to answer it by myself: both statements are false. The curve generated is $S_0(t)$ but not $S(t)$. 
The baseline survival function $S_0(t)$ will equal to $S(t)$ only when $x=0$. Therefore the curve is NOT describing the whole population or any individual.
A: Your first option is correct. Generally, $S(t) = 0.2$ indicates that 20 % of initial patients have survived till day $t$, without taking into account censoring. On censored data, it is not exactly correct to say that 20 % were still alive that day, since some were lost to follow-up earlier and their status is unknown. A better way to put it would be estimated fraction of patients still alive that day is 20 %.  
The second option (chance to survive one more day, given survival until $t$) is $1-h(t)$, with $h(t)$ denoting the hazard function.
Regarding assumptions: I thought that the usual coefficient tests in a Cox regression setting do assume independence, conditional on observed covariates? Even the Kaplan-Meier estimate seems to require independence between survival time and censoring (reference). But I might be wrong, so corrections are welcome.
