# Some doubts about survival analysis

I'm struggling to perform a correct and unbiased survival analysis, but I have some doubts.

## Data

I observe a population of posts on a blog within a temporal window $[T_{0}, T_{final}]$. Each post is characterized by:

• a $start$ date: the date in which it has been published on the blog;
• a $stop$ date: the date in which the last comment on the post has been left;
• a $event$ date: the date in which the post has been shared on Twitter for the first time.

Basically, the lifetime of each post is defined as the difference between $stop$ and $start$.

## Analysis and Related Doubts

My analysis (at least in this first part) aims at quantifying the effect on the lifetime of the post being shared on Twitter. Suppose I have data organized as follows:

id  start event stop shared
1     0   105  192      1
2     0   162  245      1
3     0     2   88      1


that is, in the case of $id = 1$, the post is originally published in $t = 0$, it is shared on Twitter after $105$ weeks, and it stopped being commented after $192$ weeks. Note that I brought back all $start$ time to zero.

Then, I re-organized data as follows:

id  start stop shared event
1     0  106      0     1
1   106  194      1     1
2     0  163      0     1
2   163  247      1     1
3     0    3      0     1
3     3   90      1     1


that is, now for each id I have two rows: the first ($shared = 0$) specifies the $start$ and $stop$ times before the post is shared on Twitter; whereas the second ($shared = 1$) specifies the $start$ and $stop$ times after the post is shared on Twitter.

At this point, I would like to perform a Kaplan-Meier estimate of the survival functions before and after the post is shared on Twitter.

s = with(surv.df, Surv(start, stop, event))
KM = survfit(s ~ shared, data = surv.df)


and then apply a Cox hazard regression to model to quantify the effect on the lifetime of the post being shared on Twitter

sCox = coxph(s ~ as.factor(shared), data=surv.df)
summary(sCox)


However, I'm not convinced about such a procedure.

Questions

• Do you see any sources of bias in the analysis I have just presented?
• Is it correct the way I organized data to perform my analysis?
• Am I actually observing the effect of the post being shared on Twitter?

I think I've found a solution to my problem. Instead of using traditional survival analysis and Cox regression models, I used multi-state models. Here two good references:

Multi-state models allow to model complex event histories, by handling censoring and left truncation, and avoiding two common kind of bias: length bias and immortal bias.

Practically, I have three states, $S = {1,2,3}$:

• $S = 1 \to$ initial state: posts are published on the blog;
• $S = 2 \to$ shared state: posts are shared on Twitter;
• $S = 3 \to$ absorbing state: posts did not receive comments in the last month;

Note that $S = 1$ is the initial state, i.e. each post starts in such a state, whereas $S = 3$ is an absorbing state, so that once the post arrives there, it remains there.

A transition matrix specifies the admissible transitions from one state to another:

$$T = \begin{pmatrix} p_{1,1}(t) & p_{1,2}(t) & p_{1,3}(t) \\ p_{2,1}(t) & p_{2,2}(t) & p_{2,3}(t) \\ p_{3,1}(t) & p_{3,2}(t) & p_{3,3}(t) \end{pmatrix}$$

In my case, I specified the following transition matrix:

$$T = \begin{pmatrix} p_{1,1}(t) & p_{1,2}(t) & p_{1,3}(t) \\ 0 & p_{2,2}(t) & p_{2,3}(t) \\ 0 & 0 & p_{3,3}(t) \end{pmatrix}$$

Note that all the probabilities are time-dependent. Such a transition matrix can be estimated by using R and the package msSurv. Once you have estimated the transition matrix, you can analyze the state entry/exit distribution in each state, the distributions of time-varying probabilities for transitions from one state to another, and the probability to be in a given state at a certain instant $t$.

In particular, by comparing the time-varying transition probabilities $p_{1,3}(t)$ and $p_{2,3}(t)$, I can understand if there is a difference between survival times of posts being or not being shared on Twitter.